Handbook of Fractional Calculus with Applications: Volume 1 Basic Theory 9783110571622, 9783110570816

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Anatoly Kochubei, Yuri Luchko (Eds.) Handbook of Fractional Calculus with Applications

Handbook of Fractional Calculus with Applications Edited by J. A. Tenreiro Machado

Volume 2: Fractional Differential Equations Anatoly Kochubei, Yuri Luchko (Eds.), 2019 ISBN 978-3-11-057082-3, e-ISBN (PDF) 978-3-11-057166-0, e-ISBN (EPUB) 978-3-11-057105-9 Volume 3: Numerical Methods George Em Karniadakis (Ed.), 2019 ISBN 978-3-11-057083-0, e-ISBN (PDF) 978-3-11-057168-4, e-ISBN (EPUB) 978-3-11-057106-6 Volume 4: Applications in Physics, Part A Vasily E. Tarasov (Ed.), 2019 ISBN 978-3-11-057088-5, e-ISBN (PDF) 978-3-11-057170-7, e-ISBN (EPUB) 978-3-11-057100-4 Volume 5: Applications in Physics, Part B Vasily E. Tarasov (Ed.), 2019 ISBN 978-3-11-057089-2, e-ISBN (PDF) 978-3-11-057172-1, e-ISBN (EPUB) 978-3-11-057099-1 Volume 6: Applications in Control Ivo Petráš (Ed.), 2019 ISBN 978-3-11-057090-8, e-ISBN (PDF) 978-3-11-057174-5, e-ISBN (EPUB) 978-3-11-057093-9 Volume 7: Applications in Engineering, Life and Social Sciences, Part A Dumitru Bǎleanu, António Mendes Lopes (Eds.), 2019 ISBN 978-3-11-057091-5, e-ISBN (PDF) 978-3-11-057190-5, e-ISBN (EPUB) 978-3-11-057096-0 Volume 8: Applications in Engineering, Life and Social Sciences, Part B Dumitru Bǎleanu, António Mendes Lopes (Eds.), 2019 ISBN 978-3-11-057092-2, e-ISBN (PDF) 978-3-11-057192-9, e-ISBN (EPUB) 978-3-11-057107-3

Anatoly Kochubei, Yuri Luchko (Eds.)

Handbook of Fractional Calculus with Applications |

Volume 1: Basic Theory Series edited by Jose Antonio Tenreiro Machado

Editors Prof. Dr. Anatoly Kochubei National Academy of Sciences of Ukraine Institute of Mathematics Tereschenkivska str., 3 Kiev 01004 Ukraine [email protected]

Prof. Dr. Yuri Luchko Beuth Hochschule für Technik Berlin FB II Mathematik-Physik-Chemie Luxemburger Str. 10 13353 Berlin Germany [email protected]

Series Editor Prof. Dr. Jose Antonio Tenreiro Machado Department of Electrical Engineering Instituto Superior de Engenharia do Porto Instituto Politécnico do Porto 4200-072 Porto Portugal [email protected]

ISBN 978-3-11-057081-6 e-ISBN (PDF) 978-3-11-057162-2 e-ISBN (EPUB) 978-3-11-057063-2 Library of Congress Control Number: 2018967839 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: djmilic / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Fractional calculus (FC) originated in 1695, nearly at the same time as conventional calculus. However, FC attracted limited attention and remained a pure mathematical exercise, in spite of the contributions of important mathematicians, physicists and engineers. FC had a rapid development during the last few decades, both in mathematics and in the applied sciences, it being nowadays recognized as an excellent tool for describing complex systems, phenomena involving long range memory effects and non-locality. A huge number of research papers and books devoted to this subject have been published, and presently several specialized conferences and workshops are being organized each year. The FC popularity in all fields of science is due to its successful application in mathematical models, namely in the form of FC operators and fractional integral and differential equations. Presently, we are witnessing considerable progress both as regards theoretical aspects and applications of FC in areas such as physics, engineering, biology, medicine, economy, or finance. The popularity of FC has attracted many researchers from all over the world and there is a demand for works covering all areas of science in a systematic and rigorous form. In fact, the literature devoted to FC and its applications is huge, but readers are confronted with a high heterogeneity and, in some cases, with misleading and inaccurate information. The Handbook of Fractional Calculus with Applications (HFCA) intends to fill that gap and provides the readers with a solid and systematic treatment of the main aspects and applications of FC. Motivated by these ideas, the editors of the volumes involved a team of internationally recognized experts for a joint publishing project offering a survey of their own and other important results in their fields of research. As a result of these joint efforts, a modern survey of FC and its applications, reflecting present day scientific knowledge, is now available with the HFCA. This work is distributed in the form of several distinct volumes, each one developed under the supervision of its editors. The first volume of HFCA is devoted to the basic theory of FC and presents both a selection of the well-known fundamental results and some modern new trends in FC. It starts with a survey of the recent history of FC, including lists of the FC books published until now, FC conferences, and journals focusing on FC and its applications. In the second part of the volume, the basic FC operators and their properties as well as their mathematical and physical interpretations are discussed. Moreover, some selected generalizations of the basic FC operators like the multiple Erdélyi–Kober operators, the fractional Laplace operator, and general FC are presented. The next part of the volume provides an overview of the FC special functions and the integral transforms that play a prominent role in FC. In particular, the fractional Fourier transform and the Mellin integral transform as well as their numerous applications in FC are discussed. The most important functions of FC—the Wright function and the MittagLeffler function—are presented along with their applications in FC and useful properhttps://doi.org/10.1515/9783110571622-201

VI | Preface ties including their asymptotical behavior. The fourth part of the volume provides a short survey of basic results in one of the modern branches of FC: fractional calculus of variations. In addition, a special class of fractional integral equations of thermistor type is discussed here. The very important link between FC and probability theory is subject of the fifth part of the volume. The chapters’ authors explain in detail how both the time- and the space-fractional partial differential equations appear in a natural way on the macro-level starting from special stochastic processes on the microlevel. Finally, the last part of the volume presents some FC results not very well known even to the FC experts, including the surveys of the spectral theory of direct sums of multiples of the Riemann–Liouville fractional integrals and fractional differentiation in p-adic analysis. Our special thanks go to the authors of the individual chapters, which are excellent surveys of selected classical and new results in several important fields of FC. The editors believe that the HFCA will represent a valuable and reliable reference work for all scholars and professionals willing to perform research in this challenging and timely scientific area. Anatoly Kochubei and Yuri Luchko

Contents Preface | V J. A. Tenreiro Machado and Virginia Kiryakova Recent history of the fractional calculus: data and statistics | 1 Anatoly N. Kochubei and Yuri Luchko Basic FC operators and their properties | 23 Rudolf Hilfer Mathematical and physical interpretations of fractional derivatives and integrals | 47 Virginia Kiryakova Generalized fractional calculus operators with special functions | 87 Anatoly N. Kochubei General fractional calculus | 111 Virginia Kiryakova and Yuri Luchko Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus | 127 Mateusz Kwaśnicki Fractional Laplace operator and its properties | 159 Yuri Luchko and Virginia Kiryakova Applications of the Mellin integral transform technique in fractional calculus | 195 Yuri Luchko Fractional Fourier transform | 225 Yuri Luchko The Wright function and its applications | 241 Rudolf Gorenflo, Francesco Mainardi, and Sergei Rogosin Mittag-Leffler function: properties and applications | 269 Richard B. Paris Asymptotics of the special functions of fractional calculus | 297

VIII | Contents Moulay Rchid Sidi Ammi and Delfim F. M. Torres Analysis of fractional integro-differential equations of thermistor type | 327 Ricardo Almeida and Delfim F. M. Torres A survey on fractional variational calculus | 347 Teodor M. Atanacković, Sanja Konjik, and Stevan Pilipović Variational principles with fractional derivatives | 361 Mark Meerschaert and Hans Peter Scheffler Continuous time random walks and space-time fractional differential equations | 385 Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy Inverse subordinators and time fractional equations | 407 Mark M. Malamud Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations | 427 Anatoly N. Kochubei Fractional differentiation in p-adic analysis | 461 Index | 473

J. A. Tenreiro Machado and Virginia Kiryakova

Recent history of the fractional calculus: data and statistics Abstract: Fractional Calculus (FC) was a bright idea of Gottfried Leibniz originating in the end of the seventeenth century. The topic was developed mainly in a mathematical framework, but during the last decades FC was recognized to represent an useful tool for understanding and modeling many natural and artificial phenomena. Scientific areas including not only mathematics and physics, but also engineering, biology, finance, economy, chemistry and human sciences successfully applied FC concepts. The huge progress can be measured by the increasing number of papers, books, and conferences. This chapter presents a brief historical sketch and some bibliographic metrics of the evolution that occurred during the previous five decades. Keywords: fractional calculus, fractional order differential equations, fractional order mathematical models MSC 2010: 26A33, 01A60, 01A61, 01A67, 34A08, 35R11, 60G22

1 Introduction In the year 1695 in a letter from Guillaume l’Hôpital to Gottfried Leibniz the question was raised if the derivative of integer order could be extended to fractional values. Leibniz studied the problem for the power function and replied “It leads to a paradox, from which one day useful consequences will be drawn.” During the succeeding centuries the topic was further developed, but it remained in the limited scope of pure mathematics. The first conference dedicated to FC was due to Bertram Ross who organized the International Conference on Fractional Calculus and its Applications at the University of New Haven, during June 1974, and who edited the proceedings [192]. The first monograph is due to Keith B. Oldham and Jerome Spanier [162]. Acknowledgement: The authors thank the colleagues that helped by providing various information data and sources to make the survey as complete as possible. Virginia Kiryakova’s work on this chapter is on the program of the projects (2017–2019) under bilateral agreements of Bulgarian Academy of Sciences with Serbian and Macedonian Academies of Sciences and Arts, and COST Action program CA 15225. J. A. Tenreiro Machado, Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal Virginia Kiryakova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., Block 8, Sofia 1113, Bulgaria https://doi.org/10.1515/9783110571622-001

2 | J. A. T. Machado and V. Kiryakova We can list (in alphabetical order) some important FC researchers since 1695, excluding alive ones. Readers can find further details in [54, 122, 116, 115, 120, 121, 114], and posters at http://www.math.bas.bg/~fcaa. – Abel, Niels Henrik (5 August 1802–6 April 1829), Norwegian mathematician – Al-Bassam, Mohamad Ali (7 December 1923–d. 2004), mathematician of Iraqi origin – Bagley, Ronald (31 May 1947–4 May 2017), American mechanical engineer – Bhrawy, Ali Hassan (11 February 1975–3 February 2017), Egyptian mathematician – Cole, Kenneth Stewart (10 July 1900–18 April 18 1984), American physicist – Cole, Robert H. (26 October 1914–17 November 1990), American physicist – Cossar, James (d. 24 July 1998), British mathematician – Davis, Harold Thayer (5 October 1892–14 November 1974), American mathematician – Djrbashjan, Mkhtar M. (11 September 1918–6 May 1994), Armenian mathematician – Erdélyi, Arthur (2 October 1908–12 December 1977), Hungarian-born British mathematician – Euler, Leonhard (15 April 1707–18 September 1783), Swiss mathematician and physicist – Feller, William (Vilim) (7 July 1906–14 January 1970), Croatian–American mathematician – Fourier, Jean Baptiste Joseph (21 March 1768–16 May 1830), French mathematician and physicist – Gelfand, Israel (Israïl) Moiseevich (2 September 1913–5 October 2009), Russian mathematician – Gemant, Andrew (27 July 1895–1 February 1983), American physicist – Gerasimov, Alexey N. (24 March 1897–14 March 1968) Russian (Soviet) physicist working in the field of mechanics – Gorenflo, Rudolf (31 July 1930–20 October 2017), German mathematician – Grünwald, Anton Karl (23 November 1838–2 September 1920), German mathematician – Hadamard, Jacques Salomon (8 December 1865–17 October 1963), French mathematician – Hardy, Godfrey Harold (7 February 1877–1 December 1947), English mathematician – Heaviside, Oliver (18 May 1850–3 February 1925), English electrical engineer, mathematician, and physicist – Holmgren, Hjalmar J. (22 December 1822–29 August 1885), Swedish mathematician – de l’Hôpital, Guillaume François Antoine (1661–2 February 1704), French mathematician – Kilbas, Anatoly A. (20 July 1948–28 June 2010), Belarusian mathematician – Kober, Hermann (1888–4 October 1973), mathematician born in Poland, studied and worked in Germany, later in Great Britain – Krug, Anton (19th–20th century), German mathematician

Recent history of the fractional calculus: data and statistics | 3

– – – – – – – – – – – – – – – – – – –

– – – – – – – –

Lacroix, Sylvestre François (28 April 1765–24 May 1843), French mathematician Lagrange, Joseph-Louis (25 January 1736–10 April 1813), Italian-French mathematician and astronomer Laplace, Pierre-Simon (23 March 1749–5 March 1827), French mathematician and astronomer Laurent, Paul Matthieu Hermann (2 September 1841–19 February 1908), French mathematician Leibniz, Gottfried Wilhelm (1 July 1646–14 November 1716), German mathematician and philosopher Letnikov, Aleksey V. (1 January 1837–27 February 1888), Russian mathematician Lévy, Paul Pierre (15 September 1886–15 December 1971), French mathematician Liouville, Joseph (24 March 1809–8 September 1882), French mathematician Littlewood, John Edensor (9 June 1885–6 September 1977), British mathematician Love, Eric Russel (31 March 1912–7 August 2001), British-Australian mathematician Marchaud, André (1887–1973), French mathematician Mikolás, Miklós (5 April 1923–2 February 2001), Hungarian mathematician Mittag-Leffler, Magnus Gustaf (Gösta) (16 March 1846–7 July 1927), Swedish mathematician Montel, Paul Antoine Aristide (29 April 1876–22 January 1975), French mathematician Nagy, Béla Szőkefalvi (29 July 1913–22 December 1998), Hungarian mathematician Nayfeh, Ali Hasan (21 December 1933–27 March 2017), Palestinian–American mechanical engineer Nekrasov, Pavel Alekseevich (13 February 1853–20 December 1924), Russian mathematician Newton, Isaac (4 January 1643–31 March 1727), English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian Nigmatullin, Rashid Shakirovich (5 January 1923–7 July 1991), Tatarstan, Russian Federation (Soviet Union) scientist in the field of radio-engineering, radioelectronics, electrochemistry, etc. Nonnenmacher, Theo F. (7 March 1933–1 March 2016), German physicist Pincherle, Salvatore (11 March 1853–10 July 1936), Italian mathematician Post, Emil Leon (11 February 1897–21 April 1954), Polish–American mathematician Rabotnov, Yury Nikolaevich (24 February 1914–15 May 1985), Russian (Soviet) scientist in the field of mechanics Riemann, Georg Friedrich Bernhard (17 September 1826–20 July 1866), German mathematician Riesz, Marcel (16 November 1886–4 September 1969), Hungarian mathematician Ross, Bertram (1917–27 October 1993), American mathematician Rossikhin, Yury (14 March 1944–29 March 2017), Russian scientist in Mechanics

4 | J. A. T. Machado and V. Kiryakova – – – – – – – – – – – –

Rozovskiy, Moses Isaakovich (12 October 1906–1994), Soviet scientist in the field of mathematics and mechanics Scott Blair, George William (23 July 1902–1987), British chemist Shilov, Georgiy Evgen’evich (3 February 1917–17 January 1975), Soviet mathematician Sneddon, Ian Naismith (8 December 1919–4 November 2000), Scottish mathematician Sonine, Nikolay Ya. (22 February 1849–27 February 1915), Russian mathematician Stankovic, Bogoljub (1 September 1924–16 May 2018), Serbian mathematician Tardy, Placido (23 October 1816–2 November 1914), Italian mathematician Wallis, John (23 November 1616–28 October 1703), English mathematician Weierstrass, Karl Theodor Wilhelm (31 October 1815–19 February 1897), German mathematician Weyl, Hermann Klaus Hugo (9 November 1885–8 December 1955), German mathematician Widder, David Vernon (25 March 1898–8 July 1990), American mathematician Zygmund, Antoni (25 December 1900–30 May 1992), Polish-born American mathematician

During the last decades applied sciences paid attention to FC and verified that it constitutes a formidable tool to describe many natural and artificial phenomena embedding long range memory effects, which classical integer order calculus neglects. This state of affairs motivated a huge development, in theoretical, numerical and applied aspects, and the emergence of a plethora of models and new proposals. This chapter presents a brief list of the contributors for the FC progress during four centuries and assembles the main bibliographic information in books available in the time of writing. Evidence for the wide spread of FC as a tool in mathematical disciplines and other related areas is the following. Since 2010, the Mathematics Subject Classification (MSC 2010) includes not only the basic index 26A33, but many other positions related to FC, like 28A80, 33C60, 33E12, 33E30, 34A08, 34K37, 35R11, 44A20, 44A40, 45E10, 60G22, 93B99, 93D05, etc. In MSC 2020, we expect yet new FC related positions to appear. Many of the open problems and hypotheses from the early years (20th century) of FC contemporary development (posed, for example, at 1st and next conferences on FC) have been solved or already out of question. However, new ones, questions about the present state of the art, and challenges for the future were discussed at recent meetings as ICFDA 2014, ICFDA 2016 and ICFDA 2018 (Section 4). Considering the successful development and ever wider popularity of FC, the last years’ flood of publications and topic exploitation could now threaten the FC prestige achieved by the serious efforts and contributions by predecessors. The Round Table sessions [112, 119, 117] aimed at putting forward new problems and challenges and they addressed the needs of the FC community.

Recent history of the fractional calculus: data and statistics | 5

The chapter is organized as follows. Section 2 lists the books edited and books with author, addressing FC. Section 3 reports a list of periodic conferences that are devoted, entirely or partly, to FC. Section 4 provides information on journals specialized on the topics of FC and its applications.

2 Books published In this section we collect the books with author(s) and books with editor(s) published since the middle of the twentieth century up to the year 2018. The list of books with author and with editor(s) is presented in Table 1. The evolution of FC in the light of indices such as the number of published books with author(s) and books edited (more than 195 and 46, respectively) may be not totally representative, as any other measure of scientific progress. Nevertheless, the data collected and represented in Figure 1 is unequivocal. The trends of the type ln N = a + bx, a, b ∈ ℝ, with N for the number of books with author and x for the year reveals a clear pattern of continuous growth (see Figure 2).

3 Conferences devoted to fractional calculus In this section we list some conferences dedicated, entirely or partly (with special sessions), to FC during the last decades. FC conferences without periodicity are not included. For more details, such as organizers, places, timetable, and other information, see the survey [114]. – Action thématique “Les systèmes à dérivée non entière” (SDNE 2001, 2002, 2003, 2004, 2005, 2006) – Conference on Non-integer Order Calculus and its Applications (Rα RNR 2009, 2010, 2011, 2012, . . . , 2016, 2017) – Fractional Calculus Day (FCDay 2009, 2013, 2015, 2016, 2017) – International Carpathian Control Conference (ICCC 2000, 2001, 2002, 2003, . . . , 2016, 2017, 2018) – International Conference on Analytic Methods of Analysis and Differential Equations (AMADE 1999, 2001, 2003, 2006, 2009, 2011, 2012, 2015, 2018) – International Conference (Workshop) on Fractional Differentiation and its Applications (FDA 2004, 2006, 2008, 2010, 2012, 2013; ICFDA 2014, 2016, 2018) – International Conference Modern Methods, Problems and Applications of Operator (OTHA 2011, 2012, . . . , 2017, 2018) – International Conference Transform Methods and Special Functions (TMSF 1994, 1996, 1999, 2003, 2010, 2011, 2014, 2017)

6 | J. A. T. Machado and V. Kiryakova Table 1: Published books with author and with editor(s) since the 60s. Year

Books with author

Books with editor

1960–1979 1980–1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

[55, 205, 38, 135, 162, 186, 136, 206, 139] [209, 210, 166, 155, 183, 202, 156, 157] [212] [159, 158, 167, 71] [208, 39] [203, 149] [242, 96, 54] [168] [160, 193] [40, 132, 91] [143] [179, 170, 224, 234]

[192] [140] [78]

2012 2013 2014 2015 2016 2017 2018

– –

–

[12, 19] [237, 177, 63, 83, 82] [233, 174, 153] [94] [84, 184, 79] [124, 95] [246, 220, 128, 134, 241, 240, 228, 20, 30, 49] [8, 102, 137, 33, 100, 87, 175] [125, 37, 152, 187, 52] [176, 163, 98, 75, 141, 105, 5, 213, 243, 88, 50, 106, 195, 101] [21, 11, 161, 2, 111, 85, 204, 127, 250, 51, 227, 244, 221] [43, 65, 172, 81, 222, 80, 219, 42, 27, 238] [76, 247, 129, 13, 14, 86, 89, 144, 169, 3, 70, 28, 64, 145] [181, 180, 164, 9, 69, 7, 225, 60, 72, 18, 131, 199, 171, 26, 108, 235, 188, 194, 25, 245, 126, 35, 146] [34, 248, 44, 59, 74, 154, 182, 109, 113, 173, 249, 147, 185] [36, 239, 61, 231, 31, 68, 66, 236, 15, 217, 218, 133, 4, 201, 200, 178, 17] [226, 230, 10, 148, 67, 1, 211, 58, 29, 223, 232, 189, 103, 130]

[90] [229] [196]

[197, 138] [77] [22]

[142, 57] [216, 198] [150, 92, 99] [123, 23] [110, 118, 73] [97, 24] [93, 191] [46, 151, 32, 107, 6] [47, 48, 45, 62] [104] [190, 53, 207, 41]

[215, 214, 56, 165, 16]

International Symposium on Fractional Signals and Systems (FSS 2009, 2011, 2013, 2015, 2017) Mini-Symposium: Fractional Derivatives and Their Applications FDTA at EUROMECH Nonlinear Dynamics Conference (FDTA-ENOC 2005, 2008, 2011, 2014, 2017) Symposium on Fractional Derivatives and Their Application (FDTA 2003, 2005, . . . , 2015, 2016, 2017).

Recent history of the fractional calculus: data and statistics | 7

Figure 1: Timeline of FC during 1650–1950.

8 | J. A. T. Machado and V. Kiryakova

Figure 2: Number N of books with author and with editor versus year x, and trends of the type ln N = a + bx, a, b ∈ ℝ during the period 1965–2018.

4 Journals specialized in fractional calculus Finally, we present a list of journals specialized in FC, in order of their appearance. Since their names are somewhat similar, it is important to distinguish these journals by exact wording and abbreviations. Journal of Fractional Calculus (JFC); ISSN 0918-5402; Publisher: Descartes Press, Japan; Ed.-in-Chief: Katsuyuki Nishimoto (Japan); Starting year: Vol. 1 (1992), info available for Vol. 21–24 (2002), next information n/a at Internet. Fractional Calculus and Applied Analysis (FCAA; Fract. Calc. Appl. Anal.); ISSN: 1311-0454 (print), ISSN: 1314-2224 (online); Publishers: Institute of Math. and Inform. – Bulg. Acad. Sci. (1998–2010), Versita and Springer (2011–2014), De Gruyter (since 2015); Website (current): http://www.degruyter.com/view/j/fca; Ed.-in-Chief: Virginia Kiryakova (Bulg. Acad. Sci., Bulgaria);

Recent history of the fractional calculus: data and statistics | 9

Starting year: Vol. 1 (1998), current Vol. 21 (2018); 6 issues / year; Impact Factor (SCI) IF (2017) = 2.865, 5-year IF = 3.323, CiteScore = 3.06. Communications in Fractional Calculus (CFC); ISSN: 2218-3892; Publisher: Asian Academic Publisher Ltd, China; Website was: http://www.nonlinearscience.com/journal_2218-3892.php (now n/a), some currently available traces: http://en.journals.sid.ir/JournalList.aspx?ID=24970 http://blog.sciencenet.cn/blog-298018-343352.html; Editor: Lan Xu; Starting year: Vol. 1 (2010), next information n/a. Fractional Differential Calculus (FDC; earlier: Fractional Differential Equations); ISSN: 1847-9677; Publisher: Ele-Math (Element d.o.o.), Croatia; Website: http://fdc.ele-math.com/; Current Eds.-in-Chief: Mokhtar Kirane (Univ. de La Rochelle, France), Josip Pečarić (Univ. of Zagreb, Croatia), Sabir Umarov (Univ. of New Haven, USA); Starting year: Vol. 1 (2011), current Vol. 8 (2018); 2 issues / year. Journal of Fractional Calculus and Applications (JFCA); ISSN: 2090-584X (print), ISSN; 2090-5858 (print); Publisher: South Valley University, Egypt; Website: http://fcag-egypt.com/journals/jfca/; Managing Eds.: A. M. A. El-Sayed (Alexandria Univ., Egypt), S. Z. Rida (South Valley Univ., Egypt); Starting year: Vol. 1 (2011), current Vol. 9 (2018); 2 issues / year. Progress in Fractional Differentiation and Applications (PFDA; Progr. Fract. Differ. Appl.); ISSN 2356-9336 (print), ISSN 2356-9344 (online); Publisher: Natural Sciences Publ., USA; Website: http://naturalspublishing.com/show.asp?JorID=48&pgid=0; Ed.-in-Chief: Dumitru Baleanu (Çankaya University, Turkey); Starting year: Vol. 1 (2015), current Vol. 4 (2018); 4 issues / year. Fractal and Fractional (FF; Fractal Fract.); ISSN 2504-3110, Open Online Access; Publisher: MDPI AG, Switzerland; Website: http://www.mdpi.com/journal/fractalfract; Ed.-in-Chief: Carlo Cattani (University of Tuscia, Italy); Starting year: Vol. 1 (2017), No 1 (Dec. 2017), current Vol. 2 (2018).

10 | J. A. T. Machado and V. Kiryakova Electronic Newsletter “FDA Express” (Fractional Derivative and Applications Express) distributed monthly by e-mail to subscribers and available online; Publisher: Institute of Soft Matter Mechanics, Hohai University, China; Website: http://em.hhu.edu.cn/fda/; Editor: Wen Chen, Team: Hongguang Sun, Guofei Pang, Xindong Hei, Yingjie Liang and Lin Chen (China); Starting year: Vol. 1 (2011), current Vol. 26 (2018); 3 vols. (12 issues) / year. Several special and topical issues of many other prestigious journals are dedicated to the topic of FC and its applications. Also, there are many websites and blogs created by colleagues, where useful information, files, links, etc. related to FC are available. More details as regards special issues of journals, courses, tutorials, resources from Matlab, Wolfram MathWorld, other packages on computational aspects for FC, patents, and others were provided in our previous survey of 2011, [122].

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S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, Implicit Fractional Differential and Integral Equations, De Gruyter, 2018. S. Abbas, M. Benchohra, and G. M. N’Guérékata, Topics in Fractional Differential Equations, Developments in Mathematics, vol. 27, Springer, New York, 2012. S. Abbas, M. Benchohra, and G. M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Mathematics Research Developments, Nova Science Publishers, 2014. B. Ahmad, A. Alsaedi, S. K. Ntouyas, and J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017. S. Al-Azawi, Some Results in Fractional Calculus, LAP Lambert Acad. Publ., 2011. A. Almeida, L. F. Castro, and F.-O. Speck (eds.), Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, Birkhäuser, Basel, 2013. R. Almeida, S. Pooseh, and D. F. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015. G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, Heidelberg, 2009. G. A. Anastassiou and I. K. Argyros, Intelligent Numerical Methods: Applications to Fractional Calculus, Studies in Computational Intelligence, Springer, Cham, 2015. G. A. Anastassiou and I. K. Argyros, Functional Numerical Methods: Applications to Abstract Fractional Calculus, Springer, Cham, 2018. M. H. Annaby and Z. S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics, vol. 2056, Springer, Heidelberg, 2012. P. Arena, R. Caponetto, and L. F. M. Porto, Nonlinear Noninteger Order Systems: Theory and Applications, Nonlinear Science, World Scientific Publishing Company, Singapore, 2001. T. M. Atanacković, S. Pilipović, B. Stanković, and D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, Mechanical Engineering and Solid Mechanics, Wiley-ISTE, Croydon, 2014.

Recent history of the fractional calculus: data and statistics | 11

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Anatoly N. Kochubei and Yuri Luchko

Basic FC operators and their properties Abstract: In this chapter, a brief introduction to the main notions and constructions of fractional calculus operators is presented. We start with the Riemann–Liouville integrals and derivatives and continue with other kinds of fractional integrals and derivatives, including the Caputo derivative, the Grünwald–Letnikov derivative, the Marchaud derivative, the Weyl integrals and derivatives of periodic functions, the Erdelyi– Kober integrals and derivatives, the Hadamard integrals and derivatives, and the Riesz and the Feller potentials and fractional derivatives. We also mention the fractional calculus operators introduced by Hilfer, Nakhushev, and Pskhu, as well as the fractional derivatives of distributed order and the general fractional integrals and derivatives. Keywords: fractional integral, fractional derivative, fractional Laplacian MSC 2010: 26A33, 35S05

1 Introduction The theory of differentiation and integration of an arbitrary (not necessarily integer) order, nowadays referred to as the Fractional Calculus (FC), is nearly as old as conventional calculus. Leibniz, L’Hôspital, Jacob Bernoulli, Euler, Laplace, Fourier and other prominent mathematicians who essentially contributed to the development of analysis, tried to interpret some of their results for integrals and derivatives of non-integer order and suggested first ideas regarding possible definitions of fractional derivatives. In particular, Euler introduced his Gamma-function as an extension of the factorial function and noticed that the formula for the nth derivative of a power function can be meaningfully interpreted also in the case of a non-integer order derivative. In Abel’s paper [1], devoted to the generalization of the tautochrone problem and published in 1823, he introduced both the fractional integrals called nowadays Riemann–Liouville fractional integrals and the fractional derivatives called nowadays Caputo fractional derivatives, and he employed their relationship to derive a closed form solution to the generalized tautochrone problem. Unfortunately, in his more famous paper [2], devoted to the same problem, he did not use notations for the

Anatoly N. Kochubei, National Academy of Sciences of Ukraine, Institute of Mathematics, Tereshchenkivska 3, 01004 Kyiv, Ukraine, e-mail: [email protected] Yuri Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110571622-002

24 | A. N. Kochubei and Yu. Luchko fractional integrals and derivatives anymore and thus his pioneering contribution to FC remained mostly unknown (see the recent paper [41] for more details). In the series of publications that started with [29], dated 1832, Liouville systematically worked out a rather complete theory of fractional integro-differentiation. His first idea was to introduce a formula for fractional differentiation of series with respect to the exponential functions. Later on, he introduced the “Liouville” form of the fractional integral and considered a series of its applications in physics. Another approach to defining the fractional derivatives was suggested in the work by Grünwald [12] and Letnikov [28]. For a given function f , they introduced the differences of an arbitrary order α > 0 with a step h ∈ ℝ and then defined the fractional derivatives as the limits of the quotients of the corresponding differences like in the case of the integer order derivatives. Later on, several other important definitions of the fractional integro-differential operators were suggested (see [46] for a very detailed and excellent written survey of the FC history). They were all motivated either by mathematical considerations (say, by extension of the existing FC operators to the new classes of functions) or by applications. In this short survey, we present the most important notions of fractional derivatives and integrals and their basic properties. The rest of the chapter is organized as follows. In Section 2, the Riemann–Liouville fractional integral defined on the finite and infinite intervals and its basic properties are discussed. Section 3 is devoted to the Caputo fractional derivative. Also the Hilfer derivative, which interpolates between the Riemann–Liouville and the Caputo derivatives, is mentioned. In Section 4, some other important fractional derivatives and integrals including the Grünwald–Letnikov derivative, the Marchaud derivative, the Weyl integrals and derivatives of periodic functions, the Erdelyi–Kober integrals and derivatives, the Hadamard integrals and derivatives, and the Riesz and the Feller potentials and fractional derivatives as well as the FC notation suggested by Nakhushev and Pskhu, the fractional derivatives of distributed order, and the general fractional integrals and derivatives are introduced and some of their most important properties are mentioned.

2 The Riemann–Liouville fractional integrals and derivatives In the mathematical treatises related to FC, the Riemann–Liouville fractional integrals and derivatives are the most widely investigated and used forms of the fractional integro-differential operators. An in-depth investigation of these operators, their properties, applications, and various generalizations was presented in [46]. In this section, their definitions and some of the basic properties are briefly discussed.

Basic FC operators and their properties | 25

2.1 Riemann–Liouville fractional integrals on a finite interval In an attempt to interpret the well-known formula for the n-fold definite integral x

x

x

x

a

a

a

a

1 ∫ dx ∫ dx⋅ ⋅ ⋅ ∫ f (x)dx = ∫(x − t)n−1 f (t) dt (n − 1)! for non-integer values of n, the main problem is how to extend the domain of the factorial function (n − 1)!, n ∈ ℕ, to arbitrary real (or even complex) values of the argument. Evidently, there are infinite many candidates for such extension and some additional conditions are required for uniqueness of solution to this problem. If we look for solutions of the functional equation (which uniquely defines the factorial function on ℕ) f (s + 1) = sf (s),

f (1) = 1,

(1)

in the class of analytical functions under the additional conditions f (s) ≠ 0, s ∈ ℂ and f = f (s) is logarithmically convex for s > 0, then the solution to (1) is unique and coincides with the Euler Gamma-function (see, e. g., [38]) defined by the convergent integral ∞

Γ(s) = ∫ e−t t s−1 ds

(2)

0

for ℜ(s) > 0 and by its analytic continuation for other values of s ∈ ℂ. It means that the Euler Gamma-function is a natural extension of the factorial function and thus plays a prominent role in FC. Moreover, its integral representation (2) read from right to left can be interpreted as the Mellin integral transform of the exponential function e−t . For the role of the Mellin integral transform in FC we refer the interested reader to [34]. Having defined an extension of the factorial function, we are ready to introduce an α-fold definite integral. Let (a, b) ⊂ ℝ be a finite interval, f ∈ L1 (a, b), and α > 0 (ℜ(α) > 0). The left- and right-hand sided Riemann–Liouville fractional integrals of order α of a function f are defined as follows: α (Ia+ f )(x) =

α (Ib− f )(x)

x

1 ∫(x − t)α−1 f (t) dt, Γ(α)

x ∈ (a, b),

(3)

x ∈ (a, b).

(4)

a

b

1 = ∫(t − x)α−1 f (t) dt, Γ(α) x

In the case α = 0, both integrals are interpreted as the identity operators: 0 (Ia+ f )(x) = f (x),

0 (Ib− f )(x) = f (x).

(5)

26 | A. N. Kochubei and Yu. Luchko The definitions given by (5) are justified by the formulas [46] α lim (Ia+ f )(x) = f (x),

α→0+

α lim (Ib− f )(x) = f (x),

α→0+

(6)

which are valid for f ∈ L1 (a, b) in every Lebesgue point of f , i. e., almost everywhere on [a, b]. Evidently, the left- and right-hand sided Riemann–Liouville fractional integrals are linear operators: α α α f2 )(x), f1 )(x) + c2 (Ia+ (c1 f1 + c2 f2 ))(x) = c1 (Ia+ (Ia+

α (c1 f1 (Ib−

+ c2 f2 ))(x) =

α f1 )(x) c1 (Ib−

+

α f2 )(x), c2 (Ib−

(7) (8)

α α α α for any c1 , c2 ∈ ℝ provided that Ia+ f1 and Ia+ f2 or Ib− f1 and Ib− f2 , respectively, exist almost everywhere on [a, b]. The left- and right-hand sided Riemann–Liouville fractional integrals are connected by means of a simple variable substitution. With the notation (Rf )(x) = f (a + b − x) we have the relations α α R, = Ib− RIa+

α α R. = Ia+ RIb−

(9)

The well-known formula for the Euler Beta-function 1

B(s, t) = ∫ τs−1 (1 − τ)t−1 dτ = 0

Γ(s)Γ(t) , Γ(s + t)

ℜ(s) > 0, ℜ(t) > 0,

and the Fubini theorem immediately lead to the important semigroup property of the Riemann–Liouville fractional integrals: β

α+β

α Ia+ Ia+ = Ia+ ,

β

α+β

α Ib− Ib− = Ib− .

(10)

For applications, a formula for integration by parts plays a significant role: b

b

a

a

α α f )(x) dx = ∫ f (x)(Ib− g)(x). ∫ g(x)(Ia+

(11)

It holds true, say, for f ∈ Lp (a, b), q ∈ Lq (a, b) under the conditions p ≥ 1, q ≥ 1, and 1/p + 1/q ≤ 1 + α (and p ≠ 1, q ≠ 1 in the case 1/p + 1/q = 1 + α). The left- and right-hand sided Riemann–Liouville fractional derivatives are introduced as the left-inverse operators to the corresponding fractional integrals. Let n−α n − 1 < α ≤ n, n ∈ ℕ and a function f satisfy the condition Ia+ f ∈ AC n [a, b], where n n−1 (n−1) AC [a, b] = {f : (f ∈ C [a, b]) ∩ (f ∈ AC[a, b])} and by AC[a, b] we denote the space of absolutely continuous functions on the interval [a, b]. The left- and right-hand

Basic FC operators and their properties | 27

sided Riemann–Liouville fractional derivatives of order α of the function f are defined as follows: dn n−α (I f )(x), x ∈ (a, b), dxn a+ dn n−α f )(x), x ∈ (a, b). (Dαb− f )(x) = (−1)n n (Ib− dx (Dαa+ f )(x) =

(12) (13)

Due to (5), the Riemann–Liouville fractional derivatives are reduced to the nth derivatives for α = n ∈ ℕ: (Dna+ f )(x) =

dn f , dx n

(Dnb− f )(x) = (−1)n

dn f , dxn

x ∈ (a, b).

(14)

Because of formulas (7) and (8), the left- and right-hand sided Riemann–Liouville fractional derivatives are linear operators. In what follows, we mostly deal with the left-hand sided Riemann–Liouville fractional integrals and derivatives. The corresponding formulas for the right-hand sided operators can easily be obtained from the ones for the left-hand sided operators and the relations (9). n−α Let us mention that the inclusion Ia+ f ∈ AC n [a, b] follows from the inclusion f ∈ AC n [a, b]. In the last case, the Riemann–Liouville fractional derivatives exist almost everywhere on the interval (a, b) and we get the formula (Dαa+ f )(x)

x

n−1

f (n) (t) dt 1 f (k) (a) . (x − a)k−α + = ∑ ∫ Γ(1 + k − α) Γ(n − α) (x − t)α−n+1 k=0

(15)

a

The integral at the right-hand side of (15) is a composition of a fractional integral and the nth derivative like in formula (12), but in reverse order: x

1 f (n) (t) dt n−α (n) = (Ia+ f )(x), ∫ Γ(n − α) (x − t)α−n+1

x ∈ (a, b).

(16)

a

In Section 2, we deal with this object that nowadays is referred to as the Caputo fractional derivative. As already mentioned, the Riemann–Liouville fractional derivative is the leftinverse operator to the corresponding Riemann–Liouville fractional integral, i. e., the relation α (Dαa+ Ia+ f )(x) = f (x)

(17)

holds true for any function f ∈ L1 (a, b). As to the composition of the Riemann– Liouville integral and the Riemann–Liouville derivative, the formula n−1

(x − a)α−k−1 dn−k−1 n−α ( n−k−1 Ia+ f )(a) Γ(α − k) dx k=0

α α (Ia+ Da+ f )(x) = f (x) − ∑

(18)

28 | A. N. Kochubei and Yu. Luchko n−α holds true for any function f ∈ L1 (a, b) that satisfies the condition Ia+ f ∈ AC n [a, b]. If a function f can be represented as a Riemann–Liouville integral of g ∈ L1 (a, b), i. e., if α f (x) = (Ia+ g)(x), then it follows from (17) that α α (Ia+ Da+ f )(x) = f (x).

(19)

The function space that consists of all functions f that can be represented as the α Riemann–Liouville fractional integrals of functions from Lp (a, b) (f (x) = (Ia+ g)(x), p α p α p g ∈ L (a, b)) is denoted by Ia+ (L ), ℜ(α) > 0. The space of functions Ib− (L ), ℜ(α) > 0 is defined as above with the corresponding adaptations. For the Riemann–Liouville fractional derivatives, there exists a formula for integration by parts similar to (11): b

b

a

a

∫ f (x)(Dαa+ g)(x) dx = ∫ g(x)(Dαb− f )(x).

(20)

α α This formula follows from (11) and holds true, say, for f ∈ Ib− (Lp ) and q ∈ Ia+ (Lp ) under conditions p ≥ 1, q ≥ 1, and 1/p+1/q ≤ 1+α (and p ≠ 1, q ≠ 1 in the case 1/p+1/q = 1+α). In some cases, it is convenient to use the same symbol for the Riemann–Liouville α fractional derivatives and integrals. In the rest of this subsection, we denote by Ia+ the Riemann–Liouville fractional integral of order α if ℜ(α) ≥ 0 and the Riemann– Liouville fractional derivative of order −α if ℜ(α) < 0. Similarly, Dαa+ stands for the Riemann–Liouville fractional derivative of order α if ℜ(α) ≥ 0 and for the Riemann– Liouville fractional integral of order −α if ℜ(α) < 0. The semigroup property (10), which was formulated for the Riemann–Liouville fractional integrals, can be extended to some compositions of the fractional integrals and derivatives [46]. More precisely, the formula β

α+β

α (Ia+ Ia+ f )(x) = (Ia+ f )(x)

(21)

is valid in the following cases: 1) ℜ(β) > 0, ℜ(α + β) > 0, f ∈ L1 (a, b); −β 2) ℜ(β) < 0, ℜ(α) > 0, f ∈ Ia+ (L1 ); −α−β 3) ℜ(α) < 0, ℜ(α) < 0, f ∈ Ia+ (L1 ). For analytical functions f and g, the semigroup formula β

α+β

(Dαa+ Da+ f )(x) = (Da+ f )(x)

(22)

is valid for α ∈ ℝ and β < 1 [46]. This property is employed to prove the Leibniz type rule for the Riemann–Liouville fractional derivative that can be written in different forms like, e. g., ∞ α (k) (Dαa+ f ⋅ g)(x) = ∑ ( )(Dα−k a+ f )(x)g (x), k k=0

α ∈ ℝ,

(23)

Basic FC operators and their properties | 29 +∞

(Dαa+ f ⋅ g)(x) = ∑ ( k=−∞

α α−β−k β+k )(Da+ f )(x)(Da+ g)(x), k+β

(24)

α, β ∈ ℝ (α ≠ −1, −2, . . . if β ∈ ̸ ℤ), Γ(α+1) with the generalized binomial coefficients ( αβ ) = Γ(β+1)Γ(α−β+1) . In particular, formulas (23) and (24) are valid for the functions f and g analytical on (a, b). Other forms of the Leibniz rule for the Riemann–Liouville fractional derivatives including the integral form were deduced in [32, 50] by using an operational method. The Leibniz rule (23) was shown to be a direct consequence of the well-known summation theorem for the Gauss hypergeometric function 2 F1 saying that 2 F1 (a, b; c; 1)

=

Γ(c)Γ(c − a − b) , Γ(c − a)Γ(c − b)

ℜ(c − a − b) > 0,

where the Gauss function 2 F1 is defined as the series 2 F1 (a, b; c; z)

=

Γ(c) ∞ Γ(a + n)Γ(b + n) z n ∑ Γ(a)Γ(b) n=0 Γ(c + n) n!

for |z| ≤ 1 and ℜ(c − a − b) > 0 and as an analytic continuation of this series for other values of z. It is worth mentioning that violation of the standard Leibniz rule is one of the characteristic properties of the fractional derivatives of any kind. More precisely, the formula Dα (f ⋅ g) = (Dα f ) ⋅ g + f (Dα g) with any fractional derivative is valid only for α = 1 [49]. For further properties of the Riemann–Liouville integrals and derivatives on a finite interval including their mapping properties on the spaces H λ (a, b) of the Hölder functions, Lp (a, b) of the Lebesque integrable functions, and on the spaces H λ (a, b) and Lp (a, b) with some weights, we refer the reader to [46].

2.2 Riemann–Liouville fractional integrals on infinite intervals The definitions (3) and (4) of the left- and right-hand sided Riemann–Liouville fractional integrals can be used in the case of the semi-infinite intervals (a, +∞) (formula (3) with b = +∞) and (−∞, b) (formula (4) with a = −∞), respectively. For a = 0, the left-hand sided Riemann–Liouville fractional integral on the semi-infinite interval (0, +∞) is defined as follows: α (I0+ f )(x)

x

1 = ∫(x − t)α−1 f (t) dt, Γ(α) 0

x > 0.

(25)

30 | A. N. Kochubei and Yu. Luchko The only differences in the theory of these fractional integrals compared to the case of the Riemann–Liouville integrals defined on the finite intervals are their domains (the integrals have to be convergent in +∞ or −∞, respectively) and the corresponding mapping properties. Usually, these integrals are treated either on Lp (ℝ) with 1 < p < 1/α or on Lp (ℝ) with some weights, or on the spaces of the Hölder functions with some weights that tend to zero at infinity. On the whole axis ℝ, the left- and right-hand sided Riemann–Liouville fractional integrals of order α > 0 (or ℜ(α) > 0) are defined as follows: (I+α f )(x)

x

1 = ∫ (x − t)α−1 f (t) dt, Γ(α)

x ∈ ℝ,

(26)

1 ∫ (t − x)α−1 f (t) dt, Γ(α)

x ∈ ℝ.

(27)

(I−α f )(x) =

−∞ +∞ x

In the case α = 0, both integrals are interpreted as the identity operators: (I+0 f )(x) = f (x),

(I−0 f )(x) = f (x).

(28)

The fractional integrals I± are well defined for f ∈ Lp (ℝ) under the conditions 0 < α < 1 and 1 ≤ p < 1/α. The properties of the Riemann–Liouville fractional integrals mentioned in the previous subsection are also valid with some suitable modifications and restrictions for the Riemann–Liouville integrals on the infinite intervals. Evidently, the left- and right-hand sided Riemann–Liouville fractional integrals on infinite intervals are linear operators. The left- and right-hand sided Riemann–Liouville fractional integrals I±α are connected by the relation (compare to formula (9)) QI±α = I∓α Q,

(Qf )(x) = f (−x),

x ∈ ℝ.

(29)

The semigroup property (10) and the integration by parts formula (11) are also valid for the Riemann–Liouville fractional integrals on the infinite intervals: β

+∞

α+β

α I0+ I0+ = I0+ ,

β

α+β

I±α I± = I± ,

(30)

+∞

α f )(x) dx = ∫ f (x)(I−α g)(x), ∫ g(x)(I0+

(31)

∫ g(x)(I+α f )(x) dx = ∫ f (x)(I−α g)(x).

(32)

0 +∞

−∞

0 +∞

−∞

The integration by part formulas hold true, e. g., for g ∈ Lp , f ∈ Lq under conditions p > 1, q > 1 and 1/p + 1/q = 1 + α [46].

Basic FC operators and their properties | 31

The left- and right-hand sided Riemann–Liouville fractional derivatives on the semi-infinite intervals (a, +∞) and (−∞, b) are defined by formula (12) with b = +∞ and by formula (13) with a = −∞, respectively. In the case of the whole real axis, the Riemann–Liouville left- and right-hand sided fractional derivatives of order α, n − 1 < α ≤ n, n ∈ ℕ, are defined as follows: dn n−α (I f )(x), x ∈ ℝ, dx n + dn (Dα− f )(x) = (−1)n n (I−n−α f )(x), x ∈ ℝ. dx (Dα+ f )(x) =

(33) (34)

Because of (28), the Riemann–Liouville fractional derivatives Dα± are reduced to the nth derivatives for α = n ∈ ℕ: (Dn+ f )(x) =

dn f , dxn

(Dn− f )(x) = (−1)n

dn f , dx n

x ∈ ℝ.

(35)

Of course, the fractional derivatives Dαa+ , Dαb− , Dα± are linear operators, too. The left- and right-hand sided Riemann–Liouville fractional derivatives defined on the infinite intervals are left-inverse operators to the corresponding Riemann– Liouville fractional integrals for the functions from L1 (ℝ). To make this property valid for Lp (ℝ), p > 1, the definition of the fractional derivatives Dα± has to be appropriately extended, namely, to the Marchaud fractional derivative, which appears to be more suitable in the case of the whole real axis. The Marchaud fractional derivative will be introduced in Section 4. The integration by parts formula (20) is valid also for Dα± with appropriate modifications. In applications, integral transforms of the fractional integrals and derivatives are often employed. In the rest of this section, we present the basic formulas for the Fourier, Laplace, and Mellin integral transforms of the Riemann–Liouville fractional integrals and derivatives. If 0 < ℜ(α) < 1 and f ∈ L1 (ℝ) then the Fourier transform of the Riemann–Liouville fractional integrals is given by the formula [46] {ℱ (I±α f )(x); κ} = {ℱ f (x); κ}/(∓iκ)α ,

(36)

where the complex valued function (∓iκ)α is understood as follows: (∓iκ)α = exp(α ln |κ| ∓

απi sign(κ)). 2

For ℜ(α) ≥ 1, the Fourier transform of the Riemann–Liouville fractional integrals at the left-hand side of formula (36) does not exist in a usual sense even for very smooth functions. However, formula (36) remains valid for all α with ℜ(α) > 0 in a special subspace Φ(ℝ) of the space S of rapidly decreasing test functions. Let us denote by V(ℝ) the set of functions v ∈ S satisfying the conditions dn v 󵄨󵄨󵄨󵄨 = 0, 󵄨 dx n 󵄨󵄨󵄨x=0

n = 0, 1, 2, . . . .

32 | A. N. Kochubei and Yu. Luchko The space of functions Φ(ℝ) is defined as the Fourier pre-image of V(ℝ) in S: Φ(ℝ) = {φ ∈ S : φ̂ ∈ V(ℝ)}. Φ(ℝ) is referred to as the Lizorkin space of functions [30, 31] and it is very convenient for problems involving fractional integrals and derivatives [46]. Under suitable conditions, the Fourier transform of the Riemann–Liouville fractional derivatives is given by the formula {ℱ (Dα± f )(x); κ} = (∓iκ)α {ℱ f (x); κ}.

(37)

While employing the operational relations (36) and (37), potential problems with different branches of the multi-valued complex function (∓iκ)α might appear. To avoid these problems and to simplify the operational relations (36) and (37), a special approach to the fractional Fourier transform was suggested [33]. With respect to this fractional Fourier transform, the operational relations of type (36) and (37) look very similar to the ones for the integer order derivatives with respect to the conventional Fourier transform [33]. α can be interpreted as the Laplace The Riemann–Liouville fractional integral I0+ convolution α f )(x) = (f ∗ (I0+

t+α−1 )(x), Γ(α)

t, t ≥ 0, ℜ(α) > 0, t+ = { 0, t < 0.

(38)

The Laplace convolution theorem leads then to the formula α f )(x); p} = p−α {ℒf (x); p}, {ℒ(I0+

ℜ(α) > 0,

(39)

which is valid for ℜ(p) > p0 for the functions f ∈ L1 (0, b), ∀b > 0, such that |f (x)| ≤ Aep0 x ,

x > b, A > 0, p0 ≥ 0.

(40)

Formula (39) holds true also for negative values of ℜ(α), i. e., for the Riemann– Liouville fractional derivative under some additional conditions. Let n − 1 < ℜ(α) ≤ n, n ∈ ℕ. If f ∈ AC n [0, b] for any b > 0, f (k) (0) = 0, k = 0, 1, . . . , n − 1 and the estimate (40) is valid, then {ℒ(Dα0+ f )(x); p} = pα {ℒf (x); p}.

(41)

If not all derivatives of f of order k = 0, 1, . . . , n − 1 at the point zero are equal to zero, we have a more general formula, n−1

dk n−α (I0+ f )(x)pn−k−1 . x→0+ dx k k=0

{ℒ(Dα0+ f )(x); p} = pα {ℒf (x); p} − ∑ lim

(42)

Basic FC operators and their properties | 33

k

d n−α It is worth mentioning that the expressions of the form limx→0+ dx k (I0+ f )(x) appear also in a natural formulation of the initial conditions for the fractional differential equations with the Riemann–Liouville fractional derivatives. The Mellin integral transform of the fractional integrals and derivatives is discussed in detail in [34]. Here we just list the formulas

Γ(1 − α − s) {ℳf (x); s + α}, ℜ(s + α) < 1, Γ(1 − s) Γ(s) {ℳf (x); s + α}, ℜ(s) > 0, {ℳ(I−α f )(x); s} = Γ(s + α)

α {ℳ(I0+ f )(x); s} =

(43) (44)

which are valid for ℜ(α) > 0 and if f (x)xs+α−1 ∈ L1 (ℝ+ ). Finally, we mention the important formula (Dα0+ xp )(x) =

Γ(p + 1) p−α x , Γ(p − α + 1)

α > 0, p > −1,

(45)

which can easily be obtained using the well-known relation between the Euler Betaand Gamma-functions. In particular, the Riemann–Liouville fractional derivative of order α of a constant C is not equal to zero unless if α = n ∈ ℕ: (Dα0+ C)(x) =

Cx −α . Γ(1 − α)

(46)

3 The Caputo fractional derivative In the theoretically oriented publications regarding FC, the Riemann–Liouville fractional derivatives as a natural generalization of the integer order derivatives are often employed. However, in applications there are some problems with these derivatives including the non-standard formulation of the initial conditions in the fractional differential equations with the Riemann–Liouville derivatives and the fact that the Riemann–Liouville fractional derivative of a constant is not equal to zero. Thus a regularized Riemann–Liouville fractional derivative was introduced not only in theoretical treatises [6, 22] but also for models of viscoelastic materials with memory [9], creep of structural members [43], seismographic waves [4], etc. Nowadays this fractional derivative is usually referred to as the Caputo (or Caputo–Djrbashian) fractional derivative. For a historical survey of publications devoted to theory and applications of this derivative we refer the interested reader to [44]. Let n − 1 < α ≤ n, n ∈ ℕ and f ∈ AC n [a, b]. Then the Caputo fractional derivative of order α of the function f is defined by (∗ Dαa+ f )(x) = (Dαa+ (f − Tn−1 (f , a)))(x),

(47)

where Dαa+ is the Riemann–Liouville fractional derivative (12) and Tn−1 (f , a) is the Taylor polynomial for f of degree n − 1 centered at the point a.

34 | A. N. Kochubei and Yu. Luchko If f (n) ∈ L1 (a, b), then it is possible to interchange the orders of differentiation and integration in (47) and we get another form of the Caputo fractional derivative: (∗ Dαa+ f )(x)

=

n−α (n) (Ia+ f )(x)

x

f (n) (t) dt 1 = . ∫ Γ(n − α) (x − t)α−n+1

(48)

a

Comparing this formula with formula (15), we arrive at the following relation between the Caputo and the Riemann–Liouville fractional derivatives: n−1

f (k) (a) (x − a)k−α , Γ(1 + k − α) k=0

(∗ Dαa+ f )(x) = (Dαa+ f )(x) − ∑

(49)

which is valid for f ∈ AC n [a, b]. Of course, this relation also follows directly from formulas (45) and (47). In particular, the relation (49) means that the Riemann–Liouville and the Caputo derivatives of order α ≠ n ∈ ℕ coincide if and only if f (k) (a) = 0, k = 0, . . . , n − 1. In the case α = n ∈ ℕ both derivatives are reduced to the nth order derivative. The Caputo fractional derivative is a linear operator: (∗ Dαa+ (c1 f1 + c2 f2 ))(x) = c1 (∗ Dαa+ f1 )(x) + c2 (∗ Dαa+ f2 )(x)

(50)

for any c1 , c2 ∈ ℝ provided that ∗ Dαa+ f1 and ∗ Dαa+ f2 exist almost everywhere on [a, b]. If f ∈ C[a, b], the Caputo fractional derivative ∗ Dαa+ is a left-inverse operator to the α Riemann–Liouville fractional integral Ia+ , i. e., α (∗ Dαa+ Ia+ f )(x) = f (x).

(51)

Formula (51) follows from the fact that the derivatives of orders k = 0, . . . , n − 1 of the α function Ia+ f at the point a are all equal to zero and thus the Caputo derivative ∗ Dαa+ of this function coincides with the Riemann–Liouville derivative Dαa+ , a left-inverse operator to the Riemann–Liouville fractional integral. As to the composition of the Riemann–Liouville fractional integral and the Caputo derivative, we have the formula [5, 17] n−1

f (k) (a) (x − a)α−k−1 , k! k=0

α (Ia+∗ Dαa+ f )(x) = f (x) − ∑

(52)

which is valid for f ∈ AC n [a, b], n − 1 < α ≤ n ∈ ℕ. One of the implications of formula (52) is the Laplace transform of the Caputo fractional derivative ∗ Dα0+ defined on ℝ+ : n−1

{ℒ(∗ Dα0+ f )(x); p} = pα {ℒf (x); p} − ∑ f (k) (0+)pα−k−1 , k=0

n − 1 < α ≤ n ∈ ℕ.

(53)

This formula is valid under conditions similar to the ones we required for the validity of the Laplace transform formula (42) for the Riemann–Liouville fractional derivative.

Basic FC operators and their properties | 35

However, it contains integer order derivatives f (k) (0+) with k = 0, 1, . . . n − 1 instead of the fractional derivatives in formula (42). This makes it possible to formulate the initial conditions in the Cauchy problems containing the Caputo derivatives in terms of the integer and not fractional order derivatives. For the Caputo derivative, a Leibniz type rule is valid, too [5]. Let 0 < α < 1 and the functions f and g be analytic in a neighborhood of a point a ∈ ℝ. Then the Leibniz rule (∗ Dαa+ f ⋅ g)(x) =

(x − a)−α g(a)(f (x) − f (a)) + (∗ Dαa+ g)(x)f (x) Γ(1 − α)

∞ α k−α + ∑ ( )(Ia+ g)(x)(∗ Dka+ f )(x) k k=1

(54)

holds true. For more advanced properties of the Caputo fractional derivative we refer the reader to [5] and [17]. Finally, we mention that the Riemann–Liouville derivative and the Caputo derivative of order α, n − 1 < α ≤ n ∈ ℕ are both compositions of the nth derivative and the Riemann-Lioville integral of order n−α, but in a different sequence. A natural question is if there exists an operator with an additional parameter (or additional parameters) that has these two derivatives as particular cases. The probably simplest construction of such an operator was suggested by Hilfer in [13, Ch. 2] (see also subsection 4.5 of this chapter for the FC notations introduced by Nakhushev and Pskhu). The so-called generalized Riemann–Liouville fractional derivative (nowadays referred to as the Hilfer fractional derivative) or order α, n − 1 < α ≤ n ∈ ℕ, and type β, 0 ≤ β ≤ 1, is defined by the following composition of three operators: α,β

β(n−α)

(Da+ f )(x) = (Ia+

dn (1−β)(n−α) f )(x). I dxn a+

(55)

For β = 0, this operator is reduced to the Riemann–Liouville fractional derivative α (Dα,0 a+ ≡ Da+ ) and the case β = 1 corresponds to the Caputo fractional derivative: α Dα,1 a+ ≡ ∗ Da+ . In [13, Ch. 2], a master equation with the time-fractional derivative in the form (55) was used to describe the continuous random walks with the Mittag-Leffler function as the waiting time probability density function. For other properties and applications of the Hilfer derivative we refer to [13, Ch. 2] and [15].

4 Other forms of fractional integro-differential operators In this section, we give a brief survey of some other notions of fractional integrals and derivatives. From a large variety of existing concepts, we choose those for which there

36 | A. N. Kochubei and Yu. Luchko is a rigorous mathematical theory, which admit an interpretation in terms of fractional powers of some operators and/or which are employed in serious applications. Note that our notations are specific for each of the subsections below.

4.1 Marchaud fractional derivative For a function f ∈ C 1 [a, b], −∞ < a < b < ∞, the Marchaud fractional derivative 𝔻αa+ f , 0 < α < 1, is an equivalent form of the Riemann–Liouville derivative: (𝔻αa+ f )(x)

x

f (x) α f (x) − f (t) = dt. + ∫ α Γ(1 − α)(x − a) Γ(1 − α) (x − t)1+α

(56)

a

Under the above assumption, 𝔻αa+ f = Dαa+ f . The expression (56) is useful in the theory of fractional partial differential equations, since it admits the maximum principle arguments. For a more general class of functions f , the Marchaud derivative is defined as a limit of restricted Marchaud derivatives, (𝔻αa+,ε f )(x) =

α f (x) ψ (x) + α Γ(1 − α)(x − a) Γ(1 − α) ε

where ε > 0, x−ε

ψε (x) = ∫ a

f (x) − f (t) dt (x − t)1+α

for x ≥ a + ε,

and the function ψε has to be defined separately for a ≤ x < a + ε. This definition can be understood in such a way that there exists an Lp -limit of 𝔻αa+,ε f , as ε → 0, provided α f = Ia+ φ, φ ∈ Lp (a, b), where 1 ≤ p < ∞ (see § 13 in [46]). On such functions, the above limit is called the Marchaud derivative, and it coincides almost everywhere with the Riemann–Liouville derivative. See [46] for further details, in particular regarding the Marchaud derivative on infinite intervals.

4.2 The Weyl fractional integrals and derivatives of periodic functions Let φ be a 2π-periodic function on ℝ, and ∞

ikx

φ ∼ ∑ φk e , k=−∞

2π

1 φk = ∫ e−ikx φ(x) dx, 2π 0

Basic FC operators and their properties | 37

be its Fourier series. In this subsection we assume that 2π

1 φ0 = ∫ φ(x) dx = 0. 2π

(57)

0

The idea of Weyl’s FC is to construct the fractional integrals I±(α) and derivatives 0 < α < 1, in such a way that

D(α) ± ,

φk ikx e α (±ik) k=−∞ ∞

I±(α) φ ∼ ∑ (which is meaningful due to (57)) and ∞

α ikx D(α) ± φ ∼ ∑ (±ik) φk e k=−∞

where (±ik)α = |k|α exp(± απi sign k). 2 The definitions are as follows: (I±(α) φ)(x)

2π

1 = ∫ φ(x − t)Ψα± (t) dt, 2π

α > 0,

0

where ∞ cos(kt ∓ απ/2) eikt =2∑ α (±ik) kα k=1 k =0 ̸

Ψα± (t) = ∑ and

(D(α) ± f )(x) = ±

d (1−α) f )(x). (I dx ±

On 2π-periodic functions from L1 (0, 2π), the Weyl fractional integral coincides with the Riemann–Liouville fractional integral on ℝ (for brevity, we consider only I+(α) ): (I+(α) φ)(x)

x

φ(t) dt 1 = , ∫ Γ(α) (x − t)1−α −∞

where the integral in the right-hand side is conditionally convergent, that is, x

∫ −∞

x

φ(t) dt φ(t) dt = lim ∫ . (x − t)1−α n→+∞ (x − t)1−α x−2nπ

There is also a representation by an absolutely convergent integral: ∞

(I+(α) φ)(x)

α−1

1 t = ∫ φ(x − t){t α−1 − (2π[ ]) Γ(α) 2π 0

where [s] means the integer part of s.

} dt,

38 | A. N. Kochubei and Yu. Luchko On “good” functions, D(α) ± coincides with the Weyl–Marchaud derivative (D(α) ± f )(x) =

2π

1 d (t) dt. ∫ {f (x − t) − f (x)} Ψ1−α 2π dt ± 0

For the proofs, further details and references, see [46].

4.3 Grünwald–Letnikov fractional derivative This notion is often used in the study of difference approximations to fractional differential equations. Here we present its simplest version, for functions defined on ℝ; for other cases and further details see [17, 40, 46]. For a function f on ℝ, the difference of order α > 0 with a step h ∈ ℝ is defined by the infinite series ∞ α (Δαh f )(x) = ∑ (−1)k ( )f (x − kh) k k=0

(58)

where α ( ) = 1, 0

α α(α − 1) ⋅ ⋅ ⋅ (α − n + 1) , ( )= n! n

n = 1, 2, . . . .

When h > 0 (h < 0), the difference (58) is called left-sided (resp. right-sided). The series (58) converges absolutely and uniformly whenever f is a bounded function. The left- and right-sided Grünwald–Letnikov derivatives f+α) and f−α) are defined as (Δαh f )(x) , h→+0 hα

f+α) (x) = lim

(Δα−h f )(x) . h→+0 hα

f−α) (x) = lim

These constructions coincide with the Marchaud fractional derivative, if f ∈ Lp (ℝ), 1 ≤ p < ∞.

4.4 Erdélyi–Kober operators These are the following modifications of the Riemann–Liouville fractional integrals and derivatives: γ,δ (Iβ f )(x)

x

β −β(γ+δ) δ−1 x = ∫(xβ − t β ) t β(γ+1)−1 f (t)dt, Γ(δ)

δ, β > 0, γ ∈ ℝ,

(59)

0

n 1 d γ,δ γ+δ,n−δ (Dβ f )(x) = (∏( x + γ + j))(Iβ f )(x), β dx j=1

(60)

Basic FC operators and their properties | 39

with n − 1 < δ ≤ n, n ∈ ℕ, β > 0, γ ∈ ℝ. There is also a right-hand sided version of this definition. The operators (59)–(60) satisfy a number of identities extending well-known properties of the Riemann–Liouville fractional integrals and derivatives. For details see, e. g., [18, 20, 46, 47, 50]. In [35], a Caputo type modification of the Erdélyi–Kober fractional derivative was introduced and studied. The left-hand sided derivative of this kind is defined as follows: γ,δ

γ+δ,n−δ

∗ Dβ f (x) = (Iβ

n 1 d + γ + j)f )(x), ∏( x β dx j=1

(61)

where n − 1 < δ ≤ n, n ∈ ℕ, β > 0, γ ∈ ℝ. For the properties and applications of this derivative we refer to [20] and [35].

4.5 FC notations by Nakhushev and Pskhu Nakhushev (see [39] and the references therein) and Pskhu [42] introduced notations for fractional integrals and derivatives covering simultaneously the Riemann– Liouville and Caputo cases, both left-sided and right-sided, as well as the more general operators by Djrbashian and Nersessian [6]. Subsequently this notation was used by other authors. The Riemann–Liouville derivative of order α with the origin at y = η is denoted (without specifying its left or right orientation) as (Dαηy f )(y) = signp (y − η)(

p

𝜕 ) (Dα−p ηy f )(y), 𝜕y

(62)

while the Caputo derivative (on a sufficiently smooth function f ) is written as α (𝜕ηy f )(y) = signp (y − η)Dα−p ηy ((

p

𝜕 ) f )(y). 𝜕y

In (62) and (63), the number p ∈ ℕ is such that p − 1 < α ≤ p, (D0ηy f )(y) = f (y), (Dμηy f )(y)

y

sign(y − η) = ∫ f (t)(y − t)−μ−1 dt, Γ(−μ)

μ < 0,

η

is the Riemann–Liouville fractional integral of order −μ with the origin at y = η. It follows from (62) and (63) that, if p ∈ ℕ, then p Dpηy f (y) = 𝜕ηy f (y) = signp (y − η)

dp f (y). dyp

(63)

40 | A. N. Kochubei and Yu. Luchko The Djrbashian–Nersessian operator of fractional differentiation associated with a finite sequence {γ0 , γ1 , . . . , γm } ⊂ (0, 1], of order α = ∑m k=0 γk − 1, is defined as {γ0 ,γ1 ,...,γm } γm −1 γm−1 (𝔻α f )(y) = (Dηy f )(y) = (Dηy Dηy ⋅ ⋅ ⋅ Dγηy1 Dγηy0 f )(y).

In particular, if {γk }m 1, . . . , 1}, 0 = {α − m + 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m

then we obtain the Riemann–Liouville derivative: = Dαηy , Dα−m+1,1,...,1 ηy

m − 1 < α ≤ m.

The Caputo derivative corresponds to the sequence {γk }m . . . , 1, α − m + 1} : ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0 = {1, m

α D{1,...,1,α−m+1} = 𝜕ηy , ηy

m − 1 < α ≤ m.

4.6 Distributed order calculus The distributed order derivative is an operator of the form 1

(ν)

(D φ)(t) = ∫(𝔻(α) φ)(t)ν(dα),

(64)

0

where 𝔻(α) is the Caputo derivative of order α, 0 < α < 1 and ν is a Borel measure on [0, 1]. The most widely investigated case is the one where ν is absolutely continuous with respect to the Lebesgue measure, that is, ν(dα) = μ(α) dα where μ is a nonnegative function. In this case, the operator (64) is denoted by D(μ) . The origin of this notion lies in physics—ultraslow diffusion, with logarithmic growth of the mean square displacement of a diffusive particle, can be described by evolution equations with the distributed order time derivative. The available results for distributed order FC include the notion of distributed order integral and a theory of “ordinary” and “partial” evolution equations with distributed order derivatives. See Chapter [24] of this handbook and the references therein.

4.7 General FC In the framework of the general FC [23, 24], one deals with the differential-convolution operators of the form t

d (D(k) u)(t) = ∫ k(t − τ)u(τ) dτ − k(t)u(0), dt 0

Basic FC operators and their properties | 41

where k is a nonnegative locally integrable function. This notion covers the Caputo fractional derivative 𝔻(α) , for which t −α , Γ(1 − α)

k(t) =

t > 0,

as well as the distributed order derivative 𝔻(μ) with 1

k(t) = ∫ 0

t −α μ(α) dα, Γ(1 − α)

t > 0.

A natural question was to determine conditions upon k, under which the relaxation equation (D(k) u)(t) = −λu(t) (λ > 0) and the diffusion equation (D(k) w)(t, x) = Δw(t, x), t > 0, x ∈ ℝn , have full collection of properties typical for conventional and fractional relaxation and diffusion equations. The answer to this question was formulated in terms of the Laplace transform of k, which should be a Stieltjes function satisfying some asymptotic relations [23, 24].

4.8 Hadamard fractional integrals and derivatives The left- and right-hand sided Hadamard fractional integrals are defined by the formulas [17, 46] (J+α φ)(x)

x

φ(t) dt 1 , = ∫ Γ(α) t(log xt )1−α

x > 0, α > 0,

φ(t) dt 1 , ∫ Γ(α) t(log xt )1−α

x > 0, α > 0.

(J−α φ)(x) =

0 ∞ x

There are related notions of the Hadamard fractional derivatives. The left-hand sided one is defined as (𝒟+α f )(x)

x

1 f (t) dt d = x ∫ Γ(1 − α) dx (log xt )α t 0

and the right-hand sided derivative is introduced with evident modifications. d . The The above operators are interpreted as fractional powers of the operator x dx latter (connected with the concept of regular singularity) is important in various applications and this explains in part the growing interest in the Hadamard fractional derivatives; see, e. g., [3, 8, 36]. Some of the results are concerned with the properties of the Caputo type modification of the Hadamard derivative introduced in [16].

42 | A. N. Kochubei and Yu. Luchko

4.9 Fractional Laplacians and the Riesz fractional integro-differentiation The idea of the multi-dimensional Riesz fractional integro-differentiation is to construct such operators I α , α > 0 (Riesz fractional potentials) and Dα , α > 0 (Riesz fractional derivatives or fractional Laplacians) that on some classes of test functions (the Lizorkin spaces) the relations Iαf , if Re(α) > 0, (−Δ)−α/2 f = ℱ −1 |x|−α ℱ f = { −α D f , if Re(α) < 0, hold true, where ℱ is the Fourier transform. The operators I α and Dα (Re(α) > 0) can in fact be defined on much wider classes of functions. Their constructions are as follows [46]: (I α f )(x) = ∫ kα (x − t)f (t) dt,

Re(α) > 0,

ℝn

where kα (x) =

|x|α−n , 1 { α−n γn (α) |x| log(

1 ), |x|

if α − n ≠ 0, 2, 4, . . . ; if α − n = 0, 2, 4, . . . ,

γn (α) are certain constants (see [46] for their values). The fractional Laplacian Dα is realized in the form of the hypersingular integral (Dα y)(x) =

(Δl y)(x) 1 ∫ t n+α dt, dn (l, α) |t|

l > α,

ℝn

where l l (Δlh f )(x) = ∑ (−1)k ( )f (x − kh), k k=0

and we assume that α is not an integer. For the values of the constants dn (l, α), see [46]. It is worth mentioning that the hypersingular integral Dα does not depend on the choice of l. The fractional Laplacian is the model example for a class of hypersingular integral operators [27, 45], as well as the class of Lévy generators (for α < 2) in the theory of stochastic processes. On the other hand, the study of partial differential equations containing fractional Laplacians is a rapidly developing branch of contemporary mathematical research (see Chapters [21, 48] of this handbook).

Basic FC operators and their properties | 43

4.10 Feller potentials and derivatives The fractional Laplacian described in the previous subsection can be interpreted as a generator of a symmetric stable process on ℝn . Feller [7] considered, for n = 1, a more general class of generators Dαθ of stable processes. These operators can be defined as the pseudo-differential operators with the symbols ̂α (ξ ) = −|ξ |α ei(sign ξ )θπ/2 , D θ

ξ ∈ ℝ, 0 < α < 2,

where |θ| < α, if 0 < α < 1, { { { |θ| ≤ 2 − α, if 1 < α < 2, { { { if α = 1, 2. {θ = 0, Here we follow the notation from [11]. The operator −Dαθ can be interpreted as the inverse of the Feller potential [46], the integral operator Iθα , which is, on “good” functions φ, of the form (Iθα φ)(x) = c+ (α, θ)(I+α φ)(x) + c− (α, θ)(I−α φ)(x), where I±α are the Riemann–Liouville fractional integrals on ℝ (see (26) and (27)), { {c+ (α, θ) = { { {c− (α, θ) =

sin[ π2 (α−θ)] , sin(απ) sin[ π2 (α+θ)] , sin(απ)

0 < α < 2, α ≠ 1. See [7] for the special case α = 1. If 0 < α < 1, 0 < β < 1, α + β < 1, we have a kind of semigroup property, that β α+β is, Iθ = Iθα Iθ . There exist versions of the above notions for a semi-axis and a finite interval [46]. The study of the multi-dimensional case was initiated in [26].

4.11 More advanced FC operators and further reading In this chapter, we restricted ourselves to definitions and essential properties of the basic FC operators. In other chapters of this handbook, both advanced properties of the operators mentioned here and more general FC operators are discussed in detail. In Chapter [19], the so-called generalized FC operators are discussed. This notion refers to the integral, differ-integral, and integro-differential operators involving special functions in the kernels, having convolutional structure, and satisfying the basic “axioms” of the classical FC. In particular, the generalized FC operators include the conventional and the multiple Erdélyi–Kober integrals and derivatives of the Riemann–Liouville and Caputo type. A theory of these operators is presented in

44 | A. N. Kochubei and Yu. Luchko Chapter [20]. FC operators with the general kernels (not necessarily in form of some elementary and special functions) are discussed in Chapter [24]. Chapter [27] is devoted to the fractional Laplace operator and its connection to the isotropic stable Lévy processes. In Chapter [25], fractional derivatives of distributed order as well as the Cauchy problem for the diffusion equation with these derivatives are considered. The Mellin integral transform approach to the FC operators is discussed in Chapter [34], whereas Chapter [33] is devoted to the fractional Fourier transform. We also refer to Chapter [14], where a survey of mathematical and physical interpretations of the FC operators is provided. The literature devoted to FC operators in different forms, their history, properties, and applications, is huge and permanently growing. In Chapter [37] of this handbook, 250 published FC books are listed and we refer the reader to the bibliography of [37] for a nearly complete register of FC monographs. Here, we just repeat references to some selected FC books already cited in the text, including [5, 10, 13, 17, 18, 39, 40, 45, 46, 50].

Bibliography [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10]

[11] [12] [13] [14]

N. H. Abel, Oplösning af et par opgaver ved hjelp af bestemte integraler, Mag. Naturvidensk., Aargang I, Bind 2, Christiania, 1823. N. H. Abel, Auflösung einer mechanischen Aufgabe, J. Reine Angew. Math., 1 (1826), 153–157. B. Ahmad, A. Alsaedi, S. K. Ntouyas, and J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017. M. Caputo, Linear models of dissipation whose q is almost frequency independent: II, Geophys. J. R. Astron. Soc., 13(5) (1967), 529–539. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. M. M. Djrbashian and A. B. Nersessian, Fractional derivatives and the Cauchy problem for differential equations of fractional order, Izv. Akad. Nauk Arm. SSR. Ser. Mat. 3(1) (1968), 1–29 (in Russian). W. Feller, On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Medd. Lunds Univ. Mat. Sem., Tome suppl. 21 (1952), 72–81. R. Garra and F. Polito, On some operators involving Hadamard derivatives, Integral Transforms Spec. Funct., 24 (2013), 773–782. A. N. Gerasimov, Generalization of laws of the linear deformation and their application to problems of the internal friction, Prikl. Mat. Meh., 12(3) (1948), 251–260 (in Russian). R. Gorenflo and F. Mainardi, Fractional calculus, integral and differential equations of fractional order, in A. Carpinteri and F. Mainardi (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276, Springer Verlag, Wien, 1997. R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1(2) (1998), 167–191. A. K. Grünwald, Über “begrenzte” Derivationen und deren Anwendung, Z. Angew. Math. Phys., 12 (1867), 441–480. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. R. Hilfer, Mathematical and physical interpretations of fractional derivatives and integrals, in Handbook of Fractional Calculus with Applications, vol. 1, 2019.

Basic FC operators and their properties | 45

[15] R. Hilfer, Yu. Luchko, and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299–318. [16] F. Jarad, T. Abdeljawad, and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142, 8 pp. [17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [18] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Sci. Tech. & J. Wiley, Harlow–N. York, 1994. [19] V. Kiryakova, Generalized fractional calculus operators with special functions, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [20] V. Kiryakova and Yu. Luchko, Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [21] V. P. Knopova, A. N. Kochubei, and A. M. Kulik, Parametrix methods for equations with fractional Laplacians, in Handbook of Fractional Calculus with Applications, vol. 2, 2019. [22] A. N. Kochubei, Fractional-order diffusion, Differ. Equ., 26 (1990), 485–492. [23] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equ. Oper. Theory, 71 (2011), 583–600. [24] A. N. Kochubei, General fractional calculus, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [25] A. N. Kochubei, Equations with general fractional time derivatives. Cauchy problem, in Handbook of Fractional Calculus with Applications, vol. 2, 2019. [26] V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. Lond. Math. Soc., 80 (2000), 725–768. [27] M. Kwaśnicki, Fractional Laplace operator and its properties, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [28] A. V. Letnikov, Theory of differentiation of arbitrary order, Mat. Sb., 3 (1868), 1–68 (in Russian). [29] J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, J. Éc. R. Polytech., 13 (1832), 1–69. [30] P. I. Lizorkin, Generalized Liouville differentiation and functional spaces Lrp (En ). Embedding theorems, Mat. Sb., 60 (1963), 325–353 (in Russian). [31] P. I. Lizorkin, Generalized Liouville differentiation and the method of multipliers in the theory of embeddings of classes of differentiable functions, Proc. Steklov Inst. Math., 105 (1969), 105–202. [32] Yu. Luchko, Theory of the Integral Transforations with the Fox H-Function as a Kernel and Some of Its Applications Including Operational Calculus, PhD. thesis, Byelorussian State University, Minsk, 1993 (in Russian). [33] Yu. Luchko, Fractional Fourier transform, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [34] Yu. Luchko and V. Kiryakova, Applications of the Mellin integral transform technique in Fractional Calculus, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [35] Yu. Luchko and J. J. Trujillo, Caputo type modification of the Erdélyi–Kober fractional derivative, Fract. Calc. Appl. Anal., 10 (2007), 249–267. [36] L. Ma and C. Li, On Hadamard fractional calculus, Fractals, 25(3) (2017), 1750033 16 pp. [37] J. A. Tenreiro Machado and V. Kiryakova, Recent history of the fractional calculus: data and statistics, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [38] O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions. Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.

46 | A. N. Kochubei and Yu. Luchko

[39] A. M. Nakhushev, Fractional Calculus and Its Applications, Fizmatlit, Moscow, 2003 (in Russian). [40] I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999. [41] I. Podlubny, R. L. Magin, and I. Trymorush, Niels Henrik Abel and the birth of Fractional Calculus, Fract. Calc. Appl. Anal., 20 (2017), 1068–1075. [42] A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 73 (2009), 351–392. [43] Yu. N. Rabotnov, Creep Problems in Structural Members, North-Holland, Amsterdam, 1969. [44] Yu. A. Rossikhin, Reflection of two parallel ways in the progress of fractional calculus in mechanics of solids, Appl. Mech. Rev., 63 (2010), 010701. [45] S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2001. [46] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. [47] I. N. Sneddon, The use in mathematical physics of Erdélyi–Kober operators and of some of their generalizations, in Proc. Internat. Conf. Fractional Calculus and Its Applications, pp. 37–79, Springer, New York, 1975. [48] P. R. Stinga, User’s guide to the fractional Laplacian and the method of semigroups, in Handbook of Fractional Calculus with Applications, vol. 2, 2019. [49] V. E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2945–2948. [50] S. Yakubovich and Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Acad. Publ., Dordrecht, 1994.

Rudolf Hilfer

Mathematical and physical interpretations of fractional derivatives and integrals Abstract: Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point. Keywords: fractional derivatives and integrals, Riemann–Liouville integrals, Weyl integrals, Riesz potentials, operational calculus, functional calculus, Mikusinski calculus, Hille–Phillips calculus, Riesz–Dunford calculus, classification, phase transitions, time evolution, anomalous diffusion, continuous time random walks MSC 2010: 26A33, 34A08, 35R11

1 Prolegomena 1.1 A multitude of mathematical and physical interpretations for fractional derivatives and integrals have been developed since Gottfried Wilhelm Leibniz [123, p. 301] first noted and then asked “...on le peut exprimer per seriem infinitam, sed quid est in Geometria?”1 Derivatives Dα and integrals I α of fractional (non-integer) order arise from viewing the symbolic relations I I n = I n+1 , n

n+1

DD = D n

DI = I

n−1

,

,

n ∈ ℕ,

(1a)

n ∈ ℕ,

(1b)

n ∈ ℕ,

(1c)

1 “... one can express that” difference (or sum) whose exponent is a fraction∗ “by an infinite series, but ∗ what is it in geometry?” inserted from context [123, p. 300–301]. Acknowledgement: The author thanks Prof. Dr. M. Zähle for comments and suggestions. Rudolf Hilfer, ICP, Fakultät für Mathematik und Physik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Deutschland https://doi.org/10.1515/9783110571622-003

48 | R. Hilfer for iterated integrals I and derivatives D as almost representing the additive semigroup (ℕ, +) of natural numbers, and extending (1) to the semigroup (ℝ+0 , +) of non-negative reals2 ℝ+0 := [0, ∞) ⊂ ℝ or to the full field (ℝ, +, ⋅ ). 1.2 Many ideas for interpreting and extending the formal relations (1) from n ∈ ℕ to n ∈ ℚ, ℝ or ℂ have been proposed over the centuries. All attempts are met squarely by some basic facts of calculus. Indeed, the geometric property D2 = 0 of (exterior) derivatives [53, p. 20] would restrict equation (1b) to n ≤ 2. Or the non-commutativity I D ≠ D I seems to prevent extension of equation (1c) to n = 0. Recall that the power functions Pn (x) :=

xn , n!

x ≥ 0, n ∈ ℕ

(2)

obey the rules I Pn = Pn+1 and D Pn = Pn−1 similar to I n in (1a) and (1c). Extension to n = 0 (or n < 0), however, is fraught with a singularity at x = 0. Many mathematical impediments arise from these basic facts. 1.3 Derivatives and integrals of fractional (non-integer) order originate from the operational analogy (“analogie merveilleuse”) observed by Leibniz [122], n n Dn (fg) = ∑ ( )(Dk f )(Dn−k g), k k=0

(3)

n n Pn (f + g) = ∑ ( )(Pk f )(Pn−k g), k k=0

(4)

between the nth derivative of a product (3) of two real-valued functions f , g, and the nth power of their sum (4), if powers Pn (f ) = f n = Pn f with f 0 = 1 are written using an operational symbol P. Extension of equations (2)–(4) from n ∈ ℕ to α ∈ ℂ, Re α > 0 (or α ∈ ℂ [162, Sec. 2.3]) requires interpolation formulae n n! from ( ) = k!(n − k)! k n

from n! = ∏ k k=1

to

Γ(α + 1) , Γ(k + 1)Γ(α − k + 1)

k ∈ ℕ,

and

(5a)

Γ(α + 1) = ∫(− log x)α dx = ∫ xα e−x dx,

(5b)

1

to

0

∞

0

for factorials and binomial coefficients. It was for this purpose that Euler solved the ‘interpolation problem’ and introduced the Γ-function in [50, § 27–29]. 1.4 Given the operational analogy between (3) and (4), Lagrange obtained the celebrated symbolic formula [61, p. 194–195] df

α

Δαh f = (eh dx − 1)

2 The notations ℝ+ := (0, ∞), ℝ+0 := [0, ∞), and ℝ− := (−∞, 0) will be used.

(6)

Mathematical and physical interpretations of fractional derivatives and integrals | 49

for general α, where (Δh f )(x) = f (x + h) − f (x) is the finite difference with shift h ≥ 0. Lagrange’s formula results from treating the symbol of differentiation systematically as an algebraic quantity with the provision of replacing its nth powers n

(

df dn f ) 󳨀󳨀󳨀→ n dx dx

(7)

with the nth derivative at the end. Other authors have subsequently tried to build differential calculus on an algebraic or algorithmic basis [63, 7, 180, 31]. Recall that Leibniz’ product rule D(fg) = g(D f ) + f (D g)

(8)

is the algebraic basis of Dn and equation (3) reduces to equation (8) for n = 1. Indeed equation (8) is the axiomatic basis for differentiation (resp. derivation) on manifolds [33] and algebras [26, 119]. A full account of Leibniz’ influential idea to generalize equation (1) is not possible in these pages. Many books and specialized treatises [125, 157, 168, 82, 193] contain a section on history (see also [134]). 1.5 The objective of this chapter is to collect a list of mathematical and physical interpretations in the sense of Definitions 1 and 2 given below. The scope is necessarily rather restricted due to the long history of fractional calculus. Only a selection of established and widely known mathematical interpretations is aimed at here. As to the physical interpretations, there will be even fewer examples, because, in the opinion of this author, most physical interpretations are still tentative.

2 Integrals and derivatives 2.1 Integrals. Integrals are (weighted) sums with infinitely many terms, or linear functionals on function spaces. More concretely, let (Ω, M , μ) be a measure space with σ-algebra M and measure (or weight) μ, and let f : Ω → ℝ be a real-valued function on Ω. Then I μ,Ω (f ) = ∫ fμ = ∫ f (x) dμ(x) = μΩ (f ) ∈ ℝ Ω

(9a)

Ω

denotes the integral of f with respect to μ. When μ = λ is the Lebesgue measure and Ω ⊂ ℝ is an interval, notations such as I Ω (f ) = ∫ f = ∫ f (x) dx = λΩ (f ) Ω

Ω

often suppress the dependence on the measure or weight μ.

(9b)

50 | R. Hilfer 2.2 Let 𝕁 = (a, b) be an interval with −∞ ≤ a < b ≤ ∞ and Ω = (a, x) an interval with a < x < b. For a locally integrable function f : 𝕁 → ℝ the notation x

x

(I a+ f )(x) := I Ω (f ) = ∫ f = ∫ f (y)dy a

(10a)

a

b

b

(I b− f )(x) := I 𝕁\Ω (f ) = ∫ f = ∫ f (y)dy x

(10b)

x

is used to emphasize the dependence on the variable upper or lower limit x. 2.3 Derivatives. Let Ω ⊂ ℝ be open and let F(Ω) be the algebra of real-valued functions f : Ω → ℝ with pointwise addition and pointwise multiplication.3 Algebraically, the derivative (or derivation) D f is defined as a linear operator D : F → F on F such that the product rule (8) holds true for all f , g ∈ F. On commutative Banach algebras every such D has a range contained in the Jacobson radical of F, i. e. in the kernel of the Gelfand isomorphism [183, 194]. 2.4 Difference quotients. Analytically, the derivative of f at x ∈ Ω is defined as the limit of finite difference quotients (h ≥ 0) Th − 1 f (x ± h) − f (x) = lim ( ± )f (x) h→0 h→0 h h Δ = lim ( ±h )f (x) h→0 h

(D f )(x) := lim

(11a) (11b)

if both limits exist, are well defined, and are equal. Here 1 is the identity (see equation (100) in the appendix), Δ±h = Th± − 1

(12)

are the right/left-sided difference operators appearing in equation (6), and (Th± f )(x) = f (x ± h)

(13)

stands for translations of f to the left (+) or right (−) by h. 2.5 Iteration. The algebra of integer powers of D and I in equation (1) is the starting point of fractional calculus. Let A : X → X be a linear operator on a linear space X with domain D(A) ⊂ X. For n ∈ ℕ the nth power of an operator An := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ A∘⋅⋅⋅ ∘ A n-times

3 Sets of functions such as F(Ω), or the Lebesgue spaces Lp (Ω) are denoted by a sans serif font.

(14)

Mathematical and physical interpretations of fractional derivatives and integrals | 51

is interpreted as its n-fold composition or iteration and defined recursively as A0 := 1,

n+1

A

D(A0 ) = X

n

f := A ∘(A f ), D(A

n+1

(15a) n

) = {f ∈ D(A) | A f ∈ D(A )}

(15b)

where 1 is the identity operator on X (see appendix). Then the law of exponents An Am = An+m holds for n, m ∈ ℕ. This shows that An+1 = A An and thus the domains are shrinking D(An+1 ) ⊂ D(An ). Moreover, An (D(Am )) ⊂ D(Am−n ) for m ≥ n. 2.6 Iterated integrals. Let 𝕁 = (a, b) ⊆ ℝ with −∞ ≤ a < b ≤ ∞ as in Paragraph 2.2, and let f ∈ L1loc (𝕁) := {f : 𝕁 → ℝ | f is integrable on all compact subsets K ⊂ Ω} be locally integrable. Iterating I a+ : D(I a+ ) → L1loc (𝕁) from equation (10) gives x y1

n

yn−1

[(I a+ ) f ](x) = ∫ ∫ . . . ∫ f (yn ) dyn . . . dy2 dy1 a a

a

(16a)

x

x

a

a

1 = ∫(x − y)n−1 f (y) dy = ∫ f (y)[Tx− (S K n )](y) dy (n − 1)!

(16b)

where n ∈ ℕ and D(I a+ ) = {f ∈ L1loc (𝕁) : I a+ f ∈ L1loc (𝕁)}. The function K n : ℝ → ℝ is defined as K n (x) := Θ(x) Pn−1 (x) = Θ(x)

xn−1 Γ(n)

(17)

where Pn was defined in equation (2), Γ is Euler’s Gamma function in equation (5b) and Θ : ℝ → ℝ, 0 x < 0, Θ(x) := { 1 x ≥ 0,

(18)

is the Heaviside step function.4 The reflection operator S : F(ℝ) → F(ℝ) is defined by (S f )(x) := f (−x) for x ∈ ℝ and the translation operators Th± : F(ℝ) → F(ℝ) with h ≥ 0 are given in equation (13) on the set F(ℝ) of real-valued functions f : ℝ → ℝ. A similar b formula, [(I b− )n f ](x) = ∫x f (y)(Tx− K n )(y) dy, holds for I b− . 2.7 Iterated derivatives. Let 𝕁 = (a, b) ⊆ ℝ with −∞ ≤ a < b ≤ ∞. Iterates of the translation operators for f : 𝕁 → ℝ, n

[(Th± ) f ](x) = [(Tnh ± )f ](x) = f (x ± nh),

n ∈ ℕ, a < x < b,

(19)

are well defined for h with |h| < min{x − a, b − x}/n. The iterated derivative (Dn f )(x) = lim ( h→0

n

n

Th − 1 Δ±h ) f (x) = lim ( ± ) f (x) h→0 h h

4 Other conventions for Θ(0) are sometimes used.

(20a)

52 | R. Hilfer 1 n n ∑ (−1)n−k ( )(Thk ± f )(x) h→0 hn k k=0

= lim

(20b)

is defined, if the limits exist and are equal. A domain for D : C(𝕁) → C(𝕁) is D(D) = {f ∈ C10 (𝕁) : D f ∈ C0 (𝕁)}

(21)

where C(𝕁) := {f : 𝕁 → ℝ : f is continuous}, Ck0 (𝕁) := {f ∈ C(𝕁) : f is k-times continuously differentiable}, and C0 (𝕁) := {f ∈ C(𝕁) : f vanishes at ∞}.

3 Mathematical interpretations 3.1

It seems pertinent to specify what is meant by an interpretation.

Definition 1. A mathematical interpretation of fractional derivatives Dα or fractional integrals I α is an incomplete mathematical definition. Interpretations are abbreviated as (A) ⋅ − (B), which is to be read as “(A) is interpreted as (B)”. 3.2 Many mathematical interpretations of fractional derivatives and integrals are based on equation (16) or (20) (see Sections 4 and 5). For more information on standard interpretations see [76] or the preceding chapter [111].

4 Standard interpretations for integrals 4.1 Riemann–Liouville Interpretation. A standard interpretation of fractional integration is I αa+ ⋅ − (I a+ )n from equation (16b)

(22)

with α ⋅ − n ∉ ℕ interpreted as a non-integer power of integration. 4.2 Riemann–Liouville integrals. Let 𝕁 = (a, b) and −∞ < a < x < b < ∞. Riemann–Liouville fractional integrals of order α > 0 are defined as (I αa+ f )(x) = (I αb− f )(x)

x

1 ∫(x − y)α−1 f (y) dy Γ(α)

(23a)

a

b

1 = ∫(y − x)α−1 f (y) dy Γ(α) x

(23b)

Mathematical and physical interpretations of fractional derivatives and integrals | 53

for f ∈ L1 (𝕁) [67, 92, 96, 171, 169, 27]. For α = 0 the specification (I 0a+ f )(x) = (I 0b− f )(x) = f (x) completes the definition.5 Some authors require piecewise continuity [145, p. 45]. For f ∈ L1 (𝕁) the fractional integral (I αa+ f )(x) exists for almost every x ∈ 𝕁. Then I αa+ f ∈ L1 ((a, c)) for every a < c < b and its L1 -norm is finite. If f ∈ Lp (𝕁) with 1 ≤ p ≤ ∞ and α > 1/p then (I αa+ f )(x) is finite for all x ∈ 𝕁. On Lebesgue spaces Lp (𝕁) for 1 ≤ p ≤ ∞ the fractional integral I αa+ is a bounded non-negative operator. It is unbounded (and non-negative) on Lp ((a, ∞)), i. e. for b = ∞. The definition of I αa+ may be generalized to α ∈ ℂ with Re α > 0. 4.3 Weyl Interpretation. The Riemann–Liouville integral I αa+ f of a periodic function f (x) ∼ ∑k ck eikx will in general not be periodic. Integration of periodic functions amounts to division of the Fourier transform with the Fourier variable. This led Weyl to interpret I α± f ⋅ − F −1 {(±ik)−α F {f }}

(24)

as a Fourier multiplication operator [205], where F {f }(k) is the Fourier transform of f (x), and f (x) ∼ ∑k ck eikx with c0 = 0 is periodic. 4.4 Weyl integrals. Let Ω = ℝ/2πℤ be the unit circle and let f ∈ Lp (Ω) for 1 ≤ p < ∞ be a 2π-periodic function such that its integral over a period vanishes. The Weyl fractional integral of order α is defined as (I α± f )(x) :=

2π

∞ 1 eik(x−y) f (y) dy ∫ ∑ 2π k=−∞ (±ik)α 0

(25)

k =0 ̸

for 0 < α < 1 [171, 27]. For such 2π-periodic f with vanishing integral over a period the relations I α+ f = I αa+ f with a = −∞ and I α− f = I αb− f with b = ∞ hold [212]. They motivate an extension of Weyl integrals as improper integrals, (I α+ f )(x) = lim (I αa+ f )(x),

f ∈ L1loc (ℝ− ),

(26a)

(I α− f )(x) = lim (I αb− f )(x)

f ∈ L1loc (ℝ+ ),

(26b)

a→−∞

b→∞

to locally integrable functions f : ℝ± → ℝ [49, 171, 145, 27]. Then I α− f ∈ L1 ([a, b]) for 0 < a < b and I α+ f ∈ L1 ([a, b]) for a < b < 0. As operators on Lp (ℝ± ) with 1 ≤ p < ∞ the domains are (see [137]) D(I α± ) = {f ∈ Lp (ℝ∓ ) : ∃ x0 s. t. (I α± f )(x) exists and I α± f ∈ Lp (ℝ∓ )}.

(27)

5 The notation for fractional integrals has varied over time. Leibniz, Lagrange and Liouville used the α α −α −α −α symbol ∫ [123, 61, 130], Grünwald wrote ∫ [dxα ]x=x x=a , while Riemann used 𝜕x [163], Most da /dx x

[151], Krug D [116] and Weyl J α [205]. The notation in (23a) is that of [171, 169, 79, 77]. Modern authors a

−α −α also use fα [67], I α [166], a Ixα [27], Ixα [39], a D−α [157] instead of I αa+ . x [145, 176, 156], or d /d(x − a)

54 | R. Hilfer 4.5 Convolution. For f ∈ L1 (ℝ± ) the Weyl fractional integral may be written as a convolution (I α± f )(x) = (K±α ∗ f )(x)

(28)

where the convolution kernels are defined as K+α := K α and K−α := S K α with K α defined by extending K n in equation (17) from n ∈ ℕ to α > 0. For α = 0 the definition is K+0 (x) = K−0 (x) = δ(x) with the Dirac distribution at 0. The convolution is defined pointwise as x

∞

(f ∗ g)(x) = ∫ f (x − y)g(y) dy

resp. (f ∗ g)(x) = ∫ f (x − y)g(y) dy

(29)

0

−∞

for functions on ℝ resp. ℝ+ for x ∈ ℝ+ . 4.6 Riesz integrals. Let f ∈ L1loc (ℝ) be locally integrable. The conjugate Riesz fractional integral of order α > 0 is defined as (Ĩα f )(x) =

∞

sgn(x − y)f (y) 1 dy ∫ 2Γ(α) sin(απ/2) |x − y|1−α

(30)

−∞

where α ≠ 2k, k ∈ ℤ. For α = 0 one sets (Ĩ0 f )(x) = f (x). Riesz fractional integration may α α ̃α ∗ f )(x) with K ̃α (x) = K+ (x)−K− (x) for α ≠ 2k, be written as a convolution (Ĩα f )(x) = (K 2 sin(απ/2)

k ∈ ℤ, and K±α from equation (28). For more information see [27], [82, Sec. 2.2.5], [111, Sec. 4.9] and Section 19.

5 Standard interpretations for derivatives 5.1 Riemann–Liouville Interpretation. Riemann [163, p. 341] and Liouville [130] suggested to interpret fractional derivatives of order n + α − m with α > 0, n ≥ m Dα−m+n ⋅ − Dn I m−α

(31)

as derivatives of integer order n ≥ m of a fractional integral of order m − α > 0. 5.2 Fractional derivatives. The Riemann–Liouville interpretation applies to all fractional integrals in Section 4. It can be generalized to all α ∈ ℂ with Re α > 0 as dn n−α f )(x), (I dxn a± dn (Dα± f )(x) := (±1)n n (I n−α f )(x), dx ± dn n−α (Dα f )(x) := n (Ĩ f )(x) dx

(Dαa± f )(x) := (±1)n

(32a) (32b) (32c)

Mathematical and physical interpretations of fractional derivatives and integrals | 55

where n = ⌈Re α⌉ := min{k ∈ ℤ : k ≥ Re α} is the smallest integer larger than Re α. These fractional derivatives are named after Riemann–Liouville, Weyl and Riesz, respectively.6 Their domains D(Dα ) = {f ∈ D(I n−α ) : ∃g ∈ D(Dn ) s. t. g = I n−α f }, where Dn stands for dn /dxn , depend on those of Dn and I n−α . 5.3 Grünwald–Letnikov. Interpretation. Already Liouville [129, p. 107] suggested to interpret fractional derivatives Dα ⋅ − Equation (20)

(33)

as a limit of nth order finite difference quotients with α ⋅ − n ∉ ℕ. The suggestion was later taken up in [62, 124, 125]. The Grünwald–Letnikov fractional derivative of order α > 0 is defined as the limit 1 (GLDα± f )(x) := lim+ α (Δα±h f )(x) (34) h→0 h

of fractional difference quotients whenever the limit exists. The Grünwald–Letnikov fractional derivative is called pointwise or strong depending on whether the limit is taken pointwise or in the norm. For periodic functions in C(ℝ/2πℤ) or Lp (ℝ/2πℤ) with 1 ≤ p < ∞ see [27], for non-periodic f see [171]. 5.4 Marchaud–Hadamard. Interpretation. Marchaud’s idea is to interpret the fractional derivative of order α > 0 directly Dα ⋅ − I −α

(35)

as a fractional integral of negative order −α by subtracting divergent parts [136, 65]. For f : ℝ → ℝ this leads to (MHDα± f )(x)

∞

f (x) − f (x ∓ y) α = dy ∫ Γ(1 − α) yα+1

(36)

0

and MHD0± f = f . Marchaud–Hadamard derivatives have a larger domain of definition than Riemann–Liouville derivatives. For the case of functions on bounded intervals and for more details, see [171, 76, 111].

6 Generalized Riemann–Liouville interpretation 6.1 Interpretation. For n = m = 1 and 0 < α < 1 the interpretation (31) has been generalized to fractional derivatives of order α Dα,β ⋅ − I β(1−α) D I (1−β)(1−α)

(37)

6 The notation is not standardized. Leibniz and Euler used dα [123, 122, 50] Riemann wrote 𝜕xα [163], (α) α x=x Liouville preferred dα /dx α [130], Grünwald used {dα f /dx α }x=x x=a or D [f ]x=a [62], Marchaud wrote Da , and Hardy–Littlewood used an index f α [67]. The notation in (32a) follows [171, 169, 79, 77]. Modern authors also use I −α [166], Ix−α [39], a Dαx [27, 145, 176], dα /dxα [210, 176], dα /d(x−a)α [157] instead of Dαa+ .

56 | R. Hilfer and type 0 ≤ β ≤ 1 in [77, p. 433]. Using equation (32) yields generalized Riemann– Liouville, generalized Weyl, and generalized Riesz derivatives. 6.2 The type β of a fractional derivative allows to interpolate continuously from α,1 ̃α ̃α Dαa± = Dα,0 a± to Da± = Da± . The fractional derivative Da± was introduced in [130, p. 10]. A relation between fractional derivatives of the same order but different types is found in [77, p. 434]. An operational calculus for generalized Riemann–Liouville derivatives was developed in [90]. 6.3 Generalized Riemann–Liouville derivatives have been further generalized in [207, 57, 102] and reformulated in [103]. They have found applications to telegraphtype equations [173], ultra-hyperbolic equations [38], nonlinear analysis in weighted spaces [55, 199], Ulam stability [198], functional differential inclusions [2], implicit differential equations [1], thermodynamics [77] and dielectric spectroscopy [86].

7 Localized Riemann–Liouville interpretation α,β

7.1 Interpretation. The Riemann–Liouville derivatives Da+ are nonlocal operators. Localization interprets α,β

α,β

(Dα,β f )(a) ⋅ − (Da+ f )(a) or (Dα,β f )(x) ⋅ − (Dx− f )(x)

(38)

as a limiting value at the boundary points of the interval (a, x). 7.2 For −∞ < x < ∞ the localized Riemann–Liouville fractional derivative of order 0 < α < 1 and type β is dα,β f α,β α,β (x) := lim (Da+ f )(x) = lim (Db− f )(x) a→x b→x dxα,β

(39)

whenever the two limits exist and are equal. Localized fractional differentiability at x is related to regular variation at x [77]. Let f : ℝ+0 → ℝ+0 be monotonously increasing with α,β f (0) = 0 and such that (Da+ f )(x) with 0 < α < 1 and 0 ≤ β ≤ 1 is also monotonously increasing in [a, a + δ] for some a ≥ 0, δ > 0. Let Λ be slowly varying near a in the sense of [179], let 0 ≤ λ < β(1 − α) + α and C ≥ 0. Then regular variation of f near a with α,β index λ is equivalent to regular variation of Da+ f near a with index λ − α [77, p. 438]. 7.3 Localized fractional derivatives were introduced in [68, 69, 77] and immediately applied to the classification of phase transitions in [68–70, 72, 71, 77] (see also Section 20). Later the localized interpretation was appropriated by [112], who claimed inappropriately the absence of the lower limit.

Mathematical and physical interpretations of fractional derivatives and integrals | 57

8 Convolution quotient interpretation 8.1 The restriction of Heaviside’s step function Θ with convention Θ(0) = 1 as defined in equation (18) to ℝ+0 is the constant function Θ(x) = 1. Inserting f = g = Θ into equation (29) shows (Θ ∗ Θ)(x) = x. Iterating yields ⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟)(x) = (Θ ∗ ⋅⏟⏟⏟ ⋅ ⋅⏟⏟⏟⏟⏟⏟⏟ ∗Θ n-times

xn−1 = K n−1 (x), (n − 1)!

n ∈ ℕ, x ∈ ℝ+0 ,

(40)

with K n from equation (17) for the nth convolution power of Θ. The convolution of Θ with f ∈ C(ℝ+0 ) x

(Θ ∗ f )(x) = ∫ f (y) dy

(41)

0

is the operator of integration. It is treated as if it were a multiplication. 8.2 Interpretation. The operational calculus of convolution quotients is based on interpreting fractional integration, I α f ⋅ − (Θ ∗ ⋅ ⋅ ⋅ ∗ Θ)f = Θn f ,

(42)

as an n-fold convolution product with Θ where α ⋅ − n ∉ ℕ. 8.3 Let (C(ℝ+0 ), +, ∗) denote the commutative ring (over ℂ) of complex-valued continuous functions f : ℝ+0 → ℂ with pointwise addition, pointwise multiplication with numbers and convolution as in (29). Then Θ ∈ C(ℝ+0 ), and Euler’s first integral implies the law of exponents Θα ∗ Θβ = Θα+β for Re α > 0, Re β > 0. 8.4 The commutative convolution ring C(ℝ+0 ) does not contain any divisors of zero, because f ∗ g = 0 implies that either f = 0 or g = 0. The ring C(ℝ+0 ) can therefore be extended to a field (Q(ℝ+0 ), +, ×) of convolution quotients in the same way as the ring of integers ℤ is extended to the field of rationals ℚ. The elements of Q(ℝ+0 ) are ordered pairs (f : g) of a convolution numerator f and a convolution denominator g ≠ 0 defined such that g ∗ (f : g) = f ,

g ≠ 0,

(43)

holds. Addition, multiplication and multiplication of elements from the field Q(ℝ+0 ) with numbers are defined as (f : g) + (h : k) = (f ∗ k + g ∗ h : g ∗ k), (f : g) × (h : k) = (f ∗ h : g ∗ k), a(f : g) = (af : g),

(44a) (44b) (44c)

58 | R. Hilfer for f , g, h, k ∈ C(ℝ+0 ), g ≠ 0, k ≠ 0, a ∈ ℂ. Note that a(f : g) ≠ a × (f : g) where a × (f : g) = aΘ × (f : g) = a(Θ ∗ f : g), i. e. multiplication with a number is not the same as multiplication with a constant function in Q(ℝ+0 ). 8.5 The neutral element for multiplication δ = (Θ : Θ) acts like a Dirac δ-function. This suggests an interpretation of convolution quotients as generalized functions. The mappings a 󳨃→ (aΘ : Θ),

a ∈ ℂ,

f 󳨃→ (Θ ∗ f : Θ),

f ∈

f 󳨃→ (Θ ∗ f : Θ),

f ∈

C(ℝ+0 ), L1loc (ℝ+0 ),

(45a) (45b) (45c)

are embeddings of ℂ, C(ℝ+0 ) resp. L1loc (ℝ+0 ) into the field Q(ℝ+0 ). 8.6 The definition of fractional integration as convolution with Θα for Re α > 0 can be extended also to all α ∈ ℂ with Re α < 0 as Θα = (Θα+n : Θn )

(46)

where n = −(⌊Re α⌋ − 1) is the smallest positive integer such that Re α + n > 1. For α = −1 one finds D = Θ−1 = (Θ : Θ2 )

(47)

and this is interpreted as the differentiation operator D. The fractional derivative operators are Dα = Θ−α with Θ0 = δ = D0 , and they obey Dα Dβ = Dα+β . 8.7 The operational calculus in Q(ℝ+0 ) is called Mikusinski calculus [144, 208]. For applications see [144, 48, 208, 60] and [133] in this handbook. An operational calculus for generalized Riemann–Liouville derivatives was given in [90].

9 Distributional interpretation 9.1 Let f (x) ∈ Lp (𝕁), g ∈ Lq (𝕁) with 1/p + 1/q ≤ 1 + α, p, q ≥ 1 and p ≠ 1, q ≠ 1 for 1/p + 1/q = 1 + α. Fractional integration by parts b

∫ f (x)(I αa+ g)(x)dx a

b

= ∫ g(x)(I αb− f )(x)dx

(48)

a

can be used to extend fractional integrals to distributions, if g is viewed as a test function from a space mapped to itself by I αb− . Let C∞ c (ℝ) denote the space of test functions, i. e. smooth functions f : ℝ → ℝ with compact support. Its topological dual space consisting of continuous linear forms u : C∞ c (ℝ) → ℝ is the space of distributions,

Mathematical and physical interpretations of fractional derivatives and integrals | 59

denoted 𝒟󸀠 (ℝ). Let 𝒟+󸀠 (ℝ) denote the set of distributions u such that there is an a ∈ ℝ with supp u ⊂ [a, ∞). 9.2 Interpretation. The distributional interpretation views differentiation Dα u ⋅ − δ(n) ∗ u

(49)

as convolution with the nth derivative δ(n) of the Dirac-δ when α ⋅ − n ∉ ℕ. 9.3 The distributional interpretation is based on the relations Dn K n = δ and D K n+1 = K n for all n ∈ ℕ in the sense of distributions, and the fact that K α ∈ 𝒟󸀠 (ℝ) K

−n

and supp K α ⊂ ℝ+0 ,

(50a) (50b)

(n)

=δ ,

K α ∗ K β = K α+β ,

(50c)

hold true for all α, β ∈ ℂ and n ∈ ℕ ∪ {0}. As a result I α0+ : 𝒟+󸀠 (ℝ) → 𝒟+󸀠 (ℝ) with I α0+ u = K α ∗ u is a bounded linear operator on 𝒟+󸀠 (ℝ) for all α ∈ ℂ. It fulfills additivity α+β β α I α0+ I 0+ = I 0+ and I −α 0+ = D0+ [178, 58, 40].

10 Functional calculus interpretation 10.1 Interpretation. Let A : X → X be a closed operator (see Definition A.4) on a Banach space X and A (σ(A)) an algebra of functions F : σ(A) → ℂ on the spectrum of A. A functional calculus for A is interpreted as a mapping A ∋ F 󳨃→ B ⋅ − F(A) ∈ 𝒞 (X)

(51)

that assigns to each F ∈ A a closed operator B : X → X interpreted as F(A) in such a way that for B ⋅ − F(A) and C ⋅ − G(A) also B ∘ C ⋅ − (F ∘ G)(A). 10.2 The natural powers An from Paragraph 2.5 suffice to define polynomial or raak z k be a polynomial of degree tional functions of A as examples. Let F(z) = ∑deg(F) k=0 deg(F) with complex coefficients ak ∈ ℂ. Then the operator deg(F)

F(A) := ∑ ak Ak , k=0

D(F(A)) = D(Adeg(F) )

(52)

is well defined. If ρ(A) ≠ 0, then F(A) is a closed operator for each polynomial F in the polynomial ring ℂ[z]. The spectral mapping theorem σ(F(A)) = F(σ(A)) holds. If a bounded operator commutes with A, then it commutes also with F(A). The mapping ℂ[z] ∋ F 󳨃→ F(A) is a functional calculus for polynomials. 10.3 Let F, G ∈ ℂ[z] be two polynomials and let G be such that its set of zeros {λ ∈ ℂ : G(λ) = 0} ⊂ ρ(A) is contained in the resolvent set of A (see Definition A.5). The rational function h(A) := F(A)G(A)−1

(53)

60 | R. Hilfer of A is well defined with domain D(Adeg(F)−deg(G) ), if deg(F) ≥ deg(G), D(h(A)) = { X otherwise.

(54)

Again h(A) is a closed operator. Its spectrum obeys h(σ(A)) ⊂ σ(h(A)) where σ(A), σ(A) = { σ(A) ∪ {∞}

if A is bounded, otherwise,

(55)

is the extended spectrum of A. The mapping h 󳨃→ h(A) is a rational calculus for linear operators on X.

11 Spectral projection interpretation 11.1 In finite dimensional spaces X an operator A : X → X is a matrix. If it can be transformed into diagonal form A = diag(λ1 , . . . , λn ), the eigenvalues λi appear on the diagonal. The fractional power Aα is then defined as Aα = diag(λ1α , . . . , λnα ). This Aα has the same eigenvectors as A, and, if λ is an eigenvalue of A, then λα is an eigenvalue of Aα . The finite dimensional calculus is extended to Hilbert spaces using the spectral theorem. 11.2 Let A : H → H denote a selfadjoint operator on a Hilbert space H with scalar product (⋅, ⋅). Its domain is denoted D(A), its spectrum σ(A) and its spectral family Eλ . Then (A u, v) = ∫ λ d(Eλ u, v)

(56)

σ(A)

holds for all u, v ∈ D(A). The fractional power Aα is defined by (Aα u, u) := ∫ λα d(Eλ u, v)

(57)

σ(A)

on the domain D(Aα ) = {u ∈ H : (Aα u, u) < ∞}. Generally, for any Borel measurable function g : σ(A) → ℂ the operator g(A) is defined by replacing the integrand in equation (57) with g(λ). The mapping g 󳨃→ g(A) is sometimes referred to as spectral calculus. 11.3 Fractional spectral calculus was studied in [127]. It was shown that the domains D(Aα ) for m-accretive operators coincide with certain real interpolation spaces constructed by the trace method.

Mathematical and physical interpretations of fractional derivatives and integrals | 61

12 Cauchy integral interpretation 12.1 Interpretation. Let F : Ω → ℂ be a holomorphic function in an open domain Ω ⊂ ℂ and let C : [0, 1] → Ω be a closed path inside Ω so that C (0) = C (1). Let a ∈ Ω be a point in the interior of the region encircled by C . Then Cauchy’s integral formula F(a) =

1 F(z) dz ∫ 2πi z − a

(58)

C

holds, where the path is traversed counter clockwise. The interpretation F(a) ⋅ − F(A) of the number F(a) as the operator F(A) is based on the interpretation (z − a)−1 ⋅ − R(z, A)

(59)

of the function (z − a)−1 as the resolvent operator of A at z defined in Definition A.5. 12.2 Let A ∈ ℬ(X) be a bounded operator on a Banach space X such that its spectrum σ(A) ⊂ Ω ⊂ ℂ is contained within an open set Ω. Let Hol(Ω) denote the algebra of holomorphic functions F : Ω → ℂ and define the mapping Φ : Hol(Ω) → ℬ(X) with Φ(F) = F(A) :=

1 ∫ F(λ)R(λ, A) dλ 2πi

(60)

C

by Cauchy’s integral. Here C is a path encircling the spectrum σ(A) of A in a positive sense. Then Φ is characterized as the unique map satisfying: (a) Φ is an algebraic homomorphism. (b) If Pα are the functions Pα (z) = z α for α ∈ ℂ, then Φ(P0 ) = 1X and Φ(P1 ) = A. (c) If a sequence Fn ∈ Hol(Ω) converges uniformly on compact sets to F ∈ Hol(Ω), then Φ(Fn ) → Φ(F) in ℬ(X). The mapping Φ is called Riesz–Dunford calculus for bounded operators. 12.3 If the domain Ω of F is such that ℂ \ Ω ∩ σ(A) ≠ 0, then the contour integral is singular. The problem can be overcome by finding a function g : Ωg → ℂ such that ℂ \ Ωg ∩ σ(A) = 0 and ℂ \ Ωh ∩ σ(A) = 0 where h = Fg −1 , and such that equation (60) can be used to define g(A) and h(A). Then F(A) is defined as F(A) := g(A)h(A). Examples where g(z) = (1 + z)n is a polynomial can be found in [9, 138]. 12.4 This Riesz–Dunford calculus for ℬ(X) can be generalized to sectorial operators [138]. Because sectorial operators (defined in Definition A.6) can have unbounded spectra, the path C , as a curve on the Riemann sphere, passes through the point at ∞. The function F therefore must decrease sufficiently rapidly at ∞ for the Cauchy integral to make sense. Suitable functions F : 𝕊ϕ → ℂ on a sector 𝕊ϕ (see equation (103)) are those for which there exist C ≥ 0 and γ > 0 such that |F(z)| ≤ C min{|z|γ , |z|−γ } for all z ∈ 𝕊ϕ . For such functions an operator F(A) ∈ 𝒞 (X) can be defined by the Cauchy

62 | R. Hilfer algorithm for a suitably restricted class of functions [5]. The complex powers defined in this way coincide with the complex powers studied in [113]. 12.5 The benefits of Cauchy integrals for fractional calculus were highlighted in [116] and an incipient functional calculus for compact operators emerged from [54, 164, 206]. Later developments [192, 131, 41] were summarized in [42] (see also [170, Chapter 10]). The application of Cauchy integrals to fractional powers of sectorial operators was studied in [137].

13 Laplace transform interpretation 13.1 Interpretation. Let ℳ(ℝ+0 ) be the set of all complex Borel measures on ℝ+0 . The Laplace transform F := L {μ} of a measure μ ∈ ℳ(ℝ+0 ) is ∞

F(u) = ∫ e−ut dμ(t)

(61)

0

for all u ∈ ℂ with Re u > 0. The interpretation F(a) ⋅ − F(−A) of the number F(a) ∈ ℂ as an operator F(−A) is now based on the interpretation e−at ⋅ − eAt ⋅ − T(t)

(62)

of the function e−at as a semigroup T(t) = eAt with infinitesimal generator A. 13.2 Let X be a Banach space with norm ‖ ⋅ ‖. A one-parameter family of operators {T(t)}t≥0 ⊂ ℬ(X) is called a bounded strongly continuous semigroup if (a) T(t) ∈ ℬ(X) for all t ≥ 0 and sup{‖T(t)‖ : t ≥ 0} < ∞; (b) T(0) = 1X and T(t)T(s) = T(t + s) for all t, s ≥ 0; (c) limt→0+ ‖T(t)f − f ‖ = 0 for all f ∈ X. The operator A := lim+ t→0

T(t)f − f t

(63)

with D(A) = {f ∈ X: the limit (63) exists}, is called an infinitesimal generator of the semigroup. Generators of strongly continuous semigroups are closed and densely defined operators that uniquely determine T(t). Their spectrum lies in the left half plane ℂ \ 𝕊π/2 [92, 28, 46]. 13.3 Denote the set of Laplace transformations of measures from ℳ(ℝ+0 ) by L {ℳ(ℝ0 )} := {F : 𝕊π/2 → ℂ | F = L {μ} for some μ ∈ ℳ(ℝ0 )} +

+

(64)

Mathematical and physical interpretations of fractional derivatives and integrals | 63

and let A be the infinitesimal generator of a bounded strongly continuous semigroup on X. If the function F ∈ Hol(σ(−A)) ∩ L {ℳ(ℝ+0 )} corresponds to the measure μ ∈ ℳ(ℝ+0 ), then the bounded linear operator ∞

f 󳨃→ ∫ T(t)f dμ(t),

f ∈X

(65)

0

defined for all f ∈ X equals the operator F(−A) obtained from the extended Riesz– Dunford calculus in Paragraph 12.4. It is written as F(−A) or Φ(μ) ∈ ℬ(X). 13.4 The triple (ℳ(ℝ+0 ), +, ∗) is an algebra with convolution of measures as multiplication. The triple (L {ℳ(ℝ+0 )}, +, ⋅ ) is the corresponding function algebra by virtue of the convolution theorem. The mapping Φ : L {ℳ(ℝ+0 )} → ℬ(X)

F 󳨃→ Φ(μ) = F(−A)

(66)

with F(−A) from equation (65) is the unique map satisfying: (a) Φ is an algebraic homomorphism; (b) if Gz (λ) = (z − λ)−1 with Re z < 0, then Φ(Gz ) = (z − A)−1 = R(z, A); (c) if the sequence Fn corresponding to the sequence of measure μn converges weakly to the limit F in L {ℳ(ℝ+0 )}, then limn→∞ ‖Φ(Fn )f − Φ(F)f ‖ = 0 for all f ∈ X. The mapping Φ is called generalized Hille–Phillips calculus. The Laplace transform interpretation was developed originally for functions in [159, 92] and then extended to distributions in [154, 203]. 13.5 Rescaling the measure μ as μ(⋅/s) or the semigroup T as T( ⋅ s) with a parameter s > 0 gives a one-parameter family of bounded linear operators ∞

∞

0

0

Φs (μ) = ∫ T(ts) dμ(t) = ∫ T(t)dμ(t/s)

(67a)

on X, which can be used to obtain several known interpretations of fractional powers via integral representations. Let δx denote the Dirac measure at x ≥ 0 and D δx its distributional derivative. Then μ = δ0

n

μ = D δ0

󳨀→ Φs (μ) = 1X ,

󳨀→ Φs (μ) = (−A)

n

(67b) (67c)

for all s > 0 and n ∈ ℕ [204, Thm 2.5], while μ = δ1

󳨀→ Φs (μ) = T(s)

(67d)

64 | R. Hilfer reproduces the semigroup for the rescaling parameter, and μ = Θ(t)e−t arg s dt

󳨀→ Φ|s| (μ) = (s − A)−1 = R(s, A)

(67e)

yields its resolvent family for s ∈ ℂ and Re s > 0. The examples n

μ = Dn (δ0 − δ1 )

󳨀→ Φs (μ) = [1X −T(s)] ,

μ = D (δ0 − e dt)

󳨀→ Φs (μ) = [1X −(1X −sA) ] ,

n

−t

−1 n

(67f) (67g)

with n ∈ ℕ were studied in [188]. 13.6 For a large set of so-called n-measures the fractional powers, when interpreted as the limit α

∞

(−A) := lim Cα,n ∫ s−α−1 Φs (μ) ds, ε→0

ε

0 < α < n, n ∈ ℕ,

(68)

all define the same operator for a suitable constant Cα,n and a domain consisting of all f ∈ X such that the limit exists [188]. Indeed equation (67f) for n = 1 and Cα,1 = Γ(α)/Γ(1 − α) was used in [159] and for general n in [128] to define fractional powers. Example (67g) was used in [114] for that purpose. Balakrishnan’s interpretation [10] −(− A)α =

∞

sin(απ) ∫ AR(s, A) sα−1 ds π

(69a)

0

is obtained from equation (67f) using the integral representation α

∞

x = ∫ (1 − e 0

∞ −xy

)μ(y)dy = ∫ 0

x dν(y) x+y

(69b)

with μ(x) = αx −1−α /(Γ(1 − α)), ν(x) = sin(απ)xα−1 /π and 0 < α < 1.

14 Integral transform interpretations 14.1 An interpretation based on the double Laplace (or Stieltjes) transforms applies ∞ ∞ to functions f representable as f (λ) = ∫0 ∫0 eλz e−zt dμ(t)dz for μ ∈ ℳ(ℝ+0 ). It has been studied in [93, 94] and is reviewed in [137, Ch. 4]. 14.2 An interpretation using Fourier transforms instead of Laplace transforms has been given in [118]. It goes beyond generators of strongly continuous semigroups and considers also weak topologies. In this way it extends and unifies earlier approaches. 14.3 An interpretation in terms of Mellin transforms was developed in [161, 197]. It is well suited for studying purely imaginary powers.

Mathematical and physical interpretations of fractional derivatives and integrals | 65

15 Stochastic interpretation of α 15.1 A stochastic process (Xt )t ≥ 0 with state space ℝd is called a Lévy process if it has stationary and independent increments, and if its sample paths are right continuous and have left limits [18]. The process is completely determined by its characteristic exponent ζ : ℝd → ℂ defined via the relation E(eiXt ⋅k ) = e−tζ (k) ,

(70)

where E denotes the operation of taking expectation values and Xt ⋅ k the scalar product. The characteristic exponent ζ is continuous and negative definite (see equation (104)). The characteristic exponent ζ contains all information about the process. For example, the stochastic process is conservative if and only if ζ (0) = 0. 15.2 Interpretation. The stochastic interpretation identifies α from the behavior of the function ζ near the origin as α ⋅−

log gλ , log λ

(71)

i. e. as the index of regular variation [104, 179]. Here |ζ (λ|k|−1 )| , |k|→∞ |ζ (|k|−1 )|

gλ := lim

(72)

provided the limit exists on a set of λ > 0 with positive measure. 15.3 The stochastic interpretation is the basis for many physical interpretations at some mesoscopic scale. It is also the basis of stochastic differential (Langevin) equations and the continuous time random walk interpretation in Section 22. Its mathematical origin is potential theory (see also Section 19). The positive hyperharmonic functions for the Laplace equation are the excessive functions of the Brownian semigroup, or, equivalently, the harmonic measures in potential theory are the hitting distributions for Brownian motion. 15.4 For diffusion with fractional Laplaceans the stochastic interpretation was pioneered in [126, 23, 24, 51], for time fractional diffusion in [89]. The interpretation has been generalized to Feller processes and pseudodifferential equations [100].

16 Geometric interpretation 16.1 Leibniz’ question “... sed quid est in Geometria?”7 from September 30th, 1695 seems unanswered to this day. Leibniz’ rule (8) has over centuries evolved into 7 “... but what is it in geometry?”.

66 | R. Hilfer the basis for the modern geometric interpretation of derivatives as vectors and tensors [33]. The modern interpretation of D applies also in cases where the classical interpretation of (D f )(x), as the slope of the tangent to the graph of f at x, fails [26, 119]. 16.2 Iteration of Leibniz’ rule (8) yields Leibniz’ formula (3) for Dn . Extending equation (3) from n ∈ ℕ to α ∈ ℂ leads to a binomial series [92] ∞ α Dα (fg) = ∑ ( )(Dk f )(Dα−k g) k k=0

(73)

where the generalized binomial coefficient is given in equation (5a). This formula appears already in [129, p. 117] and has been discussed in [158]. Because equation (73) differs from equation (3) for α ≠ n, n ∈ ℕ, a geometrical interpretation of fractional derivatives appears remote and difficult. 16.3 Notwithstanding the failure of Leibniz’ rule some authors proposed a “fractional curl operator” [47] or “fractional vector calculus” [139]. Inspecting the definition of curlα in [47, eq. (9)] reveals, however, that curl0 = 1 ≠ limα→0+ curlα rendering the definition discontinuous at α = 0. More disturbingly there is no indication that the curl of a vector field is a 2-form, i. e. a tensor of rank 2. Inspecting in turn the definitions of fractional curl, divergence and gradient operators in [139, Sec. 3] reveals that the “mixing measure” survives the limit α → 1, so that the definitions do not reduce to their vector calculus counterpart in that limit. 16.4 There exist also publications [12, 37] that speak of “fractional differential forms”. The proposed interpretation based on eq. (11) in [37] fails, because eq. (11) in [37] is not defined for functions on ℝn if n ≠ 1. As a consequence, eq. (19) in [37, p. 2006] lacks meaning. 16.5 Note that the localized Riemann–Liouville derivative dα,β f (x)/dx α,β has a certain geometrical interpretation. For 0 < α < 1 it approximates f at x by a cusp instead of a straight line. The slope of the tangent at x corresponds to the “opening coefficient” of the cusp at x. The geometric interpretation was exploited for the generalized Ehrenfest classification [68–70, 72, 71, 77] (see Section 20).

17 Type changing interpretation 17.1 Interpretation. An unusual mathematical interpretation of the fractional derivative Dα,1 0+ of order α and type β = 1 [130, p. 10] [29, 30] has become popular in the physics literature [184, eq. (22)] [34, eq. (34)] [32, eq. (2)]. It interprets fractional differential equations of type β = 1 Dα,1 0+ f = g

⋅−

D f = D1−α,0 0+ g

(74)

Mathematical and physical interpretations of fractional derivatives and integrals | 67

as equations of type β = 0 with a second differential operator D entering the type-zero equation. 17.2 To exhibit the unusual character consider Dα,1 0+ : X → Y as a linear operator α,1 between two Banach spaces X, Y. Usually its domain D(Dα,1 0+ ) and its range R(D0+ ) are the spaces given in equation (95), namely α,1 D(Dα,1 0+ ) := {f ∈ X : ∃ g ∈ Y s. t. (f , g) ∈ D0+ },

(75a)

α,1 R(Dα,1 0+ ) := {g ∈ Y : ∃ f ∈ X s. t. (f , g) ∈ D0+ }.

(75b)

17.3 In [184, eq. (22)] or [34, eq. (34)], however, the authors interpret the operator Dα,1 0+ : X → Y indirectly by the first order derivative D : X → Y and the fractional Riemann–Liouville derivative D1−α,0 : X → Y of order 1 − α and type β = 0. Indeed, 0+ equation (74) suggests that the authors interpret Dα,1 0+ as having domain and range given by α,1 D(D0+ ) := {f ∈ D(D) : ∃ g ∈ Y s. t. (D f , g) ∈ Dα,1 0+ },

R(Dα,1 0+ )

:= {g ∈

D(D1−α,0 0+ )

: ∃ f ∈ X s. t.

(f , D1−α,0 0+ g)

(76a) ∈

Dα,1 0+ },

(76b)

which is puzzling. This unusual interpretation seems to restrict the domain without any physical or mathematical motivation or justification. 17.4 In addition, as pointed out already in [82, p. 46], this interpretation gives rise to an unusual form of the eigenvalue equation. Nevertheless this interpretation is adopted by numerous authors in physics [14, 185, 174, 184, 143, 209, 43].

18 Physical interpretations 18.1 While mathematical interpretations of fractional derivatives and integrals abound, physical interpretations are often questionable. Fundamental theories of physics generally contain only integer order derivatives. This raises at least two fundamental questions discussed in [82]: (a) Are mathematical models with fractional derivatives consistent with the fundamental laws and fundamental symmetries of nature? (b) Can the fractional order α of differentiation be related to, or derived from, established theories of physics? A partially positive answer to question (a) is given in Sections 20 and 21. It suggests the following definition. Definition 2. A physical interpretation of a fractional derivative Dα or integral I α is an identification α ⋅ − (x ←→ H)

(77)

68 | R. Hilfer of the fractional order α ∉ ℕ with a quantity x that can be related to (or computed from) the energy H (Hamiltonian) of a physical system. A physical interpretation is called tentative, if an established phenomenological but nonrigorous relation x ←→ H exists that is supported by experimental or numerical evidence. Other interpretations are called questionable. 18.2 Numerous physical interpretations have been attempted in the literature. The reader might consult [76, 202, 108, 107, 196] for reviews and references reflecting their evolution with time. Physical interpretations of fractional derivatives are significantly more difficult than mathematical interpretations. Besides being well defined, a physical interpretation must not contradict established theory or experiment. Given such fundamental constraints, surprisingly few proposed interpretations mention, discuss or contemplate the basic questions above. 18.3 Fractional models that arise from reformulating models without fractional operators, will not be considered here. For examples, see [157, Ch. 10]. Also, questionable physical interpretations will mostly be left out. 18.4 An example for a questionable interpretation is “fractional duality in electromagnetism” [47]. Equations (18) and (19) in [47] would seem to be in direct conflict with electrodynamics and relativity theory, because the vectors appearing in them cannot be vectors in the usual sense (see also Paragraph 16.3). 18.5 Consider next the “fractional time Schrödinger equation” [99, eq. (8)]. While ̂ in [99, eq. (8)] is a (non-dimensional) energy, the operator the Hamiltonian H α α α α (ih) 𝜕 /𝜕t on the left hand side is a non-dimensional (energy) instead. In other words, the left hand side operator cannot be a physical interpretation of the right hand side operator for α ≠ 1. The error can be traced to [153] where, curiously, also the speed of light and the gravitational constant appear in a non-relativistic equation. 18.6 Similar problems appear in eq. (1.118) in [191, p. 39]. Using the notation of [191] and inserting ρ = R30 ρ󸀠 and x = x󸀠 /R0 , y = y󸀠 /R0 , z = z 󸀠 /R0 into eq. (1.118) in [191, p. 39], the left hand side has units of charge [C], while the right hand side has units of [Cm3−D ] where D ≠ 3. The same error appears in [191] (e. g. eq. (1.88)) and numerous other publications.

19 Nonlocal interpretations 19.1 Fractional derivatives with respect to position are nonlocal operators. Locality is a deep and well established fundamental principle of physics [190, 64, 82]. An increasing number of publications “generalize” partial differential equations for physical phenomena by replacing the local Laplace operator Δ with a nonlocal fractional power −(−Δ)α/2 . The generalization is usually accompanied by postulating a “meso-

Mathematical and physical interpretations of fractional derivatives and integrals | 69

scopic” stochastic process along the lines of Section 15. Independent experimental evidence for this process is often absent. 19.2 The mathematical background for such generalizations is fractional potential theory [166]. Let ϱ be a positive measure on ℝd that is absolutely continuous with respect to the d-dimensional Lebesgue measure dx, and let ρ = dϱ/dx be its density function ρ : ℝd → ℝ. Then for d ∈ ℕ, d > 2 the integral Φϱ,d (x) = ∫ ℝd

dϱ(y) , |x − y|d−2

x ∈ ℝd

(78)

is called the potential of ϱ (or ρ) at x, because −ΦGϱ,3 is the specific potential energy of gravitation (with units [J/kg]), if G ≈ 6.67 × 10−11 kg−1 m3 s−2 is the gravitational constant and ρ is the mass density. 19.3 Interpretation. Let Δd denote the Laplace operator in ℝd . The d-dimensional Riesz integral I α ρ (or Riesz derivative Dα ρ) of a density function ρ = dϱ/dx, Iα ρ ⋅ −

Γ((d − α)/2) α d/2

2 π

Φϱ,d+2−α ⋅ − (−Δd )−α/2 ρ − ⋅ D−α ρ,

Γ(α/2)

(79)

is interpreted as a Newtonian potential in d + 2 − α “fractional dimensions”, or as a “fractional potential” in d dimensions [166]. The parameter range 0 < α < d/2 can be analytically continued to all α ∈ ℂ with α ≠ ±(d + 2k), k ∈ ℕ ∪ {0}. 19.4 Many “generalized” partial differential equations are based on this nonlocal interpretation and thus contain the fractional Dirichlet problem as a special case. The fractional Dirichlet problem for a domain 𝔹(z, R) and fractional order 0 < α ≤ 2 is to find a suitably regular function f : ℝd → ℝ obeying [84] (−Δ)α/2 f (x) = 0,

x ∈ 𝔹(z, R),

x ∈ ℝd \ 𝔹(z, R),

f (x) = g(x),

(80a) (80b)

for suitably regular data g with ∫ ℝd \𝔹(z,R)

|g(x)| < ∞. 1 + |x|d+α

(81)

Here 𝔹(z, R) = {x ∈ ℝd : |x − z| < r} denotes a ball of radius R > 0 centered at z ∈ ℝd . The solution of the fractional Riesz–Dirichlet problem is the fractional Poisson integral [117] f (x) =

) Γ( d2 ) sin( πα 2 π

d +1 2

∫ ℝd \𝔹(z,R)

|R2

|R2 − |x − z|2 |α/2 g(y) dd y − |y − z|2 |α/2 |x − y|d

(82)

for x ∈ 𝔹(z, R). The solution reduces to the standard Poisson integral for α → 2. The crucial difference between the cases α = 2 and α < 2 is the dimensionality of the

70 | R. Hilfer domain of integration. Although it has been known for 80 years [165], the fractional Poisson formula (82) seems to have escaped the attention of many contemporary authors. 19.5 The problem with equation (82) was pointed out in [82, 84]. While for α = 2 the solution depends only on values of g on the boundary, for α < 2 it depends on all values of g in an unbounded infinite exterior domain. For this reason the pioneers never ventured into proposing physical interpretations of their generalized diffusion equations. Indeed, Bochner cautions his readers explicitly, writing “Whether this might have a physical interpretation, is not known to us.” [23, eq. (7)]. The nonlocality of −(−Δ)α/2 for α < 2 implies action at a distance and makes it impossible to isolate the physical system from influences of the environment [82, 84]. The skeptical attitude of Bochner remains adequate as long as action at a distance remains unproven in experiment. 19.6 Contrary to Bochner’s skepticism towards physical interpretations, it is claimed in “fractional quantum mechanics” [120, eq. (3), eq. (7)] that α is a “fundamental [sic!] parameter in standard quantum and classical mechanics”. In other words, page 395 in [120] claims that physical interpretation is no problem at all. The Levy exponent α is interpreted in [120] along the lines of Section 15 as α ⋅ − dfractal , the fractal dimension of random paths. But Feynman paths are neither basic nor needed in “standard” quantum and classical mechanics. They have yet to be observed in experiment. Similarly, in mathematics a joint spectral measure for two non-commuting physical observables is yet to be derived. 19.7 On top of postulating a new parameter α for quantum physics, a second fundamental constant Dα with strange dimensions is also postulated in [120]. The ad hoc postulate of α necessitates the introduction of Dα as a fundamental constant of nature, above and beyond Planck’s constant. But it is unclear how to observe, measure or interpret the fractional constant Dα in experiment. 19.8 Finally, there has been some debate whether the eigenvalues and eigenfunctions for infinite potential wells published by numerous authors are valid or not [101, 132]. Besides such problems with technical aspects, the physical interpretation suggested in [120] currently has no derivation from a Hamiltonian, no phenomenological basis in theoretical physics, and no experimental support.

20 Thermodynamic interpretation 20.1 A genuine physical interpretation of fractional derivatives in the sense of Definition 2 was established in thermodynamics [68, 69]. Fractional derivatives of thermodynamic potentials permit a generalization of Ehrenfest’s classification scheme [44] for thermodynamic phase transitions.

Mathematical and physical interpretations of fractional derivatives and integrals | 71

20.2 Interpretation. The fractional order α is interpreted as the generalized Ehrenfest order of a thermodynamic phase transition, α ⋅ − 2 − φ1 ,

(83)

where φ1 = maxi {φi } is the largest thermodynamic fluctuation exponent. 20.3 The thermodynamic fluctuation exponents characterize a thermodynamic system in the vicinity of a phase transition [189, 66, 52]. The exponents form a partially ordered set. Its maximum, denoted as φ1 , characterizes fluctuations of the order parameter [200, 160]. For a liquid–gas system the maximal fluctuation exponent is related by δ=

1 1 − φ1

(84)

to the equation of state exponent δ. In this way α is directly given by the equation of state of the physical system. 20.4 Thermodynamic equations of state are measurable experimentally [8]. Theoretically they follow from thermodynamic potentials [59]. Thermodynamic potentials are in turn obtained from the Hamiltonian H of the physical system via the basic formula U = ⟨H⟩ for the thermodynamic internal energy U. Here the expectation value map ⟨ ⋅ ⟩ : A → ℝ+ is a continuous, positive and normalized linear functional on the C*-algebra of observables of the physical system [64]. For given (inverse) temperature β ∈ ℝ+ it can be defined for all A, B ∈ A , t ∈ ℝ by the KMS-condition [26] ⟨(T t A)B⟩ = ⟨B(T t+iβ A)⟩ where the map T z : A → A , z ∈ ℂ is defined as T z A = exp(iHz)A exp(−iHz) for all A ∈ A . The KMS-characterization links Hamiltonian mechanics, equilibrium statistical mechanics and thermodynamics with each other. 20.5 Localized fractional derivatives with respect to the field h conjugate to the order parameter of the phase transition appear in the fractional Clausius–Clapeyron equation in [77, p. 458f]. Examples are derivatives of order α = 4/3 for mean-field theory, of order α = 16/15 for the two-dimensional Ising model, or α = 2d/(d + 2) for the spherical model in 2 < d < 4 dimensions. Numerical examples are derivatives of order α ≈ 1.208 for the three-dimensional Ising model or α ≈ 1.216 for the three-dimensional Heisenberg model. 20.6 The classification of phase transitions and the thermodynamic interpretation of α as α ⋅ − 1 + (1/δ) was introduced in [68] and further developed in [69, 70, 72, 71, 77]. It has subsequently been extended to topological pressure functionals for dynamical systems in [172].

21 Classification of long time limits 21.1 This section gives a partial answer to question (a) in Section 18. The classification of limits is a physical interpretation in the sense that it gives a general bound

72 | R. Hilfer 0 < α < 1 on α, and links α to a subset of measure zero of the microscopic state or phase space of a physical system. It does not establish a direct link with the Hamiltonian. Instead, the classification provides a mathematical framework for the concept of local equilibrium in nonequilibrium statistical physics. This answers a fundamental question formulated in [187, Sec. 2.4, p. 25]. 21.2 Mathematically, the problem studied is that of induced automorphisms on subsets of measure zero in ergodic theory [36]. Given the transition map T : X → X of a dynamical system between any two time instants t0 < t1 ∈ ℝ its iterates T k f , k ∈ ℕ, f ∈ X represent the state of the system on the arithmetic progression of time instants 𝔸 = {t0 + kτ : k ∈ ℕ} ⊂ ℝ

(85)

where τ = t1 − t0 > 0 and t0 is the initial instant. The classification arises from investigating the induced automorphism TY : Y → Y induced by T for k → ∞ on a subset Y ⊂ X of small or zero measure. Its iterates TYN represent the state of the system on the arithmetic progression of time instants 𝔸 = {Nt0 + kτ : k ∈ ℕ, N ∈ ℕ}

(86)

for large k → ∞. The classification emerges from the limit N → ∞ while rescaling the time axis. 21.3 The result of taking the limit and rescaling the time axis yields a one-parameter family of semigroups, Tα (t) = Φt (μα ),

t ≥ 0,

(87a)

of the form of equation (67) with parameter 0 < α ≤ 1. The probability measure dμα = hα (t) dt has the density function 0 { { { hα (x) = { 1 ∞ (−1)j x−αj { ∑ { { x j=0 j! Γ(−αj)

for x ≤ 0, for x > 0,

(87b)

with respect to Lebesgue measure. The semigroups Tα arise from subordination [167, 24] with a stable subordinator [208, Sec. IX.11]. Concrete examples are given in Sections 22 and 23. 21.4 Originally, the classification of long time limits emerged from the classification of phase transitions of Section 20. The main result (87) was described in [71, Sec. V]. Equation (87) was initially obtained for classical systems in [75, 74]. It was then rederived by interpreting convolution as time averaging in [79]. 21.5 The inequality t ≥ 0 in (87a) reflects the irreversibility of time. The result t ≥ 0 provides “a general and model-independent mechanism for the origin of macroscopic

Mathematical and physical interpretations of fractional derivatives and integrals | 73

time irreversibility” [75] as well as additional “insight into the longstanding irreversibility paradox” [74, p. 544]. This insight was enunciated as the “reversed irreversibility problem,” introduced and solved in [81, 83]. The “reversed irreversibility problem” is the problem to explain the abundance of reversible equations in theortical physics given that time evolution is always irreversible in experiment [81, p. 235], [83, 85]. 21.6 Ergodicity breaking understood as “invariance breaking” or “stationarity breaking” was introduced in [75, 74]. The phenomenon was called “fractional ergodicity” in [75] or “fractional stationarity” in [74], and it emerges spontaneously from the dynamics. Its relevance for aging phenomena in glasses and other systems has long been appreciated [135, 45, 4, 6, 186, 25, 80, 141, 86]. 21.7 More recently, the basic result (87) has been generalized to quantum systems [85, 86, 88]. The invariance breaking expressed by equation (87) resolves the fundamental puzzle [187, Sec. 2.4, p. 25] of local equilibrium in nonequilibrium statistical physics.

22 CTRW interpretation 22.1 The continuous time random walk (CTRW) interpretation, discovered in [71, 89, 73], emerged from the classification of long time limits (Section 21) and the stochastic process interpretation (Section 11). It illustrates and exemplifies equation (87) for the case of master equations and Fokker–Planck equations [56]. 22.2 Continuous time random walks are parametrized by a waiting time density ψ : ℝ+0 → ℝ+0 and a transition probability λ : Ω × Ω → [0, 1] for transitions between two states in a set of states Ω. The integral equation for CTRWs reads t

p(z, t) = δzz0 Φ(t) + ∫ ψ(t − s) ∑ λ(z, z 󸀠 )p(z 󸀠 , s) ds, 0

z 󸀠 ∈Ω

(88)

t

where z, z0 ∈ Ω, t ≥ 0, Φ(t) = 1 − ∫0 ψ(s)ds, and p(z, t) is the probability to be in state z at time t if the walker started from state z0 at time 0. 22.3 It was shown in [89] that equation (88) is exactly equivalent to the fractional master equation 󸀠 󸀠 Dα,1 0+ p(z, t) = ∑ w(z, z )p(z , t), z󸀠

p(z, 0) = δzz0 ,

(89)

for all t ≥ 0, if and only if λ = 1 + τα w and ψ = ψα /τ holds true. Here α−1

t t ψα ( ) = ( ) τ τ

Eα,α (−

tα 1 ∞ (−t/τ)α(k+1) , )= ∑ α τ t k=0 Γ(α(k + 1))

(90)

74 | R. Hilfer τ > 0 is a time constant, w : Ω×Ω → ℝ are the transition rates between two states [56], and Eα,α denotes the generalized Mittag-Leffler function [97, 3, 98, 181]. For α → 1 one has ψ1 (x) = ex and the result was known since [21]. Recently, the result was extended to composite CTRWs [87], where a binomial Mittag-Leffler function (from [90]) appears for the waiting time density ψ. 22.4 Interpretation. The CTRW interpretation α in equation (89) ⋅ − parameter α in equation (90)

(91)

holds that the fractional order is the parameter of the waiting time density. 22.5 Continuous time random walks were introduced by Montroll and coworkers into solid-state, chemical and statistical physics as an idealization on a mesoscopic level of description [149, 148, 150, 211, 201, 95]. They were later studied also in mathematics [140, 20, 19] and have found applications to exciton trapping [146, 152], β󸀠󸀠 -alumina superionic conductors [91], organic photoconductors [17], dielectric relaxation [147, 22], turbulent plasmas [13], semiconductors [121], electron transport in noncrystalline electrodes [155], or transient photocurrents in amorphous solids [175], to name but a few. 22.6 The fractional master equation for continuous time random walks was introduced only much later in [89]. Contrary to the presentation in [184, p. 50f] there is no mention of fractional derivatives in [11] and no mention of CTRWs in [177]. Although the relation between continuous time random walks and generalized master equations [105, 21], as well as the asymptotic Fourier–Laplace solution [195, eq. (21), p. 402] [182, eq. (23), p. 505] [106, eq. (29), p. 3083] were known, the connection to fractional master equations and fractional diffusion had been overlooked.

23 Anomalous diffusion interpretation 23.1 The CTRW interpretation “plays a particularly important role in the theory of fractal time processes by virtue of its universality [71, 75]” as emphasized in [73]. For lattice walks with lattice constant σ > 0, where Ω = σℤd , the CTRW interpretation leads asymptotically to the fractional diffusion equation Dα,1 0+ p(r, t) = Cα Δp(r, t),

p(r, 0) = δrr0 ,

(92)

for any waiting time density ψ from the class of regularly varying functions with index 0 < α ≤ 1 (see Definition A.8 for the definition). The term ‘asymptotically’ means that position r and time t are rescaled in a suitable way. Equivalently, the limits σ → 0, τ → 0 are taken such that σ→0 σ2 󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ Cα 2τα τ→0

is the fractional diffusion constant Cα in (92) (see [73, 80], [79, Sec. 3.4]).

(93)

Mathematical and physical interpretations of fractional derivatives and integrals | 75

23.2

Interpretation. The anomalous diffusion interpretation holds that α in equation (92) ⋅ − index of regular variation of ψ in equation (88)

(94a)

or, equivalently, α ⋅− 1+

ψ(bt) 1 log( lim ) t→∞ ψ(t) log b

(94b)

where b > 0 is arbitrary as long as the limit exists for more than countably many values of b. 23.3 Note that Proposition A “p(r, t) satisfies a fractional diffusion equation” and Proposition B “p(r, t) is the solution of a CTRW with long time tail” are not equivalent [80]. Some claims in this direction are too general [35, 184, 142, 15] and some comments in [16] are invalid. The simultaneous limits σ → 0, τ → 0 can and must be taken in various ways to explore the different asymptotic regions in the parameter space of a given lattice CTRW-model [80]. Note that [80, eq. (22)] does not imply p(r, t) = 0. As referenced in [80, p. 38f] it is well known [18, p. 202] that whenever a sequence ϕn (k) of characteristic functions converges pointwise to some function ϕ(k) for all k, then the following propositions are equivalent: (a) ϕ(k) is the characteristic function of some random variable, (b) ϕ(k) is a continuous function of k, (c) ϕ(k) is continuous at k = 0. 23.4 The fundamental solutions for the integral formulation of equation (92) were studied in [177, 109, 110]. In [73, Table 1] their stretched Gaussian asymptotic behavior, their cusp at the origin and their relation with CTRWs were found. Fundamental solutions for equation (92) with arbitrary types 0 ≤ β ≤ 1 are given in [79, Sec. 3.3]. Fractional diffusions of type β ≠ 1 are not expected to arise asymptotically from a CTRW-model, because they do not have a probabilistic interpretation (see [78] and [79, Sec. p. 116ff]). 23.5 The continuous time random walk interpretation and the anomalous diffusion interpretation of α, in equation (94) are tentative in the sense of Definition 2. Theoretical arguments for continuous time random walks [211] fall short of a rigorous derivation from the Hamiltonian of a physical system. While there is some experimental support (see Paragraph 22.5), the waiting time density ψ remains a hypothetical mesoscopic quantity. It is difficult to measure ψ directly in an experiment.

Appendix A A.1 Let X, Y, Z be Banach spaces. A linear operator A : X → Y is a linear subspace of the direct sum X ⊕ Y. The domain D(A), range R(A) and kernel (or null space) N(A) of a linear operator A are D(A) := {f ∈ X : ∃ g ∈ Y s. t. (f , g) ∈ A},

(95a)

76 | R. Hilfer R(A) := {g ∈ Y : ∃ f ∈ X s. t. (f , g) ∈ A},

N(A) := {f ∈ X : (f , 0) ∈ A},

(95b) (95c)

and A is called injective if N(A) = 0 and surjective if R(A) = Y. The set of all linear operators from X to Y is denoted 𝒜(X, Y) and 𝒜(X, X) = 𝒜(X) for short. A.2 For λ ∈ ℂ the scalar multiple λ A and the inverse A−1 of A are defined as λ A := {(f , λg) ∈ X ⊕ Y : (f , g) ∈ A},

D(λ A) = D(A)

(96)

A

D(A ) = R(A).

(97)

−1

:= {(g, f ) ∈ Y ⊕ X : (f , g) ∈ A},

−1

For A, B ∈ X ⊕ Y their sum is defined as A + B = {(f , g + h) ∈ X ⊕ Y : (f , g) ∈ A, (g, h) ∈ B}

(98)

with D(A + B) = D(A) ∩ D(B). For A ∈ X ⊕ Y, B ∈ Y ⊕ Z the composition B ∘ A : X → Z is the linear operator defined as B ∘ A := {(f , h) ∈ X ⊕ Z : ∃g ∈ Y s. t. (f , g) ∈ A and (g, h) ∈ B}

(99)

with D(B ∘ A) = {f ∈ D(A) : ∃g ∈ D(B) s. t. (f , g) ∈ A}. The identity operator 1 : X → X is defined as 1 := {(f , f ) : f ∈ X}

(100)

and its scalar multiples will be abbreviated as λ 1 = λ. A.3 An operator A is called continuous if there exists a constant c ≥ 0 such that ‖ A f ‖ ≤ c‖f ‖ for all f ∈ D(A). An operator A : X → Y is called bounded, if it is continuous and D(A) = X. The set of all bounded linear operators from X to Y is denoted ℬ(X, Y) and ℬ(X) := ℬ(X, X) for short. A.4 An operator A : X → X is called closed if its graph {(f , A f ) : f ∈ D(A)} is a closed subspace in X ⊕ X. Equivalently, if its domain D(A) endowed with the graph norm ‖f ‖A := ‖f ‖ + ‖ A f ‖ is a Banach space. The set of closed operators is denoted 𝒞 (X). A.5 For a linear operator A : X → X the set ρ(A) := {λ ∈ ℂ : (λ − A)−1 ∈ ℬ(X)}

(101)

is called the resolvent set and σ(A) := ℂ \ ρ(A) spectrum of A. The operator R(λ, A) := (λ − A)−1 is called the resolvent operator at λ.

(102)

Mathematical and physical interpretations of fractional derivatives and integrals | 77

A.6 An operator A is called non-negative if ℝ− is contained in its resolvent set and ‖λ(λ+A)−1 ‖ ≤ M for 0 < λ < ∞ [115]. An operator A is called positive if it is non-negative and 0 ∈ ρ(A). Let 𝕊θ := {z ∈ ℂ : z ≠ 0, |arg z| < θ}

(103)

denote a sector of angle θ in the complex plane and 𝕊θ its closure. An operator A is called sectorial of angle θ < π if σ(A) ⊂ 𝕊θ and sup{‖λR(λ, A)‖ : λ ∉ 𝕊θ󸀠 } < ∞ for all θ < θ󸀠 < π. A.7 A function f : ℝn → ℂ is called negative definite, if n

∑ (f (xi ) + f (xj ) − f (xi + xj ))ci cj ≥ 0

i,j=1

(104)

for all x1 , . . . , xn ∈ ℝn and c1 , . . . , cn ∈ ℂ [24]. A.8 A measurable function f : ℝ+0 → ℝ+0 is said to vary regularly at infinity with index α, if lim

x→∞

f (bx) = bα f (x)

(105)

for all b > 0 [179]. For this to hold true it suffices that the limit exists on a set of b with positive measure.

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Virginia Kiryakova

Generalized fractional calculus operators with special functions Abstract: In this chapter we present a brief history and the basic ideas of the generalized fractional calculus (GFC). Here, by this term we mean integral, differ-integral and integro-differential operators involving special functions in the kernels, having a convolutional structure and satisfying the basic “axioms” of the classical FC. The notion “generalized operators of fractional integration” appeared in the papers of Prof. S. L. Kalla in the years 1969–1979. He suggested the general form of these operators, studied some of their formal properties and examples of them when the kernels were special functions as the Gauss and generalized hypergeometric functions, including arbitrary G- and H-functions. His ideas provoked the author to choose a more peculiar case of G- and H-kernels, which allowed developing a full theory of the corresponding GFC with many applications. The known fractional integrals and derivatives and other generalized integration and differential operators used in various areas of analysis, differential equations and mathematical models happened to fall in the scheme of this GFC. Keywords: fractional calculus, generalized fractional integrals and derivatives, generalized hypergeometric functions, convolutional integral transforms involving special functions MSC 2010: 26A33, 33C60, 44A20

1 A brief introduction to the classical R-L type fractional calculus The classical Fractional Calculus (FC) can be thought of as originating as early as probably in 1695, from the correspondence between l’Hospital and Leibnitz, discussing “What if n be 1/2 in dn y/dxn ?”. Since then, many well-known analysts and applied scientists contributed to the development of this “strange” calculus, but the first book and the first conference dedicated specially to that topic took place only 279 years after the mentioned correspondence! The detailed history, theory and various applications Acknowledgement: The author’s work on this chapter is in the framework of the program of the projects under bilateral agreements (2017–2019) of the Bulgarian Academy of Sciences with the Serbian and Macedonian Academies of Sciences and Arts, and under COST Action program CA 15225. Virginia Kiryakova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str. 8, 1113 Sofia, Bulgaria, e-mail: [email protected] https://doi.org/10.1515/9783110571622-004

88 | V. Kiryakova in the years 1987–1993 can be found in the “FC Encyclopedia” [55], and more recently in our surveys [64, 65]. One of the trends of the contemporary fractional calculus, since 1960s, is the socalled Generalized Fractional Calculus (GFC). Along with the explosion of numerous and even unexpected recent applications of the operators of the classical FC, the GFC is another powerful tool stimulating the development of this field. The GFC surveyed here induced also the study of new classes of special functions (special functions of fractional calculus, abbrev. as SFs of FC) and new integral transforms (generalized ITs of convolutional type); and it provided new transmutation operators useful to solve more complicated problems in analysis and its applications, via their reduction to simpler ones with known solutions. The GFC posed also new challenges for interpretations of its operators, similar to already established ones for the classical fractional integrals and derivatives (see, e. g., [47, 48, 9], etc.), and new open problems for its applications in solving not only theoretical problems for fractional (multi-)order differential and integral equations, but in mathematical models of real phenomena and events (as has already been well illustrated for the classical FC). The classical FC is based on several (often equivalent) definitions for the operators of integration and differentiation of arbitrary order (i. e. noninteger order—including real, fractional or complex cases), as continuations of the classical integration and differentiation operators and of their integer order powers (n ∈ ℕ), namely of the n-fold integration n

z

tn−2

t1

tn−1

R f (z) := D f (z) = ∫ dt1 ∫ dt2 ⋅ ⋅ ⋅ ∫ dtn−1 ∫ f (tn )dtn −n

0

=

0

z

0

0

1 ∫(z − t)n−1 f (t)dt, (n − 1)!

(1)

0

and of the nth order derivatives Dn f (z) = f (n) (z). We need to emphasize that in this survey we give the details only for the socalled Riemann–Liouville (R-L) type integrals (left-hand sided), and the corresponding R-L and Caputo-type differentiation analogs. Thus, for the imposed limitations, we skip discussions of the Weyl-type (right-hand sided) integrals of GFC; the Riesz-, Hadamard-, Liouville-types etc. of operators of FC. We also do not discuss definitions based on discrete analogs, the multivariable cases, generalized function settings, etc. For all these operators and details, we refer for example, to [55, 47, 21], and numerous recent handbooks and websites on classical FC and its applications. The essence of the mathematical problem for defining integrals and derivatives of fractional order consists in the following, so-called “axioms” of FC (which can be found, e. g., in Dzherbashyan [6], and since very recently in [45], etc.): for each function f (z), z = x + iy, of sufficiently large functional class and for each number δ (rational, irrational, complex), to set up a correspondence to a function g(z) = Dδ f (z)

Generalized fractional calculus operators with special functions | 89

satisfying the conditions: – If f (z) is an analytic function of z, the derivative Dδ f (z) is an analytic function of z and δ. – The operation Dδ gives the same result as the usual differentiation of order n, when δ = n is a positive integer, and the same effect as the n-fold integration, if δ = −n is a negative integer (i. e. D−n = Rn ). Moreover, Dδ f (z) should vanish at the initial point z = 0 (or z = c) together with its first (n − 1) derivatives. – The operator of order δ = 0 is the identity operator. – The fractional calculus operators are linear: Dδ {af (z) + bg(z)} = aDδ f (z) + bDδ g(z). –

For integrations of arbitrary fractional orders α > 0, β > 0 (ℜα > 0, ℜβ > 0) the additive index law (semigroup property) holds: D−α D−β f (z) = D−(α+β) f (z),

Rα Rβ f (z) = Rβ Rα f (z) = Rα+β f (z);

i. e.

the denotation Rδ f (z) := D−δ f (z), ℜδ > 0 is often used in the case of a derivative of negative order (or one with negative real part). The definition for the Riemann–Liouville (R-L) fractional integral (in the whole chapter we discuss only the so-called left-hand sided variants of operators, denoted in the δ literature also as I0+ ) is z

1

0

0

1 (1 − σ)δ−1 R f (z) = D f (z) = f (zσ)dσ. ∫(z − t)δ−1 f (t)dt = z δ ∫ Γ(δ) Γ(δ) δ

−δ

(2)

It is easily seen to satisfy all mentioned conditions, and in particular, to coincide with the repeated (n-fold) integration represented by the Dirichlet formula in the form (1). If z = x+iy is a complex variable, the above representation can be modified to the Cauchy integral formula. The R-L definition (2) concerns integrations of (real part) positive orders and could not be used directly for a differentiation (ℜδ < 0). However, a little trick is helpful to set a suitable correct expression. For noninteger δ > 0 we take n := [δ] + 1 (the smallest integer greater than δ); then we can define properly the R-L fractional derivative by means of the differ-integral expression Dδ f (z) = Dn Dδ−n f (z) = (

n

d ) Rn−δ f (z) dz

n

z

d 1 =( ) { ∫(z − t)n−δ−1 f (t)dt}, dz Γ(n − δ)

(3)

0

since n − δ > 0. In suitable functional spaces, Dδ Rδ f (z) = f (z), i. e. the inversion formula holds: {Rδ }

−1

= Dδ .

90 | V. Kiryakova In view of the formula Dδ {cz α } = c

Γ(α + 1) α−δ z , Γ(α + 1 − δ)

δ > 0, α > −1,

an interesting fact, in conflict with the classical calculus, is obtained for α = 0: Dδ {c} = c

z −δ . Γ(1 − δ)

That is, a R-L type fractional derivative of a constant is not zero in general, but only for positive integer values δ = n = 1, 2, 3, . . . . This problem is avoided by using the Caputo fractional derivative (called also Dzherbashyan or Gerasimov derivative, among others; see details, e. g., in [33, 34]) δ

∗D

f (z) := I

z

1 f (n) (t)dt D f (z) = , ∫ Γ(n − δ) (z − t)δ−n+1

n−δ n

n − 1 < δ ≤ n.

(4)

0

Skipping the details on the Riesz- and Hadamard-type operators of classical FC, let us mention the so-called Hilfer (generalized R-L) derivative [10], of order α and type β, defined for α > 0, 0 ≤ β ≤ 1 and n − 1 < α ≤ n ∈ ℕ, as a combination of the fractional derivatives (3) and (4): Dα,β f (z) := I β(n−α)

dn (1−β)(n−α) I f (z). dz n

(5)

The additional parameter β allows one to interpolate continuously from the R-L derivative Dα,0 ≡ Dα to the Caputo derivative Dα,1 ≡ ∗ Dα . As mentioned, along with the R-L definitions of the fractional order integrals, several modifications and their generalizations are widely used. The most useful classical fractional integrals, however, seem to be the Erdélyi–Kober (E-K) integrals, allowing two additional parameters: γ,δ Iβ f (z)

z

=z

−β(γ+δ)

1

=∫ 0

∫ 0

(z β − τβ )δ−1 βγ τ f (τ)d(τβ ) Γ(δ)

1 (1 − σ)δ−1 σ γ f (zσ β )dσ, Γ(δ)

δ ≥ 0, γ ∈ ℝ, β > 0,

(6)

appearing first for β = 1, β = 2 in the work of Kober [36], Erdélyi [7], and further applied and studied in the general case by Erdélyi [7] and Sneddon [58]. A very close variant of the Erdélyi–Kober integral (6) (but with particular value γ = 0) is the socalled Katugampola operator [19], which, considering the presence of the additional multiplier βδ−1 in front of the integral, can encompass also the Hadamard operator as a limiting case.

Generalized fractional calculus operators with special functions | 91

The E-K fractional integrals (6) are used essentially in our work as a basis of the GFC, and in the present survey chapter. The corresponding (left-hand sided) R-L and Caputo-type Erdélyi–Kober fractional derivatives have been defined and studied in Kiryakova [25], Luchko et al. [66, 41], with n − 1 < δ ≤ n, respectively: γ,δ

γ+δ,n−δ

Dβ f (z) := Dn Iβ γ,δ ∗ Dβ f (z)

γ+δ,n−δ

:= Iβ

n 1 d γ+δ,n−δ + γ + j)]Iβ f (z), f (z) = [∏( z β dz j=1 γ+δ,n−δ

Dn f (z) = Iβ

n 1 d + γ + j)f (z). ∏( z β dz j=1

(7) (8)

As already mentioned, in this survey we discuss classical and generalized fractional calculus operators based on the Riemann–Liouville and Erdélyi–Kober fractional integrals and their compositions (for brevity of our exposition, confined only to the left-hand sided versions). Recently many so-called “new” and other “generalized” fractional derivatives have appeared in the literature. Usually, as in the classical FC, they are not related to special functions in the kernels. Thus, they are outside the scope of this survey. But for most of them, the majority of FC experts are in doubt as to what extent some of these operators can be called operators of FC, since they are either not satisfying the “axioms”, or in essence they are integer order operators, or local modifications, etc. For these reasons we do not discuss them and refer the interested reader to some polemical work as in [45, 63].

2 Generalized fractional integrals involving special functions Since the 1960s, several authors introduced and studied integral operators with some special functions in the kernels that happened to satisfy properties similar to the axioms of the classical FC, and playing similar roles in operational calculi and transmutation approaches for integration and differentiation operators more general than the conventional ones. For example, Love [37], Saxena [56], Kalla and Saxena [17], Saigo [52, 53], McBride [44], also Tricomi, Sprinkhuizen-Kuiper, Koornwinder (see in Refs. of [25]), have studied and used different modifications of the so-called hypergeometric operators of fractional integration. We have z

Hf (z) =

μ

μz −γ−1 t ∫ F (δ, β + m; η; a( ) )t γ f (t)dt, Γ(1 − δ) 2 1 z 0

involving the Gauss hypergeometric function.

(9)

92 | V. Kiryakova Another example is the fractional integration operator involving the Bessel function, introduced by Lowndes [38, 39]: z

Iλ (η, ν + 1)f (z) =

2ν+1 −(ν+η+1) 2η+1 2 2 ν2 (z − t ) Jν (λ√z 2 − t 2 )f (t)dt, z ∫t λν

(10)

0

in relation to the second order Bessel-type differential operator Bη = z −2η−1

d 2η+1 d z . dz dz

In [14, 18], Kalla, Galue and Srivastava considered the so-called Wright–Erdélyi– Kober operator of fractional integration, ∞

γ,δ

Wβ,λ f (z) = λ ∫ σ λ(γ+1)−1 Jγ+δ−λ(γ+1)/β (σ λ )f (zσ)dσ,

(11)

−λ/β

0

μ

where β > λ > 0, δ ≥ 0, γ are real parameters and the kernel Jν (σ) stands for the Wright–Bessel (called also Bessel–Maitland, or generalized Bessel) function. For β = γ,δ λ, (11) coincides with the Erdélyi–Kober integral Iβ within limits from 0 to 1, due to the properties of the kernel function; see, e. g., Kiryakova [30]. Some authors introduced generalized operators of fractional integration involving the Mittag-Leffler function, and its modifications and extensions. For example, a generalized fractional integral was considered by Prabhakar [49] in the form Eϰα,β,λ f (z)

z

ϰ = ∫(z − t)β−1 Eα,β [λ(z − t)α ]f (t)dt,

(12)

0

and the corresponding fractional derivative appeared in Kilbas–Saigo–Srivastava [20]. Γ(ϰ+k) zk ϰ Here, Eα,β (z) = ∑∞ k=0 Γ(ϰ) k!Γ(αk+β) is the generalized Mittag-Leffler function (called the Prabhakar function), which for ϰ = 1 is the Mittag-Leffler function itself, Eα,β (z). ϰ,ξ

A further extension was provided by Srivastava and Tomovski [61] with a Eα,β -function, and we refer to Atangana and Baleanu [2] among others, for work on operators with Mittag-Leffler functions. One of the most general fractional integration operators of type (2) can be obtained when the kernel function is an arbitrary Meijer G-function, as in Kalla [12], and also in Parashar [46], Rooney [51], etc.: z

IG f (z) = z

−γ−1

r

t m,n [a( ) ∫ Gp,q z 0

󵄨󵄨 (a )p 󵄨󵄨 j 1 ] t γ f (t)dt, 󵄨󵄨 󵄨󵄨 (bk )q1

(13)

or its further generalization, the Fox H-function, as in Kalla [11], also in Srivastava and Buschman [59], Saxena and Kumbhat [57], and others:

Generalized fractional calculus operators with special functions | 93 z

r p t 󵄨󵄨󵄨 (a , A ) m,n IH f (z) = z −γ−1 ∫ Hp,q [a( ) 󵄨󵄨󵄨 j j 1q ] t γ f (t)dt. z 󵄨󵄨 (bk , Bk )1

(14)

0

In this respect, we need to recall the definitions of the mentioned generalized hypergeometric functions, which happen to play important roles as special functions of (generalized) fractional calculus. By Fox’s H-function we mean the generalized hypergeometric function defined by means of the contour integral (see [43, 62, 50], [25, the appendix]), 󵄨󵄨 (a , A ), . . . , (a , A ) 󵄨 m,n m,n p p Hp,q ] (σ) = Hp,q [σ 󵄨󵄨󵄨 1 1 󵄨󵄨 (b1 , B1 ), . . . , (bq , Bq ) 󵄨󵄨 (a , A )p 1 󵄨 m,n m,n (s)σ s ds, = Hp,q [σ 󵄨󵄨󵄨 j j 1q ] = ∫ Hp,q 󵄨󵄨 (bk , Bk )1 2πi

(15)

L

where the integrand in (15) has the form m,n (s) = Hp,q

n ∏m k=1 Γ(bk − Bk s) ∏j=1 Γ(1 − aj + Aj s)

∏qk=m+1 Γ(1 − bk + Bk s) ∏pj=n+1 Γ(aj − Aj s)

and L is a suitable contour in ℂ; the orders (m, n; p, q) are nonnegative integers such that 0 ≤ m ≤ q, 0 ≤ n ≤ q; the parameters Aj , j = 1, . . . , p and Bk , k = 1, . . . , q are positive and aj , j = 1, . . . , p, bk , k = 1, . . . , q are arbitrary complex numbers such that Aj (bk + l) ≠ Bk (aj − l󸀠 − 1);

l, l󸀠 = 0, 1, 2, . . . ; j = 1, . . . , p, k = 1, . . . , q.

The Wright generalized hypergeometric functions p Ψq (z), the Mittag-Leffler function κ Eα,β (z) and its extensions like the Prabhakar function Eα,β (z) and the multi-index Mittag-Leffler functions (see [27, 66]), the Wright–Bessel (Bessel–Maitland) function μ Jν (z) and the other mentioned SFs, appear all as cases of the H-function. Especially, when all Aj = Bk = 1, we obtain the so-called Meijer G-function [8, Vol. 1], which itself includes most of the elementary functions and of the (named) special functions of mathematical physics: 󵄨󵄨 (a , 1)p 󵄨󵄨 (a )p 󵄨 󵄨 m,n m,n Hp,q [σ 󵄨󵄨󵄨 j 1q ] = Gp,q [σ 󵄨󵄨󵄨 j 1q ] , 󵄨󵄨 (bk , 1)1 󵄨󵄨 (bk )1

(16)

n 󵄨󵄨 a , . . . , a ∏m 1 k=1 Γ(bk − s) ∏j=1 Γ(1 − aj + s) 󵄨 m,n p σ s ds. Gp,q [σ 󵄨󵄨󵄨 1 ]= ∫ q 󵄨󵄨 b1 , . . . , bq 2πi ∏k=m+1 Γ(1 − bk + s) ∏pj=n+1 Γ(aj − s)

(17)

that is,

L

In his papers [12, 13] of the years 1970–1979, Kalla suggested that all the above integral operators of R-L type (2) can be considered as “generalized operators of fractional integration” of the common form (here we mention only the left-hand sided type

94 | V. Kiryakova integrations): If (z) := I[f (z), γ] = z

−γ−1

z

1

0

0

t ∫ Φ( )t γ f (t)dt = ∫ Φ(σ)σ γ f (zσ)dσ, z

(18)

where the kernel Φ(σ) can be an arbitrary continuous function so that the above integral makes sense in sufficiently large functional spaces. Kalla established a series of their general properties, analogous to those of the classical fractional integrals, and also studied some of their special cases. By suitable choices of the kernel function Φ, the operators (18) can be shown to include all the other known fractional integrals of R-L type as particular cases.

2.1 Some formal properties of the generalized fractional integration operators, as from Kalla [12, 13] For example, in the case of a real variable z = x, the operator (18) and its right-hand sided analog (denoted by K[f (x), δ]) both exist for f (x) ∈ Lp (0, ∞), for 1 ≤ p, q < ∞, 1/p + 1/q = 1, if ℜ(γ) > −1/q (resp. ℜ(δ) > −1/p). The Mellin transform property (for the theory of the Mellin transform and its applications in FC, see, e. g., Luchko and Kiryakova [40]) is M{If (x); s} = Ψ(s)M{f (x); s} with Ψ(s) = t

1+η−s

∞

∫ xs−η−2 Φ(t/x)dx; t

other formal properties are, e. g., xβ I[f (x), γ] = I[xβ f (x), γ − β];

if I[f (x)] = g(x),

then I[f (ax)] = g(ax), a = const.;

the relation with the RH sided analog K[f (x)] : ∞

1 I[f ( ), γ] = K[f (x), γ + 1]; x

∞

∫ g(x)I[f (x)]dx = ∫ f (x)K[g(x)], etc. 0

0

For the cases (13) and (14) of (18) with arbitrary H- and G-functions in the kernels, not very many more formal properties can be derived, in addition to the above; we mention for example Kalla [11, 13]. A formal inversion formula can be written then by means of the Mellin transform property, where in such a case the function Ψ(s) is a kind of H- or G-function of the Mellin variable s. More details on the generalized operators of fractional integration in the considered sense can be found in Kalla [11–13], Kiryakova [25, 29], Srivastava and Saxena [60], etc.

Generalized fractional calculus operators with special functions | 95

3 Generalized fractional calculus based on compositions of Erdélyi–Kober operators The operators (13) and (14) involve arbitrary Meijer G-functions and Fox H-functions, and thus appear as the most general operators of the form (18), being in the classes of the convolutional type G- and H-integral transforms. However, in order to develop a meaningful detailed theory with practical applications, we had the hint to choose the kernel functions Φ in (18) (resp. in (13), (14)) as suitable peculiar cases of the G- and H-functions; we namely can write 󵄨󵄨 (γ + δ )m 󵄨 m,0 [σ 󵄨󵄨󵄨 k mk 1 ] , Gm,m 󵄨󵄨 (γk )1

󵄨󵄨 (γk + δk + 1 − 1 , 1 )m 󵄨 m,0 βk βk 1 [σ 󵄨󵄨󵄨 Hm,m ]. 󵄨󵄨 (γk + 1 − 1 , 1 )m βk βk 1

(19)

Thus, in Kiryakova [25] (also in [22–24], and further in Kalla–Kiryakova [15, 16]) we defined a class of generalized fractional integrals and derivatives by means of single (differ-)integrals involving the generalized hypergeometric functions (19). This allowed one to develop a detailed theory of GFC, with operational properties analogous to those of the classical R-L and E-K fractional integrals and derivatives, with numerous applications in solving problems for differential and integral equations (including of high integer order or fractional multi-order; see, e. g., [25, Ch. 3], [1]), for classes of analytic functions in geometric functions theory (see, e. g., [35, 28]), in operational calculus and integral transforms [25, 27, 5], and with a great impact to the theory of special functions (see [25–27, 1, 30, 32], etc.). The main functional spaces discussed in our works on GFC are weighted versions of the continuously differentiable functions C (k) , Lebesgue integrable functions Lp , or analytic functions H(Ω) in the complex domain. Let α, μ, ν be arbitrary reals, k ≥ 0 and 1 ≤ p < ∞ be integers, the variables x, z be real or complex numbers, running resp. over the interval [0, ∞) or in a domain Ω ⊂ ℂ, starlike with respect to the origin z = 0. Thus, we use the notations Cα(k) := {f (x) = xp f ̃(x);

p > α, f ̃ ∈ C (k) [0, ∞)}, ∞

Cα(0) := Cα ;

1 p

Lμ,p (0, ∞) := {f (x) : ‖f ‖μ,p = [ ∫ xμ−1 |f (x)|dx] < ∞}; 0

ν

Hν (Ω) = {f (z) = z f ̃(z); f ̃(z) ∈ H(Ω)},

ν ≥ 0,

H0 (Ω) := H(Ω).

Definition 1 ([22, 25]). Let m ≥ 1 be integer, β > 0, γ1 , . . . , γm and δ1 ≥ 0, . . . , δm ≥ 0 be arbitrary real numbers. By a generalized (multiple, m-tuple) Erdélyi–Kober (E-K) operator of integration of multi-order δ = (δ1 , . . . , δm ) we mean an integral operator 1

(γ ),(δ ) Iβ,mk k f (z)

󵄨󵄨 (γ + δ )m 1 󵄨 m,0 = ∫ Gm,m [σ 󵄨󵄨󵄨 k mk 1 ] f (zσ β )dσ. 󵄨󵄨 (γk )1 0

(20)

96 | V. Kiryakova Then each operator of the form Rf (z) = z βδ0 Iβ,mk

(γ ),(δk )

f (z) with arbitrary δ0 ≥ 0

(21)

is said to be a generalized (m-tuple) operator of fractional integration of Riemann– Liouville type, or briefly: a generalized (R.-L.) fractional integral. Generalizing further the operators of fractional calculus, in Kiryakova [23, 24, 15, 16, 25] we introduced also operators involving classes of Fox H-functions instead of the G-functions in (20), (21). We used the same name, namely generalized (multiple) E-K operators (fractional integrals): 1 󵄨󵄨 1 1 m { { m,0 [ 󵄨󵄨󵄨 (γk + δk + 1 − βk , βk )1 ] { { H σ f (zσ)dσ, ∫ 󵄨 m,m (γ ),(δ ) 󵄨󵄨 I(β k ),m k f (z) = { 󵄨󵄨 (γk + 1 − β1 , β1 )m 1 k 0 { k k [ ] { { {f (z), if δ1 = δ2 = ⋅ ⋅ ⋅ = δm = 0.

m

if ∑ δk > 0, k=1

(22)

In the above more general form, along with the multi-order of integration (δ1 , . . . , δm ) and the multi-weight (γ1 , . . . , γm ), we introduced also a multi-parameter (β1 > 0, . . . , βm > 0) (with different βk ) instead of the same β > 0 in the case with a G-function. Note that due to the relation (a generalization of (16)) p 󵄨 󵄨󵄨 (a1 , 1 ), . . . , (ap , 1 ) 󵄨 β 󵄨󵄨󵄨 (aj )1 m,n m,n β β [σ ] = βG [σ 󵄨󵄨󵄨 Hp,q 󵄨 p,q 󵄨󵄨 (b )q ] , 󵄨󵄨 (b1 , 1 ), . . . , (bq , 1 ) 󵄨 k 1 β β

β > 0,

(23)

the operator (22) involving a H-function reduces to the simpler form (20), k k for β1 = β2 = ⋅ ⋅ ⋅ = βm = β > 0 : I(β,β,...,β),m = Iβ,mk

(γ ),(δ )

(γ ),(δk )

.

(24)

Further, we introduced the corresponding generalizations of the classical Riemann–Liouville fractional derivatives (3). Definition 2. With the same parameters as in Definition 1 and with the integers δk ηk = { [δk ] + 1,

if δk is integer,

if δk is noninteger,

k = 1, . . . , m,

we introduce the auxiliary differential operator, a polynomial of m

ηr

Dη = [∏ ∏( r=1 j=1

1 d z + γr + j)]. βr dz

(25)

d : dz

(26)

Then we define the multiple (m-tuple) Erdélyi–Kober fractional derivative of multiorder δ = (δ1 ≥ 0, . . . , δm ≥ 0) by means of the differ-integral operator: D(βk ),mk f (z) = Dη I(β k ),mk (γ ),(δ ) k

(γ +δ ),(ηk −δk ) k

f (z)

Generalized fractional calculus operators with special functions | 97 1

󵄨󵄨 (γk + ηk + 1 − 1 , 1 )m 󵄨 m,0 βk βk 1 = Dη ∫ Hm,m [σ 󵄨󵄨󵄨 ] f (zσ)dσ. 󵄨󵄨 (γk + 1 − 1 , 1 )m βk βk 1

(27)

0

In the case (24) of all βk equal, βk = β > 0, we obtain a simpler representation with Meijer’s G-function, corresponding to generalized fractional integral (20): k Dβ,m

(γ ),(δk )

f (z) = Dη Iβ,mk

(γ +δk ),(ηk −δk )

f (z)

η

m r 1 d (γ +δ ),(η −δ ) + γr + j)]Iβ,mk k k k f (z). = [∏ ∏( z β dz r=1 j=1

(28)

More generally, all differ-integral operators of the form k Df (z) = Dβ,m

z

(γ ),(δk ) −δ0

(γk −

f (z) = z −δ0 Dβ,m

δ0 β

),(δk )

f (z) with δ0 ≥ 0,

(29)

are called generalized (multiple, multi-order) fractional derivatives. The generalized derivatives (28), (29) are the counterparts of the generalized fractional integrals (20), (21), of the Riemann–Liouville type. Later, their Caputo-type analogs have been introduced in Kiryakova and Luchko [33], as the integro-differential operator (γk ),(δk ) ∗ D(βk ),m f (z) 1

:= I(β k ),mk

(γ +δ ),(ηk −δk ) k

Dη f (z)

󵄨󵄨 (γk + ηk + 1 − 1 , 1 )m 󵄨 m,0 βk βk 1 = ∫ Hm,m [σ 󵄨󵄨󵄨 ] Dη f (zσ)dσ 󵄨󵄨 (γk + 1 − 1 , 1 )m βk βk 1 0

1

m ηr 󵄨󵄨 (γk + ηk + 1 − 1 , 1 )m 1 d 󵄨 m,0 βk βk 1 ] [∏ ∏( x = ∫ Hm,m [σ 󵄨󵄨󵄨 + γr + j)f (zσ)]dσ, 1 1 m 󵄨󵄨 (γk + 1 − , )1 r=1 j=1 βr dx βk βk

(30)

0

with the same parameters and operator Dη as in (25). Evidently, for m = 1, (27) and (30) are reduced to the R-L and Caputo-type E-K derivatives (7) and (8). In Kiryakova [31], we noted that the hint for introducing the operators (20), (21) and corresponding derivatives (28), (29) (in Kiryakova [22] among others) came from the studies of the Bessel-type differential operators B of arbitrary (integer!) order m > 1 (we called them afterwards hyper-Bessel differential operators [25, Ch. 3]) and studies of the related integral transform of Laplace type. In a series of papers, started by [3], Dimovski introduced these operators and developed operational calculi for them, based on Mikusinski’s algebraic scheme as well as on the Obrechkoff integral transform. The continuation of these studies in the papers by Dimovski–Kiryakova, and Kiryakova provoked involving the Meijer G-function into the theory of hyper-Bessel operators and equations—as their solutions and as kernel functions of the related

98 | V. Kiryakova integral transforms and transmutation operators. Especially, the integral representations for the fractional powers Lλ of the hyper-Bessel integral operators (see Dimovski and Kiryakova [4]) resulted in integral operators of the form (20) with all δk = λ, k = 1, . . . , m, equal. Then the operator L himself appeared as an integration of multi-integer order (1, . . . , 1)! Similar results were predicted also by an earlier paper by McBride [44]. More details and references related to the theory and applications of the hyper-Bessel operators and Obrechkoff integral transform can be found in [25, Ch. 3], [5, 31]. See details in Section 6, Example 4. Furthermore, when we mention some of the basic operational rules of the operators of GFC considered in this chapter, we have in mind that the following conditions on their parameters are satisfied, depending on the mentioned functional spaces: δk ≥ 0, βk > 0, ηk the integers as in (25), k = 1, . . . , m, α ν γk ≥ − − 1 (for Cα [0, ∞)), γk > − − 1 (for Hν (Ω)), βk βk μ − 1 (for Lμ,p (0, ∞)), k = 1, . . . , m. γk > pβk

(31)

Next, we state a basic result for the generalized fractional integrals (20), (22) suggesting their alternative name “multiple (m-tuple)” fractional integrals. Theorem 1 (Composition/decomposition theorem). Under the conditions (31), the clasγ ,δ sical E-K fractional integrals of form (6): Iβ k k , k = 1, . . . , m, commute and their product k

γ ,δm

Iβ m m

γ

{Iβ m−1

,δm−1

m−1

1

=∫ 0

...

(m)

m

γ ,δ 1

1

k=1

γ

m

∫ [∏ 0

γ ,δk

⋅ ⋅ ⋅ (Iβ 1 1 f (z))} = [∏ Iβ k (1 − σk )δk −1 σkk Γ(δk )

k=1

k

]f (z)

1 β

1

]f (zσ1 1 ⋅ ⋅ ⋅ σmβm )dσ1 ⋅ ⋅ ⋅ dσm

(32)

can be represented as an m-tuple E-K operator (22): m

γ ,δk

[∏ I β k k=1

k

]f (z) = I(β k ),m k f (z),

(33)

(γ ),(δ ) k

i. e. by means of a single integral involving the H-function. Conversely, under the same conditions, each multiple E-K operator of form (22) can be represented as a product (32). A similar composition/ decomposition, under additional restrictions, holds for the generalized fractional derivatives (27), (28) as well: they can be seen as products of E-K fractional derivatives (7), or (8), a kind of “sequential” derivatives (see [47]), namely as γ ,δ

γ ,δ2

D(βk ),mk = Dβ1 1 Dβ2 (γ ),(δ ) k

1

2

γ ,δm

⋅ ⋅ ⋅ Dβm

m

.

(34)

Generalized fractional calculus operators with special functions | 99

One can consider also compositions of Erdélyi–Kober fractional integrals of both left-hand sided type as (6) and right-hand sided type, denoted by ∞

Kρτ,α f (z) = ∫ 1

1 (σ − 1)α−1 −ρ(τ+α) σ f (zσ ρ )dσ. Γ(α)

(35)

Then the generalized operators of fractional integration (18) with kernels of the form m,n Φ(σ) = Hm+n,m+n (σ) appear. Namely (Theorem 5.2.5, Kiryakova [25]), m

γ ,δk

Tf (z) := (∏ Iβ k k=1

k

n

τ ,α

)(∏ Kρjj j )f (z) j=1

1 1 n 1 1 m t 󵄨󵄨󵄨 (−τj + 1 − ρj , ρj )1 ; (γk + δk + 1 − βk , βk )1 m,n [ 󵄨󵄨󵄨 = z −1 ∫ Hm+n,m+n 1 1 n ] f (t)dt. z 󵄨󵄨 (γk + 1 − β1 , β1 )m 1 ; (−τj − αj + 1 − ρj , ρj )1 k k ∞

(36)

0

m,n The special case when all βk = ρj = 1, k = 1, . . . , m, j = 1, . . . , n, have a Gm+n,m+n -kernel function, as presented in Samko–Kilbas–Marichev [55], eq. (10.48) in Ch. 10.3: Compositions of several integrals with power weights.

4 Basic results of the GFC with multiple E-K operators The particular choice of the kernel functions (19) ensures a decomposition of the GFC operators (called also multiple Erdélyi–Kober operators) into products of commuting classical Erdélyi–Kober (E-K) operators (6), and vice versa. Thus, the complicated repeated integrals or differ-integral expressions can be represented alternatively by means of single integrals involving special functions. The beauty and succinctness of notations and properties of these functions (G- and H-functions) and their proper choice allowed for the development of a full chain of operational rules, mapping properties and convolutional structure of the generalized fractional integrals as well as an appropriate explicit definition of the corresponding generalized derivatives. On the other hand, the frequent appearance of compositions of classical Riemann–Liouville and Erdélyi–Kober fractional operators in various problems of applied analysis holds the key to the great number of applications and known special cases of these generalized fractional differ-integrals; this is the theory presented in Kiryakova [25]. To study the generalized fractional integrals, we have used essentially the theory of the G- and H-functions, appearing as kernel functions of (20), (22). To this end, we refer the reader to the books on special functions and fractional calculus, for example [25, the appendix], [21] as well as to some “classics” [8, Vol. 1, Ch. 5], [50, 62]. Note also m,0 m,0 that the Gm,m (σ)- and Hm,m (σ)-functions have three regular singular points, σ = 0, 1

100 | V. Kiryakova and ∞; they vanish for |σ| > 1 and are analytic functions in the unit disk |σ| < 1. Their asymptotic behavior near σ = 0, 1 has to be taken in mind (see, e. g., [50, 21]) and ensures the correctness of definitions (20), (22) in the mentioned functional spaces, under the conditions (31) on the parameters. Most of the basic results for the operators of the generalized fractional calculus have been stated in Kiryakova [25] separately for the cases of G- and H-functions and for all mentioned functional spaces. We repeated them briefly in Kiryakova–Luchko [33] as well as in the other chapter of this project [34]. Here we expose only some of the m,0 basic formulas in the version with the Hm,m -kernel. Because the same kind of properties but more general ones will be reconsidered in Section 6, for the so-called multiple Wright–Erdélyi–Kober operators of GFC. Theorem 2. Each multiple E-K fractional integral (22) preserves the power functions in Cα , with α ≥ maxk [−β(γk +1)] (under conditions (31), resp. in Lμ,p , Hν (Ω)) up to a constant multiplier: I(β k ),m k {z p } = cp z p , (γ ),(δ ) k

Γ(γk +

m

p > α, where cp = ∏ k=1

and it is an invertible mapping I(β k ),m k : Cα 󳨃→ Cα

Γ(γk + δk +

(η1 +⋅⋅⋅+ηm )

(γ ),(δ ) k

p βk

+ 1)

p βk

+ 1)

,

(37)

⊂ Cα .

The whole proposition for GFC-images of analytic functions of a complex variable from Hν (Ω) can be found in Kiryakova [24], [25, Ch. 5], [35, 34]. The operator (22) can be rewritten in the form z

(γ ),(δ ) I(β k ),m k f (z) k

1 m,0 t [ = ∫ Hm,m z z 0

󵄨󵄨 (γk + δk + 1 − 1 , 1 )m 󵄨󵄨 βk βk 1 ] f (t)dt, 󵄨󵄨 󵄨󵄨 (γk + 1 − 1 , 1 )m βk βk 1

and thus appears in the form of a convolutional type integral transform, namely ∞

z dt (γ ),(δ ) I(β k ),m k f (z) = ∫ k( )f (t) = (k ∘ f )(z), k t t 0

where ∘ denotes the Mellin convolution. For example, the following statement holds. Lemma 1. Multiple E-K fractional integral (22) has the following convolutional type representation in Lμ,p : 1 m 󵄨 (γ ),(δ ) m,0 1 󵄨󵄨󵄨 (γk + δk + 1, βk )1 ] ∘ f (z), I(β k ),m k f (z) = Hm,m [ 󵄨󵄨 1 m k z 󵄨󵄨 (γk + 1, β )1 k

(38)

and for 1 ≤ p ≤ 2 its Mellin transformation is given by the equality m

M{I(β k ),m k f (z); s} = [∏ (γ ),(δ ) k

k=1

Γ(γk −

s βk

Γ(γk + δk −

+ 1)

s βk

+ 1)

]M{f (z); s} = Ψ(s)M{f (z); s}.

(39)

Generalized fractional calculus operators with special functions | 101

We see here what is the particular multiplier function Ψ(s) in the formal Mellin transform property stated in Section 2 for Kalla’s operators (18). Using representation (38) and following the pattern of [55] (a lemma descending from Hardy, Littlewood and Polya), it is easy to prove the following proposition. Theorem 3. If the condition (31) on the parameters of the multiple E-K fractional integral (γ ),(δ ) (22) are satisfied, then I(β k ),m k f (z) exists almost everywhere on the real halfline (0, ∞) k and it is a bounded linear operator from the Banach space Lμ,p into itself. More exactly, 󵄩󵄩 (γk ),(δk ) 󵄩󵄩 󵄩󵄩I(β ),m f 󵄩󵄩μ,p ≤ hμ,p ‖f ‖μ,p , k μ pβk

with hμ,p = ∏m k=1 Γ(γk −

μ pβk

+ 1)/Γ(γk + δk −

󵄩󵄩 (γk ),(δk ) 󵄩󵄩 󵄩󵄩I(β ),m f 󵄩󵄩 ≤ hμ,p , k

i. e.

(40)

+ 1) < ∞.

From the properties of the H- and G-functions some immediate corollaries of definitions (20), (22) follow; in particular we can verify the “axioms” of FC. Theorem 4. Suppose that the conditions (31) hold. Then, in the mentioned functional spaces, the following basic operational rules of the multiple E-K fractional integrals (22) hold: I(β k ),m k {λf (cz) + ηg(cz)} = λ{I(β k ),m k f }(cz) + η{I(β k ),m k g}(cz) (γ ),(δ )

(γ ),(δ )

k

(γ ),(δ )

k

k

(41)

(bilinearity of (22)); s+1 I(β 1,...,βs ),m

(γ ,...,γ ,γ 1

,...,γm ),(0,...,0,δs+1 ,...,δm )

m

s+1 f (z) = I(β s+1,...,βm ),m−s

(γ

,...,γ )(δ

s+1

,...,δm )

m

f (z)

(42)

(if δ1 = δ2 = ⋅ ⋅ ⋅ = δs = 0, then the multiplicity reduces to (m − s)); (γk + βλ ),(δk )

I(β k ),m k z λ f (z) = z λ I(β (γ ),(δ )

k

k ),m

k

f (z),

λ ∈ ℝ,

(43)

(generalized commutability with power functions); I(β k ),m k I(ε j),n j f (z) = I(ε j),n j I(β k ),m k f (z) j

j

k

(44)

(τ ),(α ) (γ ),(δ )

(γ ),(δ ) (τ ),(α )

k

(commutability of operators of form (22)); ((γ )m ,(τ )n )((δ )m ,(αj )n1 )

k 1 the left-hand side of (44) = I((β k )m1 ,(ε j)n1 ),m+n k 1

j 1

f (z)

(45)

(compositions of m-tuple and n-tuple integrals (16) give (m + n)-tuple integrals of same form); I(β k ),mk

(γ +δ ),(σk ) (γk ),(δk ) I(β ),m f (z) k k

= I(β k ),m k

(γ ),(σ +δk ) k

f (z),

if δk > 0, σk > 0, k = 1, . . . , m,

(46)

(law of indices, product rule or semigroup property); {I(β k ),m k } f (z) = I(β k ),mk (γ ),(δ ) −1 k

(formal inversion formula).

(γ +δ ),(−δk ) k

f (z)

(47)

102 | V. Kiryakova The above inversion formula follows from the index law (46) for σk = −δk < 0, k = 1, . . . , m, and definition (22) for zero multi-order of integration, since I(β k ),mk

(γ +δ ),(−δk ) (γk ),(δk ) I(β ),m f (z) k k

= Iβ

(γk ),(0,...,0) f (z) k ,m

= f (z).

But the symbols in equation (22) have not yet been defined for negative multi-orders of integration −δk < 0, k = 1, . . . , m. The problem has been to propose an appropriate meaning for them and hence to avoid the divergent integrals appearing in (47). The situation is exactly the same as in the classical case when the R-L and E-K operators of fractional order δ > 0 can be inverted by appealing to an additional differentiation of suitable integer order η = [δ] + 1. In this case, we have derived and used a suitable differential formula for the kernel H-function (see Kiryakova [25, Lemma 5.1.7 or Lemma B.3, the appendix]), resp. for the G-function: 1 m 󵄨 m,0 t 󵄨󵄨󵄨 (ak , βk )1 m,0 t [ 󵄨󵄨 Hm,m [ ] = Dη Hm,m z 󵄨󵄨 (bk , β1 )m z 1 k

󵄨󵄨 (ak + ηk , 1 )m 󵄨󵄨 βk 1 ], 󵄨󵄨 1 m 󵄨󵄨 (bk , βk )1

(48)

where the differential operator Dη is a polynomial of z(d/dz) of degree η = η1 + ⋅ ⋅ ⋅ + ηm with arbitrary integers ηk ≥ 0, k = 1, . . . , m: m

ηr

Dη = ∏ ∏( r=1 j=1

1 z + ar − 1 + j). z βr dz

This formula helps to increase the parameters ak , k = 1, . . . , m, of the H-function in the upper row by arbitrary integers ηk ≥ 0, k = 1, . . . , m, by using a suitable operator Dη . Choosing appropriately the necessary parameters, as in Definition 2, we have proved that D(βk ),mk , the operator already defined in (27), is indeed a generalized frac(γ ),(δ ) k

tional derivative with a linear right inverse operator I(β k ),m k , namely (γ ),(δ ) k

D(βk ),mk I(β k ),m k f (z) = f (z), (γ ),(δ ) (γ ),(δ ) k

k

f ∈ Lμ,p , Cα or Hν .

(49)

In other words, we have, for example, in Lμ,p the following theorem. Theorem 5. Let f ∈ Lμ,p , let conditions (31) be satisfied and g(z) = I(β k ),m k f (z). Then the k following inversion formula holds with the generalized fractional derivative defined in (γ ),(δ ) (27): f (z) = D(βk ),mk g(z), i. e. (γ ),(δ )

k

f (z) = {I(β k ),m k } g(z) = D(βk ),mk g(z) for g ∈ I(β k ),m k (Lμ,p ). (γ ),(δ ) −1 k

(γ ),(δ ) k

(γ ),(δ ) k

(50)

A combination of Theorems 5 and 1 leads to the next step in clarifying the structure of a variety of known operators: generalized or classical, fractional or integer order integrations, differentiations or differ-integrations. Namely, in [25] we introduced an unified theory based on the common notion of “generalized fractional differ-integrals”.

Generalized fractional calculus operators with special functions | 103

By now, operators I(β k ),m k with all δk ≥ 0, k = 1, . . . , m, have been considered as k (fractional) integrals, while those with all δk < 0, k = 1, . . . , m, have been understood as formal denotations for the generalized fractional derivatives (cf. (47) and (γ ),(δ )

(γ 󸀠 +δ󸀠 ),(−δk󸀠 )

(50)): I(β k ),mk

(γ 󸀠 ),(δ󸀠 )

= D(βk ),mk , i. e. I(β k ),m k = D(βk ),mk (γ ),(δ )

k

k

k

(γ +δ ),(−δk ) k

. Now, having the decompo-

sition theorem in mind, we may consider both symbols I(β k ),m k , D(βk ),mk as generalk k ized fractional differ-integrals. Namely, if not all of the components of multi-order of “differ-integration” δ = (δ1 , . . . , δm ) are of the same sign, we simply interpret them as “mixed” products of E-K fractional integrals and derivatives of R-L type. For example, if δ1 < 0, . . . , δs < 0, δs+1 = ⋅ ⋅ ⋅ = δs+j = 0, δs+j+1 > 0, . . . , δm > 0, then (γ ),(δ )

(γ ),(δ )

I(β k ),m k := D(β1 ,...,β1 ),ss (γ ),(δ ) k

s

(γ +δ ,...,γ +δs ),(−δ1 ,...,−δs ) (γs+j+1 ,...,γm ),(δs+j+1 ,...,δm ) I(β ,...,β ),m−s−j 1 s s+j+1 m γ +δi ,−δi

= ∏ Dβi i=1

i

m

γ ,δk

∏ Iβ k

k=s+j+1

k

, a (m − j)-tuple fractional differ-integral.

(51)

5 Further extension of GFC The GFC operators surveyed in Sections 3 and 4 are based on compositions of Erdélyi– Kober operators (22). In their papers [14, 18], Kalla, Galue and Srivastava went further with operators of GFC, by considering compositions of the Wright–Erdélyi–Kober (W-E-K) operators of fractional integration (11). Thus, they introduced an extension of the generalized frac(γ ),(δ ) tional integration operators I(β k ),m k , defined by means of integral operators with the k

m,0 same kernel function Hm,m and the same structure, but introducing additional multiparameter (λ1 , . . . , λm ), in general different from (β1 , . . . , βm ).

Definition 3. For integer m ≥ 1 and real parameters βk > 0, λk > 0, δk ≥ 0, and γk ; k = 1, . . . , m, the multiple W-E-K fractional integral, of multi-order (δk )m 1 , is defined by ∞ 󵄨󵄨 (γ + δk + 1 − 1 , 1 )m βk βk 1 ̃If (z) = I (γk ),(δk ) f (z) := ∫ H m,0 [σ 󵄨󵄨󵄨 k ] f (zσ)dσ, m,m (βk ),(λk ),m 󵄨󵄨󵄨 (γk + 1 − 1 , 1 )m λk λk 1

(52)

0

if ∑m k=1 δk > 0. In the case when ∀δk = 0 and ∀λk = βk , k = 1, . . . , m, it is the identity operator: ̃If (z) = f (z). Observe that the “new” operator (52) extends the GF integral (22) by using two different sets of positive parameters βk > 0, λk > 0, k = 1, . . . , m, involved, respectively, m,0 in the upper and lower rows of the parameters of the kernel Hm,m -function. Here, we present some of the basic operational properties of I(β k ),(λ k),m and of the (γ ),(δ ) k

k

corresponding derivatives D(βk ),(λk ),m . This is done from one side, to facilitate the reader k k since the two cited papers [14, 18] are in publications that are not easily available, and (γ ),(δ )

104 | V. Kiryakova also because they contain a series of misprints or technical shortages. Part of them have been repeated also in Kiryakova in [26], while corrected versions of the parameters’ conditions, of the definitions of (11) and (52) and of their basic properties, have been proposed later in Kiryakova [30], as listed below. Basically, next we assume the following conditions to hold: m

m 1 1 −∑ ≥ 0, λ β k k=1 k=1 k

A=∑

δk ≥ 0, k = 1, . . . , m; p ≥ max [−λk (γk + 1)]. 1≤k≤m

(53)

5.1 Some operational properties of the multiple W-E-K operators We have

m

γ ,δ

I(β k ),(λ k),m f (z) = [∏ Wβ k,λ k ]f (z); (γ ),(δ ) k

k

k

k=1

(54)

k

((γ )m ,(τ )n ),((δ )m ,(α )n )

I(β k ),(λ k),m I(ε k),(ξ k),n f (z) = I((β k )m1 ,(ε k)n1 ),((λ k)m1 ,(ξ k)n1),m+n f (z) (γ ),(δ ) k

(τ ),(α )

k

k

k 1

k

k 1

k 1

k 1

(55)

(bilinearity, commutativity and decomposition); m

I(β k ),(λ k),m {z p } = cp z p ,

cp = ∏

(γ ),(δ ) k

k

k=1

Γ(γk + 1 + p/λk ) Γ(γk + δk + 1 + p/βk )

(56)

(incl. for p = 0, if ∀γk ≥ −1); I(β k ),(λ k),m z p f (z) = z p I(β k ),(λ

(γ +p/λk ),(δk +p/βk −p/λk ) f (z) k k ),m

(γ ),(δ ) k

k

(57)

(shift property);

I(β k ),(βk ),mk I(β k ),(λ k),m f (z) = I(β k ),(λ k),mk f (z) (γ +δ ),(σ ) (γ ),(δ ) k

k

k

(58)

(γ ),(δ +σ )

k

k

k

(generalized semigroup property; f (z);

(59)

{I(β k ),(λ k),m } f (z) = I(λ k),(βk ),m k f (z) (inversion formula).

(60)

I(λ k),(βk ),mk I(β k ),(λ k),m f (z) = I(λ k),(λ k),mk f (z) = I(λ k),m k (γ +δ ),(σ ) (γ ),(δ ) k

k

k

(γ ),(δ ) k

(γ ),(δ +σ ) k

k

−1

k

(γ ),(δ +σk ) k

(γ +δ ),(−δ )

k

k

k

Proofs) of the above relations can be found in [14, 18], with some necessary adjustments and corrections of the parameters’ conditions. All of them follow in a way quite similar to that in [25, Ch. 5] for the operators (22), and they use the same operational properties of the H-function. From the formal inversion formula (60), as in the case of GFC in Kiryakova [25], one gets a hint of how to define properly generalized fractional 󸀠 “integrals” (52) of negative multi-order (−δ1󸀠 , . . . , −δm ), δk󸀠 ≥ 0: (γ 󸀠 ),(−δk󸀠 ) f (z) I(β k ),(λ ),m k k

m

η󸀠r

= ∏ ∏[γr󸀠 − δr󸀠 + j −

with integers η󸀠k ≥ δk󸀠 ≥ 0.

r=1 j=1

1 z d (γk󸀠 ),(−δk󸀠 +η󸀠k ) f (z), + ]I βr βr dz (βk ),(λk ),m

(61)

Generalized fractional calculus operators with special functions | 105

Then we can introduce the corresponding generalized fractional derivatives (γ ),(δ ) D(βk ),(λk ),m by means of differ-integral operators, as follows. k

k

Definition 4. Under the assumptions of Definition 3, we use again the integers ηk , k = 1, . . . , m, defined in (25) and the auxiliary differential operator Dη , as in (26). Then the multiple W-E-K fractional derivatives of multi-order (δk )m 1 , corresponding to integrals (52), are defined by means of the differ-integral expressions (γ +δ ),(η −δ ) (γ ),(δ ) ̃ Df (z) = D(βk ),(λk ),m f (z) := Dη I(β i),(λi ),mi i f (z). k

i

k

i

(62)

Applying the differential relation (3.29) of p. 151, [43] for the H-function, and properties (58)–(61) of the operators, it is seen (as in [14], p. 170) that D(βk ),(λk ),m I(β k ),(λ k),m f (z) = f (z). (γ ),(δ ) k

k

(γ ),(δ ) k

k

Lemma 2. When λk = βk , k = 1, . . . , m, the operators (52), (62) coincide with the opera(γ ),(δ ) (γ ),(δ ) tors I(β k ),m k and D(βk ),mk . k

k

Proof. This is a corollary of the decomposition properties (32) and (54), both of them proved either by mathematical induction, or by the Mellin transform method. Then, to show that, for λ = β, the W-E-K integral (11) reduces to the classical E-K fractional integral (6), we have to establish the equivalence of the kernel functions of (22) and (52), in the “single” case m = 1. This can be justified by using the H-function relations, available for example in Srivastava, Gupta, Goyal [62], as formulas (2.6.1) and (2.2.4) therein, resp.: 󵄨󵄨 γ + δ + 1 − 1/β, 1/β 󵄨󵄨 γ + δ + 1 − 1/β 󵄨 󵄨 1,0 1,0 H1,1 [σ 󵄨󵄨󵄨 ] ] = βG1,1 [σ β 󵄨󵄨󵄨 󵄨󵄨 γ + 1 − 1/β, 1/β 󵄨󵄨 γ + 1 − 1/β = βσ β−1

(1 − σ)β βγ σ ; Γ(δ)

󵄨󵄨 γ + δ + 1 − 1/β, 1/β 󵄨 1,0 ] = power series, H1,1 [σ 󵄨󵄨󵄨 󵄨󵄨 γ + 1 − 1/λ, 1/λ

same as for = λσ λ(γ+1−1/λ) Jγ+δ−(λ/β)(γ+1) (σ λ ). −λ/β

(63)

(64)

A decomposition property holds also for the GF derivatives (62), as products of respecγ ,δ tive W-E-K fractional derivatives Dβk ,λk , k = 1, . . . , m. k k We emphasize the difference in the limits of integration in the integral operators (22) and (52)! In most of the operators of FC, as the R-L (2), E-K (6), operators (22), etc., the kernel functions Φ(σ) are taken to be analytic functions in the unit disk |σ| < 1 so that Φ(σ) ≡ 0 for |σ| > 1. (As H-functions, they have a parameter Δ = 0.) Therefore, the integration is taken in the limits from 0 to 1 only (for the so-called left-hand sided, R-L type), or from 1 to ∞ (for the right-hand sided, Weyl type). In the case of the W-EK operators (11) and (52), their kernel H-functions are ranged over the whole domain σ ≠ 0, since their parameter, as in (31), is Δ > 0.

106 | V. Kiryakova

6 Examples of GFC operators 1) In the case m = 1, the “multiple” E-K operators (20), (22) and (27), (34) reduce to the classical (“single”) E-K operators (6), (7); see the relation in (63). 2) For m = 2 the operators of the GFC reduce to the hypergeometric operators (9), since 2,0 2,0 are expressed via the Gauss hypergeometric function, see in [25, Ch. 1] and G2,2 H2,2 as well as formula (3.5) in [34], another chapter in this volume. Typical examples are the Saigo integral operator [52] (see also [35] and [28]) of the form (9) and its fractional derivative counterpart of R-L type, α > 0, β, η ∈ ℝ: I

α,β,η

z

z −α−β σ f (z) = ∫(z − σ)α−1 2 F1 (α + β, −η; α; 1 − )f (σ)dσ, Γ(α) z 0 n

z Dα,β,η f (z) = ( ) I n−α,−β−n,α+η−n f (z), dz

(65)

n − 1 < α ≤ n, n ∈ ℕ,

for f ∈ Cμ , μ > max(0, β − η) − 1. These operators are indeed the 2-tuple (with m = 2) E-K operators, namely I α,β,η f (z) = z −β I1,2

(η−β,0),(−η,α+η)

η−β,−η 0,α+η I1 f (z),

f (z) = z −β I1

(η−β,0),(−η,α+η) β

Dα,β,η f (z) = D1,2

z f (z).

(66)

Another, more general, integral operator of the form (9) is the Hohlov operator used in geometric function theory; see for example [35, 28]. 3,0 3) For m = 3 the kernel function G3,3 gives the so-called Horn (Appell) F3 -function; see, e. g., [25, Ch. 1]. Such operators (Marichev-Saigo-Maeda operators) of the form (20) have been considered by Marichev [42], Saigo et al. [53, 54] in the form z

Ff (z) = ∫ 0

z τ (z − τ)c−1 F3 (a, a󸀠 , b, b󸀠 , 1 − , 1 − )f (τ)dτ Γ(c) τ z

(a,b,c−a −b ),(b,c−a −b,a ) f (z). = z c I1,3 󸀠

󸀠

󸀠

󸀠

(67)

4) Let m > 1 be arbitrary, but all δk = 1, βk = β > 0, k = 1, . . . , m. Then the operators of (γk ),(1,...,1) −β (γ ),(1,...,1) z form (20)–(22), (28) of multi-order (1, 1, . . . , 1): L = cz β Iβ,mk , B = (1/c)Dβ,m (BL = Id) are hyper-Bessel integral operators, resp. hyper-Bessel differential operators B = z α0

m d α1 d d z ⋅ ⋅ ⋅ z αm = z −β ∏(z + βγk ), dz dz dz k=1

β > 0.

(68)

These differential operators of high integer (arbitrary) order were introduced by Dimovski [3] as extensions of the second order Bessel differential operator, and appear

Generalized fractional calculus operators with special functions | 107

very often in differential equations of mathematical physics. Dimovski developed operational calculus for the integral operators L, and further their fractional powers Lλ , λ > 0 were represented as fractional integrals of multi-order (λ, λ, . . . , λ) in Dimovski and Kiryakova [4]. For details on the whole theory of the hyper-Bessel operators in the frames of GFC, see Kiryakova [25, Ch. 3] and [31]. 5) A more general case than 4) gives a fractional index analog of the hyper-Bessel operators. This is provided by the Gel’fond–Leontiev (G-L) operators of generalized integration (resp. differentiation) generated by the coefficients of the multi-index Mittag-Leffler functions (introduced in Luchko and Yakubovich [66] and Kiryakova [27, 30]): σk , Γ(α1 k + β1 ) ⋅ ⋅ ⋅ Γ(αm k + βm ) k=0 ∞

(m) E(α (σ) = ∑ ,β )m i

i 1

(1) (σ) = Eα,β (σ) for m = 1 : Eα,β

k is the classical Mittag-Leffler function. Namely, for functions f (z) = ∑∞ k=0 ak z analytic in disks |z| < R, the G-L type operators are defined as

̃ (z) = ∑ ak Df

Γ(α1 k + β1 ) ⋅ ⋅ ⋅ Γ(αm k + βm ) z k−1 , Γ(α1 (k − 1) + β1 ) ⋅ ⋅ ⋅ Γ(αm (k − 1) + βm )

̃ (z) = ∑ ak Lf

Γ(α1 k + β1 ) ⋅ ⋅ ⋅ Γ(αm k + βm ) z k+1 . Γ(α1 (k + 1) + β1 ) ⋅ ⋅ ⋅ Γ(αm (k + 1) + βm )

∞

k=1 ∞

k=0

(69)

In wider starlike complex domains, the G-L operators (69) can be analytically continued to cases of multiple E-K operators (22), (28)–(30) or, more exactly, to the operators introduced in [66]; see also [27, 33, 34]: f (0) (βk −1),(αk ) f (z) ̃ (z) = zI (βk −1),(αk ) f (z), ̃ Lf Df (z) = D(1/α − Ck . (1/αk ),m k ),m z z

(70)

6) Many linear integration and differentiation operators used in geometric functions theory, in studies on classes of univalent analytic functions, are GFC operators; see, e. g., Kiryakova [25, Ch. 5], [35, 28]. For a more extensive list of other particular cases of the GFC operators, including transmutation operators, see [25, 4, 1] and our other recent papers.

Bibliography [1] [2] [3]

I. Ali, V. Kiryakova, and S. L. Kalla, Solutions of fractional multi-order integral and differential equations using a Poisson-type transform, J. Math. Anal. Appl., 269 (2002), 172–199. A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20(2) (2016), 763–769. I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulgare Sci., 19(12) (1966), 1111–1114.

108 | V. Kiryakova

[4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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[27] V. Kiryakova, Multiindex Mittag-Leffler functions, related Gelfond–Leontiev operators and Laplace type transforms, Fract. Calc. Appl. Anal., 2(4) (1999), 445–462. [28] V. Kiryakova, On two Saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal., 9(2) (2008), 159–176. [29] V. Kiryakova, A brief story about the operators of the generalized fractional calculus, Fract. Calc. Appl. Anal., 11(2) (2008), 203–220. [30] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl., 59(3) (2010), 1128–1141. [31] V. Kiryakova, From the hyper-Bessel operators of Dimovski to the generalized fractional calculus, Fract. Calc. Appl. Anal., 17(4) (2014), 977–1000. [32] V. Kiryakova, Fractional calculus operators of special functions? – The result is well predictable!, Chaos Solitons Fractals, 102 (Sept. 2017), 2–15. [33] V. Kiryakova and Yu. Luchko, Riemann–Liouville and Caputo type multiple Erdélyi–Kober operators, Cent. Eur. J. Phys., 11(10) (2013), 1314–1336. [34] V. Kiryakova and Yu. Luchko, Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [35] V. Kiryakova, M. Saigo, and H. M. Srivastava, Some criteria for univalence of analytic functions involving generalized fractional calculus operators, Fract. Calc. Appl. Anal., 1(1) (1998), 79–104. [36] H. Kober, On fractional integrals and derivatives, Q. J. Math. Oxford, ll (1940), 193–211. [37] E. R. Love, Some integral equations involving hypergeometric functions, Proc. Edinb. Math. Soc., 15(3) (1967), 169–198. [38] J. S. Lowndes, On some generalizations of Riemann–Liouville and Weyl fractional integrals and their applications, Glasg. Math. J., 22(2) (1981), 173–180. [39] J. S. Lowndes, On two new operators of fractional integration, in Proc. Fractional Calculus, Workshop held in Glasgow 1984, Res. Notes in Math., vol. 138, pp. 87–98, Pitman Publ., Boston etc, 1985. [40] Yu. Luchko and V. Kiryakova, Mellin integral transform in fractional calculus, in Handbook of Fractional Calculus with Applications, vol. 1, 2019. [41] Y. Luchko and J. Trujillo, Caputo-type modification of the Erdélyi–Kober fractional derivative, Fract. Calc. Appl. Anal., 10(3) (2007), 249–267. [42] O. I. Marichev, Volterra equation of Mellin convolutional type with a Horn function in the kernel (in Russian), Izv. Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk, 1 (1974), 128–129. [43] A. M. Mathai and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines, Wiley East. Ltd, New Delhi, 1978. [44] A. McBride, Fractional powers of a class of ordinary differential operators, Proc. Lond. Math. Soc. (III), 45 (1982), 519–546. [45] M. D. Ortigueira and J. A. Tenreiro Machado, What is a fractional derivative?, J. Comput. Phys., 293 (July 2015), 4–13. [46] B. P. Parashar, Domain and range of fractional integration operators, Math. Jpn., 12(2) (1967), 141–145. [47] I. Podlubny, Fractional Differential Equations, Academic Press, N. York, Boston, 1999. [48] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5(4) (2002), 367–386. [49] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15. [50] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 3: More Special Functions, Gordon and Breach Sci. Publ., N. York, London, Paris, Tokyo, etc., 1990.

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Anatoly N. Kochubei

General fractional calculus Abstract: We describe a kind of fractional calculus and a theory of relaxation equad t tions associated with operators in the time variable of the form (𝔻(k) u)(t) = dt ∫0 k(t − τ)u(τ) dτ − k(t)u(0) where k is a nonnegative locally integrable function. The results are based on the theory of complete Bernstein functions. As a special case, fractional derivatives and integrals of distributed order are considered. Keywords: Differential-convolution operator, complete Bernstein functions, fractional derivatives and integrals of distributed order, Cauchy problem MSC 2010: 26A33, 34A08

1 Introduction In this survey, we describe a general concept of fractional calculus introduced in [22]. Let us consider a differential-convolution operator, t

(𝔻(k) u)(t) =

d ∫ k(t − τ)u(τ) dτ − k(t)u(0) dt

(1)

0

having in mind to use it, instead of the first derivative 𝜕t𝜕 , in evolution equations. The simplest example is the Caputo–Djrbashian fractional derivative 𝔻(α) t , 0 < α < 1, corresponding to the case where k(t) =

t −α , Γ(1 − α)

t > 0.

(2)

𝔻t(α) is widely used in evolution equations describing slow, or anomalous diffusion. This fractional derivative already belongs to the basic mathematical notions of contemporary physics. A wide class of examples is given by the distributed order derivatives used for modeling ultraslow relaxation and diffusion processes; see [1, 6, 7, 12, 13, 19, 21, 23, 25–29, 36, 37] and the references therein. These fractional derivatives correspond to 1

k(t) = ∫ 0

t −α dρ(α), Γ(1 − α)

t > 0,

where ρ is a Borel measure on [0, 1]. Anatoly N. Kochubei, Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kyiv, 01004, Ukraine, e-mail: [email protected] https://doi.org/10.1515/9783110571622-005

(3)

112 | A. N. Kochubei It was natural to look at a general operator (1) and to ask the following question. Under what conditions holding for a nonnegative function k ∈ Lloc 1 (ℝ+ ) does the operator 𝔻(k) possess a right inverse (a kind of a fractional integral) and produce, as a kind of a fractional derivative, equations of evolution type? The latter means, in particular, that we have the following two versions. (A) The Cauchy problem (𝔻(k) u)(t) = −λu(t),

t > 0;

u(0) = 1,

(4)

where λ > 0, has a solution uλ , infinitely differentiable for t > 0 and completely monotone, that is, (−1)n u(n) (t) ≥ 0 for all t > 0, n = 0, 1, 2, . . . . λ (B) The Cauchy problem (𝔻(k) w)(t, x) = Δw(t, x),

t > 0, x ∈ ℝn ;

w(0, x) = w0 (x),

(5)

where w0 is a bounded globally Hölder continuous function, that is, |w0 (ξ ) − w0 (η)| ≤ C|ξ − η|γ , 0 < γ ≤ 1, for any ξ , η ∈ ℝn , has a unique bounded solution (the notion of a solution should be defined appropriately). Moreover, the equation in (5) possesses a fundamental solution of the Cauchy problem, a kernel with the property of a probability density. A desired class of kernels k was found in [22], and we describe the results related to the problem (4) below, together with properties specific for the case of distributed order derivatives. The problem (5) will be discussed in a separate chapter of this Handbook. For another application, to models of interacting particle systems, see [23]. Note that the well-posedness of the Cauchy problem for equations with the operator 𝔻(k) has been established under much weaker assumptions than those needed for (A) and (B); see [11]. Of course, the above approach does not exhaust the existing generalizations of fractional calculus, especially those motivated by probabilistic applications. See [13, 27, 36] and the references therein. Note also the possibility to consider “derivatives” distributed on larger intervals of orders, such as (0, 2]; see [10]. The above class of kernels is described in terms of the analytic properties of the Laplace transform ∞

𝒦(p) = ∫ e

−pt

k(t) dt.

(6)

0

The main technical tool (briefly recalled in Section 2) is the theory of complete Bernstein functions. The concept of general fractional calculus, the notions of generalized derivative and integral, are expounded in Section 3. Section 4 is devoted to the special case of operations of distributed order. Their operator-theoretic interpretation is given in Section 5. Finally, Section 6 contains results on the Cauchy problem (4) for various assumptions on the kernel k including the case of distributed order calculus. For further

General fractional calculus | 113

results in this direction, see [3]. Related subjects (multi-term equations, subordination, etc.) are discussed in [2, 4].

2 Preliminaries. Bernstein and Stieltjes functions In this section we collect information as regards some classes of functions important for many problems of analysis. For the details, see [35]. A real-valued function f on (0, ∞) is called a Bernstein function, if f ∈ C ∞ , f (λ) ≥ 0 for all λ > 0, and (−1)n−1 f (n) (λ) ≥ 0

for all n ≥ 1, λ > 0.

Equivalently, a function f : (0, ∞) → ℝ is a Bernstein function, if and only if ∞

f (λ) = a + bλ + ∫ (1 − e−λt ) ν(dt)

(7)

0

where a, b ≥ 0, and ν is a Borel measure on [0, ∞), called the Lévy measure, such that ∞

∫ min(1, t) ν(dt) < ∞.

(8)

0

The triplet (a, b, ν) is determined by f uniquely. In particular, a = f (0+),

f (λ) . λ→∞ λ

b = lim

(9)

A Bernstein function f is said to be a complete Bernstein function, if its Lévy measure ν has a completely monotone density m(t) with respect to the Lebesgue measure, so that (7) takes the form ∞

f (λ) = a + bλ + ∫ (1 − e−λt )m(t) dt

(10)

0

where, by (8), ∞

∫ min(1, t)m(t) dt < ∞. 0

Recall that m(t) is completely monotone, if m ∈ C ∞ (0, ∞) and (−1)n m(n) (t) ≥ 0, t > 0, for all n = 0, 1, 2, . . . .

114 | A. N. Kochubei The class of Stieltjes functions consists of functions φ admitting the integral representation ∞

a 1 φ(λ) = + b + ∫ σ(dt) λ λ+t

(11)

0

where a, b ≥ 0; σ is a Borel measure on [0, ∞), such that ∞

∫ (1 + t)−1 σ(dt) < ∞.

(12)

0

Using the identity (λ + t)−1 = ∫0 e−ts e−λs ds we find from (11) that ∞

∞

a φ(λ) = + b + ∫ e−λs g(s) ds λ

(13)

0

where ∞

g(s) = ∫ e−ts σ(dt)

(14)

0

is a completely monotone function whose Laplace transform exists for any λ > 0. We will denote the class of complete Bernstein functions by 𝒞ℬℱ , and the class of Stieltjes functions by 𝒮 . The following characterization is proved in [35]. Proposition 1. Suppose that f is a nonnegative function on (0, ∞). Then the following conditions are equivalent. (i) f ∈ 𝒞ℬℱ . (ii) The function λ 󳨃→ λ−1 f (λ) is in 𝒮 . (iii) f has an analytic continuation to the upper half-plane ℍ = {z ∈ ℂ : Im z > 0}, such that Im f (z) ≥ 0 for all z ∈ ℍ, and the real limit exists of f (0+) =

lim

(0,∞)∋λ→0

f (λ).

(15)

(iv) f has an analytic continuation to the cut complex plane ℂ \ (−∞, 0], such that Im z ⋅ Im f (z) ≥ 0, and the real limit (15) exists. (v) f has an analytic continuation to ℍ given by the expression ∞

f (z) = a + bz + ∫ 0

z σ(dt) z+t

where a, b ≥ 0, and σ is a Borel measure on (0, ∞) satisfying (12).

(16)

General fractional calculus | 115

Note that the constants a, b are the same in the representations (10) and (16). The density m(t) appearing in the integral representation (10) of a function f ∈ 𝒞ℬℱ and the measure σ corresponding to the Stieltjes function φ(λ) = λ−1 f (λ) are connected by the relation ∞

m(t) = ∫ e−ts s σ(ds).

(17)

0

The importance of complete Bernstein functions is caused by the following “nonlinear” properties [35], quite unusual and having significant applications. Proposition 2. (i) A function f ≢ 0 is a complete Bernstein function, if and only if 1/f is a Stieltjes function. (ii) Let f , f1 , f2 ∈ 𝒞ℬℱ , φ, φ1 , φ2 ∈ 𝒮 . Then f ∘φ ∈ 𝒮 , φ∘f ∈ 𝒮 , f1 ∘f2 ∈ 𝒞ℬℱ , φ1 ∘φ2 ∈ 𝒞ℬℱ , (λ + f )−1 ∈ 𝒮 for any λ > 0. It follows from Propositions 1 (ii) and 2 (i) that 0 ≢ f ∈ 𝒞ℬℱ , if and only if the function f ∗ (λ) = λ/f (λ) belongs to 𝒞ℬℱ . Let us write its representation similar to (10), ∞

f (λ) = a + b λ + ∫ (1 − e−λt )m∗ (t) dt. ∗

∗

∗

0

Then {0, a∗ = { 1 , ∞ { b+∫0 tm(t) dt

if a > 0,

(18)

if a = 0;

if b > 0, {0, b∗ = { 1 , if b = 0. ∞ { a+∫0 m(t) dt

(19)

3 General fractional derivatives and integrals Our concept of fractional calculus will be introduced under the following assumptions regarding the Laplace transform (6) of the function k. (*) The Laplace transform (6) exists for all p > 0. The function 𝒦 belongs to the Stieltjes class 𝒮 , and 𝒦(p) → ∞,

p𝒦(p) → 0,

as p → 0;

as p → 0;

𝒦(p) → 0,

p𝒦(p) → ∞,

as p → ∞;

as p → ∞.

(20) (21)

By Proposition 1, the function p 󳨃→ p𝒦(p) is a complete Bernstein function. It follows from (9), (20), and (21) that the integral representations (like (11) and (13)) of the func-

116 | A. N. Kochubei tion 𝒦 have the form ∞

𝒦(p) = ∫ 0

1 σ(dt) p+t

and ∞

𝒦(p) = ∫ e

−ps

g(s) ds

(22)

0

where ∞

g(s) = ∫ e−ts σ(dt), 0

and the measure σ satisfies (12). For the function p 󳨃→ p𝒦(p) we have ∞

p𝒦(p) = ∫ (1 − e−pt )m(t) dt, 0

and the limit relations from (20) and (21) show that ∞

∞

0

0

∫ m(t) dt = ∫ tm(t) dt = ∞.

(23)

It follows from the uniqueness theorem for the Laplace transform that g(s) = k(s), so that the assumptions (∗) imply the representation ∞

k(s) = ∫ e−ts σ(dt),

0 < s < ∞,

(24)

0

so that k is completely monotone. For each fixed s ≥ 1, the function t 󳨃→ (1 + t)e−ts is monotone decreasing on [0, ∞), and its value at the origin is 1. It follows from (12), (24), and the dominated convergence theorem that k(s) → 0, s → ∞. Note that the conditions (∗) are satisfied for the kernel (2). The case of distributed order derivatives will be considered below. On the other hand, given a function 𝒦 satisfying (∗), one can restore k by the formula (24). As a simple example, consider the complete Bernstein function p 󳨃→ log(1 + pβ ), 0 < β < 1 (see Example 15.4.59 in [35]), and set 𝒦(p) = p−1 log(1 + pβ ). Then 𝒦(p) ∼ pβ−1 , as p → 0, and 𝒦(p) ∼ βp−1 log p, as p → ∞, so that the conditions (20) and (21) are satisfied. The above asymptotic properties are different from those corresponding to the cases (2) and (3).

General fractional calculus | 117

1 By Proposition 2, the function p 󳨃→ p𝒦(p) belongs to the Stieltjes class. Using (18), (19), and (22) we find its representation similar to (22), that is, ∞

1 = ∫ e−ps ϰ(s) ds p𝒦(p) 0

where ϰ(s) is a completely monotone function, ϰ(s) → 0, as s → 0. Just as in (24), we get the representation ∞

ϰ(t) = ∫ e−λt η(dλ),

0 < t < ∞,

(25)

0 ∞ η(dλ)

where ∫0 1+λ < ∞. Let us consider the convolution t

(k ∗ ϰ)(t) = ∫ k(t − τ)ϰ(τ) dτ. 0

By the construction of ϰ, the Laplace transform of k ∗ ϰ equals p1 , so that (k ∗ ϰ)(t) ≡ 1.

(26)

In other words, k and ϰ form a pair of Sonine kernels. Such kernels have been searched for since the 19th century; see [32] for a survey. Let us study, under the assumptions (∗), the generalized fractional differentiation operator 𝔻(k) of the form (1) and the generalized fractional integration operator t

(𝕀(k) f )(t) = ∫ ϰ(t − s)f (s) ds.

(27)

0

The operator 𝔻(k) u is defined on continuous functions u, such that k∗u is almost everywhere differentiable, for example, on absolutely continuous functions u. The operator 𝕀(k) is defined on Lloc 1 (0, ∞). The following result extends a well-known property of the Caputo–Djrbashian fractional derivative (see Lemma 2.21 and Lemma 2.22 in [18]). Theorem 1. (i) If f is a locally bounded measurable function on (0, ∞), then 𝔻(k) 𝕀(k) f = f . (ii) If a function u is absolutely continuous on [0, ∞), then (𝕀(k) 𝔻(k) u)(t) = u(t) − u(0).

118 | A. N. Kochubei

4 Distributed order calculus In this section, we consider the case of kernels (3). We assume that the measure ρ has a density μ with respect to the Lebesgue measure, so that 1

k(t) = ∫ 0

t −α μ(α) dα, Γ(1 − α)

t > 0;

(28)

for some other cases see [21]. We assume, further, that μ is a continuous nonnegative function different from zero on a set of a positive measure. Then k is a positive decreasing function. We denote the operator (1) with the kernel (28) by 𝔻(μ) , and the corresponding fractional integration operator by 𝕀(μ) . Note that 1

𝔻 φ(t) = ∫(𝔻(α) φ)(t)μ(α) dα. (μ)

0

Since the kernel (28) is among the main objects of the distributed order calculus, it is important to investigate its properties. For the proofs, see [19]. Proposition 3. If μ ∈ C 3 [0, 1], μ(1) ≠ 0, then k(s) ∼ s−1 (log s)−2 μ(1), k (s) ∼ −s (log s) μ(1), 󸀠

−2

−2

s → 0, s → 0.

(29) (30)

It follows from (29) that k ∈ L1 (0, T); however, k ∉ Lβ for any β > 1. By (30), k 󸀠 has a non-integrable singularity. The Laplace transform (6) is given in terms of μ as follows: 1

α−1

𝒦(p) = ∫ p

μ(α) dα.

(31)

0

It is often useful to write (31) as 1

𝒦(p) = ∫ e

(α−1) log p

μ(α) dα.

(32)

0

Taking the principal value of the logarithm we extend 𝒦(p) to an analytic function on the whole complex plane cut along the half-axis ℝ− = {Im p = 0, Re p ≤ 0}. Proposition 4. (i) Let μ ∈ C 2 [0, 1]. If p ∈ ℂ \ ℝ− , |p| → ∞, then 𝒦(p) =

μ(1) −2 + O((log |p|) ). log p

General fractional calculus | 119

More precisely, if μ ∈ C 3 [0, 1], then 𝒦(p) =

μ(1) μ󸀠 (1) −3 + O((log |p|) ). − log p (log p)2

(ii) Let μ ∈ C[0, 1], μ(0) ≠ 0. If p ∈ ℂ \ ℝ− , p → 0, then 𝒦(p) ∼ p (log −1

1 ) μ(0). p −1

(iii) Let μ ∈ C[0, 1], μ(α) ∼ aαλ , a > 0, λ > 0. If p ∈ ℂ \ ℝ− , p → 0, then 𝒦(p) ∼ aΓ(1 + λ)p (log −1

1 ) p

−1−λ

.

In some cases it is convenient to use a rough estimate 1 󵄨 󵄨󵄨 −1 󵄨󵄨𝒦(p)󵄨󵄨󵄨 ≤ C|p| (log ) , |p| −1

|p| ≤ p0 ,

valid for any μ ∈ C[0, 1]. This estimate follows from general results about the behavior of the Laplace transform near the origin (see Chapter II, § 1 of [8]). It follows from (31) and the asymptotic relations from Proposition 4 that the kernel (28) satisfies the conditions (*), if μ ∈ C 2 [0, 1], and μ(0) ≠ 0, or μ satisfies the condition (iii) of Proposition 4, that is, μ(α) ∼ aαλ ,

α → 0,

where a > 0, λ > 0. Another possibility leading to (*) is β

μ(α) ∼ aαγ e− α ,

α → 0,

where a > 0, γ > −1, β > 0 (see [21]). We will return to these cases later. While the generalized fractional integration operator (27) exists under very general assumptions, within the distributed order calculus one can use the Laplace inversion γ+i∞

d 1 1 ept ϰ(t) = ⋅ dp, ∫ dt 2πi p p𝒦(p)

γ > 0,

γ−i∞

and the contour deformation technique. We obtain the following asymptotic properties [19].

120 | A. N. Kochubei Proposition 5. Suppose that μ ∈ C 3 [0, 1], μ(1) ≠ 0, and either μ(0) ≠ 0 or μ(α) ∼ aαν , a > 0, ν > 0. Then, for small values of t, 1 ϰ(t) ≤ C log , t 1 󵄨󵄨 󸀠 󵄨󵄨 −1 󵄨󵄨ϰ (t)󵄨󵄨 ≤ Ct log , t The existence of the kernel ϰ was proved recently [24] under weaker assumptions regarding μ, but in that case ϰ can be more singular. One of the well-known representations of fractional derivatives used in the classical fractional calculus [33] is the Marchaud representation making it possible to use in applications the maximum principle arguments. Using the results from [32] we see that the operator 𝔻(μ) , on functions u = 𝕀(μ) f , f ∈ Lp (0, T), 1 < p < ∞, can be represented in the form t

(𝔻(μ) u)(t) = k(t)u(t) + ∫ k 󸀠 (τ)[u(t − τ) − u(t)] dτ,

0 < t ≤ T,

(33)

0

where the representation (33) is understood as follows. Let t

∫ k 󸀠 (τ)[u(t − τ) − u(t)] dτ, if t ≥ ε, (Ψε u)(t) = { ε 0, if 0 < t < ε. Then 󵄩 󵄩 lim 󵄩󵄩(𝔻(μ) u)(t) − k(t)u(t) − (Ψε u)(t)󵄩󵄩󵄩L

ε→0󵄩

p (0,T)

= 0.

(34)

The precaution we made understanding (33) in terms of (34) cannot easily be avoided due to a strong singularity of k 󸀠 in accordance with the asymptotics (13).

5 An operator-theoretic interpretation In this section we clarify, following [20], the operator-theoretic meaning of distributed order differentiation and integration operators. Below we assume that μ ∈ C 3 [0, 1], μ(1) ≠ 0, and either μ(0) ≠ 0, or μ(α) ∼ aαν , a > 0, ν > 0, as α → 0. Let A be the differential operator Au = − du in Lp (0, T), 1 ≤ p < ∞, with the bounddx ary condition u(0) = 0. Its domain D(A) consists of absolutely continuous functions u ∈ Lp (0, T), such that u(0) = 0 and u󸀠 ∈ Lp (0, T). We show that on D(A) the distributed order differentiation coincides with the function ℒ(−A) of the operator −A, where ℒ(z) = z 𝒦(z), and the function of an operator is understood in the sense of the Bochner–Phillips functional calculus (see [5, 31, 34]).

General fractional calculus | 121

Moreover, if p = 2, then the distributed order integration operator 𝕀(μ) equals 𝒩 (J), t 1 where 𝒩 (x) = ℒ(x) , J is the integration operator, (Ju)(t) = ∫0 u(τ) dτ. This result is obtained within Hirsch’s functional calculus [15, 16], giving more detailed results for a narrower class of functions. As by-products, we have an estimate of the semigroup generated by −ℒ(−A), and an expression for the resolvent of the operator 𝕀(μ) . The semigroup Ut of operators on the Banach space X = Lp (0, T) generated by the operator A has the form f (x − t), (Ut f )(x) = { 0,

if 0 ≤ t ≤ x < T; if 0 < x < t,

x ∈ (0, T), t ≥ 0. This follows from the easily verified formula for the resolvent R(λ, A) = (A − λI)−1 of the operator A: x

(R(λ, A)u)(x) = − ∫ e−λ(x−y) u(y) dy;

(35)

0

see [17] for a similar reasoning for operators on Lp (0, ∞). The semigroup Ut is nilpotent, Ut = 0 for t > T; compare Sect. 19.4 in [14]. It follows from the expression (35) and the Young inequality that ‖R(λ, A)‖ ≤ λ−1 , λ > 0, so that Ut is a C0 -semigroup of contractions. In the Bochner–Phillips functional calculus, for the operator A, as a generator of a contraction semigroup, and any Bernstein function f of the form (7) where ν is a measure on (0, ∞) satisfying (8), the subordinate C0 -semigroup Utf is defined by the Bochner integral, ∞

Utf = ∫ (Us u)σt (ds) 0

where the measures σt are defined by their Laplace transforms, ∞

∫ e−sx σt (ds) = e−tf (x) . 0

The generator Af of the semigroup Utf is identified with −f (−A). On the domain D(A), ∞

Af u = −au + bAu + ∫ (Ut u − u)ν(dt), 0

u ∈ D(A).

122 | A. N. Kochubei Theorem 2. (i) If u ∈ D(A), then Aℒ u = −𝔻(μ) u. (ii) The semigroup Utℒ decays at infinity faster than any exponential function: 󵄩󵄩 ℒ 󵄩󵄩 −rt 󵄩󵄩Ut 󵄩󵄩 ≤ Cr e

for any r > 0.

The operator Aℒ has no spectrum. (iii) The resolvent R(λ, −Aℒ ) of the operator −Aℒ has the form x

(R(λ, −A )u)(x) = ∫ rλ (x − s)u(s) ds, ℒ

u ∈ X,

0

where rλ (s) =

1 d u (s), λ ds λ

and uλ is the solution of the Cauchy problem 𝔻(μ) uλ = λuλ ,

uλ (0) = 1.

(iv) The inverse (−Aℒ )−1 coincides with the distributed order integration operator 𝕀(μ) . (v) The resolvent of 𝕀(μ) has the form 1 1 (𝕀(μ) − λI)−1 u = − u − 2 r1/λ ∗ u, λ λ

λ ≠ 0.

Let p = 2, and f be a Stieltjes function. Then the function Hf (z) = f (z −1 ) belongs to 𝒞ℬℱ . If the extra-integral terms in the representation like (11) for f are absent, then ∞

Hf (z) = ∫ 0

z σ(dλ). 1 + λz

For some classes of linear operators V, the function Hf (V) is defined in Hirsch’s functional calculus as a closure of the operator ∞

Wx = ∫ V(I + λV)−1 xσ(dλ),

x ∈ D(V).

0

In particular, this definition makes sense if −V is a generator of a contraction C0 -semigroup, and in this case the above construction is equivalent to the Bochner– Phillips functional calculus [5, 9]. In addition, by Theorem 2 of [15], if (−V)−1 is also a generator of a contraction C0 -semigroup, then [Hf (V)]

−1

= H̃f (V −1 ).

(36)

General fractional calculus | 123

In order to apply the above theory to our situation, note that [15] zα =

∞

1 z λ−α dλ, ∫ Γ(α)Γ(1 − α) 1 + λz

0 < α < 1,

0

whence ∞

ℒ(z) = ∫ 0

z β(λ) dλ 1 + λz

where 1

β(λ) = ∫ 0

λ−α μ(α) dα. Γ(α)Γ(1 − α)

Thus ℒ(z) = Hf (z), with ∞

f (z) = ∫ 0

1 β(λ) dλ. z+λ

It follows from Watson’s lemma [30] that β(λ) ≤ C(log λ)−2 for large values of λ. Therefore ∞

∫ 0

Denote 𝒩 (z) = H̃f (z) =

β(λ) dλ < ∞. 1+λ

1 . ℒ(z) −1

If V = −A, then (−V) = −J, where J is the integration operator. It is easy to check that ⟨(J + J ∗ )u, u⟩ ≥ 0 (⟨⋅, ⋅⟩ is the inner product in L2 (0, T)). Therefore −J is a generator of a contraction semigroup. After these preparations, the equality (36) implies the following result.

Theorem 3. The operator 𝕀(μ) of distributed order integration and the integration operator J are connected by the relation 𝕀(μ) = 𝒩 (J).

6 Relaxation equations Let us consider the Cauchy problem (4).

124 | A. N. Kochubei Theorem 4. Under the assumption (∗), the problem (4) has a solution uλ (t), continuous on [0, ∞), infinitely differentiable and completely monotone on (0, ∞). For the proof see [22]. If ℒ(p) = p𝒦(p) is a Bernstein function, but not a complete Bernstein function, then the assertion of Theorem 4 is in general wrong. As an example, take 𝒦0 (p) = p−1 (1 − e−p ) (compare (7) and (10)). Then ∞

𝒦0 (p) = ∫ k0 (t)e

−pt

1, if 0 ≤ t ≤ 1, k0 (t) = { 0, if t > 1.

dt,

0

It is easy to check that u(t) − u(t − 1), if t > 1; (𝔻(k0 ) u)(t) = { u(t) − u(0), if 0 < t ≤ 1, and a general solution of corresponding problem (4) has the form 1, { { { u(t) = {(1 + λ)−1 , { { −t {c(1 + λ) ,

if t = 0;

if 0 < t ≤ 1;

if t > 1,

c = const. This function is not continuous at the origin. It is continuous at t = 1, if c = 1, but even in this case it is not differentiable at t = 1. For the case of a distributed order derivative 𝔻(μ) , it is possible to study [19, 21] the asymptotic behavior of the solution uλ (t), t → ∞. Theorem 5. Let μ ∈ C[0, 1]. If μ(0) ≠ 0, then uλ (t) ∼ C(log t)−1 ,

t → ∞.

If μ(α) ∼ aαν , α → 0 (a > 0, ν > 0), then uλ (t) ∼ C(log t)−1−ν ,

t → ∞.

If μ ∈ C(0, 1] and μ(α) ∼ aα−p , 0 < p < 1, then uλ (t) ∼ C(log t)−1+p ,

t → ∞.

If μ ∈ C[0, 1], and β

μ(α) ∼ aαγ e− α ,

as α → 0,

where a > 0, γ > −1, β > 0, then γ

3

1 2

uλ (t) ∼ C(log t)− 2 − 4 e−2√β(log t) ,

t → ∞.

(37)

General fractional calculus | 125

In the last case, the function uλ (t) decreases at infinity more slowly than any negative power of t, but faster than any negative power of log t. It is also seen from (37) that the decrease is accelerated if β > 0 becomes bigger, so that the smaller the weight function μ is near 0, the faster is the relaxation for large times. In [21], the asymptotics of uλ is found also for some examples of kernels (3) with step Stieltjes weights.

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[20] A. N. Kochubei, Distributed-order calculus: an operator-theoretic interpretation, Ukr. Math. J., 60 (2008), 551–562. [21] A. N. Kochubei, Distributed order derivatives and relaxation patterns, J. Phys. A, 42 (2009), 315203. [22] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equ. Oper. Theory, 71 (2011), 583–600. [23] A. N. Kochubei and Yu. Kondratiev, Fractional kinetic hierarchies and intermittency, Kinet. Relat. Models, 10 (2017), 725–740. [24] A. Kubica and K. Ryszewska, Fractional diffusion equation with the distributed order Caputo derivative, ArXiv:1706.05591. [25] Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409–422. [26] M. M. Meerschaert and H.-P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Process. Appl., 116 (2006), 1215–1235. [27] M. M. Meerschaert and B. Toaldo, Relaxation patterns and semi-Markov dynamics, ArXiv:1506.02951. [28] M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23–32. [29] A. M. Nakhushev, Fractional Calculus and its Applications, Fizmatlit, Moscow, 2003 (in Russian). [30] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. [31] R. S. Phillips, On the generation of semigroups of linear operators, Pac. J. Math., 2 (1952), 343–369. [32] S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. Math. Sci., 57 (2003), 3609–3632. [33] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. [34] R. L. Schilling, Subordination in the sense of Bochner and a related functional calculus, J. Aust. Math. Soc. A, 64 (1998), 368–396. [35] R. L. Schilling, R. Song, and Z. Vondrac̆ek, Bernstein Functions. Theory and Applications, Walter de Gruyter, Berlin, 2010. [36] B. Toaldo, Convolution type derivatives, hitting times of subordinators and time-changed C0 -semigroups, Potential Anal., 42 (2015), 115–140. [37] S. Umarov and R. Gorenflo, Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations, Z. Anal. Anwend., 24 (2005), 449–466.

Virginia Kiryakova and Yuri Luchko

Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus Abstract: In this chapter, several types of generalized operators of Fractional Calculus (FC) are defined and investigated in detail. All of them appeared to be useful in modeling various phenomena and systems in the natural and social sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (and multi-order) differential equations. We start, for background, with a short overview of the Riemann–Liouville and Caputo derivatives and the Erdélyi–Kober operators. Then the multiple Erdélyi–Kober fractional integrals and derivatives of Riemann–Liouville and Caputo type of multi-order (δ1 , . . . , δm ) are introduced and studied as their generalizations. The generalized operators of FC are all in form of integro-differential operators with certain particular cases of the H- and G-functions in the kernels. Some examples of the generalized FC operators and the Cauchy problems for the fractional differential equations that involve these operators are discussed. In particular, the hyperBessel differential operators and differential equations of Bessel type are treated as a particular case of the generalized FC operators of integer multi-order (1, 1, . . . , 1). Finally, we discuss some interpretations and open problems related to the multiple Erdélyi–Kober operators. Keywords: fractional calculus, operators of Riemann–Liouville and Caputo type, Erdélyi–Kober operators, special functions, Cauchy problems MSC 2010: 26A33, 44A20, 33C60, 33E30, 44A05, 44A10

1 Introduction Both in classical and in Fractional Calculus (FC), the notions of derivatives and integrals (of first, second, etc., or arbitrary non-integer order, respectively) are basic and

co-related. In the classical calculus, the traditional way of introducing these objects Acknowledgement: Virginia Kiryakova’s work on this chapter is in the framework of the program of the projects (2017–2019) under bilateral agreements of the Bulgarian Academy of Sciences with the Serbian and Macedonian Academies of Sciences and Arts, and COST Action program CA15225. Virginia Kiryakova, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Acad. G. Bontchev Str. 8, 1113 Sofia, Bulgaria, e-mail: [email protected] Yuri Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110571622-006

128 | V. Kiryakova and Yu. Luchko is to start with derivatives and differentiability and then to proceed with the notions of integral (primitive) and definite integral as basic construction for the primitive. In contrast, one of the most conventional approaches in FC is first to define the Riemann– Liouville (R-L) integral of fractional order and then to introduce a fractional derivative, say, in the R-L or in the Caputo sense, as a suitable composition of an integer-order differentiation operator and the R-L integral. The fractional derivative of the R-L type is a natural generalization of the integer-order derivatives, but it has some shortages in its applications, say, in interpretation of the initial conditions for the Cauchy problems for the fractional-order differential equations with the R-L derivatives (which have to be formulated in non-local form by means of fractional-order derivatives and integrals) and also because the R-L derivative of a constant is in general not equal to zero. The Caputo derivative helps to overcome these problems and to formulate mathematical models of some applied problems with physically consistent initial conditions. During the last few decades, ordinary and partial differential equations of fractional order begun to play an important role in modeling of various phenomena and systems in natural sciences, engineering, economics, medicine, etc. In particular, we mention applications of fractional integrals and derivatives in physics, chemistry, engineering, self-similar processes, astrophysics, classical mechanics, quantum mechanics, nuclear physics, hadron spectroscopy, and quantum field theory, etc. Therefore, the theory of the operators of FC and their generalizations (“generalized” or “multiple” fractional integrals and derivatives) is an important part of contemporary applied analysis. The Generalized Fractional Calculus (GFC) discussed here is based on the successful combination of the theories of integral transforms and special functions, thus encompassing a great variety of operators of generalized integration and differentiation. In this chapter, we first give an overview of the most used operators of the classical FC: R-L and Erdélyi–Kober (E-K) fractional integrals and the corresponding fractional derivatives of the R-L and Caputo types. Also, short notes are included on the special functions (Fox’s H-function and Meijer’s G-function) employed to define the operators of GFC. The so-called “generalized” fractional integrals and derivatives, or “multiple” Erdélyi–Kober operators, are then introduced as compositions of operators of classical FC but defined by means of singe integrals (or differ-integrals) involving special functions (Fox’s H- or Meijer’s G-function) as kernels, and their basic operational properties are discussed. For the GFC integrals of fractional multi-order, corresponding R-Land Caputo-type differential operators of fractional multi-order are introduced and some of their properties are presented. Finally, we list some specific particular cases of the GFC integrals and derivatives and mention some examples of the Cauchy problems for fractional differential equations involving these operators, and some related interpretations and open problems.

Multiple Erdélyi–Kober integrals and derivatives | 129

2 Preliminaries 2.1 Classical FC operators In FC, the most often used definition of integration of an arbitrary (fractional, noninteger) order comes from the extension of the known formula for the n-fold integration and is known as the Riemann–Liouville (left-hand sided) operator of fractional integration of order δ ≥ 0: x

1 I f (x) = ∫(x − t)δ−1 f (t)dt, Γ(δ) δ

for δ > 0;

I 0 f (x) := f (x).

(1)

0

For shortness, we fix the starting point as x = 0+ and omit considering the right-hand sided operators (sometimes said to be of Weyl type) which are defined similarly, but by integrals on the infinite interval [x, ∞). The R-L fractional derivative of order δ ≥ 0 is defined as a composition of the nth order derivative (n − 1 < δ ≤ n, n ∈ ℕ) and the fractional R-L integral of order n − δ, namely, in the form Dδ f (x) = Dn I n−δ f (x) = (

x

n

d f (t) dt 1 }. ) { ∫ dx Γ(n − δ) (x − t)δ−n+1

(2)

0

For δ = 0, it is the identity operator Id: D0 f (x) = f (x). The R-L operators (1) and (2) satisfy the basic axioms of calculus including the semigroup property I δ I σ = I δ+σ (δ, σ ≥ 0) and the inversion formula Dδ I δ = Id(δ ≥ 0), which follows from the semigroup property and the coincidence of I n and Dn with the n-fold integration and differentiation, respectively. Using the definitions given above and the well-known formula for the Euler Beta-function, we get one of the basic FC formulas, Dδ {xp } =

Γ(p + 1) p−δ x , Γ(p − δ + 1)

δ > 0, p > −1.

(3)

Setting p = 0 in the formula above, we observe the “strange” (from the point of view of classical analysis) fact that Dδ {const} = const ⋅

x−δ ≠ 0, Γ(1 − δ)

unless if δ = n = 1, 2, 3, . . .

For the complete theory of the R-L integrals and derivatives in different functional spaces and their numerous applications, we refer the reader to [67]. To avoid the problem with non-zero R-L fractional derivative of a constant and also to avoid difficulties with initial conditions when considering initial value problems for the fractional order differential equations with the R-L derivatives, an alternative definition of a fractional-order derivative is often used nowadays. It is usually named after

130 | V. Kiryakova and Yu. Luchko Caputo, who applied such operators in the description of seismographic waves, see, e. g., [5], although other authors, like Gerasimov [56] and Dzrbashjan [9], introduced the same kind of derivative. The so-called Caputo derivative (see [60]) is defined by interchanging the order of differentiation and fractional integration in (2): δ n−δ n D f (x) = ∗ D f (x) = I

x

1 f (n) (t) dt , ∫ Γ(n − δ) (x − t)δ−n+1

n − 1 < δ ≤ n.

(4)

0

Since in general Dn I n−α ≠ I n−α Dn , the formulas (2) and (4) define different operators. When the passage of the nth derivative under the sign of integral in (2) is possible, we have the relation n−1

Dδ f (x) = ∗ Dδ f (x) + ∑ f (j) (0+) j=0

xj−δ . Γ(j − δ + 1)

(5)

Let us compare also the Laplace integral transforms of the two fractional derivatives of order δ, n − 1 < δ ≤ n, n ∈ ℕ: δ

n−1

δ

k n−δ

ℒ{D f (x); s} = s ℒ{f (x); s} − ∑ (D I δ

δ

n−1

ℒ{∗ D f (x); s} = s ℒ{f (x); s} − ∑ f k=0

k=0

(k)

f )(0+)sn−k−1 ,

(0+)sδ−k−1 ,

n − 1 < δ ≤ n.

(6) (7)

Usually, the R-L definition (2) is preferred in the mathematically oriented papers, while the Caputo one (4) is more restrictive for the class of functions f , but it allows one to consider conventional initial conditions that are expressed in terms of integer-order derivatives with the well known physical interpretation. Moreover, like in the classical calculus, the Caputo derivative of a constant is zero: ∗ Dδ {const} = 0. These and other parallels and distinctions in the properties of the R-L and C-derivatives are discussed in detail, e. g., in [38, 25, 52, 60, 67]. Comparing the R-L and the Caputo fractional derivatives, let us mention also a more recent definition introduced by Hilfer in [23, Ch. 2]. This is the so-called generalized R-L (Hilfer) fractional derivative of order α and type β, defined for α > 0, 0 ≤ β ≤ 1 and n − 1 < α ≤ n ∈ ℕ, as follows: Dα,β f (x) = I β(n−α)

dn (1−β)(n−α) f (x). I dxn

(8)

The additional parameter β allows one to interpolate continuously between the R-L derivative Dα,0 ≡ Dα and the Caputo derivative Dα,1 ≡ ∗ Dα . In [23, Ch. 2], a master equation with the fractional time derivative in the form (8) was used to describe the continuous random walks with the Mittag-Leffler function as the waiting time density. For other definitions of fractional integrals and derivatives in the classical FC, we refer the reader to, e. g., [38, 25, 60], and [67].

Multiple Erdélyi–Kober integrals and derivatives | 131

In this chapter, we first deal with the Erdélyi–Kober (E-K) fractional integrals and derivatives and focus on the R-L and Caputo types. Then we consider compositions of a finite number of such operators known as the multiple E-K fractional integrals and derivatives that are also referred to as generalized (multiple) fractional integrals and derivatives. The R-L- and Caputo-type generalized fractional derivatives and the relationships between them are also presented.

2.2 Basic functional spaces In this chapter, we basically work in the weighted spaces of continuous or smooth functions of a real variable x > 0 introduced by Dimovski in [6] (see also [26] and [71]). Definition 1. The space of functions Cμ , μ ∈ ℝ consists of the functions f (x), x > 0 that can be represented in the form f (x) = xq ̃f (x) with q > μ and ̃f ∈ C([0, ∞)). The functional space Cμn , n ∈ ℕ consists of the functions f (x), x > 0, representable in the form f (x) = xq ̃f (x) with q > μ and ̃f ∈ C n ([0, ∞)). For the properties of these spaces see [26] or [71]. Evidently, Cμn ⊆ Cνm , for μ ≥ ν, n ≥ m. The constructions and results presented in this chapter are valid with the suitable modifications for the weighted spaces Lpμ (0, ∞), 1 ≤ p < ∞ of Lebesgue integrable

functions with the norm ‖ f ‖μ,p = [∫0 xμ−1 |f (x)|p dx]1/p < ∞, and for the spaces Hμ (Ω) of analytic functions with power weights in the complex z-plane, in disks {|z| < R}, or in domains Ω that are starlike with respect to the origin (see [26]). ∞

2.3 Some FC special functions In this subsection, we recall definitions and properties of some basic FC special functions used in this chapter. These are generalizations of the classical Special Functions (SFs) to the case of fractional (in fact, arbitrary) parameters. For more details regarding the SF of FC we refer, e. g., to [18, 44, 25], [26, App.], [52, 62], and [70]. Definition 2. The Fox H-function is defined by means of the Mellin–Barnes-type contour integral 󵄨󵄨 p 1 󵄨 (a , A ) m,n m,n Hp,q [σ 󵄨󵄨󵄨󵄨 j j 1q ] = (s)σ −s ds, ∫ ℋp,q (b , B ) 2πi 󵄨󵄨 k k 1 ℒ

σ ≠ 0,

(9)

where ℒ is a suitable contour in ℂ, the orders (m, n, p, q) are non negative integers so that 0 ≤ m ≤ q, 0 ≤ n ≤ p, the parameters Aj , j = 1, . . . , p and Bk , k = 1, . . . , q are positive real numbers, aj , bk , j = 1, . . . , p; k = 1, . . . , q are complex numbers such that

132 | V. Kiryakova and Yu. Luchko Aj (bk + l) ≠ Bk (aj − l󸀠 − 1), l, l󸀠 = 0, 1, 2, . . . ; j = 1, . . . , n; k = 1, . . . , m, and the integrand in (9) has the form m,n

ℋp,q (s) =

n ∏m k=1 Γ(bk + Bk s) ∏j=1 Γ(1 − aj − Aj s)

∏qk=m+1 Γ(1 − bk − Bk s) ∏pj=n+1 Γ(aj + Aj s)

.

(10)

Under suitable restrictions on the parameters, the H-function is an analytic function of σ in a circle domain |σ| < ρ or in some sectors of this domain, or in the whole complex plane ℂ. When A1 = ⋅ ⋅ ⋅ = Ap = 1, B1 = ⋅ ⋅ ⋅ = Bq = 1, (9) reduces to the Meijer G-function m,n Gp,q : 󵄨󵄨 󵄨󵄨 p p 󵄨 (a ) 󵄨 (a , 1) m,n m,n [σ 󵄨󵄨󵄨󵄨 j 1q ] . Hp,q [σ 󵄨󵄨󵄨󵄨 j 1q ] = Gp,q 󵄨󵄨 (bk )1 󵄨󵄨 (bk , 1)1

(11)

For the theory of the Meijer G-function we refer, e. g., to [12], also [62]. An example of a special function that is a particular case of the Fox H-function but not reducible to a Meijer G-function is the Wright generalized hypergeometric function (aj ,Aj)p1 󵄨󵄨󵄨 1,p p Ψq [ (b ,B )q 󵄨󵄨󵄨 σ] = Hp,q+1 (−σ), in the case at least some of the parameters Aj , Bk are k k 1 irrational numbers. In a sense, the function p Ψq is a fractional parameters’ counterpart of the classical SF p Fq (a1 , . . . , ap ; b1 , . . . , bq ; σ) (see [12, 62, 70] for details). One of the most popular and important SFs of FC is the following one: Definition 3. The Mittag-Leffler function, christened the “queen”-function of FC, is defined by the convergent series (α > 0, β ∈ ℝ) 󵄨 (1, 1) 󵄨󵄨󵄨 σk 󵄨󵄨 σ] , = 1 Ψ1 [ (β, α) 󵄨󵄨󵄨 Γ(αk + β) k=0 ∞

Eα,β (σ) = ∑

Eα (σ) = Eα,1 (σ).

(12)

This is an entire function of order 1/α and type 1. The Mittag-Leffler function is a natural extension of the exponential, trigonometric, hyperbolic, error, and incomplete gamma functions. For details we refer the reader to, e. g., [18, 52, 25], and [60]. Another important function of the GFC is the multi-index Mittag-Leffler function, where the indices α, β in (12) are extended to “vector” (multi-) indices, as first introduced by Luchko in [40, 41] and further studied by Kiryakova in [27, 28], and [29]. Definition 4. Let m > 1 be an integer, α1 , . . . , αm > 0 and β1 , . . . , βm be arbitrary real numbers. The multi-index Mittag-Leffler function (vector index M-L function) is defined as follows: σk . Γ(α1 k + β1 ) . . . Γ(αm k + βm ) k=0 ∞

(m) (σ) = ∑ E(αi ),(βi ) (σ) := E(α ),(β ) i

i

(13)

The multi-index Mittag-Leffler function (13) is a particular case of the Wright generalized hypergeometric function (E(αi ),(βi ) = 1 Ψm ) and of the Fox H-function

Multiple Erdélyi–Kober integrals and derivatives | 133

1,1 (E(αi ),(βi ) = H1,m+1 ). It was applied, e. g., in operational calculus for the multiple Erdélyi–Kober fractional integration and differentiation operators and for solving IVPs involving these operators in [19, 40, 41, 51], and [71] and for deriving the scaleinvariant solutions of some PDEs of the fractional order in [4, 16, 17], and [45]. Analytical properties of the multi-index Mittag-Leffler function were studied in [28, 29], and [31]. Many of its particular cases appear as solutions of some fractional-order integral and differential equations; see, e. g., [31] and [35], also of differential equations with the considered operators when the multiplicity m is equal to two (see [32]) or is arbitrary (see [3, 26]).

3 The Erdélyi–Kober operators Definition 5. The (left-hand sided) Erdélyi–Kober (E-K) fractional integral of the order δ > 0 is defined by γ,δ Iβ f (x)

x

β −β(γ+δ) δ−1 = x ∫(x β − t β ) t β(γ+1)−1 f (t)dt Γ(δ) 1

=

0

1 ∫(1 − σ)δ−1 σ γ f (xσ 1/β )dσ, Γ(δ)

β > 0, γ ∈ ℝ.

(14)

0

For δ = 0, the E-K fractional integral is defined as the identity operator. The cases of β = 1, β = 2 were introduced in [37, 10], the case of arbitrary β > 0 was treated in [68, 69, 67, 26, 71]. Evidently, for γ = 0, β = 1, (14) reduces to the R-L integral (1) with a power weight. There is also a right-hand sided E-K fractional integral, but here we restrict ourselves to the left-hand sided one. The E-K fractional integrals have been used by many authors, in particular, to obtain solutions of the single, dual, and triple integral equations involving special functions of mathematical physics as their kernels. For the theory and applications of the Erdélyi–Kober fractional integrals see, e. g., [24–26, 32, 41, 42, 50, 57, 64, 67–69], and [71] to mention only few of many relevant publications. In particular, in [24] and [26] the following important properties of the operator (14) have been proved: γ,δ

γ+λ,δ

Iβ xλβ f (x) = xλβ Iβ

f (x),

(15)

Iβ Iβ

f (x),

(16)

Iβ Iβ f (x) = Iβ Iβ f (x).

(17)

γ,δ γ+δ,α γ,δ α,η

γ,δ+α

f (x) = Iβ

α,η γ,δ

The Mellin integral transform (see [43, 47]) +∞

ℳ{f (x); s} := ∫ f (t)t 0

s−1

dt

(18)

134 | V. Kiryakova and Yu. Luchko of the E-K fractional integral is given by the formula (see [25, 26, 71]) γ,δ

ℳ{Iβ f (x); (s)} =

Γ(1 + γ − s/β) ℳ{f (x); s}. Γ(1 + γ + δ − s/β)

(19)

In the case β = 1, this formula was first proved by Kober in [37]. For μ ≥ −β(γ + 1), the Erdélyi–Kober fractional integration operator (14) is a linear map of the space Cμ into itself, i. e., γ,δ

Iβ : Cμ → Cμ .

(20)

For a proof, see, e. g., [50]. The corresponding fractional differentiation operator is defined by means of an auxiliary differential operator Dn which is a polynomial of the Euler differential operd (see [26, 51, 71]). ator x dx Definition 6. Let n − 1 < δ ≤ n, n ∈ ℕ. The differ-integral operator γ,δ

γ+δ,n−δ

Dβ f (x) := Dn Iβ

n 1 d γ+δ,n−δ + γ + j)]Iβ f (x) f (x) = [∏( x β dx j=1

(21)

is called the (left-hand sided) Erdélyi–Kober (E-K) fractional derivative of order δ of the of Riemann-Lioville type. The following result holds true (see [26, 50]). Lemma 1. For the functions from the space Cμ , μ ≥ −β(γ+1), the E-K fractional derivative (21) is a left-inverse operator to the E-K fractional integral (14), i. e., γ,δ γ,δ

Dβ Iβ f (x) ≡ f (x),

f ∈ Cμ .

(22)

More precisely, the following mappings hold true: γ,δ

Iβ

γ,δ

Dβ

Cμ 󳨀→ Cμn 󳨀→ Cμ

γ,δ

Dβ

γ,δ

Iβ

Cμn 󳨀→ Cμ 󳨀→ Cμn .

and

(23)

In the general case, the E-K fractional derivative is not a right-inverse operator to the E-K fractional integration operator (14). Instead, the following relation is valid (see [50]). Theorem 1. Let n − 1 < δ ≤ n, n ∈ ℕ, μ ≥ −β(γ + 1) and f ∈ Cμn . Then the formula γ,δ γ,δ

n−1

Iβ Dβ f (x) = f (x) − ∑ ck x−β(1+γ+k) k=0

(24)

holds true, where ck =

n−1 Γ(n − k) 1 d γ+δ,n−δ f (x)]. lim [x β(1+γ+k) ∏ ( x + γ + i + 1)Iβ Γ(δ − k) x→0 β dx i=k+1

(25)

Multiple Erdélyi–Kober integrals and derivatives | 135

The formula (24) is an analog of the corresponding formula for the R-L fractional integral (1) and derivative (2) (see [25] or [67]): n−1

xδ−k−1 lim [(Dδ−k−1 f )(x)]. 0+ x→0 Γ(δ − k) k=0

δ I0+ Dδ0+ f (x) = f (x) − ∑

(26)

The expressions limx→0 [(Dδ−k 0+ f )(x)], k = 1, . . . , n, whose physical interpretation is not clear, appear also as initial conditions in the Cauchy problems for the fractional differential equations with the R-L fractional derivatives. This is one of the reasons, especially for modeling of applied and physically consistent problems, to introduce the Caputo fractional derivative (4), for which the Laplace transform (7) and the Cauchy problems include the “reasonable” initial values f (0), . . . , f (n−1) (0). For the Caputo derivative, the relation corresponding to (26) reads (see [25, 60]) n−1

xk lim [f (k) (x)]. x→0 k! k=0

δ δ I0+ ∗ D0+ f (x) = f (x) − ∑

(27)

In [50], a Caputo-type modification of the E-K fractional derivative was introduced and studied in analogy with the Caputo derivative (4). Definition 7. Let n − 1 < δ ≤ n, n ∈ ℕ. The integro-differential operator γ,δ ∗ Dβ f (x)

γ+δ,n−δ

:= Iβ

γ+δ,n−δ

Dn f (x) = Iβ

n 1 d + γ + j)f (x) ∏( x β dx j=1

(28)

is called the (left-hand sided) Caputo-type modification of the Erdélyi–Kober fractional derivative of order δ. For a composition of the E-K fractional integral and the Caputo-type E-K fractional derivative, the following result was derived in [50]. Theorem 2. Let n − 1 < δ ≤ n, n ∈ ℕ, μ ≥ −β(γ + δ + 1) and f ∈ Cμn . Then the formula γ,δ

Iβ

γ,δ ∗ Dβ f (x)

n−1

= f (x) − ∑ pk x−β(1+γ+k) k=0

(29)

holds true, where n−1 1 d pk = lim [x β(1+γ+k) ∏ ( x + γ + i + 1)f (x)]. x→0 β dx i=k+1

(30)

In [50], the conditions under which the Caputo-type and the R-L-type E-K derivatives coincide on Cμn , μ ≥ −β(γ + δ + 1) were also given. Theorem 4.2 from [50] states that the Caputo-type and the R-L-type E-K derivatives coincide if and only if the coefficients ck and pk defined by (25) and (30), respectively, are equal for all k = 0, 1, . . . , n − 1. Moreover, it is known that the Caputo-type E-K derivative is a left-inverse to the E-K integral (see Theorem 4.3 from [50]): γ,δ γ,δ

(∗ Dβ Iβ f )(x) ≡ f (x),

f ∈ Cμ , μ ≥ −β(γ + 1).

(31)

136 | V. Kiryakova and Yu. Luchko

4 Multiple Erdélyi–Kober fractional integrals Since the 1960s, various generalizations of the R-L and E-K operators with some special functions in the kernel, like the Gauss hypergeometric function 2 F1 , the Bessel function Jν , and the G- and H-functions, have been considered. These generalized operators of fractional integration can be written in the form proposed by Kalla (see [30]) x

ℐ f (x) = x

−γ−1

t ∫ Φ( )t γ f (t)dt, x

(32)

0

where the kernel Φ is an arbitrary continuous function so that “the above integral makes sense in sufficiently large functional spaces”. A very general class of operators in the form (32) has the Fox H-function as kernel Φ, see, e. g., [26, 30]. To ensure that the transform of type (32) possesses some nice and useful properties, the kernel Φ has to be suitably chosen. For example, in [67, Sect.10], powerweighted compositions of a finite number of R-L integrals, i. e., compositions of E-K integrals (14) were considered and represented in the form (32). For such compositions, m,0 m,n the kernels are the Fox H-functions of the form Hm,m (or more generally, Hm+n,m+n , if compositions of both left- and right-hand sided R-L/E-K operators are taken). In [26], a theory of “Generalized Fractional Calculus” (GFC) has been developed that deals with generalized fractional integrals and derivatives defined as composim,0 tions of E-K operators that can be represented in the form (32) with the kernels Gm,m or m,0 Hm,m . This theory has been applied (see [26] and subsequent work) to different topics like the special functions, integral transforms, operational calculus, classes of integral and differential equations, and the geometric function theory. Definition 8. Let m ≥ 1 be an integer, δk ≥ 0, γk ∈ ℝ, βk > 0, k = 1, . . . , m and consider δ = (δ1 , . . . , δm ) as a fractional multi-order of integration, γ = (γ1 , . . . , γm ) as a multiweight, and β = (β1 , . . . , βm ) as an additional multi-parameter. For ∑m k=1 δk > 0, the integral operator defined by the formula 1

󵄨󵄨 (γ + δ + 1 − 1/β , 1/β )m (γ ),(δ ) 󵄨 m,0 k k k 1 I(β k ),m k f (x) := ∫ Hm,m [σ 󵄨󵄨󵄨 k ] f (xσ) dσ k 󵄨󵄨 (γk + 1 − 1/βk , 1/βk )m 1 0

x

1 m,0 t [ = ∫ Hm,m x x 0

󵄨󵄨 (γ + δ + 1 − 1/β , 1/β )m 󵄨󵄨 k k k k 1 ] f (t)dt 󵄨󵄨 󵄨󵄨 (γk + 1 − 1/βk , 1/βk )m 1

(33)

is called a generalized (multiple, m-tuple) Erdélyi–Kober (E-K) operator of integration of the multi-order δ. For δ1 = ⋅ ⋅ ⋅ = δm = 0, it is defined as the identity operator. The operator of the form δ

ℐ f (z) = x 0 I(β k ),m k f (x), (γ ),(δ ) k

δ0 ≥ 0,

(34)

Multiple Erdélyi–Kober integrals and derivatives | 137

is called a generalized (m-tuple) operator of fractional integration of the Riemann– Liouville type, or briefly a generalized (R-L) fractional integral (see [26, Ch. 5]). In the case βk = β > 0, k = 1, . . . , m, the multiple E-K fractional integral (33) has a simpler representation by means of the Meijer G-function (11), of the form (see [26, Ch. 1]) k k I(β,...,β),m f (x) := Iβ,mk

(γ ),(δ )

(γ ),(δk )

1 󵄨󵄨 (γ + δ )m 1 󵄨 m,0 f (x) = ∫ Gm,m [σ 󵄨󵄨󵄨 k mk 1 ] f (xσ β ) dσ. 󵄨󵄨 (γk )1

(35)

0

4.1 Basic examples For m = 1, the multiple E-K fractional integral is reduced to the classical E-K fractional integral, since the kernel function in (33) (or (35)) can be written in the form 󵄨󵄨 γ + δ + 1 − 1/β, 1/β 󵄨 1,0 1,0 H1,1 [σ 󵄨󵄨󵄨 [σ β ] = βσ β−1 G1,1 󵄨󵄨 γ + 1 − 1/β, 1/β = βσ β−1

󵄨󵄨 γ + δ 󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 γ

(1 − σ β )δ−1 βγ σ , Γ(δ)

that is, the kernel of (14). For m = 2, the operators (33), (35) are the so-called hypergeometric fractional inte2,0 2,0 grals (see [30], [26, Ch. 1]), since the G2,2 - and H2,2 -kernels include the Gauss hypergeometric function 2 F1 . For example, if β1 = β2 = β, then we get the representation (see [26, 67]) 1

󵄨󵄨 (γ + δ + 1 − 1/β, 1/β)2 󵄨 k k 1 2 ] f (xσ) dσ 󵄨 (γk + 1 − 1/β, 1/β)1

2,0

ℋf (x) = ∫ H2,2 [σ 󵄨󵄨󵄨 󵄨 0

1

=∫ 0

(1 − σ)δ1 +δ2 −1 βγ2 σ 2 F1 (γ2 − γ1 + δ2 , δ1 ; δ1 + δ2 ; 1 − σ β )f (xσ)dσ. Γ(δ1 + δ2 )

(36)

For m = 3, we have as an example, the so-called Marichev–Saigo–Maeda (M-S-M) 3,0 operators of FC. In this case, the kernel G3,3 of (35) with some special parameters can be represented in terms of the so-called Horn (Appell) F3 -function (see [26, Ch. 1], [30]): x

ℱ f (x) = ∫ 0

x τ (x − τ)c−1 F3 (a, a󸀠 , b, b󸀠 , 1 − , 1 − )f (τ)dτ Γ(c) τ x

(a,b,c−a −b ),(b,c−a −b,a ) f (x). = xc I1,3 󸀠

󸀠

󸀠

󸀠

(37)

A long list of other examples of the operators in the form (33), (35) (m = 1, 2, 3) along with their applications to various problems of analysis and mathematical

138 | V. Kiryakova and Yu. Luchko physics can be found in the references mentioned above, and in other publications. We shall comment some of applications in Section 8. Especially, in the case m > 1, βk = β > 0, δk = 1, k = 1, . . . , m, we have the socalled hyper-Bessel integral operator L, which is the right-inverse to the hyper-Bessel differential operator B, 0 < x < ∞, introduced in 1966 by Dimovski [6] as higher-order generalization of the second-order Bessel differential operator (see other details also in Sections 5, 6, 7) B = x α0

d α1 d d x . . . xαm−1 xαm , dx dx dx

β := m − (α0 + α1 + ⋅ ⋅ ⋅ + αm ) > 0.

(38)

In [6], Dimovski defined also the fractional powers of the corresponding integral operators L, namely Lλ , λ > 0, in terms of the convolution products. Later on, using two completely different approaches, McBride [54] and Dimovski–Kiryakova [7] (see [26, Ch. 3]) proposed integral representations of the operators Lλ with the Meijer G-function in the kernel: 1 󵄨󵄨 γ + λ, . . . , γ + λ 1 󵄨 m,0 m ] f (xσ β ) dσ, Lλ f (x) = ∫ Gm,m [σ 󵄨󵄨󵄨 1 󵄨󵄨 γ1 , . . . , γm

Lf (x) := L1 f (x),

(39)

0

where γk = β1 (αk + αk+1 + ⋅ ⋅ ⋅ + αm − m + k), k = 1, . . . , m. The operator (39) is a special case of (35) with all δk = λ. It is worth mentioning that the representation (39) served for Kiryakova as a hint to introduce in [26] the generalized fractional integrals (35) and (33), as described in more detail in the survey [33]. Another, more general particular case of the multiple E-K fractional integral with (λk ),(βδk ) with βk = 1/δk so that the products arbitrary m > 1 is of the form Lβ = xβ I(1/δ k ),m (βδk ) ⋅ (1/δk ) = β > 0 are all equal. For these operators in their composition form (43) as repeated E-K integrals (see Theorem 4) an operational calculus of Minusinski type has been developed in [40] and then applied for solving the Cauchy-type problems for the fractional differential equations with these operators in [40, 51], and [71]. Some more details are given in Section 7.

4.2 Operational properties of the multiple E-K operators First, let us consider the Mellin transform image and the convolutional structure of the multiple E-K fractional integrals. In what follows, we assume that the following conditions for the parameters of the operators and the indices of the functional spaces are satisfied: βk (γk + 1) >

μ p

(for f ∈ Lμ,p (0, ∞)) or βk (γk + 1) > −μ (for f ∈ Cμ [0, ∞), and for f ∈ Hμ (Ω)); and δk ≥ 0, k = 1, . . . , m.

(40)

Multiple Erdélyi–Kober integrals and derivatives | 139

Theorem 3 (Th. 5.1.5, [26]). The multiple E-K fractional integral (33) has the following convolutional-type representation, say, in Lμ,p (0, ∞): 󵄨󵄨 m (γ ),(δ ) m,0 1 󵄨󵄨 (γk + δk + 1, 1/βk )1 I(β k ),m k f (x) = Hm,m [ 󵄨󵄨󵄨 ] ∘ f (x) := k(x) ∘ f (x) k x 󵄨󵄨 (γk + 1, 1/βk )m 1

(41)

in terms of the Mellin convolution x

x dt (k ∘ f )(x) = ∫ k( )f (t) . t t 0

Moreover, under the conditions (40) and ℜ(s) < mink βk (γk + 1) (e. g., if ℜ(s) < μ/p), its Mellin integral transform satisfies the relation m

ℳ{I(β k ),m k f (x); s} = [∏ (γ ),(δ ) k

k=1

Γ(γk + 1 − s/βk ) ]ℳ{f (x); s}. Γ(γk + δk + 1 − s/βk )

(42)

Indeed, it follows from (41), the Mellin convolution theorem, and the Mellin transform formula (10) of the H-function that ℳ{I(β k ),m k f (x); s} = ℳ{(k ∘ f )(x); s} = M{k(x); s}ℳ{f (x); s} (γ ),(δ ) k

and we immediately get the relation (42). For more information regarding the role of the Mellin transform theory in FC, we refer the reader to [43, 47, 48]. Theorem 4 (Composition/Decomposition theorem). Let the conditions (40) be satisγ ,δ fied. The E-K fractional integrals Iβ k k , k = 1, . . . , m of the form (14) commute in the k space Lμ,p (resp. in Cμ , Hμ ) and their composition γ ,δm

Iβ m m

γ

{Iβ m−1 m−1

,δm−1

m

γ ,δ

γ ,δk

. . . (Iβ 1 1 f (x))} = [∏ Iβ k 1

k

k=1

1

1

]f (x)

m

= ∫ . . . ∫[∏ 0 (m) 0

k=1

γ

(1 − σk )δk −1 σkk Γ(δk )

1 β

1

]f (xσ1 1 . . . σmβm )dσ1 . . . dσm (43)

can be represented as an m-tuple E-K operator (33) involving the H-function: 1 󵄨󵄨 󵄨󵄨 (γk + δk + 1 − β1 , β1 )m 1 (γ ),(δ ) m,0 k k I(β k ),m k f (x) = ∫ Hm,m [σ 󵄨󵄨󵄨 ] f (xσ)dσ. k 󵄨󵄨 (γk + 1 − 1 , 1 )m 1 β β 󵄨 k k 0

(44)

Conversely, under the same conditions, each multiple E-K operator of the form (33) can be represented as a composition of type (43).

140 | V. Kiryakova and Yu. Luchko One of the possible ideas for the proof would be to compare the Mellin transforms γ ,δ (19) of the classical E-K integrals Iβ k k , k = 1, . . . , m and the relation (42). The coincik dence of the Mellin images m

γ ,δk

ℳ{I(β k ),m k f (x); s} = ℳ{(∏ Iβ k (γ ),(δ ) k

k=1

k

)f (x); s}

suggests the coincidence of the considered operators. Another way to prove the statement of the last theorem is by employing the principle of mathematical induction with respect to m (see [24, 26] for details). By means of Theorem 3 and a modification of the Hardy–Littlewood–Polya theorem (Th. 1.5, [67]), one easily proves that the multiple E-K fractional integrals are bounded linear operators from Lμ,p into itself under the conditions (40) (Th. 5.1.3, [26]). The mapping properties of (33) in the spaces Cμ (Definition 1) can be characterized as follows. Theorem 5. The multiple E-K fractional integral (33) preserves the power functions from Cμ , μ ≥ max1≤k≤m [−βk (γk + 1)] up to a constant multiplier: Γ(γk +

m

I(β k ),m k {x q } = cq xq , q > μ,

where cq = ∏

(γ ),(δ ) k

k=1

q βk

Γ(γk + δk +

+ 1) q βk

+ 1)

(45)

,

and maps Cμ isomorphically into itself: I(β k ),m k : Cμ 󳨃󳨀→ Cμ .

(46)

(γ ),(δ ) k

Under the conditions (40), the E-K integral (33) preserves also the class Hμ (Ω) of weighted analytic functions, say, in Ω := ΔR = {|x| < R} of the form ∞

f (x) = xμ ∑ an xn = xμ (a0 + a1 x + ⋅ ⋅ ⋅) ∈ Hμ (ΔR ), n=0

−1

with R = {lim sup √n |an |} , n→∞

(47)

mapping them into the same kind of functions ∞

m

I(β k ),m k f (x) = xμ ∑ {an ∏ (γ ),(δ ) k

n=0

k=1

n+μ + 1) βk n+μ δk + β + k

Γ(γk + Γ(γk +

1)

}xn ∈ Hμ (ΔR )

(48)

that have the same radius of convergence R > 0 and the same signs of the coefficients in their series expansions. Indeed, the image (45) of f (x) = xq can easily be calculated by means of the known properties and formulas for the H-function (as a kernel in (33)) and in particular by the integral formula (E.21) from [26]. Then the image of the power series (47) is obtained by using a legitime term-by-term fractional integration. In view of (44), the mapping

Multiple Erdélyi–Kober integrals and derivatives | 141

(46) is a result of the successive applications of the formula (20); see for details [36]. To show that (33) is an injective and surjective mapping, we observe that the unique solution of the convolution integral equation 1

󵄨󵄨 (γ + δ + 1 − 1/β , 1/β )m 󵄨 m,0 k k k 1 = ∫ Hm,m [σ 󵄨󵄨󵄨 k ] f (xσ) dσ = 0 󵄨󵄨 (γk + 1 − 1/βk , 1/βk )m 1

(γ ),(δ ) I(β k ),m k f (x) k

0

is the function f (x) ≡ 0 (see [70]). This follows, by a suitable substitution, from a theorem of Mikusinski and Ryll–Nardzewski from [55]. For treating the Cauchy problems for differential equations of fractional multiorder (see examples in Section 7), the following auxiliary result is useful. Lemma 2. For f ∈ Cμ(N) , μ ≥ max1≤k≤m [−βk (γk + 1)], N ∈ ℕ, the following relations between the values of the function f and its derivatives and the values of its multiple E-K fractional integral (33) and its derivatives at the point x = 0 are valid (j = 0, 1, 2, . . . , N): m

{I(β k ),m k f (x)} (0) = cj f (j) (0),

cj = ∏

(j)

(γ ),(δ ) k

k=1

Γ(γk + 1 + j/βk ) . Γ(γk + δk + 1 + j/βk )

(49)

The above formula follows from the properties of the H-function (eq. (E.9) in [26, App.]) and by jth differentiation of f (xσ) under the integral sign in (33). The coefficients m,0 cj can be found by calculating the integral of the Hm,m -function from 0 to 1 which is m,0 the same as the integral from 0 to ∞ (as Hm,m ≡ 0 for x > 1). Some of the operational rules for the multiple E-K integrals, analogous to (15), (16), (17), are as follows. Theorem 6 (Th. 5.1.6, [26]). Let the conditions (40) for Cμ , Hμ , or Lμ,p , respectively, hold true. Then the following operational rules are valid in these spaces of functions: s+1 I(β 1,...,βs ),m

(γ ,...,γ ,γ 1

,...,γm ),(0,...,0,δs+1 ,...,δm )

s+1 f (x) = I(β s+1,...,βm ),m−s

(γ

,...,γ )(δ

s+1

m

,...,δm )

m

f (x)

(50)

(i. e., if δ1 = δ2 = ⋅ ⋅ ⋅ = δs = 0, then the multiplicity reduces to (m − s)); (γk + βλ ),(δk )

I(β k ),m k xλ f (x) = xλ I(β (γ ),(δ )

k

k ),m

k

f (x),

λ∈ℝ

(51)

(generalized commutability with power functions); I(β k ),m k I(ε j),n j f (x) = I(ε j),n j I(β k ),m k f (x) (γ ),(δ ) (τ ),(α )

(τ ),(α ) (γ ),(δ )

j

k

j

k

(52)

(commutability); ((γ )m ,(τ )n )((δ )m ,(αj )n1 )

k 1 I(β k ),m k I(ε j),n j f (x) = I((β k )m1 ,(ε j)n1 ),m+n

(γ ),(δ ) (τ ),(α ) k

j

k 1

j 1

f (x)

(53)

(compositions of the m-tuple and the n-tuple integrals (33) give the (m+n)-tuple integrals of the same form); I(β k ),mk

(γ +δ ),(σk ) (γk ),(δk ) I(β ),m f (x) k k

= I(β k ),m k

(γ ),(σ +δk ) k

f (x),

if δk > 0, σk > 0, k = 1, . . . , m,

(54)

142 | V. Kiryakova and Yu. Luchko (indices law, product rule, or semigroup property); {I(β k ),m k } f (x) = I(β k ),mk (γ ),(δ ) −1

(γ +δ ),(−δk )

k

k

f (x)

(55)

(formal inversion formula). The proofs of these properties follow from the definition (33) and the properties of the Fox H-function (see [26, Appendix], [62, 70]). Note that the above formal inversion formula (55) follows from the semigroup property (54) with σk = −δk , k = 1, . . . , m and definition of (33) in the case of the zero multi-order of integration: I(β k ),mk

(γ +δ ),(−δk ) (γk ),(δk ) I(β ),m f (x) k k

= I(β k ),m

(γ ),(0,...,0) k

f (x) = f (x).

k However, the symbols I(β k ),mk with negative multi-orders of integration −δk < 0, k k = 1, . . . , m as in (55) are not yet defined. Our aim in Section 5 is to assign a correct (γ +δ ),(−δk ) meaning to the objects of the form I(β k ),mk to avoid divergent integrals in the repk resentation (33).

(γ +δ ),(−δ )

5 Multiple Erdélyi–Kober fractional derivatives in the Riemann–Liouville sense The situation with a suitable definition of the generalized fractional derivative that corresponds to the multiple E-K integral (33) is very similar to the case of the R-L and the E-K fractional integrals of the order δ > 0 that can be inverted using a derivative of the order n that satisfies the condition n − 1 < δ ≤ n. Say, for the R-L integral (1), the idea is to use the differential relation n

(x − t)δ−1 (x − t)δ+n−1 d =( ) , Γ(δ) dx Γ(δ + n)

i. e. Φδ (x, t) = (

n

d ) Φδ+n (x, t) dx

for the kernel Φδ (x, t) = (x − t)δ−1 /Γ(δ) of (1) which allows one to increase the exponent of the power function Φδ and to make it nonnegative even for the negative values of δ. In the case of the multiple E-K operators, the following result for the kernel funcm,0 tion Hm,m plays an important role for their inversion. Lemma 3. Let nk ≥ 0, k = 1, . . . , m be arbitrary integers. Then the following differential relation holds true: 󵄨󵄨 󵄨󵄨 m m m,0 t 󵄨󵄨 (ak + nk , 1/βk )1 m,0 t 󵄨󵄨 (ak , 1/βk )1 ], (56) Hm,m [ 󵄨󵄨󵄨 m ] = Dn Hm,m [ 󵄨󵄨󵄨 m x 󵄨󵄨 (bk , 1/βk )1 x 󵄨󵄨 (bk , 1/βk )1 d of the degree n = n1 + where the differential operator Dn stands for a polynomial in x dx ⋅ ⋅ ⋅ + nm defined by the formula

Dn = Pn (x

n

m r d 1 x ) = ∏ ∏( x + ar − 1 + j). dx β r dx r=1 j=1

(57)

Multiple Erdélyi–Kober integrals and derivatives | 143

The proof follows by m-times application of a suitable modification of the differentiation formula (20), § 8.3.2 in [62]. For the details, we refer to the proof of Corollary B.6 in [26, Appendix, pp. 326–327] in the case of the G-function and to [36] in the general case. The formula (56) means that the parameters ak , k = 1, . . . , m of the H-function or of the G-function, respectively, can be increased by applying the differential operator Dn . Let us return to the multiple E-K operator and introduce the auxiliary integer numbers η1 , . . . , ηm as follows: [δk ] + 1, for noninteger δk , ηk := { δk , for integer δk ,

k = 1, . . . , m.

(58)

Then we can assign a meaning to the formal symbols {I(β k ),m k }−1 = I(β k ),mk k k (55) by means of some suitable differ-integral expressions. (γ ),(δ )

(γ +δ ),(−δk )

in

Definition 9. Let m

ηr

Dη = [∏ ∏( r=1 j=1

1 d + γr + j)] x βr dx

(59)

be a differential operator with the same parameters as in Definition 8, and the integers ηr as in the formula (58). Then the multiple (m-tuple) Erdélyi–Kober fractional derivative of multi-order δ = (δ1 ≥ 0, . . . , δm ≥ 0) is defined by means of the differ-integral operator of the R-L type: D(βk ),mk f (x) := Dη I(β k ),mk

(γ +δ ),(ηk −δk )

(γ ),(δ ) k

1

k

f (x)

󵄨󵄨 󵄨󵄨 (γk + ηk + 1 − β1 , β1 )m 1 m,0 k k ] f (xσ) dσ. = Dη ∫ Hm,m [σ 󵄨󵄨󵄨 󵄨󵄨 (γk + 1 − 1 , 1 )m 1 βk βk 󵄨 0

(60)

If β1 = ⋅ ⋅ ⋅ = βm = β > 0, we have a simpler representation, involving the Meijer G-function, which corresponds to the generalized fractional integral (35): k Dβ,m

(γ ),(δk )

f (x) = Dη Iβ,mk

(γ +δk ),(ηk −δk )

f (x)

η

m r 1 d (γ +δ ),(η −δ ) = [∏ ∏( x + γr + j)]Iβ,mk k k k f (x). β dx r=1 j=1

(61)

More generally, we refer to the differ-integral operators of the form 𝒟f (x) = D(βk ),mk x

(γ ),(δ ) −δ0 k

(γk −

f (x) = x−δ0 D(β

δ0 β

k ),m

),(δk )

f (x),

δ0 ≥ 0

as to generalized (multiple, multi-order) fractional derivatives of the R-L type.

(62)

144 | V. Kiryakova and Yu. Luchko The generalized derivatives (60) and (62) are the counterparts of the generalized fractional integrals (33) and (35) and the analogs of the R-L and E-K fractional derivatives (2) and (21). Theorem 7. The multiple E-K fractional derivative (60) is a left-inverse operator to the multiple E-K fractional integral (33) on the functional space Cμ , μ ≥ max1≤k≤m [−βk (γk + 1)], i. e., D(βk ),mk I(β k ),m k f (x) = f (x),

f ∈ Cμ .

(γ ),(δ ) (γ ),(δ ) k

k

(63)

The same relation is valid also for the generalized fractional derivatives and integrals of the form (34), (62): 𝒟ℐ f (x) = f (x). The formula (63) can be checked, although only formally, by using Definition 9 and the operational rules (52) and (54): D(βk ),mk I(β k ),m k = Dη I(β k ),mk

(γ +δ ),(ηk −δk ) (γk ),(δk ) I(β ),m k k

(γ ),(δ ) (γ ),(δ ) k

k

= Dη I(β k ),m k I(β k ),mk

(γ ),(δ ) (γ +δ ),(ηk −δk ) k

k

= D(βk ),m k I(β k ),m k = Id . (γ ),(η ) (γ ),(η ) k

k

These symbolical manipulations can be justified with the help of the differential formula (56) (which in fact suggested the form of the multiple E-K derivative), the semigroup property (54), and the representation (33). For details see [36]. Remark 1. The relation (63) holds true also in the functional spaces Lμ,p (0, ∞) and Hμ (Ω) under the conditions (40). Remark 2. In the case of integer orders δk = ηk , k = 1, . . . , m (i. e., in the case of an (γ +δ ),(η −δ ) integer multi-order), the operator I(β k ),mk k k becomes the identity operator and so k

the derivative D(βk ),mk = D(βk ),m k = Dη is a purely differential operator that is left(γ ),(δ ) k

(γ ),(η ) k

inverse to the integral operator I(β k ),m k . (γ ),(η ) k

In the case δ1 = ⋅ ⋅ ⋅ = δm = 1, β1 = ⋅ ⋅ ⋅ = βm = β > 0, the multiple E-K derivative (61) is reduced to the hyper-Bessel differential operator B (38) that is left-inverse to the hyper-Bessel integration operator L (39) in the space Cμ , μ ≥ max1≤k≤m [−β(γk + 1)]. Indeed, the hyper-Bessel operators B and L, and their fractional powers Lλ , belong to m,0 the class of operators in the form (35), (61) with the Meijer G-function Gm,m -function in the kernel: 1 m B = x−β D(β,...,β),m

(γ ,...,γ ),(1,...,1)

,

1 m L = xβ I(β,...,β),m

(γ ,...,γ ),(1,...,1)

,

1 m Lλ = xβλ I(β,...,β),m

(γ ,...,γ ),(λ,...,λ)

.

(64)

The singular differential operator B of an arbitrary order m > 1 with variable coefficients was introduced by Dimovski [6] in the form (38) and can also be represented as follows: m 1 d dm dm−1 B = x−β ∏( x + γk ) = x−β [x m m + a1 xm−1 m−1 + ⋅ ⋅ ⋅ + am ], β dx dx dx k=1

(65)

Multiple Erdélyi–Kober integrals and derivatives | 145

with the parameters β > 0, γk , k = 1, . . . , m, as described for (38) and (39). It is a natural generalization of the second-order Bessel differential operator and appears very often in differential equations modeling problems in mathematical physics, especially in the axially-symmetric case. The theory of the hyper-Bessel operators, including operational calculi via a family of convolutions and via Laplace-type integral transform (known as the Obrechkoff transform) has been proposed by Dimovski [6] and then extended by means of the fractional calculus and the special functions in Kiryakova [26, Ch. 3], see also Luchko [40, 41, 51, 71] for an operational approach. Remark 3. We can assign a meaning to the symbols I(β k ),m k for arbitrary real δk , k = k 1, . . . , m, that is, define a fractional integration/differentiation of an arbitrary multiorder. Let the conditions (γ ),(η )

δ1 < 0, . . . , δs < 0;

δs+1 = ⋅ ⋅ ⋅ = δr = 0;

δr+1 > 0, . . . , δm > 0,

be fulfilled. Then the symbol I(β k ),m k has to be understood as a composition of an k s-tuple E-K fractional derivative (60), of (r−s) identity operators, and of an (m−r)-tuple E-K fractional integral (33), namely (γ ),(η )

s

γ +δk ,−δk

I(β k ),m k := [∏ Dβk (γ ),(δ ) k

k

k=1

m

γ ,δ

] ⋅ [ I . . . I ] ⋅ [ ∏ Iβ j j ] = D(βk ),s k j k [(r − s)] j=r+1

(γ +δ ),(−δk ) (γj ),(δj ) I(β ),m−r . j

(66)

Having in mind definition (66), both I(β k ),m k and D(βk ),mk can be called by the common k k name “multiple E-K fractional differ-integrals”. (γ ),(δ )

(γ ),(δ )

Remark 4. The properties of the multiple E-K integrals (33) and Definition 9 easily lead to the corresponding operational properties of the multiple E-K derivatives (60) that are analogous to those formulated in Theorem 5 and Theorem 6: m

D(βk ),mk {xq } = xq ∏ (γ ),(δ ) k

k=1 (γk ),(δk ) D(β ),m : k

Γ(γk + δk + 1 + q/βk ) , Γ(γk + 1 + q/βk ) Cμ(η1 +⋅⋅⋅+ηm ) 󳨃󳨀→ Cμ ,

D(βk ),mk xλ f (x) = xλ D(βk ),m (γ ),(δ ) k

(γ +λ/βk ),(δk ) k

f (x).

q > μ,

(67) (68) (69)

Like in the classical calculus, the fractional derivative is, in general, not a rightinverse operator to the fractional integral unless some initial conditions are all equal to zero (see, e. g., Theorem 1 for the E-K operators). The difference between the identity operator and the composition of an operator and its left-inverse operator is called the projector of this operator, or its operator of the initial conditions. In the GFC, we have the following theorem. Theorem 8. Let the integers ηk ∈ ℕ be defined as in (58), μ ≥ max1≤k≤m [−βk (γk + 1)] (η +⋅⋅⋅+ηm ) as in (40), and f ∈ Cμ 1 . Then the projector of the multiple E-K integral (33) of the

146 | V. Kiryakova and Yu. Luchko multi-order δ = (δ1 , . . . , δm ) has the form m ηk

Ff (x) := f (x) − I(β k ),m k D(βk ),mk f (x) = ∑ ∑ Ak,j x−βk (γk +j) , (γ ),(δ ) (γ ),(δ ) k

k

k=1 j=1

(70)

where the coefficients Ak,j are connected with the initial conditions (at x = 0+) for the fractional differ-integrals of f (x) and are given by the formula m

Ak,j = ∏ ηk −1

k=1

Bk,j := lim [x βk (γk +j) ∏ ( x→0

i=j+1

Γ(ηk + 1 − j) B , Γ(δk + 1 − j) k,j

(71)

1 d (γ +δ ),(η −δ ) + γk + i)I(β k ),mk k k f (x)]. x k βk dx

The idea of the proof is similar to one employed for the E-K operators in [50]. The case of the hyper-Bessel operators was considered in Lemma 3.2.2 from [26, Ch. 3], where the corresponding coefficients Bk,j have been derived; see also [40, 51, 71].

6 Multiple Erdélyi–Kober fractional derivatives in the Caputo sense Like in the case of the R-L and E-K operators (m = 1), Caputo-type multiple Erdélyi– Kober fractional derivative can be introduced, say, to avoid the physically unnatural initial conditions as in Theorem 8. Its definition has a form similar to (60), but the auxiliary differential operator Dη is moved under the integration sign by analogy with (4) and (28). Definition 10. The Caputo-type multiple Erdélyi–Kober fractional derivative is defined as the integro-differential operator (γk ),(δk ) ∗ D(βk ),m f (x)

:= I(β k ),mk

(γ +δ ),(ηk −δk ) k

Dη f (x)

1 󵄨󵄨 󵄨󵄨 (γk + ηk + 1 − β1 , β1 )m 1 m,0 k k ] Dη f (xσ) dσ = ∫ Hm,m [σ 󵄨󵄨󵄨 󵄨󵄨 (γk + 1 − 1 , 1 )m 1 β β 󵄨 k k 0 1

󵄨󵄨 m ηr 󵄨󵄨 (γk + ηk + 1 − β1 , β1 )m 1 d 1 m,0 k k + γr + j)f (xσ)]dσ, ][∏ ∏( x = ∫ Hm,m [σ 󵄨󵄨󵄨 1 1 m 󵄨󵄨 (γk + 1 − , )1 β r dx r=1 j=1 βk βk 󵄨 0 (72) where the parameters are the same as in Definitions 8 and 9, the integers ηk are given by (58) and the differential operator Dη by (59).

Multiple Erdélyi–Kober integrals and derivatives | 147

In what follows, we consider the operator (72) in the functional space Cμ with μ ≥ max1≤k≤m [−βk (γk + δk + 1)]. Evidently, for m = 1 (72) is reduced to the Caputo-type E-K derivative (28). (η)

Remark 5. In the case of integer multi-order of differentiation, i. e., if δk = ηk ∀k, the R-L- and the Caputo-type multiple E-K derivatives coincide and are differential operators (γ ),(η ) Dη = D(βk ),m k of integer order η = η1 + ⋅ ⋅ ⋅ + ηm , namely k

(γk ),(δk ) ∗ D(βk ),m

= D(βk ),mk = D(βk ),m k = Dη , (γ ),(η )

(γ ),(δ )

k

k

I(β k ),mk

(γ +δ ),(ηk −δk ) k

= I(β k ),mk

(γ +δ ),(0,0,...,0) k

since

(73)

= I.

If δk = ηk = 1, βk = β, k = 1, . . . , m, the multiple E-K derivatives are reduced to the hyper-Bessel differential operators (64), (65). m Theorem 9 (see [36]). Let the integers ηk ∈ ℕ be defined as in (58) and f ∈ Cμ 1 with μ ≥ max1≤k≤m [−βk (γk + δk + 1)]. Then the composition of the multiple E-K integral (33) of the multi-order δ = (δ1 , . . . , δm ) and the corresponding Caputo-type multiple E-K fractional derivative (72) can be represented as follows:

(η +⋅⋅⋅+η )

I(β k ),m k

(γ ),(δ ) k

(γk ),(δk ) ∗ D(βk ),m f (x)

= f (x) − ∗ Ff (x)

(74)

m ηk

with the projector ∗ Ff (x) = ∑ ∑ Ck,j x−βk (γk +j) , k=1 j=1

ηk −1

Ck,j = lim [x βk (γk +j) ∏ ( x→0

i=j+1

where

1 d + γk + i)f (x)]. x βk dx

(75)

The multiple analog of Theorem 4.2 from [50] reads as follows. Theorem 10 (see [36]). Let the conditions (40) and (58) be fulfilled. The Caputo- and (γ ),(δ ) (γ ),(δ ) the Riemann–Liouville-type multiple E-K fractional derivatives ∗ D(βk ),mk and D(βk ),mk η +⋅⋅⋅+ηm

coincide for a function f ∈ Cμ1

Ck,j = Ak,j ,

if and only if the equalities

k

k = 1, . . . , m; j = 1, . . . , ηk

k

(76)

are fulfilled, where Ck,j and Ak,j are defined by (75) and (71), respectively. Then it can be shown that the Caputo-type multiple E-K derivative satisfies an analog of Theorem 7 for the R-L-type multiple E-K derivative. Theorem 11 (see [36]). The Caputo-type multiple Erdélyi–Kober fractional derivative (γk ),(δk ) ∗ D(βk ),m is a left-inverse operator to the multiple E-K fractional integral for the functions from the space Cμ , μ ≥ max1≤k≤m [−βk (γk + 1)], that is, (γk ),(δk ) (γk ),(δk ) ∗ D(βk ),m I(βk ),m f (x)

≡ f (x),

f ∈ Cμ .

The detailed proofs of the theorems presented above can be found in [36].

(77)

148 | V. Kiryakova and Yu. Luchko

7 Initial value problems for equations with the GFC operators The results presented in Sections 4, 5, 6 can be employed among other things for solving initial value problems and eigenvalue problems for the fractional differential equations involving the multiple E-K derivatives or their particular cases. Such equations have been considered by several authors using either operational calculus or the FC techniques, or some suitable integral transforms. In particular, we refer to the work by Luchko [40, 41], Luchko et al. [16, 17, 45, 51, 71] (operational calculus method), Kiryakova [26, Ch. 3], Kiryakova et al. [3, 32] (solutions in terms of the hyper-Bessel, M-L functions (12) and multi-index M-L functions (13)), to mention only few of many relevant publications. To emphasize the differences in treatment of the initial value problems for equations with R-L- and Caputo-type fractional derivatives, we start with some examples involving operators of classical FC (m = 1 in the operators of GFC). Example 7.1. For the Cauchy problem for the fractional differential equation with the R-L derivative (2) Dδ y(x) − λy(x) = f (x), x > 0, { δ−i D y(x)|x=0 = bi , i = 1, 2, . . . , n,

n − 1 < δ ≤ n, λ ≠ 0,

(78)

the solution is given in terms of the Mittag-Leffler functions (12) by (Examples 42.1, 42.2 in [67]) n

y(x) = ∑ bi x i=1

δ−i

x

δ

Eδ,1+δ−i (λx ) + ∫(x − τ)δ−1 Eδ,δ [λ(x − τ)δ ]f (τ)dτ.

(79)

0

And for the same differential equation with the Caputo derivative (4) {

∗ δ

D y(x) − λy(x) = f (x), x > 0, y(i−1) (0) = ci , i = 1, 2, . . . , n,

n − 1 < δ ≤ n,

(80)

the solution, again in terms of the M-L functions, has the same structure but with a different part that involves the initial conditions (see [25, 46, 60]): n

x

i=1

0

y(x) = ∑ ci xi−1 Eδ,i (λx δ ) + ∫(x − τ)δ−1 Eδ,δ [λ(x − τ)δ ]f (τ)dτ.

(81)

Example 7.2. Several examples of explicit solutions to differential equations and to integro-differential equations involving the Erdélyi–Kober operators (14) and (21) were presented in [32]. The Cauchy problem analogous to (78) in Example 7.1, for the equation x−βδ Dα,δ β y(x) − λy(x) = f (x),

with arbitrary δ > 0, β > 0, α ∈ ℝ

(82)

Multiple Erdélyi–Kober integrals and derivatives | 149

was shown to have a solution in the form n

y(x) = ∑ bi xβ(δ−i) Eδ,α+2δ−i+1 (λx βδ ) i=1

+x

x

−β(α+δ)

δ−1

∫(x β − τβ )

δ

Eδ,δ [λ(xβ − τβ ) ]τβ(α+δ) f (τ)dτ.

(83)

0

Another example treated in [32] was the hypergeometric integral equation y(x) − λ ℋy(x) = f (x) with ℋ := xδ+ν I1α+δ+ν,δ I1α−ν,ν being a multiple (m = 2) E-K integral (33). As a consequence of this example, an explicit solution of the differ-integral equation ν α−ν,ν x−δ Dα,δ y(x) = f (x) was derived, again in terms of M-L functions (see 1 y(x) − λx I1 eq. (37), Th. 4, [32]). Fractional differential equations in the form 𝒟β y(x) − λy(x) = f (x), x > 0 with the γ,β

Caputo-type E-K derivative 𝒟β := xβ ∗ Dβ/δ , β > 0, δ > 0 have been treated in [20] by the method of operational calculus. Their solutions were represented in terms of the M-L functions, but in a form more complicated than (81) and (83).

Example 7.3. An example of a fractional differential equation with the so-called weighted sequential fractional derivatives has been recently provided in [13], where a nonlinear Cauchy-type problem of the form α,β {𝒟r y(x) = f (x, y(x)), x > 0, 0 ≤ r < α < 1, 0 < β < 1, { lim I 1−β y(x) = c , lim I 1−α [xr Dβ y(x)] = c , 0 1 x→0+ { x→0+

(84)

was considered. In particular, some bounds and conditions for the existence and the uniqueness of solution to this problem were derived. In (84), I δ and Dδ denote the α,β R-L fractional integral (1) and R-L derivative (2), respectively, and the operator 𝒟r is defined as follows: α,β

α r

β

𝒟r f (x) := D x D f (x).

(85)

Note that for r = 0 the above 𝒟-derivative reduces to a sequential derivative introα,β duced by Miller and Ross (see also [60]). Its right-inverse integral operator ℐr f (x) = β −r α I x I f (x) is a hypergeometric fractional integral of the form (36). Thus, the operators α,β α,β 𝒟r and ℐr are examples of the multiple E-K operators with the multiplicity m = 2. In the case of a linear problem with the r. h. s. not depending on y(x), its solution can be expressed by means of the multi-index M-L functions (13) with m = 2. Example 7.4. The Saigo operators have been introduced in [66] for studying some boundary value problems for the Euler–Darboux partial differential equations and later used by many authors for the purposes of the FC and the geometric function

150 | V. Kiryakova and Yu. Luchko theory. The Saigo integral operator of the form (36) and its fractional derivative counterpart of R-L type are defined as (α > 0, β, η ∈ ℝ): I

α,β,η

x

x−α−β σ f (x) = ∫(x − σ)α−1 2 F1 (α + β, −η; α; 1 − )f (σ)dσ, Γ(α) x

Dα,β,η f (x) = (

0 n

d ) I n−α,−β−n,α+η−n f (x), dx

n − 1 < α ≤ n, n ∈ ℕ.

(86) (87)

These operators are the multiple E-K operators with m = 2: I α,β,η f (x) = x−β I1,2

(η−β,0),(−η,α+η)

η−β,−η 0,α+η f (x), I1

f (x) = x−β I1

(η−β,0),(−η,α+η) β

Dα,β,η f (x) = D1,2

x f (x).

(88)

Recently, also a Caputo-type modification of the Saigo fractional derivative has been introduced in [63] in the form α,β,η f (x) = I n−α,−β−n,α+η−n ( ∗D

n

d ) f (x). dx

(89)

Following the lines of [50], the Saigo operators I α,β,η , Dα,β,η and the Caputo-type Saigo fractional derivative ∗ Dα,β,η were investigated in detail in [63]. Further, in [8] and other related work the following Cauchy problem for the equation with the Caputo-type derivative (89) was considered: Dα,β,η y(x) − λy(x) = f (x, y(x), ∗ Dσ,δ,η y(x)), {∗ (k) y (0) = ck ≠ 0, k = 0, 1, . . . , n − 1,

x ∈ (0, 1],

(90)

where α ∈ (n − 1, n), σ ∈ (l − 1, l), n, l ∈ ℕ, α > σ; β, η, δ are real. Sufficient conditions for the existence and the uniqueness of solution to the problem (90), mainly in the case λ = 0, were given in [8]. Also some examples of the closed form solutions in terms of generalizations of the M-L function were presented in the case f ≡ 0, λ ≠ 0. Example 7.5. In [26, Ch. 3], the differential equations of the form By(x) = λy(x), By(x) = f (x), and By(x) = λy(x) + f (x) with the hyper-Bessel differential operator B (38), (64)–(65) were solved in terms of the Meijer G-functions. In particular, let β = m > 1, one of the γ-parameters in (65) be zero, and γ1 < γ2 < ⋅ ⋅ ⋅ < γm−1 < γm = 0 < γ1 + 1. Then, for the Cauchy problems with λ = −1 and f (t) = 0 in the form By(x) = −y(x),

y(0) = 1,

y󸀠 (0) = ⋅ ⋅ ⋅ = y(m−1) (0) = 0,

and By(x) = −y(x),

(91)

Multiple Erdélyi–Kober integrals and derivatives |

lim Bk y(x) = bk = 0, k = 1, 2, . . . , m − 1;

x→+0

lim Bm y(x) = bm = 1,

x→+0

151 (92)

the initial conditions of Caputo and resp., of R-L type are equivalent (see Remark 5, Sect. 5). Thus they have the same (unique) solution in terms of the normalized hyperBessel function: y(t) = 0 Fm−1 ((1 + γj )m−1 1 ; −(

m

x (x). ) ) := jγ(m−1) 1 ,...,γm −1 m

Also, the fundamental system of solutions of this hyper-Bessel differential equation of multi-order m = (1, 1, . . . , 1) has been found in [26, Ch. 3]. It consists of the following Meijer’s G-functions, which can be represented in terms of the hyper-Bessel functions, too: 1,0 yk (x) = G0,m [

xβ βm

󵄨󵄨 󵄨󵄨 󵄨󵄨 −γk , γ1 , . . . , −γk−1 , −γk+1 , . . . , −γm ], 󵄨󵄨

k = 1, . . . , m.

Example 7.6. In a series of works by Luchko et al. (see, e. g., [1, 19, 40, 41, 49, 51, 71]), the multiple E-K fractional integral in the form β

λ ,βδm

m ℒβ f (x) := x (I1/δ

m

λ

m−1 (I1/δ

,βδm−1

m−1

λ ,βδ

1 1 )))f (x) . . . (I1/δ 1

(93)

was investigated. Evidently, (93) is a variant of the multiple E-K integral (33) of the multi-order (βδ1 , . . . , βδm ) with βk = 1/δk , written in the form (43) as a composition of the E-K integrals. Note that for its indices the products (βδk ) ⋅ (1/δk ) = β > 0 are equal for all k = 1, . . . , m. The corresponding fractional derivative 𝒟β was introduced in the form of the multiple E-K derivative (60) in the R-L sense. In the work mentioned above, some results similar to those presented in Sections 4 and 5 were derived for the fractional integrals (93) and the corresponding fractional derivatives. Moreover, a family of convolutions and an operational calculus of the Mikusinski type were provided for the operator ℒβ . Some basic elements of the field of the convolution quotients were then interpreted in terms of the multi-index Mittag-Leffler functions (13) and of their particular cases. For example, the operational relations of the type (m) (λx β ) and more complicated ones were deduced. This I/(S − λ) = xβ−ν E(βδ k ),(λk +δk (β−ν)) operational calculus was applied in [1, 19, 40, 51, 49], and [71] for solving Cauchy-type problems for the fractional differential equations with a polynomial P of the generalized fractional differentiation operator 𝒟β of the form: P(𝒟β )y(x) = f (x), { ℱ𝒟βk y(x) = fk (x), k = 0, 1, . . . , m − 1, fk ∈ ker 𝒟β ,

(94)

where ℱ = I − ℒβ 𝒟β is the projector of the operator ℒβ (as in Th. 8). The unique solution of the problem (94) was represented in convolution form involving the multi-index M-L functions (13) (see, e. g., [41] or Th. 20.1 from [71]).

152 | V. Kiryakova and Yu. Luchko As in Definition 10, the Caputo-type generalized fractional derivative that corresponds to the fractional integral (93) can be introduced in the form ∗ 𝒟β f (x)

m

λ ,nk −βδk

k = (∏ I1/δ

k=1

k

)x−β Dn f (x)

(95)

with nk ∈ ℕ, so that nk − 1 < βδk ≤ nk , k = 1, . . . , m, and with m

nr

Dn := ∏ ∏(δr x r=1 j=1

d − βδr + λr + j). dx

Then all results presented in Section 6 are valid for the fractional derivative (95), too. For m = 1, this is the operator introduced in [20] and mentioned at the end of Example 7.2. If δk = 1/β, λk = γk , ∀k = 1, . . . , m, this derivative (which is then of both R-L and C type) reduces to the hyper-Bessel differential operator (64)–(65): 𝒟β f (x) ≡ Bf (x), see [20] for details. Example 7.7. In the work by Kiryakova (see, e. g., [27] or [28]), the so-called Gelfond– Leontiev (G-L) operators of generalized integration and differentiation, generated by the multi-index M-L function (13), were considered. For m = 1, the G-L operators with the M-L function (12) as the generating function, are reduced to the E-K operators, studied in [26, Ch. 2] together with the corresponding Borel–Dzrbashjan integral transform (see [2]). If the generating function is the exponential function, we get the classical differentiation and integration; cf. [67]. Definition 11. Let f (x) be an analytic function in a disk ΔR = {|x| < R} and αi > 0, k βi ∈ ℝ, i = 1, . . . , m be arbitrary parameters. The correspondences f (x) = ∑∞ k=0 ak x 󳨃󳨀→ ̃ ̃ (x) = ℐ(α ),(β ) f (x) defined by the formulas Df (x) = 𝒟(αi ),(βi ) f (x), Lf i i ̃ Df (x) = ∑ ak

Γ(α1 k + β1 ) . . . Γ(αm k + βm ) xk−1 , Γ(α1 (k − 1) + β1 ) . . . Γ(αm (k − 1) + βm )

̃ (x) = ∑ ak Lf

Γ(α1 k + β1 ) . . . Γ(αm k + βm ) xk+1 , Γ(α1 (k + 1) + β1 ) . . . Γ(αm (k + 1) + βm )

∞

k=1 ∞

k=0

(96)

are called multiple Dzrbashjan–Gelfond–Leontiev (D-G-L) differentiation and integration operators, or the G-L-type operators that are generated by the multi-index M-L functions (13). For the functions f = f (x) defined in wider complex domains (Ω ⊃ ΔR ), the G-L operators (96) can be analytically continued to the multiple E-K operators (33), (60) or, more exactly, to the operators introduced in [71]: 𝒟β , ℒβ defined in Example 7.6 with β := 1 and δk = αk , λk = βk − 1, k = 1, . . . , m (see, e. g., [28, 29]): ̃ (x) = xI (βk −1),(αk ) f (x), Lf (1/α ),m k

f (0) (βk −1),(αk ) f (x) ̃ − Ck . Df (x) = D(1/α k ),m x x

(97)

Multiple Erdélyi–Kober integrals and derivatives | 153

The following result relates the multiple E-K operators (96)–(97) with the multi-index Mittag-Leffler functions (13) as their eigenfunctions, i. e., as solutions of the fractional multi-order equation (see [29, 35]): ̃ DE(αk ),(βk ) (λx) = λE(αk ),(βk ) (λx),

λ ≠ 0, x > 0.

(98)

For the proof see, e. g., [3] and [29], where the formula (98) was proved by the transmutation method using a Poisson-type transformation. But once found, equality (98) can be directly verified also by substituting (13) into (96). For related integral transform of Laplace type, the multi-index Borel–Dzrbashjan transform, see [2].

8 Concluding remarks and open problems Many other forms of fractional integrals and derivatives that are particular cases of the multiple E-K fractional integrals and derivatives were introduced and investigated by different authors. In this chapter, we could discuss only the most interesting and important ones and ask the authors of publications not mentioned for their understanding. In what follows, we point out some other important applications, interpretations, and open problems related to the considered GFC operators.

8.1 Applications to improper integrals involving SF Recently, a lot of papers devoted to evaluation of improper integrals containing SFs or products of SFs were published. The methods used there are mainly based on the SF series expansions and the standard technique of exchanging the order of integration and summation, integration of the power functions, and then representation of the obtained power series as another special function. Thus, a tremendous pile of publications has been produced, one or even several publications for every particular integral. However, all these integrals can easily be evaluated by employing another technique based on the Mellin integral transform and representations of the SFs as Mellin–Barnes-type contour integrals of functions in the form of quotients of products of Gamma functions, well described in the handbook [53] by Marichev. Another general method for evaluation of improper integrals uses the representations of the generalized fractional integrals and derivatives (33), (60) and a known formula for the improper integral of product of two arbitrary H-functions (9) combined with the fact that most of the SFs are cases of the generalized Wright function p Ψq [

󵄨 ∞ Γ(a + kA ) . . . Γ(a + kA ) k (a1 , A1 ), . . . , (ap , Ap ) 󵄨󵄨󵄨 1 1 p p x 󵄨󵄨 x] = ∑ 󵄨 (b1 , B1 ), . . . , (bq , Bq ) 󵄨󵄨 Γ(b1 + kB1 ) . . . Γ(bq + kBq ) k! k=0 󵄨󵄨 󵄨 (1 − a1 , A1 ), . . . , (1 − ap , Ap ) 1,p ] . (99) = Hp,q+1 [−x 󵄨󵄨󵄨󵄨 󵄨󵄨 (0, 1), (1 − b1 , B1 ), . . . , (1 − bq , Bq )

154 | V. Kiryakova and Yu. Luchko Then the general results from Kiryakova [34], for example, the formula (see [34, Th. 1]) I(β k ),m k {xc p Ψq [ (γ ),(δ ) k

󵄨 (a1 , A1 ), . . . , (ap , Ap ) 󵄨󵄨󵄨 μ 󵄨󵄨 λx ]} (b1 , B1 ), . . . , (bq , Bq ) 󵄨󵄨󵄨

= xc p+m Ψq+m [

(ai , Ai )p1 , (γi + 1 +

(bj , Bj )q1 , (γi + δi + 1

c μ m , ) βi βi 1 μ + βc , β )m 1 i i

󵄨󵄨 󵄨󵄨 μ 󵄨󵄨 λx ] 󵄨󵄨 󵄨󵄨

(100)

can be used to quit all the special cases treated by many authors (say for evaluating the R-L and E-K integrals, Saigo operators (86), the MSM operators (37), etc.). Similarly, one can obtain many of the differential formulas for SFs or products of SFs by using a (γ ),(δ ) formula from [34, Th. 2] for evaluation of D(βk ),mk {xc p Ψq (λx μ )}. k

8.2 Some applications of the E-K operators The integral operators in the form (14) were introduced by Kober in [37] and by Erdélyi in [10]. They also suggested some applications of these operators. In [11], Erdélyi utilized his integral operator (in the special case β = 2) to represent the D’Alembert solution of the wave and Euler–Poisson–Darboux equations, while Sneddon applied it to derive solutions of dual integral equations in [68]. Besides, in [68] Sneddon applied the E-K operators to some problems in potential theory and in his survey paper [69] to several problems in mathematical analysis. Recently, some other applications of the E-K operators were suggested, namely, in the framework of the E-K fractional diffusion equation (see [57]), for the timevarying fractional-relaxation equations (see [14, 15]), in numerical schemes for the time-fractional differential equations with the E-K operators (see [58, 59]), and in the context of employing the Lie group methods to nonlinear fractional differential equations (see [39, 65]).

8.3 Interpretations of the E-K operators and open problems For the classical FC operators (the Riemann–Liouville fractional integrals and derivatives, the Caputo fractional derivatives, the Riesz potentials, and the Feller potentials), some physical and geometrical interpretations have been provided by Podlubny in [60, 61] and also by other authors, against the pessimistic conjecture made by the 1st Conference on FC in New Haven in 1975: “Is there a geometrical representation of a fractional derivative? If not, can one prove that a graphical representation of a fractional derivative does NOT exist?… The consensus of the experts … is that there is, in general, NO geometrical interpretation of a derivative of fractional order … It can be asked, however at least for a geometrical meaning or a physical phenomena that can be represented by means of equations involving a derivative of a particular order such as 1/2 …”.

Multiple Erdélyi–Kober integrals and derivatives |

155

As to the Erdélyi–Kober operators, in Herrmann [21], a family of generalized E-Ktype fractional integrals was interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of the generalized Cassini ovaloids on ℝN . Based on this geometric point of view, several extensions were discussed in [21]. Then, in [22], the reflection symmetric E-K-type fractional integral operators were used to construct fractional quasi-particle generators. A set of fractional creation- and annihilation-operators was defined and the properties of the corresponding free Hamiltonian were investigated. Analogously to the classical approach for the interacting multi-particle systems, these results were then interpreted as a fractional quantum model for a description of residual interactions of pairing type. –

– – – –

Finally, let us list some of the open problems related to the multiple E-K operators: The Hilfer derivative (8) interpolates between the R-L and the C-derivatives. It would be interesting to introduce and to study a multiple fractional derivative that interpolates between the multiple fractional derivatives of the R-L and C types. What are geometrical and physical interpretations of the Hilfer derivative? What are the properties, representations, and applications of the multiple compositions of the Hilfer derivatives with different parameters? What are geometrical and/or physical interpretations of the multiple fractional derivatives of the R-L and C types? The multiple fractional integrals and derivatives are closely connected with the multi-index M-L functions (13). Their properties (like, e. g., asymptotic behavior, complete monotonicity) and algorithms for their numerical evaluation are among important open problems, too.

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Mateusz Kwaśnicki

Fractional Laplace operator and its properties Abstract: The fractional Laplace operator appears in various areas of pure and applied mathematics. This survey collects several equivalent definitions of this operator, most known explicit expressions, a sample of regularity results and a number of restrictions of this operator to a domain. Connections to isotropic stable Lévy processes are also discussed. Keywords: Fractional Laplace operator, Riesz potential, isotropic stable Lévy process MSC 2010: 35R11, 35S05, 47G30, 60J35

1 Introduction If α ∈ (0, 2), the fractional Laplace operator is the singular integral operator −(−Δ)α/2 f (x) =

2α Γ( n+α ) 2

π n/2 |Γ(− α2 )|

pv ∫ ℝn

f (y) − f (x) dy. |y − x|n+α

(1)

The kernel of −(−Δ)α/2 appears frequently in the text, so we denote να (x) =

) 2α Γ( n+α 2

1 . π n/2 |Γ(− α2 )| |x|n+α

(2)

We always assume that ∫ℝn (1 + |x|)−n−α |f (x)|dx < ∞, so that the integral in (1) is absolutely convergent away from y = x. The principal value in (1) is understood as −(−Δ)α/2 f (x) = lim+ r→0

∫

(f (y) − f (x))να (y − x)dy,

(3)

ℝn \B(x,r)

where the limit can be considered in various senses. In this work we focus on pointwise, uniform or Lp limits, and we discuss several equivalent definitions of −(−Δ)α/2 (see Sections 4.1, 4.4 and 5). Unless explicitly stated otherwise, we always understand −(−Δ)α/2 in the pointwise sense. We sometimes write −(−Δ)α/2 [f (x)] for −(−Δ)α/2 f (x); if more variables are involved, −(−Δ)α/2 always acts on variable x. In some results we allow α = 2, which corresponds to the Laplace operator Δ in ℝn , or even α > 2, in which case −(−Δ)α/2 is a hypersingular integral; see [89]. However, if not explicitly stated otherwise, we always consider α ∈ (0, 2). Mateusz Kwaśnicki, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland, e-mail: [email protected] https://doi.org/10.1515/9783110571622-007

160 | M. Kwaśnicki The operator −(−Δ)α/2 is sometimes denoted by Δα/2 or −|∇|α . The name fractional Laplace operator reflects the fact that (−Δ)α/2 is the fractional power of −Δ (in the sense of spectral theory or Bochner’s subordination); see Section 4.4 and Theorem 5.1. The operator −(−Δ)α/2 is also known as the Riesz fractional derivative. In partial differential equations it is common to use s = α2 instead of α as a parameter. We note that, by Taylor’s formula, −(−Δ)α/2 f (x) is well defined if, for example, f is twice differentiable at x and f (y)(1+|y|)−n−α is absolutely integrable. As a consequence of Taylor’s formula, for every r > 0, 󵄨 󵄨 2 󵄨 󵄨󵄨 α/2 󵄨󵄨−(−Δ) f (x)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨D f (y)󵄨󵄨󵄨 w(y − x)dy + cn,α B(x,r)

∫ ℝn \B(x,r)

|f (y) − f (x)| dy, |y − x|n+α

(4)

where |D2 f (y)| denotes the operator norm of the Hessian of f at y, and w is an integrable r function on B(0, r), given by w(z) = cn,α |z|2−α−n when α ∈ (1, 2), w(z) = cn |z|1−n log |z|

when α = 1 and w(z) = cn,α r 1−α |z|1−n when α ∈ (0, 1). Widespread interest in the fractional Laplace operator among the partial differential equations community was initiated by the work of Caffarelli and Silvestre, starting with the PhD thesis of the latter [93]. There are at least three recent surveys on the fractional Laplace operator from this perspective: [45] is closely related to our work, [8] focuses on variational problems, while [22] puts more emphasis on applications. Recent developments related to the fractional Laplace operator can be found in [79]. Riesz potentials and their inversions, as well as regularity results for −(−Δ)α/2 in full space, are discussed in [88, 89]. For the definition of −(−Δ)α/2 for general α > 0, we refer to [89]. Various equivalent definitions of the fractional Laplace operator in ℝn are given in [65]. A review of local regularity results can be found in [86]. For a survey of spectral theory of the fractional Laplace operator in domains, see [41]. Classical results in potential theory for −(−Δ)α/2 are discussed in [9, 69]. More refined potentialtheoretic analysis of the fractional Laplace operator started with the PhD thesis of Bogdan [11]; see [17–19, 28]. For a survey of various restrictions of −(−Δ)α/2 to domains see [2, 35, 70]; [70] additionally discusses applications and numerical methods. An elementary introduction to the fractional Laplace operator in one, two and three dimensions is given in [81]. Many of the problems considered below have long and interesting history, which we do not discuss here due to space constraints. For the same reason, a number of aspects is either only mentioned or completely omitted in this work. Throughout this work, n ≥ 1 is the dimension of the Euclidean space ℝn , and B(x, r) denotes a ball. All sets and functions that appear in this chapter are assumed to be Borel measurable. By Lp (D) we denote the space of p-integrable functions on D. The class of continuous functions on D is denoted by C(D), while Cbu (D), C0 (D) and Cc (D) are sub-classes of C(D) consisting of: bounded and uniformly continuous functions; functions that converge to zero on the boundary of D and (if D is unbounded) at infinity; compactly supported functions. When D = ℝn , we simply write C, Cbu , C0

Fractional Laplace operator and its properties | 161

and Cc . By C k (D) we denote the class of functions whose partial derivatives of order up to k belong to C(D), and Cck (D) is defined in a similar way. The Fourier transform of an absolutely integrable function f is defined by ℱ f (ξ ) = ∫ℝn f (x)e−iξx dx, and we extend ℱ to Lp for p ∈ [1, 2] by continuity. Occasionally we write a+ = max(a, 0), with the convention that ap+ = 0 when a ≤ 0 even if p ≤ 0. By cn,α etc. we denote generic positive constants that depend only on the parameters listed in the subscript; f ≍ cn,α g −1 is a shorthand notation for cn,α g ≤ f ≤ cn,α g.

2 Riesz potentials In 1938, M. Riesz [83, 84] introduced what is now called the Riesz potential operator: a convolution operator ℐα defined by ℐα f (x) =

Γ( n−α ) 2

2α π n/2 Γ( α2 )

∫ ℝn

f (y) dy, |y − x|n−α

(5)

where α ∈ (0, n). Note that the convolution kernel of ℐα , called the Riesz potential kernel or simply Riesz kernel, is formally equal to ν−α (x). The function ℐα f is said to be the Riesz potential of f . We understand that ℐα f (x) in (5) is defined pointwise, whenever the integral is well defined. The Riesz potential operator ℐα is, under the assumptions given in Theorem 2.4 below, the inverse of the fractional Laplace operator (−Δ)α/2 , and it was historically the first motivation to introduce the operator −(−Δ)α/2 ; see [85]. We list some basic properties of Riesz potentials. Theorem 2.1 (formula (1.1.12) in [69]). If α, β, α + β ∈ (0, n) and f is a non-negative function, then ℐα ℐβ f (x) = ℐα+β f (x).

(6)

Theorem 2.2 (Hardy–Littlewood–Sobolev inequality; Theorem V.1 in [97]). If α ∈ (0, n) and p, q ∈ [1, ∞], then ℐα maps continuously Lp into Lq if and only if p ∈ (1, αn ) and 1 = p1 − αn . q Theorem 2.3 (Lemma V.2 in [97]). If α ∈ (0, n) and f ∈ Cc∞ , then the function |ξ |−α ℱ f (ξ ) is absolutely integrable and ℐα f = ℱ [|ξ | ℱ f (ξ )]. −1

−α

(7)

162 | M. Kwaśnicki Theorem 2.4 (Theorems 3.22 and 7.18 in [89]; Proposition 7.1 in [65]). (a) If α ∈ (0, n), p ∈ [1, αn ) and f , g ∈ Lp , then (−Δ)α/2 f = g

if and only if

f = ℐα g,

(8)

where (−Δ)α/2 f is defined by a limit in Lp ; in this case, the limit in (3) exists also almost everywhere. (b) If α ∈ (0, n) ∩ (0, 2], g is continuous at x (or, more generally, x is a Lebesgue point of g) and the integral in the definition of ℐα g(x) is absolutely convergent, then (−Δ)α/2 ℐα g(x) = g(x). Similar results are available for general α ∈ (0, n), and also for α ≥ n if Riesz potentials are understood in an appropriate way; see [89]. To some extent, restrictions on α and p can be removed by considering Bessel potential operator 𝒥α , which is a Fourier multiplier with symbol (1 + |ξ |2 )−α/2 , and a convolution operator with kernel 1

2(n+α)/2−1 π n/2 Γ( α2 )

K(n−α)/2 (|x|) |x|(n−α)/2

,

(9)

where Kν denotes the modified Bessel function of the second kind. The operator 𝒥α is closely related to the 1-resolvent operator for −(−Δ)α/2 . Theorem 2.5 (Lemma V.2 in [97]; formulas (7.38) and (7.39) in [89]). If α ∈ (0, ∞), then there are absolutely integrable functions ηα , ϱα such that, for every f , g, h ∈ Cbu , f = 𝒥α g

if and only if (−Δ)α/2 f = g − g ∗ ηα ,

(10)

and f + (−Δ)α/2 f = h

if and only if

f = 𝒥α (h − h ∗ ϱα ).

(11)

A similar result holds if p ∈ [1, ∞) and f , g, h ∈ Lp , if (−Δ)α/2 f is defined by a limit in Lp .

3 Explicit expressions 3.1 General identities In the previous section we found that if α ∈ (0, n), then the Riesz potential kernel uα (x) = ν−α (x) is the fundamental solution for (−Δ)α/2 . When α > n, the fundamental solution is given be the same expression, while if α = n, the fundamental solution is the logarithmic kernel.

Fractional Laplace operator and its properties | 163

Theorem 3.1. Let α ∈ (0, ∞), and define uα (x) =

Γ( n−α ) 2

1 2α π n/2 Γ( α2 ) |x|n−α

(12)

if α ≠ n, and uα (x) =

1

2n−1 π n/2 Γ( n2 )

log

1 |x|

(13)

if α = n. Then (−Δ)α/2 uα (x) = 0 for all x ≠ 0,

(14)

(−Δ)α/2 (uα ∗ f ) = f .

(15)

and for every f ∈ Cc , For α ∈ (0, 2), formula (14) is a consequence of formulas (2.2), (2.5), (2.9) in [10] and Lemma 3.3 in [65]; see Theorem 3.6 below for a much more general result. Formula (15) follows from Theorem 7.1 after expressing uα ∗f as a solution of an appropriate Poisson problem; see also Lemma 5.3 in [16]. A variant of Theorem 3.1 for general α ∈ (0, ∞) is given in Lemma 2.15 in [89]. Translation invariance and scaling property for −(−Δ)α/2 follow directly from the definition. The third property listed in the next result involves the Kelvin transformation: the Kelvin transform of f is defined by 𝒦f (x) =

f (x∗ ) , |x|n−α

(16)

where x∗ = |x|−2 x is the image of x under inversion with respect to the unit sphere. Note that if α ≠ n, then 1/|x|n−α = cn,α uα (x). Proposition 3.2 (Proposition A.1 in [87]). If α ∈ (0, 2], r > 0 and b ∈ ℝn , then (−Δ)α/2 [f (x + b)] = ((−Δ)α/2 f )(x + b), (−Δ)α/2 [f (rx)] = r α ((−Δ)α/2 f )(rx).

(17)

If in addition x ≠ 0 and f is differentiable at x ∗ = |x|−2 x (or, more generally, limy∗ →x∗ |y∗ − x∗ |1−α |f (y∗ ) − f (x∗ )| = 0), then also (−Δ)α/2 𝒦f (x) = |x|−2α 𝒦(−Δ)α/2 f (x).

(18)

A variant of (18) for Riesz potentials was already developed by M. Riesz in [83]; see formula (IV.5.4) in [69]. Kelvin transformation is also discussed in [10, 16, 21]. By translation invariance and scaling, it is possible to transform problems involving −(−Δ)α/2 in an arbitrary ball B(x0 , r) into problems in the unit ball B(0, 1). Additionally, the Kelvin transform often allows one to translate local results into statements that describe large-scale behavior of solutions of appropriate problems involving the fractional Laplace operator.

164 | M. Kwaśnicki

3.2 Ball and half-space Explicit expressions for the Poisson kernel and the Green function of a ball for the fractional Laplace operator −(−Δ)α/2 with α ∈ (0, 2) date back to the original article of M. Riesz [83] when α < n, and to [10, 56] when α ≥ n = 1. The original formulation, however, involved Riesz potentials rather than the fractional Laplace operator. A function f is said to be α-harmonic in a (non-empty) open set D if it is continuous in D and −(−Δ)α/2 f (x) = 0 for x ∈ D (in a pointwise sense; see Section 4.3 for further discussion). Finding an α-harmonic function with prescribed values in ℝn \D is known as the Dirichlet problem for −(−Δ)α/2 in D. If D = B(0, r) is a ball, the solution is given in terms of the Poisson kernel for −(−Δ)α/2 , which is defined by PB(0,r) (x, z) =

Γ( n2 )

α/2

r 2 − |x|2 ) π n/2 Γ(1 + α2 )|Γ(− α2 )| |z|2 − r 2 (

1 |x − z|n

(19)

if x ∈ B(0, r) and z ∈ ℝn \ B(0, r); otherwise PB(0,r) (x, z) = 0. This expression and the following result were already given in Section V.16 in [83] when α < n, and in Theorem A in [10] when α ≥ n, with an equivalent notion of α-harmonicity; see also Section V.4 in [9] and Section IV.5 in [69]. Theorem 3.3. Suppose that r > 0, f (x) = f0 (x) for x ∈ ℝn \ B(0, r) and f (x) =

∫

PB(0,r) (x, z)f0 (z)dz

for x ∈ B(0, r),

(20)

ℝn \B(0,r)

where f0 is a function on ℝn \B(0, r) such that the above integral is absolutely convergent for all x ∈ B(0, r) (or, equivalently, for some x ∈ B(0, r)). Then f is α-harmonic in B(0, r), and f is continuous at z ∈ 𝜕B(0, r) whenever f0 is continuous at z. The Poisson problem for (−Δ)α/2 in D asks for a function f such that (−Δ)α/2 f (x) = g(x) for x ∈ D and f (x) = 0 otherwise. This problem is solved in terms of the Green function for (−Δ)α/2 in D, which can be defined using the fundamental solution and the Poisson kernel; see Section 4.2. For a ball, this construction was carried out in Section IV.15 in [83] when α < n, and the expression was simplified in Corollary 4 in [10] in the general case. For x, y ∈ B(0, r) the Green function of a ball B(0, r) is defined by Γ( n2 )

1 GB(0,r) (x, y) = α n/2 α 2 2 π (Γ( 2 )) |x − y|n−α =

Γ( n2 )

2α π n/2 Γ( α2 )Γ(1 + α2 )

S(r,x,y)

∫ 0

sα/2−1 ds (1 + s)n/2

α/2

(S(r, x, y)) |x − y|n−α

2 F1

n (2

1+

α 2 α 2

󵄨󵄨 󵄨󵄨 󵄨󵄨 −S(r, x, y)) , 󵄨󵄨

(21)

where 2 F1 is the Gauss hypergeometric function and S(r, x, y) =

(r 2 − |x|2 )(r 2 − |y|2 ) . r 2 |x − y|2

We let GB(0,r) (x, y) = 0 when x ∈ ℝn \ B(0, r) or y ∈ ℝn \ B(0, r).

(22)

Fractional Laplace operator and its properties | 165

Theorem 3.4. Suppose that r > 0, f (x) = 0 for x ∈ ℝn \ B(0, r) and f (x) = ∫ GB(0,r) (x, y)g(y)dy

for x ∈ B(0, r),

(23)

B(0,r)

where g is a function on B(0, r) such that the above integral is absolutely convergent for almost all x ∈ B(0, r) (or, equivalently, for some x ∈ B(0, r)). Then f solves the Poisson problem for (−Δ)α/2 in B(0, r) with zero external data: for x ∈ B(0, r) we have (−Δ)α/2 f (x) = g(x) if g is continuous at x (or, more generally, if x is a Lebesgue point of g), and if g is bounded in a neighborhood of x ∈ 𝜕B(0, r), then f is continuous at x. Theorems 3.3 and 3.4 are particular cases of Theorem 7.1 below. Further properties of the Poisson kernel and the Green function in a general open set D are discussed in Section 4.2. Except α-harmonic functions described in Theorem 3.3, there are non-zero α-harmonic functions in B(0, r) equal to zero in ℝn \ B(0, r). Their characterization involves the Martin kernel for −(−Δ)α/2 in a ball, defined by MB(0,r) (x, z) =

1 (r 2 − |x|2 )α/2 r α |x − z|n

(24)

if x ∈ B(0, r) and z ∈ 𝜕B(0, r), and MB(0,r) (x, z) = 0 otherwise. Theorem 3.5 ([53]). (a) If r > 0, x ∈ B(0, r) and z ∈ 𝜕B(0, r), then MB(0,r) (x, z) = lim

y→z y∈B(0,r)

GB(0,r) (x, y) = GB(0,r) (0, y)

lim

y→z y∈ℝn \B(0,r)

PB(0,r) (x, y) . PB(0,r) (0, y)

(25)

(b) Suppose that r > 0, μ is a finite signed measure on 𝜕B(0, r), f (x) =

∫ MB(0,r) (x, z)μ(dz) for x ∈ B(0, r)

(26)

𝜕B(0,r)

and f (x) = 0 for x ∈ ℝn \ B(0, r). Then f is α-harmonic in B(0, r) with zero external data. Conversely, every non-negative α-harmonic function in B(0, r) which is equal to zero in ℝn \ B(0, r) has the form described above. (c) A non-negative function f which is equal to zero in ℝn \B(0, r) is α-harmonic in B(0, r) if and only if (r 2 − |x|2 )1−α/2 f (x) is harmonic in B(0, r). Functions described by Theorem 3.5 are often called singular α-harmonic functions, while those identified in Theorem 3.3 are said to be regular α-harmonic functions. Further discussion of α-harmonic functions is given in Section 4.3. By taking μ to be a uniform measure on 𝜕B(0, r), one can show that α/2−1

−(−Δ)α/2 [(r 2 − |x|2 )+

]=0

(27)

166 | M. Kwaśnicki for x ∈ B(0, r). On the other hand, α/2

−(−Δ)α/2 [(r 2 − |x|2 )+ ] = −

2α Γ(1 + α2 )Γ( n+α ) 2 Γ( n2 )

(28)

for x ∈ B(0, r). The latter formula was first proved in [46]. Both are special cases of Theorem 3.6 below. An appropriate application of Kelvin transformation (or a limiting procedure) allows one to prove the analogs of Theorems 3.3, 3.4 and 3.5 for half-spaces. We record the explicit expressions for the Poisson kernel, the Green function and the Martin kernel of the half-space H = {x ∈ ℝn : x1 > 0}: PH (x, z) = GH (x, y) =

Γ( n2 )

α/2

π n/2 Γ(1 + α2 )|Γ(− α2 )|

(

x1 ) −z1

Γ( n2 )

1 2α π n/2 (Γ( α2 ))2 |x − y|n−α

S(x,y)

(x )α/2 MH (x, z) = 1 n , |x − z|

∫ 0

1 , |x − z|n

(29)

sα/2−1 ds, (1 + s)n/2

(30) (31)

for x ∈ H, z ∈ ℝn \ H in (29), x, y ∈ H in (30), and x ∈ H, z ∈ 𝜕H in (31); here S(x, y) = x1 y1 /|x − y|2 . Furthermore, for x ∈ H we have ] = 0, −(−Δ)α/2 [(x1 )α/2−1 +

−(−Δ)α/2 [(x1 )α/2 + ] = 0;

(32)

see Example 2 in [12]. Similarly, if α ≤ n, Kelvin transformation provides the expressions for the Poisson kernel, the Green function and the Martin kernel of ℝn \ B(0, r); namely, Pℝn \B(0,r) (x, z) =

Γ( n2 )

α/2

|x|2 − r 2 ) π n/2 Γ(1 + α2 )|Γ(− α2 )| r 2 − |z|2 Γ( n2 )

(

1 Gℝn \B(0,r) (x, y) = α n/2 α 2 |x − y|n−α 2 π (Γ( 2 )) Mℝn \B(0,r) (x, z) =

1 (|x|2 − r 2 )α/2 , r α |x − z|n

S(r,x,y)

∫ 0

1 , |x − z|n

(33)

sα/2−1 ds, (1 + s)n/2

(34) (35)

where S(r, x, y) = (|x|2 − r 2 )(|y|2 − r 2 )/(r 2 |x − y|2 ). Appropriate corrections are required when α > n; we refer to [10] for the details, and to [21, 67] for closely related calculations.

3.3 Complement of a sphere or a hyperplane Explicit expressions for the Poisson/Martin kernel and Green’s function for the complement of a sphere were found in [80] for α ∈ (1, 2) (for α ≤ 1 this problem is not

Fractional Laplace operator and its properties | 167

well-posed; see Section 4.2). The Poisson/Martin kernel is given by Mℝn \𝜕B(0,r) (x, z) =

22−α π 1/2 Γ( n+α−2 ) 2 Γ( n2 )Γ( α−1 ) 2

r 1−α

|r 2 − |x|2 |α−1 |x − z|n+α−2

(36)

for x ∈ ℝn \ 𝜕B(0, r), z ∈ 𝜕B(0, r); the expression for the Green function is more complicated and we refer to Theorem 4.1 in [80]. The solution of the Dirichlet problem for −(−Δ)α/2 in ℝn \ 𝜕B(0, r) is given in terms of the Poisson/Martin kernel, just as it is expressed in B(0, r) using the Poisson kernel in Theorem 3.3. Furthermore, for x ∈ ℝn \ 𝜕B(0, r), 󵄨α−1 󵄨 −(−Δ)α/2 [󵄨󵄨󵄨r 2 − |x|2 󵄨󵄨󵄨 ] = 0.

(37)

Similar results can be obtained for the complement of a hyperspace ℝn∗ = {x ∈ ℝn : x1 ≠ 0} by an appropriate application of Kelvin transformation; see Section 4 in [71]. In this case Mℝn∗ (x, z) =

22−α π 1/2 Γ( n+α−2 ) 2 Γ( n2 )Γ( α−1 ) 2

|x1 |α−1 |x − z|n+α−2

(38)

for x, z ∈ ℝn , x1 ≠ 0, z1 = 0. In addition, for x ∈ ℝn∗ , −(−Δ)α/2 [|x1 |α−1 ] = 0,

−(−Δ)α/2 [x1 |x1 |α−3 ] = 0.

(39)

3.4 Special functions Most known explicit expressions for the fractional Laplace operator applied to a particular function can be deduced from the general result stated below by appropriate choice of parameters and, optionally, Kelvin transformation. Recall that V is a solid harmonic polynomial of degree ℓ if V is a homogeneous polynomial of degree ℓ and ΔV(x) = 0 for all x ∈ ℝn . The space of solid harmonic polynomials of degree ℓ = 0 consists of constant functions, while ℓ = 1 corresponds to the space of linear functionals on ℝn , spanned by x1 , x2 , . . . , xn . There are no other solid harmonic polynomials when n = 1. If n = 2 and ℓ > 0, the space of solid harmonic polynomials of degree ℓ is spanned by ℜ[(x1 + ix2 )ℓ ] and ℑ[(x1 + ix2 )ℓ ]. For n = 3, solid harmonic polynomials of degree ℓ ≥ 0 are of the form Yℓ (x/|x|)|x|ℓ , where Yℓ is a spherical harmonic of degree ℓ. The definition and basic properties of Meijer G-functions can be found in [37]. The assumptions on the parameters in the following result are minimal conditions which assert that the left-hand side of (40) is well defined; their exact form is, however, rather complicated, and we again refer to [37] for details. For brevity, we write a + t = (a1 + t, a2 + t, . . . , ap + t) and Γ(a) = Γ(a1 )Γ(a2 ) . . . Γ(ap ) when a = (a1 , a2 , . . . , ap ) ∈ ℂp .

168 | M. Kwaśnicki Table 1: Explicit expressions for the fractional Laplace operator in full space. In the second column, we assume that x ≠ 0. (−Δ)α/2 f (x)

f (x) |x|p−n

n+α−p 2 ) n−p

|x|p−n−α

2

1 (1+|x|2 )(n−α)/2

2α

[p > − α2 ]

|x|2p (1+|x|2 )(n−α)/2+p 1 F ( |x|n−α 2 1

p−α

Γ( 2 )Γ( 2 ) α n α Γ( 2 )Γ( 2 ) 1 −2 Γ( n−α ) |x|α

log(|x|2 )

1 (1+|x|2 )p

p

Γ( 2 )Γ(

2α

[0 < p < n + α]

2α

[p > − 2n ]

n−α n−α 2 ,p+ 2 q+ n−α 2

2α

󵄨󵄨 1 󵄨󵄨 − 2 ) 󵄨 |x|

[p > − 2n ]

2α

log(1 + |x|2 )

Γ( n+α 2 )

1

2 (n+α)/2 Γ( n−α 2 ) (1+|x| ) α α n+α Γ( n+α 2 2 )Γ(p+ 2 ) 2 ,p+ 2 n 2 1 Γ( 2n )Γ(p) 2 n n n+α Γ( n+α 1 1 2 )Γ(p+ 2 ) 2 ,p+ 2 n |x|n+α 2 1 |x|2 ) Γ( 2n )Γ(p+ n−α 2 2 n n−α n+α n Γ( n+α 1 2 )Γ(p+ 2 )Γ(q+ 2 ) 2 ,p+ 2 n n−α n |x|n+α 2 1 q+ )Γ(p+ )Γ(q+ ) Γ( n−α 2 2 2 2 α n+α n+α α 2 α Γ( 2 )Γ( 2 ) 2 ,2 n n 2 1 Γ( 2 ) 2 n+α Γ( n+α 2 2 ) 2 n n 1 1 Γ( 2 ) 2 n+α Γ( n+α 2 2 ) 2 Γ( 2n )Γ(ν+1+ α2 ) 1 2 2n ,ν+1+ α2

F (

F (

F (

−2

2

e−|x|

2α

|x|−ν Jν (2|x|) [ν > −α − 12 ]

2α

󵄨󵄨 󵄨󵄨 −|x| ) 󵄨 󵄨󵄨 󵄨󵄨 − 󵄨

F (

F (

) 󵄨󵄨 1 󵄨󵄨 − 2 ) 󵄨 |x|

󵄨󵄨 󵄨󵄨 −|x| ) 󵄨

󵄨󵄨 󵄨󵄨 −|x| ) 󵄨

󵄨󵄨 󵄨󵄨 −|x| ) 󵄨

F (

Theorem 3.6 (Theorem 1 in [37]). If V is a solid harmonic polynomial of degree ℓ, then,

under appropriate conditions on parameters α, ℓ, p, q, j, k and a = (a1 , a2 , . . . , ap ), b =

(b1 , b2 , . . . , bq ),

󵄨 jk a 󵄨󵄨󵄨 ( 󵄨󵄨 |x|2 )] (−Δ)α/2 [V(x)Gpq b 󵄨󵄨 α

=2

j+1,k+1 1 V(x)Gp+2,q+2 (

−

n+2ℓ+α , 2

0,

a − α2 , b − α2 ,

− α2 1 − n+2ℓ 2

󵄨󵄨 󵄨󵄨 2 󵄨󵄨 |x| ) . 󵄨󵄨

(40)

The above result includes the expressions found in [7, 36, 89]; see also [40]. Many

simpler functions are particular instances of Meijer G-functions; we refer to the mon-

umental Table 8.4 in [82], which contains several hundred examples. A few explicit expressions are given in Tables 1 and 2. Two corollaries are discussed in more detail: the first one describes the action of −(−Δ)α/2 on generalized hypergeometric functions

p Fq , while the other one provides an explicit system of eigenfunctions of the operator μ,ν α/2 (1 − |x|2 )α/2 in terms of Jacobi polynomials Pk . + (−Δ)

Theorem 3.7 (Theorem 2 in [37]). Suppose that α ∈ (0, ∞), V is a solid harmonic polynomial of degree ℓ, p and q are non-negative integers satisfying |p − q| ≤ 1, a = (a1 , a2 , . . . , ap ) ∈ ℂp , b = (b1 , b2 , . . . , bq−1 ) ∈ ℂq−1 and ℜaj >

ℓ−α 2

for j = 1, 2, . . . , p.

Fractional Laplace operator and its properties | 169 Table 2: Explicit expressions for the fractional Laplace operator in the unit ball B(0, 1). Here f (x) = 0 when |x| ≥ 1, and in the second column we assume that 0 < |x| < 1. The result of Theorem 3.8 is not listed here. f (x)

(−Δ)α/2 f (x)

(1 − |x|2 )α/2−1

0

(1 − |x|2 )α/2

2α

(1 − |x|2 )p

2α

[p > −1]

|x|2p (1−|x|2 )p+1−α/2

[− 2n < p < α2 ] −p, α2 −q 󵄨󵄨 1 󵄨󵄨 1 − |x|2 ) F ( α −p 󵄨 (1−|x|2 )p+1−α/2 2 1 2

Γ(1+ α2 )Γ( n+α 2 ) Γ( 2n )

n+α α Γ(p+1)Γ( n+α , −p 2 ) F ( 2 n2 Γ(p+1− α2 )Γ( 2n ) 2 1 2

p+1,p+ 2n Γ( α2 −p)Γ(p+ 2n ) 1 α−2p 2 F1 ( p+ n−α |x| Γ(−p)Γ(p+ n−α ) 2 2 p+1,q+ 2n Γ( α −p)Γ(q+ 2n ) 1 2α 2 α−2q 2 F1 ( q+ n−α |x| Γ(−p)Γ(q+ n−α ) 2 2

2α [p
− 2n ]

󵄨󵄨 2 󵄨󵄨 |x| ) 󵄨 󵄨󵄨 2 󵄨󵄨 |x| ) 󵄨

Then (−Δ)α/2 [V(x)p Fq ( =

b, n+2ℓ 2

2 Γ(a + α2 )Γ(b) Γ(a)Γ(b + α2 ) α

a

󵄨󵄨 󵄨󵄨 2 󵄨󵄨 −|x| )] 󵄨󵄨

V(x)p Fq (

a + α2 b + α2 , n+2ℓ 2

󵄨󵄨 󵄨󵄨 2 󵄨󵄨 −|x| ) . 󵄨󵄨

(41)

Theorem 3.8 (Theorem 3 in [37]). Suppose that α ∈ (0, ∞), V is a solid harmonic polynomial of degree ℓ and k ≥ 0. Then α/2

(−Δ)α/2 [(1 − |x|2 )+ V(x)Pk(α/2,(n+2ℓ)/2−1) (2|x|2 − 1)] =

2α Γ(k +

α 2

+ 1)Γ( n+2ℓ+α + k) 2

k! Γ( n+2ℓ + k) 2

V(x)Pk(α/2,(n+2ℓ)/2−1) (2|x|2 − 1)

(42)

for x ∈ B(0, 1).

4 Isotropic stable Lévy processes 4.1 Fractional Laplace operator as a generator The Brownian motion in ℝn is a Markov process whose infinitesimal generator is the Laplace operator Δ. Similarly, Theorem 4.1 asserts that, for α ∈ (0, 2), the fractional Laplace operator −(−Δ)α/2 generates a certain Markov process Xt : the isotropic α-stable Lévy process. This stochastic process Xt is characterized (up to linear time-change) by the following properties: (a) X0 = 0; (b) Xt has independent and stationary increments; (c) Xt has càdlàg paths (right-continuous with left limits); (d) the distribution of t −1/α Xt does not depend on t; (e) the distribution of Xt is invariant under rotations.

170 | M. Kwaśnicki Properties (a)–(c) characterize Lévy processes, (d) states that Xt is a self-similar process, while (e) corresponds to Xt being isotropic. We use ℙ and 𝔼 to denote the probability and expectation; ℙx and 𝔼x correspond to the process with X0 = x instead of condition (a). We also write Xt− = lims→0− Xs . For a general account on Lévy processes, we refer to [3, 6, 90]; more information on isotropic stable Lévy processes can be found in [17, 68]. For α ∈ (0, 2), the isotropic stable process Xt is a pure-jump Lévy process with Lévy measure να (z)dz, where να is the kernel of −(−Δ)α/2 given by (2). The Lévy measure describes the intensity of jumps of Xt : for every set A ⊆ ℝn \ {0} and T > 0, the number NT (A) of times t ∈ (0, T] such that Xt − Xt− ∈ A is a random variable with Poisson distribution and mean T ∫A να (z)dz. By the Lévy–Itô decomposition, we have XT = lim+ r→0

∑

t∈(0,T] : |Xt −Xt− |>r

(Xt − Xt− ) = lim+ r→0

xNT (dx),

∫

(43)

ℝn \B(0,r)

where we understood the limit in the L2 (Ω) sense: if the expression under the limit is denoted by XT(r) , then 𝔼(XT(r) − XT )2 converges to 0 as r → 0+ . The distribution of Xt is known to be absolutely continuous with respect to the Lebesgue measure when t > 0, with density function pt (x) satisfying pt (x) ≍ cn,α min{

1

t n/α

,

t } |x|n+α

and

lim

t −1/α |x|→∞

pt (x) = 1. tνα (x)

(44)

Self-similarity of Xt implies that pt (x) = t −n/α p1 (t −1/α x). The transition operators of the process Xt are defined by Pt f (x) = 𝔼x f (Xt ) = 𝔼f (x + Xt ) = ∫ f (x + z)pt (z)dz.

(45)

ℝn

Here f is an arbitrary function for which the integrals are absolutely convergent. The operators Pt obey the Chapman–Kolmogorov equation Pt+s f = Pt Ps f , and they form a strongly continuous semigroup of contractions on C0 (thus Pt is a Feller semigroup), Cbu and Lp for every p ∈ [1, ∞). The following theorem identifies the fractional Laplace operator defined by (1) with the generator of the semigroup Pt . For sufficiently smooth functions, this is a standard result, valid for arbitrary Lévy processes; see [3, 6, 90]. Extension to the entire domain of −(−Δ)α/2 was given in [88] for Lp with p ∈ [1, αn ), and in [65] in the general case. Theorem 4.1 (Lemma 4.2 in [65]). If f ∈ Cbu , −(−Δ)α/2 f (x) is well defined for all x ∈ ℝn and −(−Δ)α/2 f ∈ Cbu , then −(−Δ)α/2 f (x) = lim

t→0

Pt f (x) − f (x) t

(46)

Fractional Laplace operator and its properties | 171

for all x ∈ ℝn . In this case the limits in (3) and (46) are in fact uniform in x ∈ ℝn . Conversely, if f ∈ Cbu , the limit in the right-hand side of (46) exists for all x ∈ ℝn and it defines a function in Cbu , then −(−Δ)α/2 f (x) is well defined for all x ∈ ℝn . Similarly, for every p ∈ [1, ∞) and f ∈ Lp , the limit in (1) exists in Lp if and only if the limit in (46) exists in Lp . In this case, both limits exist almost everywhere, and (46) holds for almost every x ∈ ℝn . In particular, 𝜕t pt (x) = −(−Δ)α/2 pt (x) for t > 0 and x ∈ ℝn ; that is, pt (x) is the heat kernel for −(−Δ)α/2 (the fundamental solution for 𝜕t + (−Δ)α/2 ). The set of all f ∈ C0 for which the limit in (46) exists uniformly in x ∈ ℝn is called the C0 -domain (or the Feller domain) of −(−Δ)α/2 . The Cbu and Lp -domains of −(−Δ)α/2 are defined in a similar way. By the general theory of strongly continuous semigroups of operators, for every λ > 0 the λ-resolvent or λ-potential operator ∞

Rλ f (x) = ∫ e−λt Pt f (x)dt

(47)

0

is the inverse of λ Id +(−Δ)α/2 . In particular, Rλ is a bijection between Cbu and the Cbu -domain of −(−Δ)α/2 . Similar statement is true when Cbu is replaced by C0 or by Lp with p ∈ [1, ∞). The 1-resolvent operator R1 is closely related to the Bessel potential operator 𝒥α ; see Theorem 2.5. If α < n, then the 0-resolvent operator R0 defined in a similar way is precisely the Riesz potential operator ℐα . In this case R0 is the right inverse of (−Δ)α/2 on Cc , and it is the inverse of (−Δ)α/2 on Lp whenever p ∈ [1, αn ); see Theorems 2.4 and 3.1.

4.2 Dynkin’s formula For an open set D ⊆ ℝn , the first exit time is defined by the formula τD = inf{t > 0 : Xt ∉ D}.

(48)

This is used to define the α-harmonic measure (or the exit distribution of Xt ) PD (x, A) = ℙx (τD < ∞, XτD ∈ A),

(49)

where x ∈ D and A ⊆ ℝn , as well as the Green function for (−Δ)α/2 (or the occupation density of Xt ). If D is bounded, or if α < n, then the Green function is given by GD (x, y) = uα (y − x) − ∫ uα (y − z)PD (x, dz)

(50)

ℝn \D

for x, y ∈ D, with the kernel uα defined in Theorem 3.1. We define GD (x, y) = 0 if x ∈ ℝn \ D or y ∈ ℝn \ D. In case of unbounded D and α ≥ n = 1, we let GD (x, y) to be the limit of GB(0,r)∩D (x, y) as r → ∞.

172 | M. Kwaśnicki The above definitions agree with the explicit expressions given in the previous section for particular domains D: when D is a ball, a half-space or the complement of a ball, then PD (x, dz) is absolutely continuous with respect to the Lebesgue measure, with density function given by the Poisson kernel PD (x, z), while if D is the complement of a sphere or the complement of a hyperspace, then PD (x, dz) is absolutely continuous with respect to the surface measure on 𝜕D, with density function given by the Poisson/Martin kernel MD (x, z). If α < n, then GD (x, y) is finite for every open D ⊆ ℝn except when x = y. If α > n = 1, then GD (x, y) is infinite if and only if D = ℝ. Finally, if α = n = 1, then G is finite when x ≠ y and ℙx (τD < ∞) > 0 for some x ∈ D; see [21] for further discussion. The Green function satisfies the occupation time formula τD

∫ GD (x, y)f (y)dy = 𝔼x ∫ f (Xs )ds

(51)

0

D

for every x ∈ D and every non-negative function f ; in particular, 𝔼x τD = ∫D GD (x, y)dy if x ∈ D. It is known that GD (x, y) is continuous on D × D and GD (x, y) = GD (y, x) for x, y ∈ ℝn . If x ∈ 𝜕D and ℙx (τD = 0) = 1, then x is said to be a regular boundary point for (the Dirichlet problem for) −(−Δ)α/2 in D. If x ∈ D and z ∈ 𝜕D, then GD (x, y) is a continuous function of y at z if and only if z is a regular boundary point. We remark that with probability one the process Xt never hits any irregular boundary point of D; that is, the set of irregular boundary points of any open set D is polar. Recall that a set A is polar if Xt ∉ A for all t > 0 with probability one. It is known that A is polar if its Hausdorff dimension is less than n − α and it is not polar if the Hausdorff dimension of A is greater than n − α; we refer to [57] for discussion, further results and references. Spheres and hyperplanes are polar when α ∈ (0, 1] and non-polar when α ∈ (1, 2). As a consequence of the strong Markov property, whenever B and D are open sets, B ⊆ D, x ∈ B, y ∈ D \ 𝜕B and A ⊆ ℝn \ D, we have PD (x, A) = PB (x, A) + ∫ PD (z, A)PB (x, dz),

(52)

D\B

GD (x, y) = GB (x, y) + ∫ GD (z, y)PB (x, dz).

(53)

D\B

If Bn is an ascending sequence of open sets that converges to D, x, y ∈ D and A ⊆ ℝn \ D, then PBn (x, dz) converges vaguely to PD (x, dz), PBn (x, A) increases to PD (x, A), and GBn (x, y) increases to GD (x, y); see [19] for further discussion. Dynkin’s formula, given as formula (5.8) in [38], asserts that ∫ f (z)PD (x, dz) = f (x) − ∫ GD (x, y)(−Δ)α/2 f (y)dy; ℝn \D

D

(54)

Fractional Laplace operator and its properties | 173

here we assume that f is in the Cbu -domain of −(−Δ)α/2 and ∫D GD (x, y)dy = 𝔼x τD is finite. Note that 𝔼x τD < ∞ if, for example, D is bounded. Dynkin’s formula is closely related to the Dynkin characteristic operator, ℒα f (x) = lim+ r→0

= lim+ r→0

f (XτB(x,r) ) − f (x)

𝔼x B(x, r) ∫ℝn \B(x,r) (f (z) − f (x))PB(x,r) (x, dz) ∫B(x,r) GB(x,r) (x, y)dy

(55) .

Dynkin proved that f is in the C0 -domain of −(−Δ)α/2 if and only if f ∈ C0 , the above limit exists for every x ∈ ℝn , and it defines a C0 function ℒα f ; in this case −(−Δ)α/2 f = ℒα f . The same statement is true for Cbu and Lp with p ∈ [1, ∞): an analog of Theorem 4.1 holds with −(−Δ)α/2 replaced by ℒα ; see Lemma 4.2 in [65]. Interestingly, there is a corresponding pointwise result: as it was observed in Lemma 3.3 in [65], for a given point x ∈ ℝn and a function f such that (1 + |z|)−n−α f (z) is absolutely integrable, the existence of the limit in (55) implies the existence of the limit in (3), which in turn implies the existence of the limit in (46).

4.3 Harmonic functions for −(−Δ)α/2 Recall that a function f is α-harmonic in an open set D if f is defined on ℝn , continuous in D, and −(−Δ)α/2 f (x) = 0 for x ∈ D, with −(−Δ)α/2 f (x) defined pointwise by (3). This is equivalent to the probabilistic definition of α-harmonicity: by Theorem 3.9 in [14], f is α-harmonic in D if and only if f (x) = ∫ f (z)PB (x, dz) = 𝔼x f (XτB )

(56)

ℝn \B

for all bounded open sets B such that B ⊆ D and all x ∈ B. Yet another characterization requires that f (Xmin(t,τB ) ) is a martingale with respect to ℙx for every B as above and every x ∈ B; we refer to [26] for a general discussion. We say that f is a regular α-harmonic function in D if (56) holds for all open B ⊆ D, or, equivalently, for just B = D. If D is bounded and the restriction of f to ℝn \ D is continuous, then f is a regular α-harmonic function in D if and only if it is bounded in D, α-harmonic in D and continuous at every regular boundary point; see Theorem 7.2. If f is α-harmonic in D and f (x) = 0 for all x ∈ ℝn \ D, then f is said to be a singular α-harmonic function. The Green function is defined in terms of the kernel uα (x) and the α-harmonic measure. Conversely, the absolutely continuous part of the α-harmonic measure is described by the Green function and the kernel να through the Ikeda–Watanabe formula, as it is formally stated in the following result.

174 | M. Kwaśnicki Theorem 4.2 (Proposition 1 in [19]). If D ⊆ ℝn is an open set and x ∈ D, then the absolutely continuous part of the α-harmonic measure PD (x, dz) has density function given by the Poisson kernel PD (x, z) = 1ℝn \D (z) ∫ GD (x, y)να (z − y)dy.

(57)

D

n

Furthermore, for every A ⊂ ℝ \ D, ∫ PD (x, z)dz = ℙx (τD < ∞, XτD ∈ A, XτD ≠ XτD − ).

(58)

A

The singular part of PD (x, dz) is concentrated on the set 𝜕m D = {z ∈ 𝜕D : PD (x, z) = ∞} of zero Lebesgue measure, and it describes the distribution of XτD when XτD = XτD − . For z ∈ ℝn \ D, formula (57) was first established (for general Lévy processes) in [54], and in this case it is a simple consequence of Dynkin’s formula (54). When D satisfies the exterior cone condition at every boundary point, then the α-harmonic measure has no singular part and every boundary point of D is regular; see Lemma 17 in [11]. However, if ℝn \ D is a non-polar set of zero Lebesgue measure (for example, when D = ℝn \𝜕B(0, r) and α ∈ (1, 2)), the α-harmonic measure is non-zero and singular. The Poisson kernel and the Martin kernel always describe all non-negative α-harmonic functions in D. Theorem 4.3 (Theorems 2 and 3 in [19]). Suppose that D ⊆ ℝn is a bounded open set and x0 ∈ D is an arbitrary reference point. Then the Martin kernel MD (x, z) = lim y→z y∈D

GD (x, y) GD (x0 , y)

(59)

exists for every x ∈ D and z ∈ 𝜕D. Furthermore, MD (x, z) = lim

y→z y∈ℝn \D

PD (x, y) PD (x0 , y)

(60)

whenever additionally z ∈ 𝜕(ℝn \ D). Every α-harmonic function f in D which is nonnegative in ℝn is given by f (x) = ∫ PD (x, z)f (z)dz + ∫ MD (x, z)μ(dz) ℝn \D

(61)

𝜕m D

for a unique non-negative measure μ on 𝜕m D = {z ∈ 𝜕D : PD (x0 , z) = ∞}. Conversely, the right-hand side of the above Martin representation defines a non-negative α-harmonic function whenever f and μ are non-negative and both integrals are absolutely convergent for some x ∈ D. The same statement holds for unbounded open sets D if we agree that in this case z = ∞ is included in 𝜕D, and ∞ ∈ 𝜕m D if and only if ∫D GD (x0 , y)dy = 𝔼x0 (τD ) is finite.

Fractional Laplace operator and its properties | 175

The above theorem is a consequence of the boundary Harnack inequality, stated in the next result. Note that, unlike the classical case of harmonic functions, no regularity assumptions are imposed on the open set D. Theorem 4.4 (Theorem 4 and Lemma 8 in [19]). Suppose that D ⊆ ℝn is an open set, z ∈ 𝜕D, 0 < r < R, and f is a regular harmonic function in D which is non-negative on ℝn and equal to zero on B(z, R) \ D. Then f (x) ≍ cn,α,r/R (

∫

ℝn \B(z,r)

f (y) dy)( |y − z|n+α

∫

GB(z,R)∩D (x, y)dy)

(62)

B(z,R)∩D

for x ∈ B(z, r) ∩ D. Furthermore, if f and g satisfy the above assumptions, and g is not zero everywhere in D, then the boundary limit lim

y→z y∈D

f (x) g(x)

(63)

exists. If D is bounded, then the above limit exists also when g(x) = ∫D GD (x, y)dy = 𝔼x τD . Furthermore, if R > 0, then lim sup{

r→0+

f (x)g(y) : x, y ∈ B(z, r) ∩ D, g(x)f (y) z ∈ ℝn , D ⊆ ℝn , f , g ∈ ℋ(D, z, R)} = 1,

(64)

where ℋ(D, z, R) is the class of functions f satisfying the conditions listed above. Theorem 4.4 was first proved in [11] for Lipschitz domains D, and in [96] for general open sets, but with a constant depending on the boundary point. For Lipschitz domains, a more probabilistic proof was given in [15], and a purely analytical argument can be found in [23]. For extensions of the above results to more general non-local operators, we refer to [20, 55, 59–61, 86].

4.4 Subordination For α ∈ (0, 2), the isotropic α-stable Lévy process is a subordinate Brownian motion: if Bt is the Brownian motion in ℝn , if St is an increasing α/2-stable Lévy process, and if Bt and St are independent processes, then Xt = BSt defines the isotropic α-stable Lévy process in ℝn . To have Xt normalized as in the original definition, one assumes here that the covariance matrix of Bt is 2t Id, and that the Lévy measure of St is |Γ(− α2 )|−1 s−1−α/2 1(0,∞) (s)ds. In particular, the transition density pt (x) of Xt is a mixture of Gaussians: if qt (x) = (4πt)−n/2 exp(−|x|2 /(4t)) is the Gauss–Weierstrass kernel, then pt (x) = ∫ qs (x)ℙ(St ∈ ds). (0,∞)

(65)

176 | M. Kwaśnicki Similar expressions are available for the Lévy measure, ∞

qs (x) 1 να (x) = α ∫ 1+α/2 ds, |Γ(− 2 )| s

(66)

0

and, if α < n, for the potential kernel, uα (x) =

∞

q (x) 1 ds. ∫ s Γ( α2 ) s1−α/2

(67)

0

Analogous identities hold at the operator level. Formula (65) asserts that the semigroup of operators Pt is subordinate in the sense of Bochner to the standard heat semigroup Qt ; here Qt is the convolution operator with kernel qt (x). Furthermore, if f is in the Cbu -domain of −(−Δ)α/2 , then −(−Δ)α/2 f is given by the Bochner integral with values in Cbu : −(−Δ)

α/2

∞

1 1 f (x) = α ∫ 1+α/2 (Qs f (x) − f (x))ds. |Γ(− 2 )| s

(68)

0

Conversely, if f ∈ Cbu and the above integral exists, then f is in the Cbu -domain of −(−Δ)α/2 . Similar result is true also for C0 and Lp for every p ∈ [1, ∞). A closely related expression for −(−Δ)α/2 f is given by Balakrishnan’s formula. For a detailed discussion, we refer to [65, 72].

5 Fourier transform and distribution theory Subordination formula (68) provides perhaps a shortest way to prove that the fractional Laplace operator is a Fourier multiplier with symbol −|ξ |α . Recall that f is in the Cbu -domain of −(−Δ)α/2 if f ∈ Cbu and the limit in (3) (or, equivalently, in (46)) exists uniformly in x; C0 and Lp -domains are defined in a similar way. Theorem 5.1 (Lemma 5.2 in [65]). Suppose that α ∈ (0, 2] and p ∈ [1, 2]. Then f is in the Lp -domain of −(−Δ)α/2 if and only if f ∈ Lp and |ξ |α ℱ f (ξ ) is the Fourier transform of an Lp function. In this case, α/2

ℱ (−Δ)

f (ξ ) = |ξ |α ℱ f (ξ ). α

(69)

Correspondingly, one can show that ℱ pt (ξ ) = e−t|ξ | and ℱ uα (ξ ) = |ξ |−α . The action of −(−Δ)α/2 on C0 , Cbu and Lp spaces for all p ∈ [1, ∞) can be described in terms of tempered distributions. We denote by 𝒮 the Schwartz class, and by 𝒮 󸀠 the space of tempered distributions. Two tempered distributions f and g are said to be

Fractional Laplace operator and its properties | 177

𝒮 󸀠 -convolvable if the functions f ∗ φ and g ∗ ψ are convolvable in the usual sense for every φ, ψ ∈ 𝒮 . In this case there is a unique tempered distribution f ∗ g such that

(f ∗ φ) ∗ (g ∗ ψ) = (f ∗ g) ∗ (φ ∗ ψ). The Fourier transform and the inverse Fourier transform extend continuously to operators on 𝒮 󸀠 . The function −|ξ |α defines a tempered distribution. We denote the inverse Fourier transform of this tempered distribution by Lα . Theorem 5.2 (Lemma 5.3 in [65]). Suppose that α ∈ (0, 2]. Then f is in the Cbu -domain of −(−Δ)α/2 if and only if f ∈ Cbu , f and Lα are 𝒮 󸀠 -convolvable, and f ∗ Lα is a tempered distribution associated to a Cbu function. In this case −(−Δ)α/2 f = f ∗ Lα . Similar characterization holds for C0 and Lp for every p ∈ [1, ∞). For a detailed discussion of the distribution Lα , we refer to [69, 89]. The fractional Laplace operator does not act on the class of Schwartz functions. However, several classes of test functions are invariant under the action of −(−Δ)α/2 . Lemma 5.3. The fractional Laplace operator −(−Δ)α/2 is a continuous map on each of the spaces: (a) 𝒟Lp = {f ∈ C ∞ : Dν f ∈ Lp for every ν}, with topology determined by convergence of Dν f in Lp for every ν; here p ∈ [1, ∞]; (b) 𝒟C0 = {f ∈ C ∞ : Dν f ∈ C0 for every ν}, with topology determined by uniform convergence of Dν f for every ν; (c) 𝒟α = {f ∈ C ∞ : (1 + |x|)n+α Dν f (x) is bounded for every ν}, with topology determined by uniform convergence of (1 + |x|)n+α Dν f (x) for every ν; (d) Φ = {f ∈ 𝒮 : ∫ℝn f (x)P(x)dx = 0 for every polynomial P}, the Lizorkin space, with topology inherited from 𝒮 . The first three statements of the above result follow easily from the estimate (4) and we omit the details; see [89, 94] for further discussion. For the proof of the last part, note that Dν ℱ f (0) = 0 for every ν, and therefore |ξ |α ℱ f (ξ ) is a Schwartz class function with the same property; see Chapter 2 in [89]. Suppose that p ∈ [1, ∞) and p1 + q1 = 1. The dual space of 𝒟Lp consists of distribu-

tional derivatives of Lq functions, and therefore it is denoted by 𝒟L󸀠 q . Equivalently, 𝒟L󸀠 q is the class of distributions f such that f ∗ φ ∈ Lq for every φ ∈ Cc∞ . Similarly, the dual space of 𝒟C0 is the class of distributional derivatives of L1 functions, denoted by 𝒟L󸀠 1 . This class consists of distributions f such that f ∗ φ ∈ L1 for every φ ∈ Cc∞ . We refer to [91] for a detailed discussion of these classes of distributions. The class 𝒟α and its dual 𝒟α󸀠 have been introduced in Section 2.1 in [94]. The space 󸀠 𝒟α can be characterized as the space of all distributional derivatives of functions f such that (1 + |x|)−n−α f (x) is absolutely integrable. Equivalently, f ∈ 𝒟α󸀠 if f is a distribution such that (1 + |x|)−n−α |f ∗ φ(x)| is absolutely integrable for every φ ∈ Cc∞ . The proof of these results is similar to that of Théorème VI.XXV in [91], and therefore we omit it.

178 | M. Kwaśnicki Finally, the dual of the Lizorkin space is the quotient space of tempered distributions, where we identify distributions whose difference is a polynomial; see Proposition 2.5 in [89]. By duality, Lemma 5.3 implies the following result. Theorem 5.4. Suppose that α ∈ (0, 2]. Then −(−Δ)α/2 extends to a continuous linear operator on each of the spaces 𝒟L󸀠 p for p ∈ [1, ∞], 𝒟α󸀠 and Φ󸀠 . The space 𝒟α󸀠 seems to be the largest space of distributions to which the fractional Laplace operator −(−Δ)α/2 extends in a natural way. Further extensions are possible, but they lead to unexpected behavior; for example, every tempered distribution f is associated to an element of Φ󸀠 , and so −(−Δ)α/2 f can be defined as an element of Φ󸀠 , that is, as a tempered distribution modulo polynomials. In L2 the distributional definition of −(−Δ)α/2 given in Theorem 5.2 is closely related to the weak definition of the fractional Laplace operator in terms of the corresponding quadratic form ℰα . This form is defined by ℰα (f , g) =

2α−1 Γ( n+α ) 2

∫∫

π n/2 |Γ(− α2 )| n n ℝ ℝ

(f (y) − f (x))(g(y) − g(x)) dxdy |y − x|n+α

1 = ∫ ∫ (f (y) − f (x))(g(y) − g(x))να (y − x)dxdy, 2

(70)

ℝn ℝn

and f is in the domain of ℰα if f ∈ L2 and ℰα (f , f ) < ∞. We remark that ℰα is a Dirichlet form, and we refer to [42] for the general theory of Dirichlet forms. The classical theory of self-adjoint operators on Hilbert spaces leads to the following result. Theorem 5.5. A function f ∈ L2 is in the domain of ℰα if and only if |ξ |α/2 ℱ f (ξ ) is in L2 . For f and g in the domain of ℰα , we have α

ℰα (f , f ) = ∫ |ξ | ℱ f (ξ )ℱ g(ξ )dξ .

(71)

ℝn

For every f in the L2 -domain of −(−Δ)α/2 and g in the domain of ℰα , α/2

ℰα (f , g) = ∫ (−Δ)

f (x)g(x)dx.

(72)

ℝn

Finally, f is in the L2 -domain of −(−Δ)α/2 if and only if f is in the domain of ℰα and ℰα (f , g), initially defined for g in the domain of ℰα , extends to a continuous linear functional on L2 .

6 Harmonic extension technique The operator −(−Δ)α/2 for α = 1 is the Dirichlet-to-Neumann map for the half-space ℝn × (0, ∞). This classical fact can be extended to general α ∈ (0, 2). A standard reference

Fractional Laplace operator and its properties | 179

is now [23], and the method is known as the Caffarelli–Silvestre extension technique. However, the same idea was applied in some earlier work, including [76], and there is a probabilistic reformulation in [73]. We refer to Section 10 in [45] for further discussion; see also [4, 5, 43, 65, 66, 98]. Theorem 6.1. (a) Suppose that f ∈ Cbu , and define u(x, 0) = f (x) and u(x, y) = f ∗ p̃ y (x)

(73)

for x ∈ ℝn , y > 0, where p̃ y (x) =

2α Γ( n+α ) 2

π n/2 |Γ(− α2 )|

(|x|2

yα . + y2 )(n+α)/2

(74)

Then u is a bounded continuous function on ℝn × [0, ∞), twice differentiable on ℝn × (0, ∞), satisfying ∇x,y ⋅ (y1−α ∇x,y u)(x, y) = 0 for x ∈ ℝn , y > 0, { u(x, 0) = f (x) for x ∈ ℝn .

(75)

(b) Conversely, if f ∈ Cbu , u is a bounded continuous function on ℝn × [0, ∞), twice differentiable on ℝn × (0, ∞), and u is a solution of (75), then (73) holds for x ∈ ℝn , y > 0. (c) Suppose that f ∈ Cbu and u(x, y) is defined by (73) for x ∈ ℝn and y > 0. Then f is in the Cbu -domain of −(−Δ)α/2 if and only if any of the limits in 2α Γ( α2 )

lim+ (y1−α 𝜕y u(x, y)) =

α|Γ(− α2 )| y→0

2α Γ( α2 )

lim |Γ(− α2 )| y→0+

u(x, y) − u(x, 0) yα

(76)

exists for all x ∈ ℝn and it defines a Cbu function of x ∈ ℝn ; in this case both sides of the above formula are well defined and are equal to −(−Δ)α/2 f (x). (d) Parts (a)–(c) remain valid when Cbu is replaced by Lp with p ∈ [1, ∞). In this case we require that y 󳨃→ u(⋅, y) is a continuous and bounded map from [0, ∞) to Lp , and that the convergence in (76) is in the Lp sense. Part (a) of the above theorem follows from [23], while part (c) is given in Lemma 5.4 in [65]. Part (b) is proved using the maximum principle; for completeness, we sketch the argument. Sketch of the proof of Theorem 6.1(b). By considering u(x, y)−f ∗ p̃ y (x), we may assume that f (x) = 0 for all x ∈ ℝn . Let m = sup{|u(x, y)| : x ∈ ℝn , y > 0}. We claim that m = 0. Suppose, contrary to our claim, that u(x0 , y0 ) ≠ 0 for some (x0 , y0 ) ∈ ℝn × (0, ∞). With no loss of generality we may assume that u(x0 , y0 ) > 0. We fix large R > 0, and we define v(x, y) = 2m1ℝn \B(0,R) ∗ p̃ y (x). It is easy to see that v(x, y) > m outside a compact subset of ℝn × [0, ∞). Therefore, u(x, y) − v(x, y) < 0 outside a compact subset

180 | M. Kwaśnicki of ℝn × [0, ∞). On the other hand, if R is sufficiently large, then u(x0 , y0 ) − v(x0 , y0 ) > 0. Thus, u − v attains a positive global maximum in ℝn × [0, ∞). Since u(x, 0) − v(x, 0) ≤ 0 for x ∈ ℝn , in fact u − v attains a global maximum in ℝn × (0, ∞). However, u − v is a solution of (75), and thus the strong maximum principle (Theorem 3.5 in [47]) implies that u − v is constant, a contradiction. The L2 variant of Theorem 6.1(a)–(c) is given in [98]. The extension of parts (a) and (c) to Lp for general p is proved in a much more general context in [43]. In order to prove a variant of part (b) for the Lp space, simply apply the Cbu result to the convolution u(⋅, y) ∗ φ for an appropriate mollifier φ. This establishes part (d) of the theorem. Finally, we remark that the L2 variant of Theorem 6.1(a)–(c) can be extended to other Bernstein functions of −Δ; see Theorem 4.3 in [66]. The quadratic form of (−Δ)α/2 admits a similar characterization. Theorem 6.2 (formula (3.7) in [23]). Suppose that f ∈ L2 and that u(x, y) = f ∗ p̃ y (x) for x ∈ ℝn and y > 0. Then ℰα (f , f ) =

2α Γ( α2 )

∞

∫∫ α|Γ(− α2 )| 0 ℝn

󵄨2 󵄨 y1−α 󵄨󵄨󵄨∇x,y u(x, y)󵄨󵄨󵄨 dxdy;

(77)

in particular, u is in the domain of ℰα if and only if the integral in the right-hand side is finite. Furthermore, if v is a weakly differentiable function on ℝn × [0, ∞) such that the map y 󳨃→ v(⋅, y) is a continuous and bounded function from [0, ∞) to L2 and v(x, 0) = f (x) for almost all x ∈ ℝn , then ℰα (f , f ) ≤

2α Γ( α2 )

α|Γ(− α2 )|

∞

󵄨2 󵄨 ∫ ∫ y1−α 󵄨󵄨󵄨∇x,y v(x, y)󵄨󵄨󵄨 dxdy. 0

(78)

ℝn

7 Regularity results 7.1 Existence of solutions We consider the Poisson problem for (−Δ)α/2 of the form (−Δ)α/2 f (x) = g(x) for x ∈ D, { f (x) = f0 (x) for x ∈ ℝn \ D,

(79)

where D ⊆ ℝn is an open set. We first discuss the case when D is bounded. Theorem 7.1. Let D be a bounded open set. Define f (x) = ∫ GD (x, y)g(y)dy + ∫ f0 (z)PD (x, dz) D

ℝn \D

(80)

Fractional Laplace operator and its properties | 181

for x ∈ D and f (x) = f0 (x) for x ∈ ℝn \ D. (a) If g is a continuous function on D and both integrals in (80) are absolutely convergent for all x ∈ D (or, equivalently, for some x ∈ D), then f is a solution of (79). (b) More generally, for a given point x ∈ D, (−Δ)α/2 f (x) = g(x) if the integrals in the definition (80) are absolutely convergent and x is a Lebesgue point of g. (c) If p ∈ [1, ∞), g ∈ Lploc (D) and both integrals in (80) are absolutely convergent for almost all x ∈ D (or, equivalently, for some x ∈ D), then f is a solution of (79); here (−Δ)α/2 f is defined by a limit in Lp (K) for every compact K ⊆ D. (d) If z ∈ 𝜕D is a regular boundary point, f0 is continuous at z, g is bounded in a neighborhood of z and the integrals in (80) are absolutely convergent for some x ∈ D, then f is continuous at z. Part (a) of the above result is given in Lemma 5.7 in [38] for the Dynkin characteristic operator ℒα defined in (55), and Lemma 3.3 in [65] identifies this operator with −(−Δ)α/2 , defined by the singular integral (3). Part (b) is a minor refinement. Part (c) is very similar, but apparently it is not available in the literature, so we sketch the argument below. Part (d) follows from the general theory; see Section VII.3 in [9]. Sketch of the proof of Theorem 7.1(a)–(c). Suppose that the integrals in the definition (80) are absolutely convergent for a fixed x ∈ D. By (52) and (53), ∫B(x,r) (f (z) − f (x))PB(x,r) (x, dz) ∫B(x,r) GB(x,r) (x, y)dy

=−

∫B(x,r) GB(x,r) (x, y)g(y)dy ∫B(x,r) GB(x,r) (x, y)dy

(81)

whenever B(x, r) ⊆ D. Since GB(x,r) (x, y) = GB(0,r) (0, y − x), the right-hand side is a convolution of −g with an approximate identity as r → 0+ . This proves that the limit in the definition (55) of the Dynkin characteristic operator ℒα f (x) exists and it is equal to −g(x) if x is a Lebesgue point of g. Furthermore, if g ∈ Lploc (D), then the limit in the definition of ℒα f exists also in Lp (K) for every compact K ⊆ D. The argument used in Lemma 3.7 in [65] allows one to switch to the original definition (3) of −(−Δ)α/2 f , and parts (a), (b) and (c) of the theorem follow. Uniqueness of solutions requires additional conditions near the boundary. Theorem 7.2. Let D be a bounded open set. Suppose that f is a (pointwise) solution of (79) and that f ̃ is given by (80); in particular, we assume that the integrals in (80) are absolutely convergent for almost all x ∈ D. (a) If f − f ̃ ∈ C0 (D), then f = f ̃. More generally, f = f ̃ if f − f ̃ is continuous and bounded in D and continuous at every regular boundary point of D. (b) If D satisfies the exterior cone condition and f − f ̃ is continuous and bounded in D, then f = f ̃. The same statement is true if D is an arbitrary open set such that the α-harmonic measure PD (x, dz) is absolutely continuous for all x ∈ D.

182 | M. Kwaśnicki Part (a) of the above result is given in Lemma 11 in [19]. We note that if h = f − f ̃ ∈ C0 (D), this is a standard consequence of the maximum principle: if h ∈ C0 (D) and h is not identically equal to zero, then at least one of the functions h and −h attains a positive global maximum at some x0 ∈ D, and by (3) we have −(−Δ)α/2 h(x0 ) < 0 or −(−Δ)α/2 (−h)(x0 ) < 0, respectively. Part (b) of Theorem 7.2 is given in Lemma 17 in [11] when D satisfies the exterior cone condition. The proof carries over to arbitrary open sets D, provided that ℙx (XτD = XτD − ) = 0 for x ∈ D. By Theorem 4.2, this condition is equivalent to absolute continuity of PD (x, dz). The existence of solutions in ℝn is a delicate question due to integrability issues. Theorems 2.4 and 3.1 contain sample results in this direction. On the other hand, global uniqueness is a direct consequence of the following Liouville theorem. A variant for non-negative functions is given in Theorem 1.30 in [69]. Theorem 7.3 (Theorem 1.1 in [39]). If f is α-harmonic in ℝn , then f is a constant function if α ∈ (0, 1] and an affine function if α ∈ (1, 2).

7.2 Global regularity By definition, for α ∈ [0, ∞) and p ∈ (1, ∞), the Bessel potential space H α,p is the isometric image of Lp under the Bessel potential operator 𝒥α . When α is an integer, then H α,p coincides with the standard Sobolev space W α,p ; see [25]. Smooth, compactly supported functions are dense in H α,p for all α ∈ [0, ∞) and p ∈ (1, ∞), and the norm of H α,p can be compared with more explicit Besov norms, similar to the norm of Λα discussed below; see Section 7.3 in [89] or Section V.5 in [97]. The proof of the following result is similar to the proof of Corollary 7.6 below and therefore it is omitted. Proposition 7.4. Suppose that α ∈ (0, ∞), β ∈ [0, ∞) and p ∈ (1, ∞). If f ∈ Lp , g ∈ H β,p and −(−Δ)α/2 f = g, then f ∈ H α+β,p ; here −(−Δ)α/2 f is defined by a limit in Lp . The picture is more complicated when p = 1 or p = ∞; the latter case is related to Hölder continuity of solutions. For α ∈ (0, 1), let Λα denote the class of bounded, Hölder continuous functions on ℝn with exponent α. This is a Banach space, with norm ‖f ‖Λα = ‖f ‖∞ + sup{

|f (x) − f (y)| : x, y ∈ ℝn }. |x − y|α

(82)

For α = 1 the definition is slightly more complicated and it involves second order differences: f ∈ Λ1 if f is continuous and bounded, and the norm ‖f ‖Λ1 = ‖f ‖∞ + sup{

|f (x + z) + f (x − z) − 2f (x)| : x, z ∈ ℝn } |z|

(83)

Fractional Laplace operator and its properties | 183

is finite. By Λα for general α ∈ (0, ∞) we denote the class of functions f on ℝn such that all partial derivatives of f of order up to k = ⌈α⌉ − 1 exist and belong to Λα−k . Note that Λβ is a proper subspace of Λα whenever 0 < α < β. For notational convenience, we define Λ0 = Cbu . For α ∈ (0, ∞), the space Λα is often called the Zygmund–Hölder space and sometimes denoted C∗α . We remark that when α is not an integer, then Λα is often denoted by C α or C k,β , where α = k + β, k ≥ 0 is an integer and β ∈ (0, 1). Note, however, that, for a positive integer α = k + 1, the space Λα is essentially larger than the space C k,1 of functions f such that all partial derivatives of f of order up to α − 1 are Lipschitz continuous. Theorem 7.5 (Theorem V.4 in [97]). Suppose that α, β ∈ (0, ∞). Then 𝒥α is a continuous, invertible map from Λα onto Λα+β . Furthermore, 𝒥α continuously maps L∞ into Λα . The following result is essentially contained in Section 2 in [94]. We provide a short proof in order to highlight utility of Theorems 7.5 and 2.5. Corollary 7.6. Suppose that α ∈ (0, ∞), β ∈ [0, ∞), f , g ∈ Cbu and (−Δ)α/2 f (x) = g(x) for all x ∈ ℝn . If g ∈ Λβ , then f ∈ Λα+β . Conversely, if f ∈ Λα+β and β > 0, then g ∈ Λβ . Proof. Suppose that g ∈ Λβ and define h = f + g = f + (−Δ)α/2 f . Since f , h ∈ Cbu , by Theorem 2.5 we have f = 𝒥α (h − h ∗ ϱα ) = 𝒥α (f − f ∗ ϱα ) + 𝒥α (g − g ∗ ϱα ), where ϱα is an absolutely integrable function. Clearly, g − g ∗ ϱα ∈ Λβ , and so, by Theorem 7.5, 𝒥α (g − g ∗ ϱα ) ∈ Λα+β . By the same argument, if f ∈ Λγ for some γ ≥ 0, then f − f ∗ ϱα ∈ Λγ and 𝒥α (f − f ∗ ϱα ) ∈ Λα+γ . It follows that if f ∈ Λγ for some γ ∈ [0, β], then f = 𝒥α (f − f ∗ ϱα ) + 𝒥α (g − g ∗ ϱα ) ∈ Λα+γ . Since f ∈ Λγ for γ = 0, this self-improving estimate implies that f ∈ Λα+β . To prove the converse, suppose that f ∈ Λα+β . By Theorem 7.5, f = 𝒥α h for some h ∈ Λβ , and by Theorem 2.5, g = (−Δ)α/2 f = h − h ∗ ηα for some absolutely integrable function ηα . It follows that g ∈ Λβ , as desired.

7.3 Interior regularity Global results can easily be localized. For a compact set K ⊆ ℝn , α ∈ [0, ∞) and p ∈ (1, ∞), we denote by H α,p (K) the set of restrictions to K of functions from H α,p , endowed with the quotient norm: the H α,p (K) norm of f is the infimum of the set of H α,p norms of extensions of f to ℝn . The space Λα (K), where α ∈ (0, ∞), is defined in an analogous way in terms of Λα . Similarly, for an open set D ⊆ ℝn , α ∈ [0, ∞) and p ∈ (1, ∞), we α,p denote by Hloc (D) the class of functions which belong to H α,p (K) for all compact K ⊆ D; α Λloc (D) is the class of functions that belong to Λα (K) for all compact K ⊆ D. Part (a) of the following result is a direct consequence of the explicit expression for the Poisson kernel of a ball and formula (56).

184 | M. Kwaśnicki Theorem 7.7 (Theorem 3.12 in [14]). Let D be an open subset of ℝn . (a) If f is an α-harmonic function in D, then f is real-analytic in D. (b) If f is a weakly α-harmonic function in D, in the sense that f corresponds to a tempered distribution, f and Lα (defined in Theorem 5.2) are 𝒮 󸀠 -convolvable, and f ∗ Lα is zero in D, then f is equal almost everywhere in D to an α-harmonic function. We remark that f is weakly α-harmonic in D if, for example, for some p ∈ [1, ∞] and all compact K ⊆ D the limit in the definition (3) of −(−Δ)α/2 f (x) exists in Lp (K) and it is equal to zero almost everywhere. By Theorem 4.2 in [30], if f is an α-harmonic function in a (non-empty) open set D, then the values of f in D determine the values of f everywhere. More precisely, if two α-harmonic functions in D are equal in D, they are necessarily equal almost everywhere in ℝn \ D. Interestingly, it is proved in [34] that every function f ∈ C k (B(0, r)) can be approximated in C k (B(0, r)) by (compactly supported) α-harmonic functions. When β and α + β are not integers and α < n, the second part of the following theorem is given in Section 2 in [94]; see also Propositions 2.2 and 2.3 in [87]. However, it is instructive to include a short proof. Theorem 7.8. Let D be an open subset of ℝn . β,p (a) Suppose that β ∈ [0, ∞) and p ∈ (1, ∞). If f ∈ Lploc (D), g ∈ Hloc (D) and (−Δ)α/2 f = g α+β,p

in D, then f ∈ Hloc (D); here, for α ∈ (0, 2), (−Δ)α/2 f is defined by a limit in Lp (K) for every compact subset K of D. β (b) Suppose that β ∈ [0, ∞). If f ∈ C(D), g ∈ Λloc (D) and (−Δ)α/2 f (x) = g(x) for all x ∈ D, α+β

then f ∈ Λloc (D).

Proof. We only prove the second statement, the proof of the first one being similar. With no loss of generality we may assume that α < n: otherwise we artificially extend f and g to functions of more variables, which depend only on the first n coordinates. Fix a compact set K ⊆ D, and let g̃ ∈ Λβ be equal to g in K. Since Λβ is closed under multiplication by smooth, compactly supported functions, with no loss of generality we may assume that g̃ has compact support. Let f ̃ = ℐα g,̃ so that (−Δ)α/2 f ̃ = g̃ by Theorem 3.1. Corollary 7.6 implies that f ̃ ∈ Λα+β . Clearly, (−Δ)α/2 (f − f ̃) = g − g̃ = 0 in K. It follows that f − f ̃ is an α-harmonic function in the interior of K, and hence it is smooth in the interior of K. We conclude that f is in Λα+β (K)̃ for every compact set K̃ contained in the interior of K. It remains to note that for every compact subset K̃ of D there is a compact set K ⊆ D such that the interior of K contains K.̃

7.4 Boundary regularity Regularity results near the boundary are well-studied when the Dirichlet condition f = 0 is imposed near the corresponding part of the boundary. Recall that an open

Fractional Laplace operator and its properties | 185

set D is a Lipschitz set, if the boundary of D can be locally represented as a graph of a Lipschitz function. More formally, D is a Lipschitz set if there is R0 > 0 such that for ̃ every z ∈ 𝜕D the set B(z, R0 )∩D is an isometric image of the set {x ∈ B(0, R0 ) : xn > φ(x)} for some Lipschitz continuous function φ, with Lipschitz constant uniformly bounded with respect to z; here we write x = (x1 , x2 , . . . , xn ) and x̃ = (x1 , x2 , . . . , xn−1 ). Similarly, D is a C k,β domain, where k ≥ 0 is an integer and β ∈ (0, 1], if the function φ is assumed to be of class C k,β , with the corresponding norm uniformly bounded with respect to z. We remark that Lipschitz domains satisfy the interior and exterior cone condition (but not vice versa), and C 1,1 domains are precisely the domains that satisfy the interior and exterior ball conditions. Theorem 7.9 (Proposition 1.1 in [87]). Suppose that D is an open C 1,1 subset of ℝn , z ∈ 𝜕D and 0 < r < R. If g is a bounded function on D, f0 is equal to zero in B(z, R) and f is a solution of (79) given by (80), then f ∈ Λα/2 (B(z, r) ∩ D). For related results when D is merely a Lipschitz set, we refer to [11]. By the boundary Harnack inequality, the boundary decay of all functions f in Theorem 7.9 is at least as fast as the boundary decay of f ̃(x) = ∫D GD (x, y)dy = 𝔼x τD ; in fact, the ratio f /f ̃ extends continuously to the boundary for general bounded open sets D. If D is regular enough, f ̃ can be replaced by an explicit expression. Theorem 7.10. Suppose that D is a bounded open subset of ℝn , z ∈ 𝜕D and 0 < r < R. Suppose that f0 is equal to zero in B(z, R) and f is a solution of (79) given by (80). (a) Let f ̃(x) = ∫D GD (x, y)dy. If g is zero in D, then f /f ̃ extends to a continuous function

on B(z, r) ∩ D, and the modulus of continuity of this extension has an explicit bound in terms of α, n and r/R. If D is a Lipschitz set, then this extension is in fact Hölder continuous. (b) Let δD (x) = dist(x, ℝn \ D). If D is a C 1,1 set and g is bounded on D, then f /δDα/2 ∈ Λα/2−ε (B(z, r) ∩ D) for every ε > 0. If D is a C ∞ set, γ ∈ (0, ∞) and g ∈ Λγ (D), then f /δDα/2 ∈ Λα/2+γ (B(z, r) ∩ D).

Part (a) of the above result is given in Lemma 8 in [19] and Theorem 1 in [11]. Part (b) is a compilation of Theorem 1.2 in [87] and Theorem 2.5 in [48]. For a discussion of these and related results, we refer to [86].

8 Localization In order to restrict the Laplace operator to a domain D ⊆ ℝn , one needs to specify appropriate boundary conditions. For the fractional Laplace operator −(−Δ)α/2 with α ∈ (0, 2) the situation is more complicated: the value of −(−Δ)α/2 f (x) for a single point

186 | M. Kwaśnicki x essentially depends on the values of f in all of ℝn . A discussion of various restrictions of −(−Δ)α/2 can be found in [35, 44, 70].

8.1 Dirichlet fractional Laplace operator One approach is to extend a function f defined on a domain D to all of ℝn by setting f (x) = 0 for x ∈ ℝn \ D. There are at least two equivalent rigorous definitions of the corresponding operator ℒDα . In L2 it is convenient to use quadratic forms. The space Cc∞ (D) is contained in the domain of ℰα . We define ℰαD to be the Friedrichs extension of the restriction of ℰα to Cc∞ (D). In other words, f is in the domain of ℰαD with ℰαD (f , f ) = a if (f , a) belongs to the closure in L2 × [0, ∞) of the set {(f , ℰα (f , f )) : f ∈ Cc∞ (D)}. The Dirichlet fractional Laplace operator ℒDα is the self-adjoint operator associated to ℰαD . It is easy to see that D

ℰα (f , g) =

1 ∫ ∫(f (y) − f (x))(g(y) − g(x))να (y − x)dxdy 2 D D

+ ∫ f (x)g(x)( ∫ να (y − x)dy)dx, D

(84)

ℝn \D

with f in the domain of ℰαD if and only if f ∈ L2 (D) and ℰαD (f , f ) < ∞. An equivalent probabilistic definition involves the killed process XtD , defined by XtD = Xt when t < τD and XtD = 𝜕 when t ≥ τD ; here 𝜕 is an auxiliary cemetery state, Xt is the isotropic α-stable Lévy process in ℝn and τD is the first exit time from D. Transition operators of this process, PtD f (x) = 𝔼x (1{t 1. For further discussion, we refer to [50–52, 75]. The regional fractional Laplace operator is meant to be a non-local analog of the Laplace operator with Neumann boundary condition. Two other plausible definitions are proposed in [48] and [33].

8.5 Fractional Laplace operator with harmonic condition Yet another extension can be obtained by requiring that −(−Δ)α/2 f (x) = 0 for x ∈ ℝn \D. When α < n, this condition corresponds to inversion of Riesz potentials of measures supported in D. A rigorous definition of the corresponding operator ℒH α again involves quadratic forms: when f is defined on D, we let H

ℰα (f , f ) = inf{ℰα (f ̃, f ̃) : f ̃(x) = f (x) for x ∈ D}.

(88)

As was the case with previous restrictions of −(−Δ)α/2 to a domain D, the operator ℒH α is the generator of a Markov process XtH . This process can be obtained from the isotropic α-stable Lévy process Xt by removing those parts of paths which lie outside D, and gluing the remaining pieces together. For a rigorous discussion in a general context, we refer to [27]; see also [58]. When α = 1, the operator ℒH α appears in the theory of linear water waves; see [63].

8.6 Harmonic extension technique viewpoint The operator −(−Δ)α/2 can be represented as a Dirichlet-to-Neumann map for a certain harmonic extension problem, as discussed in Section 6. Similar results are available for some of the restrictions to a domain described above; we refer to [43, 98] for a detailed discussion, and to [24, 78, 99] for further developments. The Dirichlet fractional

Fractional Laplace operator and its properties | 189

Laplace operator ℒDα corresponds to the problem ∇x,y ⋅ (y1−α ∇x,y u)(x, y) = 0 for x ∈ ℝn , y > 0, { { { u(x, 0) = f (x) for x ∈ D, { { { for x ∈ ℝn \ D; {u(x, 0) = 0

(89)

namely, in this case ℒDα f (x) = (2α Γ( α2 ))/(α|Γ(− α2 )|) limy→0+ y1−α 𝜕y u(x, y) for x ∈ D. Similarly, the fractional Dirichlet Laplace operator ℒSα corresponds to ∇x,y ⋅ (y1−α ∇x,y u)(x, y) = 0 for x ∈ D, y > 0, { { { u(x, 0) = f (x) for x ∈ D, { { { for x ∈ 𝜕D, y > 0. {u(x, y) = 0

(90)

Replacing the Dirichlet condition u(x, y) = 0 by the Neumann condition 𝜕n u(x, y) = 0 for x ∈ 𝜕D, y > 0, one obtains a representation of ℒNα . The fractional Laplace operator with α-harmonic condition ℒH α is associated with ∇x,y ⋅ (y1−α ∇x,y u)(x, y) = 0 for x ∈ ℝn , y > 0, { { { u(x, 0) = f (x) for x ∈ D, { { { 1−α n {limy→0+ (y 𝜕y u(x, y)) = 0 for x ∈ ℝ \ D.

(91)

Finally, the regional fractional Laplace operator ℒRα does not seem to correspond to any extension problem.

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Yuri Luchko and Virginia Kiryakova

Applications of the Mellin integral transform technique in fractional calculus Abstract: In this chapter, basic properties of the Mellin integral transform are first briefly discussed. Then we proceed with some selected applications of the Mellin integral transform in Fractional Calculus (FC). We start with a demonstration of a convenient method for calculating improper integrals containing the FC special functions and derivation of basic properties of the Erdélyi–Kober fractional integrals and derivatives. Then we proceed with more advanced applications including derivation of a modified Post–Widder formula for the inversion of the Laplace transform, investigation of the completely monotone functions and their connection with the probability density functions (pdfs), derivation of some subordination formulas for the multidimensional space-time fractional diffusion equations, and analytical treatment of the dual integral equations with the Meijer G-functions in the kernel. Keywords: Mellin integral transform, H- and G-functions, Mittag-Leffler function, Wright function, Erdélyi–Kober integrals and derivatives, Post–Widder formula, completely monotone functions, probability density functions, fundamental solution, Mellin–Barnes integrals, subordination formulas, dual integral equations MSC 2010: 26A33, 44A20, 33C60, 33E30, 44A05, 44A10

1 Introduction Along with the Fourier and Laplace integral transforms, the Mellin integral transform is one of the most important and most used integral transforms both in mathematics and in numerous applications. In fact, the Mellin integral transform is closely connected with the Fourier transform, but it is more convenient for some applications including the theory of special functions of the hypergeometric type, integral transforms with the functions of the hypergeometric type in the kernels, and Fractional Calculus (FC). The reason for this situation is that all special functions of the hyperAcknowledgement: Virginia Kiryakova’s work on this chapter is in the framework of the program of the projects (2017–2019) under bilateral agreements of the Bulgarian Academy of Sciences with the Serbian and Macedonian Academies of Sciences and Arts, and program of COST Action CA15225. Yuri Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected] Virginia Kiryakova, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Acad. G. Bontchev Str., Bl. 8, 1113 Sofia, Bulgaria, e-mail: [email protected] https://doi.org/10.1515/9783110571622-008

196 | Yu. Luchko and V. Kiryakova geometric type can be represented in a consistent form in terms of the inverse Mellin integral transform of a special kind, called the Mellin–Barnes integrals [4] or the Fox H-function [11]. The kernels of these Mellin–Barnes integrals are in the form of quotients of products of the Euler Gamma-functions with linear arguments. Thus very different elementary and special functions are particular cases of the Fox H-function and can be treated in a uniform way [34, 36, 45, 49]. The Mellin convolutions of functions of hypergeometric type also belong to the class of hypergeometric functions. In particular, this can be used for calculation of the improper integrals of products of elementary and special functions. Furthermore, most of the integral transforms with the special functions in the kernels including different forms of fractional integrals can be classified as integral transforms of the Mellin convolution type. By employing the Parseval equality for the Mellin integral transform, these integral transforms can be represented as the Mellin–Barnes integrals and then studied in a uniform way [49]. In particular, their compositions and different forms of inverse operators can be deduced just by applying the known formulas for the Gamma-function. In a sense, one has a kind of operational method that translates the formulas involving the Gamma-function into the ones for the integral transforms. As an example, let us mention that the known summation theorem for the Gauss hypergeometric function 2 F1 saying that 2 F1 (a, b; c; 1)

=

Γ(c)Γ(c − a − b) , Γ(c − a)Γ(c − b)

ℜ(c − a − b) > 0,

where the Gauss function is defined as the series 2 F1 (a, b; c; z)

=

Γ(c) ∞ Γ(a + n)Γ(b + n) z n ∑ Γ(a)Γ(b) n=0 Γ(c + n) n!

for |z| ≤ 1 and ℜ(c − a − b) > 0 and as an analytic continuation of this series for other values of z, leads to the Leibniz-type formula ∞ n−1 δ 1 d α +n,δ−n γ,δ − α2 − j)g(x) (Dβ f ⋅ g)(x) = ∑ ( )(Dβ1 f )(x) ∏( x β dx n n=0 j=0

(1)

for the Erdélyi–Kober fractional derivatives, where α1 , α2 are arbitrary real numbers Γ(δ+1) satisfying the relation α1 − α2 = γ, ( nδ ) = n!Γ(δ−n+1) is the generalized binomial coefγ,α

ficient, and Dβ stands for the fractional Erdélyi–Kober derivative (47) if α > 0, for the Erdélyi–Kober integral (36) of order −α if α < 0, and for the identity operator if α = 0. For a derivation of formula (1) and many other formulas of this type, we refer the reader to [21, 30, 46], or [49]. Another example is the Post–Widder formula [15], n+1

(−1)n n ( ) n→∞ n! t

f (t) = ℒ−1 {F(p); t} = lim

n F (n) ( ), t

(2)

Applications of the Mellin integral transform technique in fractional calculus | 197

for the inverse Laplace transform of F(p) = ℒ{f (t); p}, which can easily be deduced from the known representation (in fact, one of the definitions) of the Gamma-function (see [30] for details): n!ns , n→∞ s(s + 1) ⋅ ⋅ ⋅ (s + n)

Γ(s) = lim

s ≠ 0, −1, −2, . . . .

(3)

One of the most important applications of the Mellin integral transform in FC is the derivation of the closed form formulas for the fundamental solutions to the Cauchy problem for the space-time fractional diffusion-wave equations. In the onedimensional case, this approach was first suggested in [13] and then worked out in detail in [32]. In the multi-dimensional case, the Mellin integral transform technique for a treatment of the Cauchy problems for the space-time fractional diffusion-wave equations was employed in [6, 7, 23–27]. It is worth mentioning that the Mellin integral transform was also successfully employed for the derivation of several subordination formulas that connect the fundamental solutions to the Cauchy problems for the space-time fractional diffusion-wave equations with fractional derivatives of different orders (see [33] for the treatment of the one-dimensional case and [28] for the multidimensional case). For a survey of several other applications of the Mellin integral transform in FC we refer the reader to [30]. The rest of the chapter is organized as follows. In Section 2, basic properties of the Mellin integral transform are briefly discussed. Section 3 is devoted to the Mellin– Barnes integral representations of the FC special functions including the Mittag-Leffler function, the Wright function, and the Fox H-function, as well as to a description of a convenient method for calculating improper integrals containing these special functions. In Section 4, we discuss representations of the Erdélyi–Kober fractional integrals and derivatives in terms of the Mellin integral transform and its inverse transform. Section 5 deals with some more advanced applications of the Mellin integral transform in FC including derivation of a modified Post–Widder formula for the inversion of the Laplace transform, investigation of the completely monotone functions and pdfs, a derivation of subordination formulas for the fundamental solutions to the multi-dimensional space-time fractional diffusion equations, and explicit solutions of some dual integral equations.

2 Basic properties of the Mellin integral transform In this section, some basic facts from the theory of the Mellin integral transform that are used in the further discussions are presented. For more information regarding the Mellin integral transform including its properties and particular cases we refer the interested reader to [3, 8, 10, 18, 22, 34, 38–40], and [47].

198 | Yu. Luchko and V. Kiryakova For a sufficiently well-behaved function f : ℝ+ → ℝ, its Mellin integral transform at the point s, s ∈ ℂ is defined as the improper integral +∞

M{f (t); s} = f (s) = ∫ f (t)t s−1 dt ∗

(4)

0

and the inverse Mellin integral transform as the improper integral γ+i∞

1 f (t) = M {f (s); t} = ∫ f ∗ (s)t −s ds, 2πi −1

∗

t > 0, γ = ℜ(s)

(5)

γ−i∞

in the sense of the Cauchy principal value. The Mellin integral transform of a function f is well defined under the following sufficient conditions: 1) f ∈ Lc (ϵ, E), 0 < ϵ < E < ∞; 2) f ∈ C(0, ϵ] and f ∈ C[E, +∞); 3) |f (t)| ≤ Mt −γ1 for 0 < t < ϵ and |f (t)| ≤ Mt −γ2 for t > E, where M is a constant and γ1 < γ2 . If the conditions 1)–3) are valid, the Mellin transform f ∗ (s) of the function f (t) exists in the vertical strip γ1 < ℜ(s) < γ2 of the complex plane and it is an analytical function in this strip. As to the inverse Mellin integral transform, formula (5) is valid in particular for a piecewise differentiable function f that satisfies the condition f (t)t γ−1 ∈ Lc (0, +∞) at all points, where it is continuous. In our further considerations, the Mellin convolution given by the formula +∞

(f ∗ g)(x) = ∫ f (x/t)g(t) M

0

dt t

(6)

plays a very essential role. If the functions f and g satisfy the conditions f (t)t γ−1 ∈

L(0, ∞) and g(t)t γ−1 ∈ L(0, ∞), then the Mellin convolution h = (f ∗ g) is well defined and fulfills the condition h(x)xγ−1 ∈ L(0, ∞), and the following important property (convolution theorem) is valid: M

M{(f ∗ g)(t); s} = M{f (t); s} ⋅ M{g(t); s}. M

(7)

Combining the convolution theorem with formula (5) for the inverse Mellin integral transform, we get the well-known Parseval equality for the Mellin integral transform: +∞

∫ f (x/t)g(t) 0

γ+i∞

dt 1 = ∫ f ∗ (s)g ∗ (s)x−s ds. t 2πi γ−i∞

(8)

Applications of the Mellin integral transform technique in fractional calculus | 199

In the further discussions, we often use elementary properties of the Mellin inteM

gral transform, which are summarized below. Denoting by ↔ the juxtaposition of a function f with its Mellin transform f ∗ , the main rules are M

f (at) ↔ a−s f ∗ (s),

a > 0,

t α f (t) ↔ f ∗ (s + α), M 1 ∗ f (t α ) ↔ f (s/α), α ≠ 0, |α| M Γ(n + 1 − s) ∗ f (n) (t) ↔ f (s − n) Γ(1 − s) M

(9) (10) (11) (12)

if lim t s−k−1 f (k) (t) = 0, k = 0, 1, . . . , n − 1, t→0

n

M Γ(1 + α + n − s/β) ∗ 1 d f (s), ∏(α + j + t )f (t) ↔ β dt Γ(1 + α − s/β) j=1

(13)

n−1 M Γ(α + n + s/β) ∗ 1 d f (s). ∏(α + j − t )f (t) ↔ β dt Γ(α + s/β) j=0

(14)

Let us mention that the variable substitution t = 1/τ in the Parseval equality (8) along with the properties (10), (11) of the Mellin integral transform leads to a very useful formula +∞

γ+i∞

0

γ−i∞

1 ∫ f (xt)g(t)dt = ∫ f ∗ (s)g ∗ (1 − s)x−s ds. 2πi

(15)

Tables of the Mellin transforms of the elementary and many special functions can be found in [10, 34, 38], and [40]. Below we present the Mellin transform formulas of some elementary functions that have a close relation to the classical FC operators (see Section 3 for Mellin transforms of some basic FC special functions): α

M

e−t ↔

1 Γ(s/α) if ℜ(s/α) > 0, |α|

(1 − t β )α−1 M Γ(s/β) + ↔ Γ(α) |β|Γ(s/β + α)

(t β − 1)α−1 M Γ(1 − α − s/β) + ↔ Γ(α) |β|Γ(1 − s/β)

if ℜ(α) > 0, ℜ(s/β) > 0, if 0 < ℜ(α) < 1 − ℜ(s/β).

In formulas (17), (18), we used the notation t α , t > 0, t+α = { 0, t ≤ 0, for the truncated power function.

(16) (17) (18)

200 | Yu. Luchko and V. Kiryakova

3 Mellin transforms of the FC special functions and evaluation of improper integrals We start this section with a list of the basic formulas for the Mellin integral transforms of the FC special functions including the Mittag-Leffler function and the Wright function ([34, 40], and [49]), Γ(s)Γ(1 − s) if 0 < ℜ(s) < 1, 0 < α < 2, β ∈ ℂ, Γ(β − αs) M Γ(s) Wa,μ (−t) ↔ if 0 < ℜ(s), −1 < μ < 1, a ∈ ℂ, Γ(a − μs) M

Eα,β (−t) ↔

m,n ( Hp,q

(α, a)p (β, b)q

n 󵄨󵄨 ∏m M j=1 Γ(βj + bj s) ∏j=1 Γ(1 − αj − aj s) 󵄨󵄨 󵄨󵄨 t) ↔ p 󵄨󵄨 ∏j=n+1 Γ(αj + aj s) ∏qj=m+1 Γ(1 − βj − bj s)

(19) (20) (21)

if − min ℜ(βj )/bj < ℜ(s) < min (1 − ℜ(αj ))/aj , 1≤j≤m

1≤j≤n

n

p

m

q

i=1

i=n+1

i=1

i=m+1

∑ ai − ∑ ai + ∑ bi − ∑ bi > 0. In the formulas above, Eα,β stands for the Mittag-Leffler function zn , Γ(αn + β) n=0 ∞

Eα,β (z) = ∑

α > 0, β ∈ ℂ,

(22)

Wa,μ for the Wright function zn , n!Γ(a + μn) n=0 ∞

Wa,μ (z) = ∑

μ > −1, a ∈ ℂ,

(23)

m,n and Hp,q for the Fox H-function. It is well known that all elementary functions and most of the special functions including the special functions of FC are particular cases of the Fox H-function. Because the Mellin transform (21) of the H-function is a quotient of some products of the Gamma-functions, this is the case for the FC special functions, too. This fact makes the Mellin transform an extremely powerful technique in FC. For the reader’s convenience, the Fox H-function as the basic function of FC is briefly discussed below. It is defined by means of a contour integral of the Mellin– Barnes type [11, 16, 18, 31, 36, 40, 45, 49], m,n Hp,q (

(α1 , a1 ), . . . , (αp , ap ) (β1 , b1 ), . . . , (βq , bq )

󵄨󵄨 1 󵄨󵄨 ∫ Φ(s)z −s ds, 󵄨󵄨 z) = 󵄨󵄨 2πi

(24)

L

where z ≠ 0, 0 ≤ m ≤ q, 0 ≤ n ≤ p, αi ∈ ℂ, ai > 0, 1 ≤ i ≤ p, βi ∈ ℂ, bi > 0, 1 ≤ i ≤ q, and (compare with (21)) Φ(s) =

n ∏m i=1 Γ(βi + bi s) ∏i=1 Γ(1 − αi − ai s) , q p ∏i=n+1 Γ(αi + ai s) ∏i=m+1 Γ(1 − βi − bi s)

(25)

Applications of the Mellin integral transform technique in fractional calculus | 201

an empty product, if it occurs, is taken to be one, and the infinite contour L that sepn arates the poles of ∏m i=1 Γ(βi + bi s) from the poles of ∏i=1 Γ(1 − αi − ai s) can be of the following three types: 1) L = Li∞ if a∗ > 0, |arg z| < a∗ π/2 or a∗ ≥ 0, |arg z| = a∗ π/2, γδ < −ℜμ. The contour Li∞ lies in a vertical strip |ℜ(s)| ≤ C and goes from the point −∞ + iy1 to the point −∞ + iy2 , y1 < y2 . 2) L = L−∞ if δ > 0, 0 < |z| < ∞ or δ = 0, 0 < |z| < β or δ = 0, |z| = β, a∗ ≥ 0, ℜ(μ) < 0. The contour L−∞ lies in a horizontal strip |ℑ(s)| ≤ C and goes from the point −∞ + iy1 to the point −∞ + iy2 , y1 < y2 . 3) L = L+∞ if δ < 0, 0 < |z| < ∞ or δ = 0, |z| > β or δ = 0, |z| = β, a∗ ≥ 0, ℜ(μ) < 0. The contour L+∞ lies in a horizontal strip |ℑ(s)| ≤ C and goes from the point +∞ + iy1 to the point +∞ + iy2 , y1 < y2 . The parameters μ, γ, β, a∗ , and δ, mentioned in the conditions above, are given by the formulas μ=

q

p

p−q + 1 + ∑ βi − ∑ αi , 2 i=1 i=1 n

p

m

i=1

i=n+1

i=1

γ = lim ℜ(s), s→∞ s∈Li∞

q

a∗ = ∑ ai − ∑ ai + ∑ bi − ∑ bi , i=m+1

p

−ai

β = ∏ ai q

i=1

p

q

i=1

δ = ∑ bi − ∑ ai . i=1

i=1

b

∏ bi i ,

(26) (27)

For some values of the parameters of the H-function, several conditions out of those given in 1)–3) are simultaneously satisfied. In this case, the corresponding integrals are equal to each other due to the Cauchy integral theorem. A prominent particular case of the H-function, encompassing lists of elementary and special functions (SFs) of mathematical physics (named SFs, or rather classical SFs—to distinguish from the SFs of FC), is the G-function of Meijer. We get it by setting ai = 1, i = 1, . . . , p, bi = 1, i = 1, . . . , q, in formulas (24)–(25): m,n Gp,q (

α1 , . . . , αp β1 , . . . , βq

󵄨󵄨 1 󵄨󵄨 ∫ Φ(s)z −s ds 󵄨󵄨 z) = 󵄨󵄨 2πi

(28)

L

with the Mellin image Φ(s) =

n ∏m i=1 Γ(βi + s) ∏i=1 Γ(1 − αi − s) . q p ∏i=n+1 Γ(αi + s) ∏i=m+1 Γ(1 − βi − s)

(29)

For more details and results regarding the H-function and the G-function, we refer the reader to [16, 18, 35, 36, 40], and [45]. More advanced properties and applications of the Mellin transform technique can be found in [3, 8, 22, 24, 28, 30, 32–34, 39, 49]. In the rest of this section, we discuss a simple but very powerful method for evaluation of some improper integrals involving the FC special functions. This method was

202 | Yu. Luchko and V. Kiryakova first described in detail in [34] and then employed both to produce some tables of integrals [40, 41] and for implementation of calculation of some improper integrals in the computer algebra system Mathematica and in the online integral calculator of WolframAlpha. The idea of the method is very simple and consists in employing the Parseval formula (8) for the Mellin integral transform or its modification (15). To evaluate the integral at the left-hand side of formula (8) or (15), one uses the right-hand side of this formula and thus the evaluation algorithm consists of three steps: 1) Calculation of the Mellin integral transforms f ∗ and g ∗ of the functions f and g. 2) Construction of the Mellin–Barnes integral at the right-hand side of formula (8) or (15). 3) Representation of the Mellin–Barnes integral at the right-hand side of (8) or (15) in terms of known elementary or special functions. Let us consider an example and evaluate the integral [14] ∞

I(x) = ∫ Wa,μ (−xt)Eα (−t α )dt,

0 < α < 2, −1 < μ < 1

0

with the Mittag-Leffler function Eα (z) = Eα,1 (z) and the Wright function Wa,μ (z) by employing the algorithm presented above. 1) The Mellin integral transform of the Wright function f (t) = Wa,μ (−t) is given by formula (20): M

Wa,μ (−t) ↔

Γ(s) . Γ(a − μs)

The Mellin integral transform of the function g(t) = Eα (−t α ) is determined by using formula (19) and the property (11) of the Mellin integral transform: Eα (−t α ) ↔ M

1 Γ(s/α)Γ(1 − s/α) . α Γ(1 − s)

2) The Mellin–Barnes integral at the right-hand side of the Parseval equality (15) is constructed as follows: γ+i∞

1 Γ(s) 1 Γ((1 − s)/α)Γ(1 − (1 − s)/α) −s I(x) = x ds ∫ 2πi Γ(a − μs) α Γ(s) γ−i∞

γ+i∞

1 Γ((1 − s)/α)Γ(1 − (1 − s)/α) −s 1 x ds. = ∫ 2πi α Γ(a − μs) γ−i∞

Applications of the Mellin integral transform technique in fractional calculus | 203

The variables substitution 1−(1−s)/α = p in the last integral gives us the representation I(x) = xα−1

γ+i∞

1 Γ(1 − p)Γ(p) −p (xα ) dp. ∫ 2πi Γ(a + μα − μ − μαp)

(30)

γ−i∞

3) Formula (30) leads to the Mellin integral transform pair x

1−α α

1

M

I(x α ) ↔

Γ(1 − p)Γ(p) ; Γ(a − μ + μα − μαp)

then from formula (19) we can now immediately read off the relation x

1−α α

1

I(x α ) = Eμα, a−μ+μα (−x)

and then get the final result: I(x) = xα−1 Eμα, a−μ+μα (−x α ). Alternatively, we can first determine a series representation of the Mellin–Barnes integral (30). According to the general theory of the Mellin–Barnes integrals, the integration contour in the integral (30) can be transformed to the loop L−∞ starting and ending at −∞ and encircling all poles pn = −n, n = 0, 1, 2, . . . of the function Γ(p). Taking into account the Jordan lemma and the formula for the residuals of the Gammafunction, the Cauchy residue theorem leads to the following series representation: Γ(1 + n) (−1)n αn x , Γ(a − μ + μα + μαn) n! n=0 ∞

I(x) = xα−1 ∑ which can be rewritten as

(−xα )n = xα−1 Eμα, a−μ+μα (−x α ). Γ(a − μ + μα + μαn) n=0 ∞

I(x) = xα−1 ∑

It is worth mentioning that the Laplace integral transform ∞

f ̃(p) = L{f (t); p} = ∫ f (t)e−pt dt,

ℜ(p) > cf ,

(31)

0

of a function f can be interpreted as a modified Mellin convolution of the function f with the exponential function g(t) = exp(−t). According to the Parseval equality (15), it can be rewritten in the form of the following Mellin–Barnes integral: γ+i∞

f ̃(p) = L{f (t); p} =

1 ∫ f ∗ (1 − s)Γ(s)p−s ds 2πi γ−i∞

(32)

204 | Yu. Luchko and V. Kiryakova because of the Mellin transform formula (16) for the exponential function. The representation (32) can be employed to determine the Laplace transforms of the FC special functions. Let us consider an example and calculate the Laplace transform of the Wright function with a power weight f (t) = t μ−1 Wλ,μ (−t λ ), −1 < λ < 0. We start with the Mellin transform correspondence t μ−1 Wλ,μ (−t λ ) ↔ − M

1 Γ((s + μ − 1)/λ) , λ Γ(1 − s)

(33)

which easily follows from formulas (10), (11), and (20). Thus the Mellin–Barnes integral (32) takes the form γ+i∞

Γ((μ − s)/λ) 1 1 f ̃(p) = − Γ(s)p−s ds ∫ λ 2πi Γ(s) γ−i∞

or γ+i∞

f ̃(p) = −

1 1 ∫ Γ((μ − s)/λ)p−s ds. λ 2πi

(34)

γ−i∞

The representation (34) means that the Mellin integral transform of f ̃(p) is just the function − λ1 Γ((μ − s)/λ). To recover the original function, i. e., the Laplace transform f ̃(p) by itself, we employ the Mellin transform formula (16) along with formulas (10) and (11) and get the final result [14]: f ̃(p) = L{t μ−1 Wλ,μ (−t λ ); p} = p−μ exp(−p−λ ),

−1 < λ < 0.

(35)

Of course, according to the algorithm for evaluation of the improper integrals presented above, we could first determine a series representation of the Mellin–Barnes integral (34) and then interpret it as the function at the right-hand side of (35). An alternative approach for evaluation of improper integrals of special functions and of products of two special functions has been provided in the recent paper [19] based on the theory of generalized fractional calculus [18]. There, such integrals are interpreted as multiple Erdélyi–Kober operators (represented by means of single integrals with Fox’s H-functions as kernels) evaluated for the generalized Wright function p Ψq , which is also a particular case of the H-function. Using the known formula for integral of product of two H-functions [18, 36, 45], which is obtained based on the Mellin transform technique, the result comes as another H-function, usually again a generalized Wright hypergeometric function. Thus, a large variety of results found sporadically by many authors for integrals of different special functions are covered by this formula in one single stroke. For more details regarding the integration technique by employing the Mellin integral transform and many other examples, we refer the reader to [34].

Applications of the Mellin integral transform technique in fractional calculus | 205

4 Fractional integrals and derivatives as Mellin convolution type transforms From the viewpoint of the Mellin integral transform, the most suitable form of the left- and right-hand sided fractional integrals is the Erdélyi–Kober (EK) integrals [18, 20, 49]: x

β −β(γ+δ) δ−1 x ∫(x β − t β ) t β(γ+1)−1 f (t)dt, Γ(δ)

γ,δ

(Iβ f )(x) = (Kβτ,α )f (x) =

∞

β > 0, ℜ(δ) > 0,

(36)

β > 0, ℜ(α) > 0.

(37)

0

β βτ α−1 x ∫ (t β − xβ ) t −β(τ+α−1)−1 f (t)dt, Γ(α) x

For δ = 0 or α = 0, respectively, these operators are defined as the identity operator. The Erdélyi–Kober fractional integrals (36), (37) are connected with the Riemann– Liouville fractional integrals. Indeed, for β = 1 we get the representations γ,δ

δ γ (I1 f )(x) = (x −γ−δ I0+ x f )(x) =

(K1τ,α )f (x) = (xτ I−α x−τ−α f )(x) =

x

1 −γ−δ x ∫(x − t)δ−1 t γ f (t)dt, Γ(δ) ∞

(38)

0

1 τ x ∫ (t − x)α−1 t −τ−α f (t)dt. Γ(α)

(39)

x

Evidently, the Erdélyi–Kober fractional integrals (36), (37) are in the form of the Mellin convolutions (6) with the power functions: γ,δ

(Iβ f )(x) = (k1 ∗ f )(x), where k1 (x) =

M

β −β(γ+δ) β δ−1 x (x − 1)+ , Γ(δ)

(Kβτ,α f )(x) = (k2 ∗ f )(x), M

k2 (x) =

β βτ α−1 x (1 − xβ )+ . Γ(α)

(40)

(41)

The Parseval equality (8) for the Mellin transform along with the properties (10), (17), and (18) leads to the representations of the Erdélyi–Kober fractional integrals in the form of the Mellin–Barnes integrals [18, 49]: γ,δ (Iβ f )(x)

γ+i∞

Γ(1 + γ − s/β) ∗ 1 = f (s)x −s ds, ∫ 2πi Γ(1 + γ + δ − s/β)

(Kβτ,α )f (x)

(42)

γ−i∞

γ+i∞

Γ(τ + s/β) ∗ 1 f (s)x−s ds. = ∫ 2πi Γ(τ + α + s/β)

(43)

γ−i∞

The right-hand sides of these representations allow us to treat the Erdélyi–Kober fractional integrals and in particular to get their inversion formulas and formulas for their compositions [18, 49].

206 | Yu. Luchko and V. Kiryakova To proceed, we notice that the Parseval equality (8) can be employed to get a simple formula for a composition of several operators of the Mellin convolution type. Let us consider two operators of this kind with the kernels k1 and k2 in the form +∞

(K1 f )(x) = ∫ k1 (x/t)g(t) 0

+∞

dt , t

(K2 f )(x) = ∫ k2 (x/t)g(t) 0

dt . t

Applying the Parseval equality (8) twice, we can represent the composition K2 ∘ K1 as the Mellin–Barnes integral: γ+i∞

1 ∫ k2∗ (s)k1∗ (s)f ∗ (s)x−s ds. 2πi

(K2 ∘ K1 )(x) =

(44)

γ−i∞

Let us now apply this procedure to a composition of two left-hand sided Erdélyi– Kober fractional integrals: γ+δ,α (Iβ

∘

γ,δ Iβ f )(x)

γ+i∞

Γ(1 + γ + δ − s/β) Γ(1 + γ − s/β) ∗ 1 f (s)x−s ds = ∫ 2πi Γ(1 + γ + δ + α − s/β) Γ(1 + γ + δ − s/β) γ−i∞

γ+i∞

=

Γ(1 + γ − s/β) 1 γ,δ+α f )(x). f ∗ (s)x−s ds = (Iβ ∫ 2πi Γ(1 + γ + δ + α − s/β) γ−i∞

Thus we easily derive the well-known semigroup property for the left-hand sided Erdélyi–Kober fractional integrals: γ+δ,α

Iβ

γ,δ

∘ Iβ = I γ,δ+α .

To derive a formula for the left-hand sided Erdélyi–Kober fractional derivative, let us γ,δ denote (Iβ f )(x) by g(x). The representation (42) leads to the following formula for the Mellin integral transform of g: g ∗ (s) =

Γ(1 + γ − s/β) ∗ f (s). Γ(1 + γ + δ − s/β)

The last equation can be easily solved with respect to f ∗ : f ∗ (s) =

Γ(1 + γ + δ − s/β) ∗ g (s). Γ(1 + γ − s/β)

(45)

Thus we obtained the Mellin transform of the left-hand sided Erdélyi–Kober fractional derivative. To represent it in the time domain, let us first rewrite (45) in the form f ∗ (s) = with n ∈ ℕ, n − 1 < δ ≤ n.

Γ(1 + γ + δ − s/β) Γ(1 + γ + n − s/β) ∗ g (s) Γ(1 + γ + n − s/β) Γ(1 + γ − s/β)

(46)

Applications of the Mellin integral transform technique in fractional calculus | 207

The representation (44) for a composition of two operators of the Mellin convolution type, along with formulas (13) and (42), leads to the inversion of the Mellin transform (46) in the form n 1 d γ+δ,n−δ γ,δ g)(x), f (x) = (Dβ g)(x) = ∏(γ + j + x )(Iβ β dx j=1

n ∈ ℕ, n − 1 < δ ≤ n.

(47)

γ,δ

The operator Dβ is called the left-hand sided Erdélyi–Kober fractional derivative [18, 49]. Of course, one can apply the same method for inversion of the right-hand sided Erdélyi–Kober fractional integral Kβτ,α . In this case, we apply formulas (14) and (43) and get n−1 1 d f (x) = (Pβτ,α g)(x) = ∏(τ + j − x )(Kβτ+α,n−α g)(x) β dx j=0

(48)

with n ∈ ℕ, n − 1 < α ≤ n. An advantage of our approach is that not only the Erdélyi–Kober fractional integrals but also the Erdélyi–Kober fractional derivatives can be represented in the form of the Mellin–Barnes integrals (see the derivations above): γ,δ (Dβ f )(x)

γ+i∞

Γ(1 + γ + δ − s/β) ∗ 1 = f (s)x−s ds, ∫ 2πi Γ(1 + γ − s/β)

(Pβτ,α )f (x)

(49)

γ−i∞

γ+i∞

Γ(τ + α + s/β) ∗ 1 f (s)x−s ds. = ∫ 2πi Γ(τ + s/β)

(50)

γ−i∞

Moreover, the representations of the Erdélyi–Kober fractional integrals and derivatives in form of the Mellin–Barnes integrals can be used among other things for derivation of the closed form formulas for their convolutions and the Leibniz-type formulas for the fractional derivatives of the products of two functions. For details we refer the reader to [49].

5 More advanced applications of the Mellin transform in FC 5.1 Post–Widder inversion formulas for convolution type integral transforms We start this section with an example of application of the Mellin transform technique for derivation of a new formula for inversion of the Laplace transform of the Post– Widder type [30].

208 | Yu. Luchko and V. Kiryakova Our starting point is a simple consequence from formula (3) for the Gammafunction in the form 1 (1 − s)(2 − s) ⋅ ⋅ ⋅ (n − s) , = lim Γ(1 − s) n→∞ n!n−s

s∈ℂ

(51)

and the representation (32) of the Laplace transform (31) as a Mellin–Barnes integral that we repeat here with a slightly different notation: γ+i∞

F(p) = L{f (t); p} =

1 ∫ f ∗ (1 − s)Γ(s)p−s ds. 2πi

(52)

γ−i∞

Application of the Mellin integral transform to (52) leads to the formula F ∗ (s) = f ∗ (1 − s)Γ(s),

(53)

which connects the Mellin transforms of a function f and its Laplace transform F. This relation can be rewritten in the form f ∗ (s) =

F ∗ (1 − s) , Γ(1 − s)

(54)

which can be interpreted as an inversion of the Laplace transform F of the function f in the Mellin integral transform domain. To get the original of this inversion, the inverse Mellin integral transform has to be applied to the relation (54). First we employ the representation (51) and rewrite (54) in the form f ∗ (s) =

(1 − s)(2 − s) ⋅ ⋅ ⋅ (n − s) ∗ F ∗ (1 − s) = lim F (1 − s). n→∞ Γ(1 − s) n!n−s

(55)

Let us now introduce an auxiliary function H that satisfies the relation H ∗ (s − n) =

F ∗ (1 − s) . n−s

(56)

Employing formula (12) and applying the inverse Mellin integral transform to formula (55), we arrive at the representation 1 dn H(t) n→∞ n! dt n

f (t) = lim

(57)

of the inverse Laplace transform in terms of the auxiliary function H. Now let us connect the function H with the Laplace transform F of f . It follows from equation (56) that H ∗ (s) =

F ∗ (1 − n − s) . n−n−s

Applications of the Mellin integral transform technique in fractional calculus | 209

The inverse Mellin integral transform of the last relation leads to the formula γ+i∞

γ+i∞

γ−i∞

γ−i∞

1 1 F ∗ (1 − n − s) −s H(t) = t ds. ∫ H ∗ (s)t −s ds = ∫ 2πi 2πi n−n−s Now we employ the variables substitution s1 = 1 − n − s in the integral at the right-hand side of the last representation and get the desired relation: γ+i∞

−s1

n nt n−1 H(t) = ∫ F ∗ (s1 )( ) 2πi t γ−i∞

n ds1 = nt n−1 F( ). t

Inserting this representation into formula (57), we obtain a new Post–Widder type formula for the inverse Laplace transform, namely, dn n−1 n 1 (t F( )). n→∞ (n − 1)! dt n t

f (t) = lim

(58)

The technique for derivation of the real inversion formulas of Post–Widder type that was illustrated above can also be applied for other integral transforms of the Mellin convolution type. In particular, we refer to [1, 9, 22], and [49], where the cases of the Obrechkoff and the generalized Obrechkoff transforms have been considered.

5.2 Completely monotone functions and pdfs Completely monotone functions and pdfs play a very essential role in different areas of FC. Several new pdfs in terms of the Mittag-Leffler function and the Wright function were introduced in the literature and actively used nowadays especially in connection with the stochastic processes employed for microscopic justification of the FC models of anomalous diffusion processes. The basic property of the completely monotone functions is given by the Bernstein theorem [37, 44], claiming that a function φ : (0, ∞) → ℝ is completely monotone if and only if it can be represented as the Laplace transform of a non-negative measure (non-negative function or generalized function). Because the Laplace transform is a Mellin convolution type integral transform (see Section 3), the technique of the Mellin transform can be applied for studying the completely monotone functions. In this subsection, we discuss some known and new examples of the completely monotone functions and pdfs given in terms of the FC special function. For the reader’s convenience, we recall the definition of the completely monotone functions: A non-negative function ϕ : (0, ∞) → ℝ is called a completely monotone function if it is of class C ∞ (0, ∞) and (−1)n ϕ(n) (λ) ≥ 0 for all n ∈ ℕ and λ > 0. The well-known examples of completely monotone functions are the exponential α function e−aλ , a ≥ 0, α ≤ 1 and the Mittag-Leffler function Eα,β (−λ), 0 < α ≤ 1, α ≤ β.

210 | Yu. Luchko and V. Kiryakova Let us suppose that the representation ∞

ϕ(λ) = ∫ e−λt Φ(t) dt,

λ>0

(59)

0

is valid for a non-negative function Φ with a known Mellin transform. According to the Bernstein theorem, the function ϕ is completely monotone and its Mellin transform is given by the formula (see (53)) ϕ∗ (s) = Φ∗ (1 − s)Γ(s),

(60)

which can be transformed to the form Φ∗ (s) =

ϕ∗ (1 − s) . Γ(1 − s)

(61)

Let us now consider the function Φγ,β (t) = t γ Φ(t −β ), which is evidently non-negative for any γ, β ∈ ℝ. Then the Bernstein theorem ensures that the function ∞

ϕγ,β (λ) = ∫ e−λt Φγ,β (t) dt,

λ > 0,

(62)

0

is completely monotone. Now we represent ϕγ,β in terms of the Mellin–Barnes integrals. From the relation (60) it follows that ϕ∗γ,β (s) = Γ(s)Φ∗γ,β (1 − s).

(63)

Using the operational rules (10)–(11) for the Mellin integral transform, the Mellin transform of Φγ,β (t) = t γ Φ(t −β ) can be written in the form Φ∗γ,β (s) =

1 ∗ γ s Φ (− − ). |β| β β

Thus we get the following formula for the Mellin integral transform ϕ∗γ,β (s) defined by (63): ϕ∗γ,β (s) =

1 s 1+γ Γ(s)Φ∗ ( − ). |β| β β

The completely monotone function ϕγ,β can then be represented as the Mellin–Barnes integral (inverse Mellin integral transform of ϕ∗γ,β (s)): γ+i∞

ϕγ,β (λ) =

1 s 1 + γ −s 1 Γ(s)Φ∗ ( − )λ ds. ∫ 2πi |β| β β γ−i∞

(64)

Applications of the Mellin integral transform technique in fractional calculus | 211

In many cases the function ϕ (and thus the function Φ) is a particular case of the Fox H-function and then Φ∗ is represented as a quotient of products of the Gammafunctions. This means that the new completely monotone function ϕγ,β given by (64) is a particular case of the H-function, too. Formula (64) can be thus employed for a derivation of the new completely monotone functions from the known ones. Let us demonstrate this procedure on a simple example. It is known that the exponential function ϕ(λ) = exp(−λα ), 0 < α ≤ 1 is completely monotone. Its Mellin integral transform is given by the formula (see (16)) α

M

e−λ ↔

1 Γ(s/α), α

ℜ(s/α) > 0.

The function Φ∗ from (61) has the form Φ∗ (s) =

1 s ϕ∗ (1 − s) 1 Γ( α − α ) = . Γ(1 − s) α Γ(1 − s)

The representation (64) of a new completely monotone function is as follows: γ+i∞

ϕγ,β (λ) =

β+γ+1

s 1 1 Γ(s)Γ( αβ − αβ ) −s λ ds. ∫ 2πi α|β| Γ( β+γ+1 − s ) β β γ−i∞

The function (65) is a particular case of the Fox H-function. In the case β > be represented as the convergent series [30, 34]

(65) 1 α

− 1, it can

β+γ+1

k 1 ∞ Γ( αβ + αβ ) (−λ)k . ϕγ,β (λ) = ∑ α|β| k=0 k!Γ( β+γ+1 + k ) β

(66)

β

We can easily recognize the last series as a particular case of the generalized Wright hypergeometric function defined by the series p

∞ ∏ Γ(ai + Ai k) z k (a1 , A1 ), . . . , (ap , Ap ) ; z] = ∑ i=1 p Ψq [ q k! (b1 , B1 ) . . . (bq , Bq ) k=0 ∏i=1 Γ(bi + Bi k)

(67)

for the z-values where the series converges, and by the analytic continuation of this series for other z-values. Thus, we have proved that the generalized Wright function ϕγ,β (λ) =

β+γ+1

1 ) ( αβ , αβ 1 [ ; −λ] Ψ 1 1 β+γ+1 1 α|β| ( , ) β β ] [

(68)

is completely monotone under the conditions 0 < α ≤ 1, α1 − 1 < β. In particular, let us set the following parameter values: β = α1 , γ = − α1 . Then the series (66) (and thus the function (68)) takes the form ∞ Γ(1 + k) (−λ)k (−λ)k = ∑ , k!Γ(α + αk) Γ(α + αk) k=0 k=0 ∞

ϕγ,β (λ) = ∑

(69)

212 | Yu. Luchko and V. Kiryakova which defines the Mittag-Leffler function Eα,α (−λ), known to be completely monotone for 0 < α ≤ 1. Another simple but important observation from the discussions presented above is that the Mellin integral transforms of the non-negative and completely monotone functions are connected by formulas (60) and (61). If a function ϕ is completely monotone then the function Φ with the Mellin integral transform given by formula (61) is non-negative. Reversely, if a function Φ is non-negative then the function ϕ with the Mellin integral transform given by formula (60) is completely monotone. Let us illustrate this observation by an example. We start with a pair of FC special functions, namely, with the Mittag-Leffler function Eα,β (−t) and the Wright function Wa,μ (−t). As already mentioned, the function ϕ(λ) = Eα,β (−λ) is completely monotone provided the conditions 0 < α ≤ 1, α ≤ β are fulfilled. The Mellin integral transform of the function ϕ is given by (19). Then the function Φ with the Mellin integral transform Φ∗ (s) =

ϕ∗ (1 − s) Γ(1 − s)Γ(s) Γ(s) = = Γ(1 − s) Γ(1 − s)Γ(β − α + αs) Γ(β − α + αs)

is non-negative. Comparing this formula with (20), we conclude that the inverse Mellin integral transform of Φ∗ (s) is the Wright function Wβ−α,−α (−t). Thus it is non-negative under the conditions 0 < α < 1, α ≤ β, i. e., Wβ−α,−α (−t) ≥ 0,

t > 0, 0 < α < 1, α ≤ β.

(70)

Let us now check that the function pα,β (t) = Γ(β)Wβ−α,−α (−t) is a pdf on ℝ+ . Indeed, it is non-negative because of (70). To calculate the integral of pα,β over ℝ+ let us mention that it can be interpreted as the Mellin integral transform of pα,β at the point s = 1. Formula (20) leads now to the following chain of equalities: ∞

∞

∫ pα,β (t) dt = ∫ Γ(β)Wβ−α,−α (−t) dt = 0

0

Γ(β)Γ(s) 󵄨󵄨󵄨󵄨 Γ(β) = 1. 󵄨󵄨 = 󵄨 Γ(β − α + αs) 󵄨s=1 Γ(β)

In the second example, we verify that the function, defined in terms of the Mellin– Barnes integral, γ+i∞

Γ( α2 − α2 s)Γ(1 − α2 + α2 s) −s 2 1 t ds Φα,β (t) = ∫ 2β 2β α 2πi Γ(1 − α + α s)Γ(1 − s) γ−i∞

(71)

can be interpreted as a pdf on ℝ+ for 0 < β ≤ 1 and 0 < α ≤ 2 when α + 2β < 4. According to the general theory of Mellin–Barnes integrals [34], the Mellin– Barnes integral (71) exists for α2 − 1 < ℜ(s) < α2 under the conditions 0 < β, 0 < α, and α + 2β < 4 and its Mellin transform can be calculated as follows: Φ∗α,β (s) =

2 2 2 2 2 Γ( α − α s)Γ(1 − α + α s) . α Γ(1 − 2β + 2β s)Γ(1 − s) α

α

(72)

Applications of the Mellin integral transform technique in fractional calculus | 213

Now we construct the function ϕ∗ (s) given by formula (60): ϕ∗ (s) = Γ(s)Φ∗α,β (1 − s) = Γ(s)

2 2 2 2 2 Γ( α s)Γ(1 − α s) 2 Γ( α s)Γ(1 − α s) = . α Γ(1 − 2β s)Γ(s) α Γ(1 − 2β s) α α

(73)

The function ϕ = ϕ(λ) can be then represented as the following Mellin–Barnes integral: γ+i∞

ϕ(λ) =

2 2 1 2 Γ( α s)Γ(1 − α s) −s λ ds. ∫ 2πi α Γ(1 − 2β s) α

γ−i∞

The variable substitution α2 s → s leads to the formula γ+i∞

ϕ(λ) =

Γ(s)Γ(1 − s) α2 −s 1 (λ ) ds. ∫ 2πi Γ(1 − βs) γ−i∞

Comparing this formula with the Mellin transform (19) of the Mittag-Leffler function, we arrive at the representation α

α

ϕ(λ) = Eβ,1 (−λ 2 ) = Eβ (−λ 2 ).

(74)

The Mittag-Leffler function f (λ) = Eβ (−λ) is known to be completely monotone for 0 < β ≤ 1. Thus for α = 2 the function ϕ(λ) defined by (74) is completely monotone. α Now let α satisfy the inequalities 0 < α < 2. Then the function g(λ) = λ 2 is a Bernstein α function because its derivative g 󸀠 (λ) = α2 λ 2 −1 is completely monotone. But a composition of a completely monotone function and a Bernstein function is completely monotone [44]. Thus the function ϕ(λ) = f (g(λ)) is completely monotone for 0 < α < 2, too. Because ϕ∗ (s) and Φ∗α,β (s) are connected by formula (60) and the function ϕ is completely monotone, it follows now that Φα,β (t) is non-negative, i. e., Φα,β (t) ≥ 0,

t > 0, 0 < β ≤ 1, 0 < α ≤ 2, α + 2β < 4.

To evaluate the integral of Φα,β (t) over ℝ+ we again use the technique of the Mellin integral transform: ∞

∫ Φα,β (t) dt = lim 0

s→1

2 2 2 Γ( α2 (1 − s)) 2 Γ( α (1 − s))Γ(1 − α + α s) 2 = lim = 1. α Γ(1 − 2β + 2β s)Γ(1 − s) α s→1 Γ(1 − s) α

α

5.3 Subordination formulas for the fractional diffusion equations Subordination principles play an important role in the theory of the fractional partial differential equations. From the analytical viewpoint, they make it possible to express

214 | Yu. Luchko and V. Kiryakova a solution of the Cauchy problem for an equation of this kind via a solution of an equation of a higher order. A probabilistic interpretation of the subordination principles is in the representation of the stochastic processes related to the fractional diffusion equations via an appropriate random time change in the Brownian motion. In [42], a subordination principle for completely positive measures was applied for constructing new resolvents for the abstract Volterra integral equations based on the known ones. In [2] and [5], this subordination principle was discussed for the case of the abstract fractional evolution equations in the form Dβ u(t) = Au(t)

(75)

subject to the initial conditions u(k) (0) = 0,

u(0) = x,

k = 0, . . . , n − 1, x ∈ X,

(76)

where Dβ is the Caputo fractional derivative of order β, n − 1 < β ≤ n, n ∈ ℕ, and A is a linear closed unbounded operator densely defined in a Banach space X. If we denote by Sβ (t) a solution operator to the abstract initial-value problem (75)–(76), then the subordination formula ∞

Sβ (t)x = ∫ t −γ W1−γ,−γ (−st −γ )Sδ (s)x ds,

t > 0, x ∈ X,

(77)

0

with 0 < β < δ ≤ 2, γ = β/δ is valid under some conditions on the operator A (see [2] and [5] for details). The function W1−γ,−γ (−τ) from (77) is a special case of the Wright function, it is non-negative for τ ∈ ℝ+ and can be interpreted as a probability density function. In this subsection, we discuss how the subordination formula (77) with the fracα tional Laplacian operator A = −(−Δ) 2 can be derived using the technique of the Mellin integral transform. For derivation of other subordination formulas for the equations of type (75) we refer to [33] in the one-dimensional case and to [28] in the multi-dimensional case. Following [28], we consider the linear multi-dimensional space-time fractional diffusion equation in the following form: α

β

Dt u(x, t) = −(−Δ) 2 u(x, t),

x ∈ ℝn , t > 0, 0 < α ≤ 2, 0 < β ≤ 2.

(78)

β

In (78), Dt denotes the Caputo time fractional derivative of order β, β > 0, defined by the formula β

n n−β 𝜕 u )(t), 𝜕t n

Dt u(x, t) = (It

n − 1 < β ≤ n, n ∈ ℕ,

(79)

Applications of the Mellin integral transform technique in fractional calculus | 215

γ

where It is the Riemann–Liouville fractional integral: t

1 ∫ (t − τ)γ−1 u(x, τ) dτ γ (It u)(t) = { Γ(γ) 0 u(x, t)

for γ > 0, for γ = 0.

α

The fractional Laplacian (−Δ) 2 from equation (78) is defined as a pseudo-differential operator with the symbol |κ|α [43]: α

(ℱ (−Δ) 2 f )(κ) = |κ|α (ℱ f )(κ),

(80)

where (ℱ f )(κ) is the Fourier transform of a function f at the point κ ∈ ℝn defined by (ℱ f )(κ) = f ̂(κ) = ∫ eiκ⋅x f (x) dx.

(81)

ℝn

In this subsection, we consider the Cauchy problem for the space-time fractional diffusion equation (78) with the Dirichlet initial conditions: u(x, 0) = φ(x),

x ∈ ℝn ,

(82)

if the order β of the time fractional derivative satisfies the condition 0 < β ≤ 1 or 𝜕u (x, 0) = 0, 𝜕t

u(x, 0) = φ(x),

x ∈ ℝn ,

(83)

if 1 < β ≤ 2. Because the initial-value problem (78), (82) or (78), (83), respectively, is a linear one, its solution can be represented in the form u(x, t) = ∫ Gα,β,n (ζ , t)φ(x − ζ ) dζ ,

(84)

ℝn

where Gα,β,n is the so-called first fundamental solution to the fractional diffusionwave equation (78) (which corresponds to the initial condition in the form of the Dirac δ-function) and the function φ is determined by the initial condition. Thus the behavior of the solutions to the problem (78), (82) or (78), (83), respectively, is determined by the fundamental solution Gα,β,n and the focus of this subsection is on the derivation of some subordination formulas for the fundamental solution Gα,β,n in the form ∞

Gα,β,n (x, t) = ∫ Φ(s, t)Gα,̂ β,n ̂ (x, s) ds,

(85)

0

where the kernel function Φ = Φ(s, t) can be interpreted as a probability density function in s, s ∈ ℝ+ , for each value of t, t > 0.

216 | Yu. Luchko and V. Kiryakova The subordination formulas for Gα,β,n can be deduced based on the Mellin–Barnes representations of the fundamental solution that were derived in [24, 25] in the case β = α, in [7] in the case β = α/2, and in [6, 27] in the general case. It is worth mentioning that the main tool employed in all these publications was the Mellin integral transform technique like the one we illustrated in Section 2 for the evaluation of improper integrals. The Mellin–Barnes integral we employ in further derivations has the form γ+i∞

βn

Gα,β,n (x, t) =

1 t− α 1 ∫ Kα,β,n (s)z −s ds, n α (4π) 2 2πi

z=

γ−i∞

|x| β

2t α

(86)

,

with Kα,β,n (s) =

Γ( 2s )Γ( αn − αs )Γ(1 − β

Γ(1 − α n +

n + αs ) α . β n s s)Γ( − ) α 2 2

(87)

For β = 1, we get the fundamental solution of the space-fractional diffusion equation in the form γ+i∞

n

1 t− α 1 Gα,1,n (x, t) = ∫ Kα,1,n (s)z −s ds, n α (4π) 2 2πi γ−i∞

z=

|x| 1

2t α

(88)

,

with Kα,1,n (s) =

Γ( 2s )Γ( αn − αs ) Γ( n2 − 2s )

(89)

.

The key point for derivation of a subordination formula for Gα,β,n with 0 < β < 1 is in the observation that the kernel function Kα,β,n in the Mellin–Barnes integral (86) can be represented as a product of two factors: Kα,β,n (s) = Kα,1,n (s) × Φ∗α,β,n (s),

(90)

where Kα,1,n (s) is the kernel function in the Mellin–Barnes integral (88) for the fundamental solution Gα,1,n and Φ∗α,β,n (s) =

Γ(1 − β

n α

+ αs ) β

Γ(1 − α n + α s)

(91)

.

Due to the convolution formula (7) for the Mellin transform, the product formula (90) in the Mellin domain leads to an integral representation of Gα,β,n in the form βn

∞

1 t− α z dτ ̃ Gα,β,n (x, t) = , n ∫ Φα,β,n (τ)Gα,1,n ( ) α (4π) 2 τ τ 0

z=

|x| β

2t α

,

(92)

Applications of the Mellin integral transform technique in fractional calculus | 217

where Φα,β,n (τ) is the inverse Mellin integral transform of Φ∗α,β,n (s) given by (91) and γ+i∞

1 G̃ α,1,n (τ) = ∫ Kα,1,n (s)τ−s ds 2πi

(93)

γ−i∞

is a slightly modified fundamental solution Gα,1,n : n

Gα,1,n (x, t) =

1 t− α ̃ (z), n G α (4π) 2 α,1,n

z=

|x| 1

2t α

.

(94)

Formula (92) is a subordination formula for the fundamental solution Gα,β,n and now we put it into the standard form. To do this, let us derive an explicit representation for the kernel function Φα,β,n (τ) that is defined as the Mellin–Barnes integral (under the condition 0 < β < 1) γ+i∞

γ+i∞

Γ(1 − αn + αs ) −s 1 1 Φα,β,n (τ) = τ ds. ∫ Φ∗α,β,n (s)τ−s ds = ∫ β β 2πi 2πi Γ(1 − α n + α s) γ−i∞ γ−i∞

(95)

The general theory of the Mellin–Barnes integrals [34] says that the contour of integration in the integral at the right-hand side of (88) can be transformed to the loop L−∞ starting and ending at −∞ and encircling all poles of the function Γ(1 − αn + αs ). Taking into account the Jordan lemma and the formula for the residuals of the Gammafunction, the Cauchy residue theorem leads to the following series representation of Φα,β,n (for details we refer the reader to [6] or [27]): α(−1)k 1 k+1− αn , (τα ) k! Γ(1 − β − βk) k=0 ∞

Φα,β,n (τ) = ∑

(96)

which can be recognized to be a special case of the Wright function: Φα,β,n (τ) = ατα−n W1−β,−β (−τα ),

0 < β < 1.

(97)

Putting now formulas (94) and (97) into the integral representation (92) and substituting the variables τα → τ, we first get the formula ∞

Gα,β,n (x, t) = ∫ W1−β,−β (−τ)Gα,1,n (x, t β τ) dτ,

(98)

0

which can be transformed into the well-known subordination formula (see (77) with δ = 1) ∞

Gα,β,n (x, t) = ∫ t −β W1−β,−β (−st −β )Gα,1,n (x, s) ds, 0

by the variable substitution t β τ → s.

0 1 2 {{ π k=0

(101) if

β α

= 21 .

In formula (101), W(a,μ),(b,ν) stands for the four parameter Wright function in the form zk , Γ(a + μk)Γ(b + νk) k=0 ∞

W(a,μ),(b,ν) (z) = ∑

μ, ν ∈ ℝ, a, b, z ∈ ℂ.

(102)

This function was introduced in [48] for positive values of the parameters μ and ν. When a = μ = 1 or b = ν = 1, respectively, the four parameter Wright function is reduced to the Wright function (23). In [29], the four parameter Wright function was investigated in the case when one of the parameters μ or ν is negative. In particular, it was proved there that the function W(a,μ),(b,ν) (z) is an entire function provided that 0 < μ + ν, a, b ∈ ℂ.

5.4 Applications to dual integral equations There are many techniques for solving mixed boundary value problems (BVPs) arising in mathematical physics. One of the approaches to such problems is to reduce them to the dual integral equations, that is, to a pair of equations in the form ∞

∫ ω(u)K(x, u)f (u)du = φ(x),

0 < x < 1,

0 ∞

∫ K(x, u)f (u)du = ψ(x), 0

(103) x > 1,

Applications of the Mellin integral transform technique in fractional calculus | 219

where the kernel K(x, u), the weight ω(u) and the BV conditions φ(x), ψ(x) are known functions and f (u) is to be determined. A well-known example of dual integral equations is a pair of equations with the Bessel function K(x, u) = Jν (xu) and ω(u) = uλ . In particular, for ν = 0 and λ = −1 these are the so-called Titchmarsh equations. In view of the comments given in Section 3, a rather general class of the dual integral equations corresponds to the case when the kernels are taken in form of the Fox H-functions (24)–(25) 󵄨󵄨 n+p 󵄨 (α , a ) m,n (xu 󵄨󵄨󵄨󵄨 j j 1m+q) , K1 (x, u) = ω(u)K(x, u) = Hp+n,q+m 󵄨󵄨 (βk , bk )1 󵄨󵄨 n+p 󵄨 (ρ , r ) m,n (xu 󵄨󵄨󵄨󵄨 j j 1m+q) , K2 (x, u) = K(x, u) = Hp+n,q+m 󵄨󵄨 (υk , vk )1

(104)

and the role of the weight function ω(u) is replaced by the fact that these H-functions have different parameters (see the property [36, (1.2.4)], [45, (2.3.6)], [18, (E.9)] of the H-function). In this way, by any specific choice of the orders and parameters of the H-functions, we can obtain the solutions of dual integral equations with various special functions in the kernels. Applying the theory of the Mellin transform, namely, the Mellin images (21) of the H-functions and the Parseval equality (8) as well as the operators of the generalized fractional calculus (compositions of the Erdélyi–Kober integrals (36),(37)), we can derive explicit solutions of the dual integral equations with the kernels in the form of (104) and thus of a wide range of equations (103). To simplify and shorten the notations, in this subsection we restrict ourselves to the case of the kernel functions being two Meijer’s G-functions that we get by setting aj = rj = 1, j = 1, . . . , n+p and bk = vk = 1, k = 1, . . . , m + q in the above H-functions. Then, consider the pair of dual integral equations ∞

󵄨󵄨 α1 , . . . , αn+p

󵄨 m,n (xu 󵄨󵄨󵄨󵄨 ∫ Gp+n,q+m 0 ∞

󵄨󵄨 β1 , . . . , βm+q

) f (u)du = ∫ G1 (xu)f (u)du = φ(x),

0 < x < 1,

0 ∞

󵄨󵄨 ρ1 , . . . , ρn+p ) f (u)du = ∫ G2 (xu)f (u)du = ψ(x), 󵄨󵄨 υ1 , . . . , υm+q

󵄨 m,n (xu 󵄨󵄨󵄨󵄨 ∫ Gp+n,q+m 0

∞

(105) x > 1,

0

whose kernels are Meijer’s G-functions (28) with the Mellin images (29), briefly denoted, respectively, Gi (x) =

Γ1,i Γ2,i 1 , ∫ χi (s)x −s ds and χi := 2πi Γ3,i Γ4,i

i = 1, 2,

(106)

L

the Γ#,i standing for the corresponding product of Γ-functions in (29). Applying the Mellin transform to both equations in (105), we reduce them to a pair of equations

220 | Yu. Luchko and V. Kiryakova whose kernels are the Mellin transforms χ1 , χ2 and, by the Parseval equality, they can be represented in the form 1 ∫ χ1 (s)x−s F(1 − s)ds = φ(x), 2πi

0 < x < 1,

1 ∫ χ2 (s)x−s F(1 − s)ds = ψ(x), 2πi

x > 1,

L

(107)

L

with a new unknown function F(s) = M{f (u); s}. Now the main idea is to transform each of the kernels χi (s), i = 1, 2, of the above equations to a common kernel χ(s) of the same structure (involving four groups of Γ-functions). To this end, a composition of suitable left-hand sided EK fractional integrals (36) is used as a transmutation operator I to transform the quotient Γ2,1 /Γ3,1 of Γ-functions with the arguments containing (−s) for χ1 (s) into the corresponding quotient Γ2 /Γ3 := Γ2,2 /Γ3,2 for χ2 (s). Similarly, a transmutation operator K in the form of a composition of right-hand sided EK operators (37) is used to transform the quotient Γ1,2 /Γ4,2 with the arguments containing (+s) for χ2 into Γ1 /Γ4 := Γ1,1 /Γ4,1 for χ1 (s). As a result, the pair of equations (107) is transformed to a pair of equations with the common kernel χ(s) = ΓΓ1 ΓΓ2 . 3 4 As proved in earlier work [17, Th. 1], [18, Lemma 5.3.2, Th. 1.2.10], the needed transmutation operator I is the left-hand sided generalized fractional integral n+q

γ ,δk

Iφ(x) = [∏ I1 k k=1

(γ )n+q ,(δk )n+q 1

k 1 ]φ(x) := I1,n+q

φ(x),

(108)

which is a composition of the EK left-hand sided integrals (36) with parameters γk = −ρk , δk = ρk − αk , k = 1, . . . , n; γn+k = −βm+k , δn+k = βm+k − υm+k , k = 1, . . . , q. According to the theory of generalized fractional calculus [18] and using the Mellin transform’s tools, this operator also has an explicit representation by a single integral involving a n+q,0 -function: Gn+q,n+q 1 󵄨󵄨 q 󵄨 (−αk )n1 ; (−υm+k )1 n+q,0 Iφ(x) = ∫ Gn+q,n+q (u 󵄨󵄨󵄨󵄨 q ) φ(xu)du. n 󵄨󵄨 (−ρk )1 ; (−βk )1 0

(109)

As mentioned above, the operator I reduces the first of equations in (107) into an integral equation of the same form but with the kernel function χ(s) =

∏m Γ(βk + s) ∏nk=1 Γ(1 − ρk − s) Γ1 Γ2 = qk=1 , Γ3 Γ4 ∏k=1 Γ(1 − υk − s) ∏pk=1 Γ(αk + s)

(110)

namely, into the equation 1 ∫ χ(s)x−s F(1 − s)ds = Iφ(x), 2πi L

0 < x < 1.

(111)

Applications of the Mellin integral transform technique in fractional calculus | 221

Similarly, a transmutation operator K defined as a composition of suitable right-hand sided EK fractional integrals (37) and representable by a single integral with limits 1 and ∞, is applied to transform the second of equations in (107) into an integral equation with the same kernel function χ(s) as in (110) [18, Lemma 5.3.3]: 1 ∫ χ(s)x−s F(1 − s)ds = Kψ(x), 2πi

x > 1.

(112)

L

Therefore, the pair of the dual integral equations (107) is reduced to a single integral equation with the common kernel χ(s) as in (110): 1 ∫ χ(s)x−s F(1 − s)ds = g(x), 0 < x < ∞, 2πi

and g(x) = {

L

Iφ(x), Kψ(x),

0 < x < 1, x > 1.

(113)

Its solution can easily be determined by the Mellin transform techniques: f (x) =

x−s G(1 − s) 1 ds, ∫ 2πi χ(1 − s)

(114)

L

where f = f (x) is the original unknown function and G(s) = M{g(x); s}. To get a representation of the solution f via the BV functions φ(x) and ψ(x), let us denote 1/χ(1 − s) by ℋ(s). Then its original h(x) is represented in the form 󵄨󵄨 p 󵄨 (−αn+k )k=1 ; (−ρk )nk=1 q,p ). (x 󵄨󵄨󵄨󵄨 h(x) = M−1 {ℋ(s); x} = Gn+p,m+q q 󵄨󵄨 (υm+k )k=1 ; (−βk )m k=1

(115)

According to the Parseval equality, the solution f of the original pair of dual integral equations (105) then has the form 1

∞

∞

f (x) = ∫ h(xu)g(u)du = ∫ h(xu)Iφ(u)du + ∫ h(xu)Kψ(u)du. 0

0

1

Replacing h(x) by the G-function (115) and the generalized fractional integrals Iφ(u) and Kψ(u) by their single integral representations ((109) for Iφ(u) and similar one for Kψ(u)), the above formula gives the explicit solution of (105): 1

󵄨󵄨 p 󵄨 (−αn+k )1 ; (−ρk )n1 q,p ) du f (x) = ∫ Gn+p,m+q (xu 󵄨󵄨󵄨󵄨 q 󵄨󵄨 (−υm+k )1 ; (−βk )m 1 0

1 󵄨󵄨 q 󵄨 (−αk )n1 ; (−υm+k )1 n+q,0 × ∫ Gn+q,n+q (v 󵄨󵄨󵄨󵄨 q ) φ(uv)dv 󵄨󵄨 (−ρk )n1 ; (−βm+k )1 0 1

󵄨󵄨 p 󵄨 (−αn+k )1 ; (−ρk )n1 q,p + ∫ Gn+p,m+q ) du (xu 󵄨󵄨󵄨󵄨 q 󵄨󵄨 (−υm+k )1 ; (−βk )m 1 0

222 | Yu. Luchko and V. Kiryakova ∞ 󵄨 p 1 󵄨󵄨 (υ + 1)m m+p,0 1 ; (αn+k + 1)1 ( 󵄨󵄨󵄨󵄨 k × ∫ Gm+p,m+p p ) ψ(uv)dv. v 󵄨󵄨 (βk + 1)m 1 ; (ρn+k )1 1

(116)

Let us note that the idea of applying the Mellin integral transform for deriving explicit formulas for solutions to the dual integral equations was suggested in [35]. Because no explicit representations of the transmutation operators were employed in [35], no closed form solutions like those presented above could be derived there. The solutions of the dual integral equations with the H-functions as in (104) were first deduced in [12]. For the details on functional spaces, conditions on the parameters, and derivation of the formulas we refer the interested reader to [12, 17], [18, Ch.5, § 5.3].

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[41] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Evaluation of Integrals and the Mellin Transforms, EN transl. from: Itogi Nauki i Tekhniki, Ser. Mat. Analiz 27 (1989), 3–146, Plenum Publ. Corp., 1991. [42] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. [43] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. [44] R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions. Theory and Applications, De Gruyter, Berlin, 2010. [45] H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982. [46] H. M. Srivastava, S. B. Yakubovich, and Yu. F. Luchko, New families of Leibniz type rules for fractional calculus and their integral analogues, in R. N. Kalia (ed.), Recent Advances in Fractional Calculus, pp. 248–291, Global Publishing Co., Sauk Rapids, Minnesota, 1993. [47] E. C. Titchmarsh, Introduction to Theory of Fourier Integrals, Oxford Univ. Press, Oxford, 1937. [48] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. Lond. Math. Soc., 10 (1935), 287–293. [49] S. B. Yakubovich and Yu. F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Acad. Publ., Dordrecht, 1994.

Yuri Luchko

Fractional Fourier transform Abstract: This chapter deals with the fractional Fourier transform (FrFT) in the form introduced a little while ago by the chapter’s author and his co-authors. An advantage of this definition of the FrFT is that it is closely connected with the Fractional Calculus operators and plays the same role for the fractional derivatives as the Fourier transform does for the integer order derivatives. The chapter starts with definitions of the used fractional derivatives, their properties, and the appropriate spaces of functions. Then a survey of the basic properties of the FrFT, including an inversion formula and the operational relations for the fractional derivatives, is presented. In the last section, some applications of the FrFT for analytical treatment of the partial fractional differential equations are discussed. Keywords: fractional Fourier transform, operational relations, fractional derivatives and integrals, partial differential equations of fractional order MSC 2010: 26A33, 33E12, 42A38, 44A40, 45K05

1 Introduction The probably first attempt to introduce a fractional Fourier transform was started by Wiener in the paper [31] published as early as 1929. The main contribution of [31] was in a discussion of a relation between an expansion of a function in a series of orthogonal Hermit polynomials and its Fourier transformation whereas the introduced FrFT was just a byproduct of the used method. The FrFT introduced in [31] did not become very popular until it was rediscovered and applied in quantum mechanics in [22] and received a strong mathematical background in [20]. From the mathematical viewpoint, the FrFT considered in [20, 22, 31] and in many other publications was based on the fractional powers of the Fourier transform defined via the set of its eigenfunctions given by the Gauss–Hermite functions. The same idea of fractionalization was employed both for different systems of eigenfunctions of the Fourier transform [5] and for other integral transforms [32]. Moreover, the FrFT of this kind permits also the following interpretation useful for its applications in applied mathematics and physics, and especially in filter design, signal analysis, and pattern recognition. The conventional Fourier transform can be construed as a linear mapping from the time- to the frequency domain that are considered to be orthogonal to each other. If we imagine a

Yuri Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110571622-009

226 | Yu. Luchko signal represented along the time axis and its Fourier transform represented along the frequency axis, then the conventional Fourier transform operator can be interpreted as a change in representation of the signal corresponding to a rotation of the time axis to the frequency axis by the angle π/2. In this interpretation, the FrFT is a linear operator that corresponds to a rotation of the signal on the time-frequency plane through an angle α, α ∈ ℝ. For details and applications of the FrFTs of this kind we refer the reader to, e. g., [2, 21, 23], and [32]. The FrFT that was mentioned above corresponds to the fractional powers of the Fourier transform, but it is not convenient for dealing with the derivatives and integrals of fractional order. In [17], a new approach to definition of the FrFT based on an extension of the operational relations for the conventional Fourier transform and derivatives of integer orders was suggested. The main operational property of this FrFT denoted by ℱα , α ≥ 0 is as follows: (ℱα Dα u)(ω) = (−icα ω)(ℱα u)(ω),

(1)

where Dα is a suitably defined fractional derivative and cα is a constant depending on its order α. In contrast to the well-known operational relation [29] (ℱ Dα± u)(ω) = (∓iω)α (ℱ u)(ω),

α ≥ 0,

(2)

where ℱ is the conventional Fourier transform and Dα± are the Riemann–Liouville leftand right-hand sided fractional derivatives, the operational relation (1) for the FrFT avoids potential problems with different branches of the multi-valued complex function (∓iω)α that might appear while employing the operational relation (2). Moreover, the operational relation (1) can be applied for treating the linear fractional differential equations with the fractional derivative Dα . We mention here the results presented in [12], where the FrFT was employed for derivation of a closed form solution0 to the Cauchy problem for a partial space-time-fractional differential equation with the Caputo time-fractional derivative and a linear combination of the spatial left- and right-hand sided Riemann–Liouville fractional derivatives. In [1], the FrFT technique was employed to obtain a solution to the Cauchy problem for the fractional Schrödinger equation with the quantum Riesz–Feller derivative in terms of the Fox H-function. Finally, let us mention the recent paper [16], where some new operational relations for the FrFT were suggested and applied for deriving closed form solutions to the fractional differential equations with both left- and right-hand sided fractional derivatives without and with delays. This kind of fractional differential equations is especially important for the fractional calculus of variations, where the necessary optimality conditions of the Euler–Lagrange type are often formulated in form of some fractional differential equations that involve both the left- and the right-hand sided fractional derivatives (see, e. g., the recent book [19] and numerous references therein).

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| 227

In this chapter, we give a survey of basic results regarding the new FrFT and its applications. For details and proofs we refer the reader to [1, 12, 16], and [17].

2 Preliminaries First let us remind the reader of the definition of the conventional Fourier transform. For a function f ∈ S, S being the space of rapidly decreasing test functions on the real axis ℝ [7], the Fourier transform ℱ and the inverse Fourier transform ℱ −1 are defined as follows: +∞

f ̂(ω) = (ℱ f )(ω) = ∫ f (x)eiωx dx,

ω ∈ ℝ,

(3)

−∞ +∞

f (x) = (ℱ −1 f ̂)(x) =

1 ∫ f ̂(ω)e−iωx dω, 2π

x ∈ ℝ.

(4)

−∞

For the theory of the Fourier transform on the space S󸀠 of tempered distributions and on the spaces Lp (ℝ), 1 ≤ p ≤ 2 we refer the reader to [7] and [30], respectively. In [29], the role of the Fourier transform in the framework of fractional calculus is discussed in detail. In this chapter, we employ the Lizorkin space of functions [14, 15, 26–29] as a basic space, where the presented definitions, operational relations, and other formulas are valid. The Lizorkin space of functions can be considered as a generalization of the Sobolev spaces to the non-integer orders of derivatives and therefore it is very convenient for problems involving fractional integrals and derivatives. Otherwise, the Lizorkin-type spaces are dense in Lp [27, 28] and thus the results obtained for these spaces can be extended to the Lp -spaces following the standard procedure. Definition 1. The Lizorkin space of functions Φ(ℝ) is defined as the Fourier pre-image of V(ℝ) in the space S of rapidly decreasing test functions, i. e., Φ(ℝ) = {φ ∈ S : φ̂ ∈ V(ℝ)}, where by V(ℝ) the space of functions v ∈ S satisfying the conditions dn v 󵄨󵄨󵄨󵄨 = 0, n = 0, 1, 2, . . . 󵄨 dxn 󵄨󵄨󵄨x=0 is denoted.

According to Definition 1, a function φ ∈ Φ(ℝ) satisfies the orthogonality conditions +∞

∫ xn φ(x)dx = 0,

−∞

n = 0, 1, 2, . . . .

228 | Yu. Luchko A very useful property of the Lizorkin space Φ(ℝ) is that it is invariant with respect to the fractional integration and differentiation operators [29]. As is well known, this property does not hold true for the whole space S of the rapidly decreasing test functions because the fractional integrals and derivatives of the functions from S do not always belong to S. Now we introduce definitions of the fractional integrals and derivatives we deal with in this chapter and then provide a precise formulation of the above statement. The main results of this chapter will be formulated for the fractional derivative Dαβ that is a linear combination of the left- and right-hand sided Liouville fractional derivatives (in the further discussions we refer to Dαβ as to the Liouville fractional mixed derivative) in the form Dαβ u = (1 − β)Dα+ u − βDα− u,

(5)

where β ∈ ℝ and Dα+ u, Dα− u are the left- and right-hand sided Liouville fractional derivatives of order α, α > 0, respectively, that are defined by the formulas [29] (Dα+ u)(x) = (

n

d ) (I+n−α u)(x), dx

(Dα− u)(x) = (−

n

d ) (I−n−α u)(x), dx

x ∈ ℝ, α > 0, n = [α] + 1,

(6)

x ∈ ℝ, α > 0, n = [α] + 1,

(7)

[α] being an integral part of α ∈ ℝ. The operators I+α and I−α from formulas (6) and (7), respectively, are the left- and right-hand sided Liouville fractional integrals of order α that for α > 0 are defined by (I+α u)(x) = (I−α u)(x) =

x

u(t)dt 1 , ∫ Γ(α) (x − t)1−α

x ∈ ℝ, α > 0,

(8)

u(t)dt 1 , ∫ Γ(α) (t − x)1−α

x ∈ ℝ, α > 0.

(9)

−∞ ∞ x

If α = 0, we set I+α and I−α to be identity operators, i. e., I+0 u = u and I−0 u = u. With these definitions in mind, for α = m ∈ ℕ the Liouville fractional derivatives (6) and (7) are just usual derivatives of order m: (m) (Dm (x), + u)(x) = u

m (m) (Dm (x), − u)(x) = (−1) u

m ∈ ℕ.

(10)

If α = 1, the Liouville fractional mixed derivative Dαβ defined by (5) is the usual derivative of the first order for any β ∈ ℝ (D1β u)(x) = (1 − β)(D1+ u)(x) − β(D1− u)(x) = (1 − β)

du du du +β = . dx dx dx

The most interesting particular cases of the Liouville fractional mixed derivative Dαβ are the following ones:

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| 229

1) β = 0: Dα0 ≡ Dα+ ; 2) β = 1: Dα1 ≡ −Dα− ; 3) β = 1/2: Dα1/2 ≡ 21 (Dα+ − Dα− ). The operator Dα1/2 can be interpreted as an inversion of the fractional Riesz potential in the one-dimensional case and thus it is important for applications. The Liouville fractional integrals and derivatives I+α , I−α , Dα+ , and Dα− , respectively, are well-defined in the Lizorkin space of functions as stated in the following theorem (see [29] for a proof). Theorem 1. The space Φ(ℝ) is invariant with respect to the fractional integration operators I+α , I−α and fractional differentiation operators Dα+ , Dα− of any positive order α > 0. The fractional differentiation operators are inverse to the corresponding fractional integration operators from the left Dα+ I+α φ = φ,

Dα− I−α φ = φ,

φ ∈ Φ(ℝ), α > 0,

(11)

and they are connected by the following relations: QI+α φ = I−α Qφ,

QDα+ φ

=

Dα− Qφ,

QI−α φ = I+α Qφ,

QDα− φ

=

Dα+ Qφ,

φ ∈ Φ(ℝ), α > 0,

φ ∈ Φ(ℝ), α > 0,

(12) (13)

where Q is the reflection operator: (Qφ)(x) = φ(−x),

x ∈ ℝ.

(14)

Similar results are valid for the Liouville fractional mixed derivative Dαβ defined by (5) and the corresponding Liouville fractional mixed integral Iβα defined for α ≥ 0 and β ∈ ℝ by the relation Iβα u = (1 − β)I+α u − βI−α u,

(15)

where I+α u and I−α u are the left- and right-hand sided Liouville fractional integrals defined by (8) and (9), respectively. As a direct consequence of Theorem 1, we get the following statements. Theorem 2 ([12]). Let α > 0, β ∈ ℝ, and let Q be the reflection operator defined by (14). The Lizorkin space of functions Φ(ℝ) is invariant with respect to the Liouville fractional mixed integral Iβα and derivative Dαβ and these operators are connected with each other by the following relations: QIβα u = Iβα Qu,

u ∈ Φ(ℝ), α > 0, β ∈ ℝ,

=

u ∈ Φ(ℝ), α > 0, β ∈ ℝ.

QDαβ u

Dαβ Qu,

230 | Yu. Luchko The Lizorkin space of functions Φ(ℝ) is also invariant with respect to the Fourier transform (3) and the inverse Fourier transform (4) and these transforms are inverse to each other: ℱ ℱ u = u, −1

ℱℱ u = u, −1

u ∈ Φ(ℝ).

(16)

Another useful tool for our treatment is the well-known formula for integration by parts for the fractional Liouville derivatives that is valid for the functions u, v from the Lizorkin space Φ(ℝ) (see [29] for a proof in the space Lp ; Φ(ℝ) ⊂ Lp ): +∞

+∞

−∞

−∞

∫ v(x)(Dα+ u)(x) dx = ∫ (Dα− v)(x) u(x) dx.

(17)

In the derivation of the operational relations for the FrFT and the Liouville fractional mixed integrals and derivatives, the formulas for the fractional Liouville integrals and derivatives acting on the kernel of the Fourier transform play a very essential role. They are presented in the next two theorems ([12] and [17]). Theorem 3. Let ω ∈ ℝ (ω ≠ 0) and α > 0. Then the relations (I+α eiωt )(x) = eiωx |ω|−α (cos(

απ απ ) − i sign(ω) sin( )) 2 2

(18)

(I−α eiωt )(x) = eiωx |ω|−α (cos(

απ απ ) + i sign(ω) sin( )) 2 2

(19)

and

hold true. Theorem 4. Let ω ∈ ℝ (ω ≠ 0) and α > 0. Then the relations (Dα+ eiωt )(x) = eiωx |ω|α (cos(

απ απ ) + i sign(ω) sin( )) 2 2

(20)

(Dα− eiωt )(x) = eiωx |ω|α (cos(

απ απ ) − i sign(ω) sin( )) 2 2

(21)

and

hold true. Remark 1. Formulas (18)–(21) can be considered as an extension of the well-known formulas [11]: (I+α eλt )(x) = λ−α eλx ,

(I−α e−λt )(x) = λ−α e−λx , (Dα+ eλt )(x) = λα eλx , (Dα− e−λt )(x) = λα e−λx , to the case ℜ(λ) = 0, ℑ(λ) ≠ 0, α > 0.

ℜ(λ) > 0, α > 0, ℜ(λ) > 0, α > 0, ℜ(λ) > 0, α > 0, ℜ(λ) > 0, α > 0,

Fractional Fourier transform

| 231

3 Definition and properties of the fractional Fourier transform We start with a definition of the FrFT. Definition 2. The fractional Fourier transform fα̂ (ω) of order α (α > 0) of a function f ∈ Φ(ℝ) at a point ω ∈ ℝ is defined as follows: +∞

fα̂ (ω) = (ℱα f )(ω) = ∫ f (x)eα (ω, x) dx,

(22)

−∞

where eα (ω, x) = e

1

i sign(ω)|ω| α x

1/α

e−i|ω| x , ω ≤ 0, = { i|ω|1/α x e , ω ≥ 0.

(23)

Evidently, for α = 1 the kernel eα defined by formula (23) coincides with the kernel of the conventional Fourier transform and thus the fractional Fourier transform of the order α = 1 is just the conventional Fourier transform, i. e., ℱ1 ≡ ℱ . For an arbitrary α, α > 0, the relation between the fractional and conventional Fourier transforms is given by the following simple formula: fα̂ (ω) = (ℱα f )(ω) ≡ (ℱ f )(ω1 ) = f ̂(ω1 ),

(24)

where 1

ω1 = sign(ω)|ω| α .

(25)

Formulas (24), (25) allow us to use the well-known properties of the Fourier transform for derivation of the properties of the FrFT. Let us start with an example and determine the FrFT of the pulse signal A, |x| ≤ T, f (x) = { 0, |x| > T. The well-known formula for the Fourier transform of the function f and the relation (24) immediately lead us to the formula (ℱα f )(ω) = (ℱ f )(ω1 ) =

A sin(Tω1 ) πω1

1/α

) , ω ≤ 0, A sin(T|ω|1/α ) { A sin(−T|ω| −π|ω|1/α = { A sin(T|ω| = . 1/α ) π|ω|1/α , ω ≥ 0, { π|ω|1/α

232 | Yu. Luchko Now we proceed with basic properties of the FrFT and first introduce the operators of translation τh and dilatation Πδ : (τh φ)(x) = τ(x − h),

(Πλ φ)(x) = φ(λx),

x, h ∈ ℝ,

(26)

x ∈ ℝ, λ > 0.

(27)

Then we have the following results. Theorem 5 ([12]). Let u ∈ Φ(ℝ) and α > 0. Then for ω ∈ ℝ the relations (ℱα τh u)(ω) = ei sign(ω)|ω| (ℱα Πλ u)(ω) =

1/α

h

(ℱα u)(ω),

1 ω (ℱ u)( α ), λ α λ

h ∈ ℝ,

(28)

λ > 0,

(29)

hold true. For α = 1, Theorem 5 entails the well-known properties of the Fourier transform: (ℱ τh u)(ω) = eiωh (ℱ u)(ω),

h ∈ ℝ,

1 ω (ℱ u)( ), λ λ

(ℱ Πλ u)(ω) =

λ > 0.

The next result yields a formula for the FrFT of the Fourier convolution ∞

(k ∗ u)(x) = ∫ k(x − t)u(t)dt. −∞

Theorem 6 ([12]). Let k, u ∈ Φ(ℝ) and α > 0. Then for ω ∈ ℝ the formula (ℱα (k ∗ u))(ω) = (ℱα k)(ω)(ℱα u)(ω) holds true. In particular, we get the well-known formula (ℱ (k ∗ u))(ω) = (ℱ k)(ω)(ℱ u)(ω),

ω ∈ ℝ.

Because the FrFT is connected with the conventional Fourier transform via formulas (24), (25), the inverse fractional Fourier transform (IFrFT) can easily be determined based on the well-known formula for the inverse Fourier transform. Theorem 7 ([12]). For u ∈ Φ(ℝ) and α > 0, the operator ∞

(ℱα−1 φ)(x) =

1 1/α 1 ∫ e−i sign(ω)|ω| x |ω| α −1 φ(ω) dω, 2πα

x ∈ ℝ,

−∞

is a left inverse to the fractional Fourier transform ℱα , i. e., the relation (ℱα−1 ℱα u)(x) = u(x), holds true.

x ∈ ℝ,

(30)

Fractional Fourier transform

| 233

In the final part of this section, we deal with the operational relations for the Liouville fractional mixed integrals and derivatives (see [12] and [17] for details and proofs) and the Riesz–Feller fractional derivative (see [1] for details and proofs). We start with the Liouville fractional mixed integrals and derivatives and recall that the Liouville fractional mixed derivative Dαβ and the Liouville fractional mixed integral Iβα are defined by formulas (5) and (15), respectively. Theorem 8. Let α > 0, β ∈ ℝ and a function u belong to the Lizorkin space Φ(ℝ). Then the following operational relation holds true for ω ∈ ℝ (ω ≠ 0): (ℱα Iβα u)(ω) =

dα (β)i (ℱα u)(ω), ω

α > 0, β, ω ∈ ℝ, ω ≠ 0,

(31)

where dα (β) = sin(

απ απ ) − i sign(ω)(1 − 2β) cos( ). 2 2

(32)

In particular, for β = 1/2, we have a simpler operational relation: α u)(ω) = (ℱα I1/2

i απ sin( )(ℱα u)(ω), ω 2

α > 0, ω ∈ ℝ.

(33)

As to the Liouville fractional mixed derivative Dαβ , we have the following result. Theorem 9. Let α > 0, β ∈ ℝ, and a function u belong to the Lizorkin space Φ(ℝ). Then the following operational relation holds true: (ℱα Dαβ u)(ω) = −icα (β)ω(ℱα u)(ω),

α > 0, β, ω ∈ ℝ,

(34)

where cα (β) = sin(

απ απ ) + i sign(ω)(1 − 2β) cos( ). 2 2

(35)

In particular, for the fractional derivative 1 Dα1/2 u = (Dα+ u − Dα− u), 2

α > 0,

(36)

we have the operational relation (ℱα Dα1/2 u)(ω) = −i sin(

απ )ω(ℱα u)(ω), 2

α > 0, ω ∈ ℝ.

(37)

Remark 2. For α = 1 and β ∈ ℝ, the operational relation (34) is reduced to the wellknown operational relation for the conventional Fourier transform and the first order derivative: (ℱ1 D1β u)(ω) = (ℱ

d u)(ω) = (−iω)(ℱ u)(ω), dx

ω ∈ ℝ.

(38)

234 | Yu. Luchko Let us now introduce the one-dimensional Riesz–Feller fractional derivative. For a sufficiently well-behaved function f : ℝ → ℝ, the Riesz–Feller space-fractional derivative x Dαθ of order α, 0 < α ≤ 2 and skewness θ, |θ| ≤ min{α, 2 − α} is defined as a pseudo-differential operator (ℱ x Dαθ f )(ω) = −ψθα (ω)̂f (ω),

ψθα (ω) = |ω|α ei sign(ω)θπ/2 .

(39)

When θ = 0, the fractional Riesz–Feller derivative is called the Riesz derivative. In the next section, we employ the so-called quantum fractional Riesz–Feller derivative Dαθ of order α, 0 < α ≤ 2 and skewness θ, |θ| ≤ min{α, 2 − α} that is defined as a pseudo-differential operator with the symbol ψθα given in formula (39): (ℱ Dαθ f )(ω) = ψθα (ω)̂f (ω).

(40)

According to (39) and (40), the quantum Riesz–Feller derivative is just the Riesz–Feller derivative multiplied by −1. For 0 < α < 2 and |θ| ≤ min{α, 2 − α}, the quantum Riesz–Feller derivative can be represented in the following form (see [18, 29]): Dαθ f (x)

∞

f (x + ξ ) − f (x) Γ(1 + α) π =− dξ {sin((α + θ) ) ∫ π 2 ξ 1+α 0

∞

f (x − ξ ) − f (x) π + sin((α − θ) ) ∫ dξ }. 2 ξ 1+α

(41)

0

For 0 < α < 2, α ≠ 1 and θ in its range, this formula can be rewritten as [8, 9] Dαθ f (x) = (c+ Dα+ + c− Dα− )f (x),

(42)

where the coefficients c± are given by c+ = c+ (α, θ) =

sin((α − θ)π/2) , sin(απ)

c− = c− (α, θ) =

sin((α + θ)π/2) , sin(απ)

(43)

and Dα+ and Dα− are the left- and right-sided Liouville fractional derivatives of order α, defined by (6) and (7), respectively. Let us note that the representation (42) allows one to define the quantum Riesz–Feller derivative on the Lisorkin space of functions Φ(ℝ) and it follows from Theorem 1 that Φ(ℝ) is invariant with respect to the quantum Riesz–Feller derivative. For α = 1, the representation (42) is not valid anymore and has to be replaced by the formula D1θ f (x) = (cos(θπ/2)D10 − sin(θπ/2)D)f (x),

(44)

Fractional Fourier transform

| 235

where the operator D10 is related to the Hilbert transform as first noted by Feller in 1952 in his pioneering paper [6], D10 f (x)

+∞

f (ξ ) 1 d = dξ ∫ π dx x−ξ −∞

and D stands for the first derivative. As proved in [1], the following operational relation is valid for the quantum Riesz– Feller derivative. Theorem 10. Let 1 < α < 2 and f ∈ Φ(ℝ). Then the operational relation θπ

(ℱα Dαθ f )(ω) = |ω|ei sign(ω) 2 (ℱα f )(ω),

ω ∈ ℝ,

(45)

holds true for the fractional Fourier transform of the quantum Riesz–Feller derivative.

4 Applications of the fractional Fourier transform The operational relations for the fractional derivatives of different types that we presented in the previous section can be applied to treating the linear fractional differential equations with these derivatives. In particular, in [12] the FrFT was employed for a derivation of the closed form solutions to the Cauchy problem for a partial space-timefractional differential equation with the Caputo time-fractional derivative and a linear combination of the spatial left- and right-hand sided Riemann–Liouville fractional derivatives. In [1], the FrFT technique was used to obtain a solution to the Cauchy problem for the fractional Schrödinger equation with the quantum Riesz–Feller derivative in terms of the Fox H-function. In [16], the operational relations for the FrFT were applied for deriving the closed form solutions of some fractional differential equations with both left- and right-hand sided fractional derivatives without and with delays. In this section, two examples are discussed (for details see [17] for the first example and [1] for the second example). We start with the space-time-fractional diffusion equation in the form α x D1/2 u(x, t)

= t Dβ∗ u(x, t),

x ∈ ℝ, t ∈ ℝ+ ,

(46)

where α, β are real parameters that satisfy the conditions 0 < α ≤ 1, 0 < β ≤ 2, 1 α α α x D1/2 = 2 (x D+ − x D− ) is the Liouville space-fractional mixed derivative of order α and β t D∗ is the Caputo time-fractional derivative of order β (m − 1 < β ≤ m, m ∈ ℕ) defined as follows [4]: (Dβ∗ f )(t)

1

t f (m) (τ) dτ , (t−τ)β+1−m

{ Γ(m−β) ∫0 ={ m d { dt m f (t),

m − 1 < β < m, β = m.

(47)

236 | Yu. Luchko For a sufficiently well-behaved function f , the property m−1

(ℒDβ∗ f )(s) = sβ ̃f (s) − ∑ sβ−1−k f (k) (0+ ),

m − 1 < β ≤ m,

k=0

(48)

holds true [18, 24], (ℒf )(s) being the Laplace transform, ∞

̃f (s) = (ℒf )(s) = ∫ e−st f (t) dt,

ℜ(s) > af ,

0

of the function f at the point s ∈ ℂ. A sufficient condition for existence of the Laplace transform of a function f is that it is at most of exponential growth as t → +∞, i. e., there exists a constant af such that e−af t |f (t)| is bounded as t → +∞. Then the Laplace transform ̃f (s) exists and is analytic in the half plane ℜ(s) > af . For equation (46), we consider the Cauchy problem with the initial condition u(x, 0) = φ(x),

x ∈ ℝ,

(49)

where φ(x) ∈ Lc (ℝ) is a sufficiently well-behaved function. If 1 < β ≤ 2, the second initial condition ut (x, 0) = 0 is added to (49), where ut (x, t) = 𝜕t𝜕 u(x, t). In addition, the “boundary” conditions at ±∞ u(±∞, t) = 0,

t > 0,

are supposed to be fulfilled, too. By a solution of the Cauchy problem for equation (46) we mean a function u which satisfies both equation (46) and the initial condition (49). The fundamental solution of the Cauchy problem is a function Gα,β that solves it for the initial condition φ(x) = δ(x), δ being the Dirac delta function. To determine the fundamental solution, the method of the integral transforms is employed with the FrFT with respect to the spatial variable and the Laplace transform with respect to the time variable. Applying the Laplace transform and the FrFT to equation (46) and the initial condition (49) with φ(x) = δ(x) and taking into account the operational relation (37) for the Liouville space-fractional mixed derivative of the order 1/2 and the Laplace transform formula (48) for the Caputo time-fractional derivative, we arrive at the following formula for the Laplace transform and the fractional Fourier transform of order α of the fundamental solution: ̂ ̂ ̃ (ω, s) = sβ G ̃ (ω, s) − sβ−1 . −i sin(απ/2)ωG α,β α,β This equation can easily be solved: ̂ ̃ (ω, s) = G α,β

sβ

sβ−1 . + i sin(απ/2)ω

(50)

Fractional Fourier transform

| 237

To recover the fundamental solution, let us first apply the inverse Laplace transform to formula (50). For this purpose, we recall the well-known Laplace transform pair (see [18] and the references therein): Eβ (ct β ) ↔ ℒ

sβ−1 , sβ − c

ℜ(s) > |c|1/β ,

(51)

with c ∈ ℂ, 0 < β ≤ 2, where Eβ denotes the Mittag-Leffler function of order β, defined by the convergent series zn , Γ(βn + 1) n=0 ∞

Eβ (z) = ∑

β > 0, z ∈ ℂ.

We compare now (50) and (51) and arrive at a formula for the fractional Fourier transform of the fundamental solution ̂ (ω, t) = E (−i sin(απ)ωt β ), G α,β β

ω ∈ ℝ, t ≥ 0.

(52)

Applying the inverse FrFT, the fundamental solution Gα,β of the Cauchy problem (46), (49) can be expressed in the form Gα,β (x, t) = (ℱα−1 Eβ (−i sin(απ)ωt β ))(x),

(53)

ℱα−1 being the inverse fractional Fourier transform given by (30). Let us note here that

the right-hand side of formula (53) can be represented in the form of the Mellin–Barnes integral [3, 25] by using the same technique that was employed in [18] for the case of the space-time-fractional diffusion equation with the Riesz–Feller fractional derivative instead of the Liouville fractional mixed derivative x Dα1/2 , but we omit the further details here. In the rest of this section, we consider the fractional Schrödinger equation for a free particle in the form iℏ

𝜕Ψ(x, t) = Cα Dαθ Ψ(x, t), 𝜕t

(54)

where the quantum fractional Riesz–Feller derivative Dαθ is defined by formula (40). For the Cauchy problem for this equation, the following result holds true. Theorem 11 ([1]). Let 1 < α < 2, |θ| ≤ 2 − α, and f ∈ Φ(ℝ). Then the Cauchy-type problem for equation (54) subject to the initial condition Ψ(x, 0) = f (x),

x ∈ ℝ,

(55)

and the boundary conditions Ψ(x, t) → 0,

as x → ±∞,

(56)

238 | Yu. Luchko possesses a unique solution given by the formula +∞

Ψ(x, t) = ∫ G(x − ξ , t)f (ξ ) dξ ,

(57)

−∞

where for x ≠ 0 the fundamental solution G is represented in terms of the Fox H-function, 1

󵄨󵄨 (1, 1 ), (1, ρ) 1 1,1 ℏ α −i 2απ 󵄨 α G(x, t) = ] H2,2 [( ) e |x| 󵄨󵄨󵄨 󵄨󵄨 (1, 1), (1, ρ) α|x| Cα t with ρ =

α−sign(x)θ 2α

(58)

and 1

1 1 ℏ α πθ −i 2απ G(0, t) = ( ) Γ( ) cos e . πα Cα t α 2α For a proof of this theorem we refer the reader to [1]. It is worth mentioning that the solution method employed in [1] substantially uses the technique of the fractional Fourier transform and in particular the operation relation (45) for the fractional Fourier transform of the quantum Riesz–Feller derivative. Remark 3. In the case θ = 0, the quantum Riesz–Feller derivative is reduced to the quantum Riesz derivative and the fractional Schrödinger equation (54) becomes the fractional Schrödinger equation with the quantum Riesz derivative that has been already studied in the literature. In particular, the fractional Schrödinger equation with the quantum Riesz derivative was solved for a free particle and its fundamental solution was presented in the form [10, 13] 1

1 1 1 1,1 ℏ α 󵄨󵄨󵄨󵄨 (1, α ), (1, 2 ) ], G(x, t) = H [( ) |x| 󵄨󵄨 󵄨󵄨 (1, 1), (1, 1 ) α|x| 2,2 iCα t 2

(59)

which for θ = 0 coincides with formula (58). Another important remark is that the relations (20) and (21) can be interpreted as the formulas for the eigenfunctions of the Liouville fractional derivatives Dα+ and Dα− in the form of the exponential functions eiωx , ω ≠ 0. Because of the relation (42), they are also eigenfunctions of the quantum Riesz–Feller fractional derivative for 0 < α < 2, α ≠ 1 and θ in its range. The eigenvalues that correspond to the eigenfunctions eiωx , ω ≠ 0 can be obtained using formulas (20), (21), and (42): (Dαθ eiωξ )(x) = λω eiωx ,

(60)

where λω = |ω|α (cos(

απ απ )(c+ + c− ) + i sign(ω) sin( )(c+ − c− )). 2 2

(61)

Fractional Fourier transform

| 239

The eigenvalues (61) are real numbers only when c+ = c− , i. e., when the quantum Riesz–Feller fractional derivative is reduced either to the quantum Riesz derivative or to the Laplace operator. Formulas (60), (61) mean that the quantum Riesz–Feller fractional derivative has a continuous spectrum. For a given θ, |θ| ≤ min{α, 2−α}, the eigenvalues of the quantum Riesz–Feller fractional derivative of order α, 0 < α ≤ 2, α ≠ 1 are located on two infinite with the positive real rays that start from the coordinates origin and build the angle θπ 2 axis. If we fix the order α, 0 < α ≤ 2, α ≠ 1, of the Riesz–Feller fractional derivative and let θ vary in its range |θ| ≤ min{α, 2 − α} then the eigenvalues fill the complete θ π angle between the two rays arg(z) = ± 02 , θ0 = min{α, 2 − α}, as can be seen from the following chain of equalities: sin( απ sin( απ )(c+ − c− ) )(sin( π2 (α − θ)) − sin( π2 (α + θ))) ℑ(λω ) 2 2 = ± =± ℜ(λω ) )(c+ + c− ) )(sin( π2 (α − θ)) + sin( π2 (α + θ))) cos( απ cos( απ 2 2 =±

2 sin( απ ) cos( απ ) sin(− θπ ) 2 2 2

) sin( απ ) cos(− θπ ) 2 cos( απ 2 2 2

= ∓ tan(

θπ ). 2

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B. Al-Saqabi, L. Boyadjiev, and Yu. F. Luchko, Comments on employing the Riesz–Feller derivative in the Schrödinger equation, Eur. Phys. J. Spec. Top., 222 (2013), 1779–1794. [2] L. B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process., 42 (1994), 3084–3093. [3] E. W. Barnes, The asymptotic expansion of integral functions defined by Taylor’s series, Philos. Trans. R. Soc. Lond. A, 206 (1906), 249–297. [4] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. R. Astron. Soc., 13 (1967), 529–539. [5] G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, Multiplicity of fractional Fourier transforms and their relationships, IEEE Trans. Signal Process., 48 (2000), 227–241. [6] W. Feller, On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié à M. Riesz, pp. 73–81, Lund, 1952. [7] I. M. Gel’fand and G. E. Shilov, Generalized Functions, vol. 2, Academic Press, New York and London, 1968. [8] R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167–191. [9] R. Gorenflo and F. Mainardi, Approximation of Lévy–Feller diffusion by random walk, J. Anal. Appl., 18 (1999), 231–246. [10] X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104. [11] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, Elsevier, Amsterdam, 2006.

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[12] A. A. Kilbas, Yu. F. Luchko, H. Martinez, and J. J. Trujillo, Fractional Fourier transform in the framework of fractional calculus operators, Integral Transforms Spec. Funct., 21 (2010), 779–795. [13] N. Laskin, Principles of fractional quantum mechanics, in J. Klafter, S. C. Lim, and R. Metzler (eds.) Fractional Dynamics: Recent Advances, pp. 393–429, World Scientific, Singapore, 2012. [14] P. I. Lizorkin, Generalized Liouville differentiation and functional spaces Lrp (En ). Embedding theorems, Mat. Sb., 60 (1963), 325–353 (in Russian). [15] P. I. Lizorkin, Generalized Liouville differentiation and the method of multipliers in the theory of embeddings of classes of differentiable functions, Proc. Steklov Inst. Math., 105 (1969), 105–202. [16] Yu. Luchko and M. M. Rodrigues, Some new properties and applications of a fractional Fourier transform, J. Inequal. Spec. Funct., 8 (2017), 13–27. [17] Yu. F. Luchko, H. Martinez, and J. J. Trujillo, Fractional Fourier transforms and some of its applications, Fract. Calc. Appl. Anal., 11 (2008), 457–470. [18] F. Mainardi, Yu. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153–192. [19] A. B. Malinowska, T. Odzijewicz, and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer, Berlin, 2015. [20] A. C. McBride and F. H. Kerr, On Namias’s fractional Fourier transform, IMA J. Appl. Math., 39 (1987), 159–175. [21] D. Mendlovic, Advances in Imaging and Electron Physics, vol. 106, Academic Press, New York, 1999. [22] V. Namias, The fractional order Fourier transform and its application in quantum mechanics, J. Inst. Math. Appl., 25 (1980), 241–265. [23] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier Transform, Wiley, Chichester, 2001. [24] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [25] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 3, Gordon and Breach Science Publ., New York, 1990. [26] B. Rubin, Fractional Integrals and Potentials, Longman, London, 1996. [27] S. G. Samko, Denseness of Lizorkin-type spaces ΦV in Lp , Math. Notes, 31 (1982), 432–437. [28] S. G. Samko, Denseness of the spaces ΦV of Lizorkin type in the mixed Lp (Rn )-spaces, Stud. Math., 113 (1995), 199–210. [29] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publ., New York, 1993. [30] E. C. Titchmarsh, Introduction to Theory of Fourier Integrals, Oxford Univ. Press, Oxford, 1937. [31] N. Wiener, Hermitian polynomials and Fourier analysis, J. Math. Phys., 8 (1929), 70–73. [32] A. I. Zayed, A class of fractional integral transforms: a generalization of the fractional Fourier transform, IEEE Trans. Signal Process., 50 (2002), 619–627.

Yuri Luchko

The Wright function and its applications Abstract: In this chapter, basic properties of the Wright function and some of its important generalizations and applications are discussed. We start with the classical Wright function and continue with the four parameter Wright function and the generalized Wright function. Both elementary properties and some advanced features of these functions, including distribution of their zeros, Laplace and Mellin integral transforms, and the Mellin–Barnes representations, are mentioned. Some applications of the Wright type functions in fractional calculus are presented, too. Keywords: Wright function, four parameter Wright function, generalized Wright function, H-function, completely monotone functions, probability density functions, fundamental solution, subordination principle, Mellin–Barnes integrals MSC 2010: 26A33, 33E20, 30C15, 30D15, 45J05, 45K05, 44A20

1 Introduction A basis of any calculus including fractional calculus is built by the functions that it deals with. Whereas the calculus operations and constructions are introduced on general spaces of functions, these are some elementary or special functions that play a significant role for concrete examples, calculations, and algorithms. Thus, in the conventional calculus, it is the exponential function and its properties including the law of exponents exp(x) ⋅ exp(y) = exp(x + y), differentiation rule (exp(x))󸀠 = exp(x), and the Euler formula exp(iϕ) = cos(ϕ) + i sin(ϕ) that are central while dealing with many, at least basic, calculations and examples. As a logical continuation of this actual situation, the exponentiation function is also heavily employed for treating both ordinary and partial differential equations. Because of its low of exponents and differentiation properties, it naturally appears in the kernel of the Fourier transform and thus plays a central role in applied mathematics. Due to the fact that the fundamental solution to the conventional diffusion equation is expressed in terms of the exponential function, it is connected with the Gaussian distribution and becomes a basic function in the probability theory. The list of remarkable properties and applications of the exponential function can be continued and it shows that the exponential function is a unique queen of calculus and related areas.

Yuri Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110571622-010

242 | Yu. Luchko Contrary to the conventional calculus, in fractional calculus two different functions play the role of the exponential function, namely, the Mittag-Leffler function and the Wright function. Both functions are generalizations of the exponential function and inherit some of its remarkable properties, but for different types of applications. Because a separate chapter of this book is devoted to the Mittag-Leffler function, its generalizations, and applications, the focus of this chapter will be on the Wright type functions that are especially important while treating partial fractional differential equations and in the theory of the fractional stochastic processes. The function zk , k!Γ(ρk + β) k=0 ∞

ϕ(ρ, β; z) = ∑

z ∈ ℂ, ρ > −1, β ∈ ℂ

(1)

named after the British mathematician E. M. Wright, was introduced by him for the first time in [58] in connection with his investigations in the asymptotic theory of partitions. The notation ϕ(ρ, β; z) is due to Wright himself and we employ it here even if nowadays some other notations are often used in the literature. Let us note that in [58] Wright restricted himself to the case ρ > 0 (Wright function of the first kind) he needed for the problem he dealt with. In [61], Wright investigated the function ϕ(ρ, β; z) from the mathematical viewpoint and extended the range of the parameter ρ to its natural domain ρ > −1. Following [40], we refer to the Wright function with the negative parameter ρ (0 > ρ > −1) as to the Wright function of the second kind. Later on, the Wright function appeared in many other applications including the Mikusiński operational calculus, integral transforms of the Hankel type, fractional differential equations, and stochastic processes. For a historical overview regarding the Wright function and some of its applications we refer the reader to Appendix F of the book [40]. Let us note that the series at the right-hand side of (1) is convergent also in the case ρ = −1 for |z| < 1 and |z| = 1 under the additional condition ℜ(β) > 1. The rest of this chapter is organized as follows. In Section 2, we survey some of the results presented in [9, 10, 15, 18, 28, 33, 39, 43, 49, 51, 52, 56, 59], and [61] and recall the main properties of the Wright function including its integral representations, special cases in terms of the hypergeometric type functions, the Laplace transform pairs related to the Wright function, and its Mellin–Barnes integral representation. We also discuss the distribution of zeros of the Wright function, its order, type and indicator function and show that this function is an entire function of completely regular growth for every ρ > −1 [28]. It is worth mentioning that from the viewpoint of the theory of analytic functions it is the Wright function and not the Mittag-Leffler function that is a natural fractional generalization of the exponential function. As to the asymptotic properties of the Wright function and its generalizations, they are presented in a separate chapter of this book and we do not repeat them here. In Section 3, some basic results regarding the four parameter Wright function and the generalized Wright function or the Fox–Wright function are presented following [24, 25, 33, 34],

The Wright function and its applications | 243

and [46–48]. In Section 4, we outline some applications of the Wright function and its generalizations, starting with the original results by Wright [58] in the asymptotic theory of partitions. Special attention is given to the key role of the Wright function in the theory of fractional partial differential equations [4, 5, 17, 29–32, 35–44]. We also mention some results from [6, 19], and [33] concerning an extension of the methods of the Lie groups to the partial fractional differential equations, where the Wright function and its generalizations play an important role. For applications of the Wright function in the Mikusiński operational calculus and for integral transforms of the Hankel type that are not included into this chapter we refer the reader to [26, 27, 49], and [56].

2 Analytical properties of the Wright function 2.1 The Wright function as a function of the hypergeometric type The function ϕ(ρ, β; z) defined by (1) was interpreted by Wright as a generalization of the Bessel function Jν (z) because of the formula −ν

1 z ϕ(1, ν + 1; − z 2 ) = ( ) Jν (z) 4 2

(2)

that immediately follows from the series representations of the functions ϕ(ρ, β; z) and Jν (z). In this subsection, several other particular cases of the Wright function will be presented. The main analytical tool for derivation of such formulas for the Wright function is its Mellin–Barnes representation or its representation in terms of the Fox H-function [14, 17, 19, 26], which is the most general function of the hypergeometric type. Because the parameter ρ can be both positive, negative, or equal to zero, we have to distinguish between three cases while representing the Wright function as a particular case of the H-function: 󵄨󵄨 1,0 − 󵄨󵄨 −z), 0 < ρ, H0,2 ( (0,1), (1−β,ρ) { 󵄨󵄨 { { { { 1,0 (β,−ρ) 󵄨󵄨 ϕ(ρ, β; z) = {H1,1 ( (0,1) 󵄨󵄨󵄨 −z), −1 < ρ < 0, { 󵄨 { { { 1 1,0 − 󵄨󵄨 󵄨 ρ = 0. { Γ(β) H0,1 ( (0,1) 󵄨󵄨󵄨 −z),

(3)

In all cases, the representation (3) can be rewritten in terms of the same Mellin–Barnes integral [41, 42, 46]: ϕ(ρ, β; z) =

Γ(s) 1 (−z)−s ds ∫ 2πi Γ(β − ρs) L

with the contour L as in the definition of the Fox H-function [14, 17, 26].

(4)

244 | Yu. Luchko Evidently, for ρ = 0 the Wright function is reduced to the exponential function: ϕ(0, β; z) =

1 exp(z). Γ(β)

Another important integral representation of the Wright function (that is employed, say, for derivation of its asymptotic behavior) is as follows: ϕ(ρ, β; z) =

−ρ dσ 1 , ∫ eσ+zσ 2πi σβ

ρ > −1, β ∈ ℂ,

(5)

Ha

where Ha denotes the Hankel path in the complex plane. This integral representation easily follows from the series representation (1) and the Hankel formula for the Gamma function: 1 1 = ∫ eu u−ζ du, Γ(ζ ) 2πi

ζ ∈ ℂ.

Ha

If ρ = n/m is a positive rational number, the Wright function can be represented in terms of the generalized hypergeometric function p Fq ((a)p ; (b)q ; z) as follows [18]: ϕ(

m−1 n zp , β; z) = ∑ m p=0 p!Γ(β +

0 Fn+m−1 (−; Δ(n, n p) m

Δ(k, a) = {a, a +

1 k−1 ,...,a + }, k k

β p p+1 zm + ), Δ∗ (m, ); m n ), n m m m n Δ∗ (k, a) = Δ(k, a) \ {1}.

A similar representation is valid for a negative rational ρ but under the condition that the parameter β is also a rational number. In particular, we have the relations (−1)n+1 z 3 3 3 z2 1 Γ( + n) 1 F1 ( + n; ; − ), ϕ(− , −n; z) = 2 π 2 2 2 4 1 1 (−1)n 1 1 1 z2 ϕ(− , − n; z) = Γ( + n) 1 F1 ( + n; ; − ), 2 2 π 2 2 2 4

n = 0, 1, 2, . . . ,

(6)

n = 0, 1, 2, . . . .

(7)

Setting n = 0 in the formulas above, we get the well-known particular cases of the Wright function [43, 56] 1 z −z 2 /4 ϕ(− , 0; z) = − e , 2 2√π

1 1 1 −z 2 /4 ϕ(− , ; z) = e . √π 2 2

(8)

It is worth mentioning that formulas (6) and (7) can be rewritten in the form [18] 2 1 ϕ(− , −n; z) = e−z /4 zPn (z 2 ), n = 0, 1, 2, . . . , 2 2 1 1 ϕ(− , − n; z) = e−z /4 Qn (z 2 ), n = 0, 1, 2, . . . , 2 2

(9) (10)

The Wright function and its applications | 245

where Pn (z), Qn (z) are polynomials of degree n defined as follows: (−1)n+1 z Γ(3/2 + n) 1 F1 (−n; 3/2; ), π 4 z (−1)n Γ(1/2 + n) 1 F1 (−n; 1/2; ). Qn (z) = π 4

Pn (z) =

We also refer here to an important recurrence relation and a basic differentiation formula for the Wright function ([10], Vol. 3, Ch. 18): ρzϕ(ρ, ρ + β; z) = ϕ(ρ, β − 1; z) + (1 − β)ϕ(ρ, β; z), d ϕ(ρ, β; z) = ϕ(ρ, ρ + β; z). dz

As already mentioned, the Wright function and its generalizations appear while treating some linear partial fractional differential equations. In the one-dimensional case, the fundamental solution to the Cauchy and signaling problems for the time-fractional diffusion-wave equation can be expressed in terms of two auxiliary functions of the Wright type nowadays referred to as the Mainardi functions [36, 40, 43]: Fν (z) := ϕ(−ν, 0; −z),

Mν (z) := ϕ(−ν, 1 − ν; −z),

0 < ν < 1,

(11)

interrelated through the formula Fν (z) = νzMν (z). The series representations of the functions Fν and Mν immediately follow from the series representation (1) of the Wright function and the Euler reflection formula for the Gamma-function: Fν (z) =

1 ∞ (−z)n−1 Γ(νn + 1) sin(πνn), ∑ π n=1 n!

Mν (z) =

1 ∞ (−z)n−1 Γ(νn) sin(πνn). ∑ π n=1 (n − 1)!

In [43], several formulas for particular cases of the functions Fν and Mν were derived and we list here some of them: M1/2 (z) =

1 exp(−z 2 /4), √π

M1/3 (z) = 32/3 Ai(z/31/3 ),

where Ai denotes the Airy function. For more details regarding the functions Fν and Mν including their asymptotics and plots we refer the reader to [40], Appendix F.

246 | Yu. Luchko

2.2 Laplace transform pairs related to the Wright function In the case ρ > 0, the Wright function is an entire function of order less than one [28]. Its Laplace transform can thus be obtained by transforming term-by-term its series representation (1) and we get the formula 1 ϕ(ρ, β; ±t) ÷ Eρ,β (±s−1 ), s

ρ > 0, β ∈ ℂ,

(12)

̃ where ÷ denotes the juxtaposition of a function φ(t) with its Laplace transform φ(s) and zk , Γ(αk + β) k=0 ∞

Eα,β (z) = ∑

α > 0, β ∈ ℂ,

(13)

is the generalized Mittag-Leffler function. The Laplace transform of the Wright function is an analytic function that vanishes at infinity and exhibits an essential singularity at the point s = 0. For −1 < ρ < 0, the term-by-term integration of the series does not work because in this case the Wright function is an entire function of order greater than one [28, 61]. Still the Laplace transform of the function ϕ(ρ, β; −t), t > 0 exists because ϕ(ρ, β; z) is exponentially small for large z in a sector of the complex plane containing the negative real semi-axis [61]. Then we have [18, 39]: ϕ(ρ, β; −t) ÷ E−ρ,β−ρ (−s),

−1 < ρ < 0.

(14)

To derive the formula above, the integral representation [7, 16] Eα,β (z) =

eζ ζ α−β 1 dζ ∫ α 2πi ζ −z

(15)

Ha

of the generalized Mittag-Leffler function is employed. Formula (14) was provided in [9] (and in [8] in the case β ≥ 0) as a representation of the generalized Mittag-Leffler function as a Laplace integral of an entire function and without identifying this function as the Wright function. In [9], a more general representation ∞

Eα2 ,β2 (z) = ∫ Eα1 ,β1 (zt α1 )t β1 −1 ϕ(−α2 /α1 , β2 − β1 0

α2 ; −t) dt α1

was also derived for α1 > α2 > 0, β1 , β2 > 0 in slightly different notations. An important particular case of the Laplace transform pair (14) is given by Mν (t) ÷ Eν (−s),

0 < ν < 1,

(16)

The Wright function and its applications | 247

where Mν is defined by (11) and zk , Γ(αk + 1) k=0 ∞

Eα (z) = Eα,1 (z) = ∑

α>0

(17)

is the Mittag-Leffler function. As to the function Fν defined by (11), we get the following Laplace transform pair [18, 40, 49, 52]: 1 νλ F (λt −ν ) = ν+1 Mν (λt −ν ) ÷ exp(−λsν ), t ν t

0 < ν < 1, 0 < λ.

(18)

The Laplace transform correspondence 1 M (λt −ν ) ÷ sν−1 exp(−λsν ), tν ν

0 < ν < 1, 0 < λ,

(19)

easily follows from (18) by applying the Laplace transform differentiation formula. Finally, we mention the following Laplace transform pairs related to the Wright function that were derived in [56]: t β−1 ϕ(ρ, β; −λt ρ ) ÷ s−β exp(−λs−ρ ), β

ρ

t 2 −1 ϕ(ρ, β; −t 2 ) ÷

−1 < ρ < 0, 0 < λ,

ρ √π − β ρ β+1 s 2 ϕ( , ; −2−ρ s− 2 ), 2 2 2β

−1 < ρ < 0,

t −β exp(−t −ρ cos(ρπ)) sin(βπ − t −ρ sin(ρπ)) ÷ πsβ−1 ϕ(ρ, β; −sρ ),

−1 < ρ < 0, β < 1.

2.3 The Wright function as an entire function In [59, 61], the Wright function (1) was shown to be an entire function for all values of the parameters ρ > −1 and β ∈ ℂ. In [8, 9] the order and the type of the Wright function as well as an estimate for its indicator function were derived in the case −1 < ρ < 0. Regarding the distribution of zeros of the Wright function, Wright himself already knew [61] that zeros of his function are located near the positive real semi-axis if −1/3 ≤ ρ < 0 and near the rays arg z = ± 21 π(3ρ + 1) if −1 < ρ < −1/3. A complete description of location and asymptotic behavior of zeros of the Wright function for all values of its parameters was presented in [28]. Here we give a short survey of the results obtained in [28]. The order and the type of the Wright function can be calculated in a straightforward way by using the standard formulas for order and type of an entire function defined by a power series and the Stirling asymptotic formula for the Gamma-function. Theorem 1. The Wright function ϕ(ρ, β; z), ρ > −1, β ∈ ℂ (β ≠ −n, n = 0, 1, . . . if ρ = 0) is an entire function of finite order p and type σ given by p=

1 , 1+ρ

ρ

− 1+ρ

σ = (1 + ρ)|ρ|

.

(20)

248 | Yu. Luchko Remark 1. For ρ = 0, the Wright function is reduced to the exponential function with the constant factor 1/Γ(β) that vanishes identically for β = −n, n = 0, 1, . . . . For all other values of the parameter β and ρ = 0, the formulas for the order and type presented in ρ

− 1+ρ

Theorem 1 (with σ = limρ→0 (1 + ρ)|ρ|

= 1) are still valid.

As is well known, the growth of an entire function f (z) of finite order p in different directions of the complex plane is described by its indicator function h(θ), |θ| ≤ π, defined by the formula h(θ) = lim sup r→+∞

log |f (reiθ )| . rp

(21)

Theorem 2 ([28]). Let ρ > −1, β ∈ ℂ (β ≠ −n, n = 0, 1, . . . if ρ = 0). Then the indicator function of the Wright function is given by the formulas hρ (θ) = σ cos pθ,

|θ| ≤ π, ρ ≥ 0,

−σ cos p(π + θ), hρ (θ) = { −σ cos p(θ − π),

−π ≤ θ ≤ 0, 0 ≤ θ ≤ π,

(22) (23)

in the cases (a) −1/3 ≤ ρ < 0, (b) ρ = −1/2, β = −n, n = 0, 1, . . . and (c) ρ = −1/2, β = 1/2 − n, n = 0, 1, . . . , −σ cos p(π + θ), −π ≤ θ ≤ 32 πp − π, { { { hρ (θ) = {0, |θ| ≤ π − 32 πp , { { 3π {−σ cos p(θ − π), π − 2 p ≤ θ ≤ π,

(24)

in the case −1 < ρ < −1/3 (β ≠ −n, n = 0, 1, . . . and β ≠ 1/2 − n, n = 0, 1, . . . if ρ = −1/2), where p and σ are the order and type of the Wright function, respectively, given by (20). Remark 2. It follows from formulas (22), (23) that the indicator function hρ (θ) of the Wright function ϕ(ρ, β; z) coincides with the indicator function cos θ of the exponential function ez for ρ → 0. This property does not hold true for the Mittag-Leffler function Eα defined by (17). Even though E1 (z) = ez , the indicator function of the Mittag-Leffler function Eα with 0 < α < 2, α ≠ 1 given by the formula (see [11, Chapter 2.7]) cos θ/α, h(θ) = { 0,

|θ| ≤ πα 2

πα , 2

≤ |θ| ≤ π,

does not tend to the indicator function cos θ of ez for α → 1. In this sense, it is the Wright function and not the Mittag-Leffler function that can be considered to be a natural fractional generalization of the exponential function from the viewpoint of theory of entire functions.

The Wright function and its applications | 249

A complete description of the distribution of zeros of the Wright function in the case β ∈ ℝ was derived in [28]. Depending on values of the parameters ρ and β, location and asymptotic behavior of zeros of the Wright function follow five different scenarios: 1) ρ > 0: all zeros with large enough absolute values are simple and located on the negative real semi-axis; 2) ρ = 0: the Wright function is the exponential function with a constant factor (equal to zero if β = −n, n = 0, 1, . . . ) and has no zeros; 3) −1/3 ≤ ρ < 0: all zeros with large enough absolute values are simple and located on the positive real semi-axis; 4) ρ = −1/2, β = −n, n = 0, 1, . . . or ρ = −1/2, β = 1/2 − n, n = 0, 1, . . . : the Wright function has exactly 2n + 1 or 2n zeros, respectively; 5) −1 < ρ < −1/3 (excluding the case 4): all zeros with large enough absolute values are simple and located in the neighborhoods of the rays arg z = ± 21 π(−1 − 3ρ). The detailed results regarding asymptotic behavior of zeros of the Wright function are presented in the following two theorems. Theorem 3 ([28]). Let {γk }∞ 1 be the sequence of zeros of the function ϕ(ρ, β; z), ρ ≥ −1/3, ρ ≠ 0, β ∈ ℝ, where |γk | ≤ |γk+1 | and each zero is counted according to its multiplicity. Then the following results are valid: A. In the case ρ > 0 all zeros with large enough k are simple and located on the negative real semi-axis. The asymptotic formula γk = −(

πk + π(pβ −

σ sin πp

p−1 ) 2

1 p

) {1 + O(k −2 )},

k → +∞,

(25)

holds true. Here and in the next formulas p and σ are the order and type of the Wright function given by (20), respectively. B. In the case −1/3 ≤ ρ < 0, all zeros with large enough k are simple and located on the positive real semi-axis and the asymptotic formula γk = (

πk + π(pβ −

p−1 ) 2

−σ sin πp

1 p

) {1 + O(k −2 )},

k → +∞,

(26)

holds true. Remark 3. Combining the representation (2) with the asymptotic formula (25) we get the well-known formula (see, for example [57]) for asymptotic expansion of the large zeros rk of the Bessel function Jν (z): 1 1 rk = π(k + ν − ) + O(k −1 ), 2 4

k → ∞.

Remark 4. In the cases ρ = −1/2, β = −n, n = 0, 1, . . . and ρ = −1/2, β = 1/2 − n, n = 0, 1, . . . , the Wright function can be represented by the formulas (9), (10) and, consequently, has exactly 2n + 1 and 2n zeros in the complex plane, respectively.

250 | Yu. Luchko For −1 < ρ < −1/3, all zeros of the Wright function ϕ(ρ, β; z) with large enough absolute values are located inside of the angular domains 󵄨󵄨 3π 󵄨󵄨󵄨󵄨 󵄨󵄨 Ω(±) )󵄨 < ϵ}, ϵ = {z : 󵄨󵄨󵄨arg z ∓ (π − 2p 󵄨󵄨󵄨 󵄨 where ϵ is any number of the interval (0, min{π − 3π , 3π }). Consequently, the function 2p 2p ϕ(ρ, β; z) has only finitely many zeros on the real axis. Let ∞

{γk(+) }1 ∈ G(+) = {z : ℑ(z) > 0},

∞

{γk(−) }1 ∈ G(−) = {z : ℑ(z) < 0}

be sequences of zeros of the function ϕ(ρ, β; z) in the upper and lower half-plane, re(+) (−) spectively, such that |γk(+) | ≤ |γk+1 |, |γk(−) | ≤ |γk+1 |, and each zero is counted according to its multiplicity. Theorem 4 ([28]). In the case −1 < ρ < −1/3 (β ≠ −n, n = 0, 1, . . . and β ≠ 1/2 − n, n = 0, 1, . . . if ρ = −1/2) all zeros of the function ϕ(ρ, β; z), β ∈ ℝ with large enough k are simple and the asymptotic formula γk(±)

=e

±i(π− 3π ) 2p

1

2πk p log k ) {1 + O( )}, ( σ k

k → +∞

(27)

holds true. Summarizing the results regarding the asymptotic behavior of the Wright function, its indicator function and the distribution of its zeros, we get the following important statement. Theorem 5 ([28]). The Wright function ϕ(ρ, β; z), ρ > −1 is an entire function of completely regular growth. For more details regarding the Wright function as an entire function of completely regular growth including a discussion of the angular density of its zeros and its connection with the indicator function h(θ) we refer the interested reader to [28].

3 Generalized Wright function and its properties 3.1 Four parameter Wright function The four parameter Wright function is defined by the series (in the case it is a convergent one) zk , Γ(a + μk)Γ(b + νk) k=0 ∞

ϕ((μ, a), (ν, b); z) := ∑

μ, ν ∈ ℝ, a, b ∈ ℂ.

(28)

The Wright function and its applications | 251

Wright himself investigated this function in the cases μ > 0, ν > 0 in [60] and b = ν = 1, −1 < μ < 0 in [61]. In the case 0 < −μ < ν ≤ 2, this function was introduced, investigated, and employed for a treatment of several fractional calculus problems in [19, 32, 33]. The series from the right-hand side of (28) is absolutely convergent for all z ∈ ℂ if μ + ν > 0. If μ + ν = 0, the series is absolutely convergent for |z| < |ν|ν |μ|μ and for |z| = |ν|ν |μ|μ under the condition ℜ(a + b) > 2. Here we present some of the basic properties of the four parameter Wright function that were proved in [33]. Theorem 6 ([33]). Let L− be a cut in the complex ζ -plane along the negative real semiaxis, γ(ε; φ) (ϵ > 0, 0 < φ ≤ π) be a contour with nondecreasing arg ζ consisting of the following parts: 1) the ray arg ζ = −φ, |ζ | ≥ ε; 2) the arc −φ ≤ arg ζ ≤ φ of the circumference |ζ | = ε; 3) the ray arg ζ = φ, |ζ | ≥ ε. Then the integral representation ϕ((μ, a), (ν, b); z) =

1 ∫ eζ ζ −a Eν,b (zζ −μ ) dζ 2πi

(29)

γ(ε;φ)

with the generalized Mittag-Leffler function Eν,b in the kernel holds true under the conditions 0 < ε, π2 < φ ≤ π, −μ < ν, 0 < ν. Remark 5. In the case ν = b = 1, the four parameter Wright function is reduced to the Wright function and the integral representation (29) takes the form (5) that was obtained and used by Wright in [59, 61] for derivation of the asymptotic behavior of his function. He proved, in particular, that for −1 < μ < −1/3 the function ϕ(μ, a; z) possesses an “algebraic” asymptotic expansion on the positive real semi-axis: L−1

ϕ(μ, a; z) = ∑

l=0

z (a−1−l)/(−μ) (−μ)Γ(l + 1)Γ(1 + (a − l − 1)/(−μ))

+ O(z (a−1−L)/(−μ) ),

z → +∞, L = 0, 1, 2, . . . .

(30)

For the four parameter Wright function, a similar result is valid. Theorem 7 ([33]). Let 0 < ν/3 < −μ < ν ≤ 2, L = 0, 1, 2, . . . , P = 0, 1, 2, . . . . Then L−1

ϕ((μ, a), (ν, b); z) = ∑

l=0

z (a−1−l)/(−μ) (−μ)Γ(l + 1)Γ(b + ν(a − l − 1)/(−μ)) P

z −k + O(z (a−1−L)/(−μ) ) Γ(b − νk)Γ(a − μk) k=1

−∑

+ O(z −1−P ),

z → +∞.

(31)

252 | Yu. Luchko

3.2 Generalized Wright function Let us now discuss some basic properties of the generalized Wright function defined by the following series (in the case it is a convergent one): p Ψq [

∞ ∏p Γ(a + A k) k (a1 , A1 ), . . . , (ap , Ap ) z i i ; z] = ∑ i=1 , q k! (b1 , B1 ) . . . (bq , Bq ) Γ(b + B k) ∏ i i k=0 i=1

(32)

where ai ∈ ℂ, Ai ∈ ℝ, i = 1, . . . , p, bi ∈ ℂ, Bi ∈ ℝ, i = 1, . . . , q. Asymptotic expansion of this function (called generalized hypergeometric function) was treated by Fox in [13] and by Wright in [60] in the case of positive real parameters Ai , i = 1, . . . , p and Bi , i = 1, . . . , q. The convergence conditions and convergence radius of the series at the right-hand side of (32) immediately follow from the known asymptotics of the Euler Gammafunction (see, e. g., [20, 24, 25]). To formulate the results, let us first introduce the following notations: q

p

i=1

i=1

Δ = ∑ Bi − ∑ Ai ,

p

q

i=1

i=1

δ = ∏ |Ai |−Ai ∏ |Bi |Bi ,

q

p

i=1

i=1

μ = ∑ bi − ∑ ai +

p−q . 2

(33)

Then the series at the right-hand side of (32) is absolutely convergent for all z ∈ ℂ if Δ > −1. If Δ = −1, it is absolutely convergent for |z| < δ and for |z| = δ under the condition ℜ(μ) > 1/2 (see [25] for details). The Wright function (1), the generalized Mittag-Leffler function (13), and the four parameter Wright function (28) are particular cases of the generalized Wright function (32): ϕ(ρ, β; z) = 0 Ψ1 [

− ; z], (β, ρ)

Eα,β (z) = 1 Ψ1 [

ϕ((μ, a), (ν, b); z) = 1 Ψ2 [

(1, 1) ; z] (β, α)

(1, 1) ; z]. (μ, a), (ν, b)

Like in the case of the Wright function, the generalized Wright function can be represented in terms of the Fox H-function. Once again, the form of this representation depends on signs of the parameters Ai ∈ ℝ, i = 1, . . . , p, and Bi ∈ ℝ, i = 1, . . . , q. In the case all of them are positive, we have the following representation [25]: p Ψq [

(1 − ai , Ai )p1 (a1 , A1 ), . . . , (ap , Ap ) 1,p ( ; z] = Hp,q+1 (b1 , B1 ) . . . (bq , Bq ) (0, 1), (1 − bi , Bi )q1

󵄨󵄨 󵄨󵄨 󵄨󵄨 −z). 󵄨󵄨

For some advanced properties of the generalized Wright function including its new integral representations and the Laplace and Stieltjes transforms we refer the reader to the recent papers [46–48]. Moreover, in these papers, the Luke inequalities and some new Turán type inequalities for the generalized Wright function were derived, too.

The Wright function and its applications | 253

3.3 Wright type functions as pdfs The fundamental solutions to the Cauchy problems for some fractional diffusion equations can be expressed in terms of the Wright function and the generalized Wright function (see the Section 4). Because these equations describe the anomalous diffusion processes, their solutions are expected to be nonnegative and the fundamental solutions are expected to be probability density functions (pdf) in space evolving in time. Moreover, for some special parameter values, the Wright type functions are completely monotone. For the reader’s convenience, we first recall the definition of the completely monotone functions: A non-negative function f : (0, ∞) → ℝ is called a completely monotone if f ∈ C ∞ (0, ∞) and (−1)n f (n) (s) ≥ 0 for all n ∈ ℕ and s > 0. The basic property of the completely monotone functions is given by the Bernstein theorem (see, e. g., [55]), which says that a function f : (0, ∞) → ℝ is completely monotone if and only if it can be represented as the Laplace transform of a non-negative measure (non-negative function or generalized function). Well-known examples of completely monotone functions are the power funcα tion sλ , λ < 0, the exponential function e−as , 0 ≤ a, α ≤ 1, and the Mittag-Leffler function Eα,β (−s), 0 < α ≤ 1, α ≤ β. Using the known completely monotone functions, the technique of the Laplace transform, and the Bernstein theorem, one can prove non-negativity of some Wright type functions. Say, the function pα,β (t) = Γ(β)ϕ(−α, β − α; −t) can be interpreted as a one-sided pdf for 0 < α ≤ 1, α ≤ β [34]. To show this, we use the Laplace transform pair (14) that we rewrite in the form ϕ(−α, β − α; −t) ÷ Eα,β (−s),

0 0 of the pdf pα,β on ℝ+ : ∞

s

∞

∫ pα,β (t)t dt = ∫ Γ(β)ϕ(−α, β − α; −t)t s+1−1 dt = 0

0

Γ(β)Γ(s + 1) . Γ(β + αs)

(34)

254 | Yu. Luchko For β = 1, the pdf pα,β can be expressed in terms of the function Mα (t), 0 < α < 1 defined by (11). As is well known (see, e. g., [40]), Mα (t) can be interpreted as a one-sided pdf on ℝ+ with the moments given by the formula ∞

∫ Mα (t)t s dt = 0

Γ(s + 1) , Γ(1 + αs)

s > 0.

Let us now apply the same method to the Mittag-Leffler function α

f (s) = Eβ (−s 2 ),

0 < β ≤ 1, 0 < α ≤ 2, α + 2β < 4.

(35)

The Mittag-Leffler function h(s) = Eβ (−s) is completely monotone for 0 < β ≤ 1. Thus for α = 2 the function f (s) defined by (35) is completely monotone. Now let α α satisfy the inequalities 0 < α < 2. Then the function g(s) = s 2 is a Bernstein function α because its derivative g 󸀠 (s) = α2 s 2 −1 is completely monotone. But a composition of a completely monotone function and a Bernstein function is completely monotone (see, e. g., [55]). Thus the function f (s) = h(g(s)) is completely monotone for 0 < α < 2, too. In [32], a closed form formula for the inverse Laplace transform of the MittagLeffler function given by (35) was derived by employing the Mellin integral transform technique. The result is the following Laplace transform correspondence: α

(36)

Φα,β (t) ÷ Eβ (−s 2 ), α −1 2

α 2

β α

{ t ϕ((−β, 1 − β), ( α2 , α2 ); −t ) if < 21 , { { { { { { −1 β α −α { if α > 21 , { {t ϕ((β, 1), (− 2 , 0); −t 2 ) Φα,β (t) = { α −1 α { ∞ t2 πα k { { { { π ∑k=0 sin( 2 (k + 1))(−t 2 ) if 0 < t < 1, { { { { { α { t −1 ∞ {{ if t > 1, k)(−t − 2 )k − ∑ sin( πα 2 {{ π k=0

(37) if

β α

= 21 .

In formula (37), ϕ((a, μ), (b, ν); z) stands for the four parameter Wright function defined by (28). According to the Bernstein theorem, the function Φα,β (t) is nonnegative. It is also normalized (see [32] for a proof) and thus it can be interpreted as a one-sided pdf on ℝ+ . β

Remark 6. In the case α = 21 , 0 < β < 1, the pdf Φα,β (t) defined by the 3rd line of (37) has an integrable singularity at the point t = 1 [32]. Remark 7. Under the conditions 0 < β ≤ 1, 0 < α ≤ 2, α + 2β < 4, the pdf Φα,β given by (37) can be represented as the following Mellin–Barnes integral [32]: γ+i∞

Φα,β (t) =

Γ( α2 − α2 s)Γ(1 − α2 + α2 s) −s 2 1 t ds. ∫ 2β 2β α 2πi Γ(1 − + s)Γ(1 − s) α α γ−i∞

The Wright function and its applications | 255

In some cases, the Wright type functions can be represented as the Laplace transforms of the non-negative functions and thus they are not only non-negative, but completely monotone. In particular, in [34], the Mellin transform technique was employed to prove that the generalized Wright function β+γ+1

1 ( αβ , αβ ) ϕα,β,γ (s) = 1 Ψ1 [ ; −s] β+γ+1 1 ( , ) β β [ ]

(38)

is completely monotone under the conditions 0 < α ≤ 1, α1 − 1 < β, γ ∈ ℝ. In particular, for β = α1 , γ = − α1 formula (38) takes the form ∞

Γ(1 + k) (−s)k = Eα,α (−s) k!Γ(α + αk) k=0

ϕα,β,γ (s) = ∑

(39)

with the Mittag-Leffler function Eα,α (−s), known to be completely monotone for 0 < α ≤ 1. In [46–48], a series of results regarding non-negativity and complete monotonicity of the generalized Wright functions were derived. In particular, it was shown there that the following Wright functions are completely monotone: p Ψq [

s−λ p+1 Ψq [

(a1 , A), . . . , (ap , A) ; −s], (b1 , A) . . . (bq , A)

s > 0,

(λ, 1), (a1 , A), . . . , (ap , A) 1 ; − ], s (b1 , A) . . . (bq , A)

s > 0,

under the conditions 0 < a1 ≤ ⋅ ⋅ ⋅ ≤ ap , 0 < b1 ≤ ⋅ ⋅ ⋅ ≤ bq , 0 ≤ ∑kj=1 bj − ∑kj=1 aj , k = 1, . . . , p−1, 0 < ∑pj=1 bj −∑pj=1 aj . For more results regarding non-negativity and complete monotonicity of the generalized Wright function we refer the reader to [46–48].

4 Applications of the Wright function and the generalized Wright function 4.1 Asymptotic theory of partitions The Wright function appeared for the first time in Wright’s investigations related to the asymptotic theory of partitions. In [23], Hardy and Ramanujan derived a formula for the asymptotic expansion of the number of partitions of n, n ∈ ℕ for n → +∞. In [58], Wright considered a more general problem, namely, determination of an asymptotic expansion for the number of partitions of n, n ∈ ℕ into perfect kth powers, which he

256 | Yu. Luchko denoted by pk (n), k, n ∈ ℕ. Following Hardy and Ramanujan, Wright considered the generating function for the sequence {pk (1), pk (2), . . . } in the form ∞

k

fk (z) = ∏(1 − z l )

−1

∞

= 1 + ∑ pk (n)z n , n=1

l=1

|z| < 1.

Then pk (n) =

f (z) dz 1 ∫ k n+1 , 2πi z Cn

where the contour Cn is a circle with the center at the point z = 0 and the radius r = 1 − n1 . Let the contour be divided into a large number of small arcs, each associated with a point αp,q = exp(2pπi/q),

p, q ∈ ℕ.

Taking the arc associated with α0,1 = 1 as typical, it can be shown that on this arc the generating function fk (z) has the following representation: zr

1

Γ(1 + (1/k))ζ (1 + (1/k)) 1 2 (log ) exp( fk (z) ∼ ), 1 (k+1) z (log(1/z))1/k (2π) 2

z → 1,

(40)

where r is a real number depending on k and ζ (z) is the Riemann zeta-function. Then, on this arc, fk (z) can be approximated by an auxiliary function Fk (z) with a singularity at the point z = 1 of the type given by the right-hand side of the formula (40). If the z-plane is cut along the infinite ray (1, ∞), Fk (z) is regular and one-valued for all values of z except those on the cut. The power series for the auxiliary function Fk (z) has coefficients given in terms of an entire function that Wright denoted by ϕ(ρ, β; z) and, by using this power series, an asymptotic expansion can be found for the partition number pk (n). In [58], Wright derived some properties of the function ϕ(ρ, β; z) for the case ρ > 0 including its integral representation (5) and asymptotic behavior. Using these results he proved the following two theorems. Theorem 8. Let α, β, γ ∈ ℂ, α ≠ 0, ρ > 0, m ∈ ℕ, m > ℜ(γ) and F = F(z) be defined by ∞

F(z) = F(ρ, α, β, γ; z) := ∑ (n − γ)β−1 ϕ(ρ, β; α(n − γ)ρ )z n . n=m

(41)

Then the function F = F(z) is regular and one-valued in the interior of the domain ℂ \ (1, ∞). Theorem 9. Let G(z) = F(z) − χ(z),

The Wright function and its applications | 257

where F(z) is defined by (41) and χ(z) =

zγ α ). exp( β (log(1/z))ρ (log(1/z))

Then the function G = G(z) is regular and one-valued in the interior of the domain ℂ \ (−∞, 0). As we see, the function F = F(z) has a singularity of type χ(z) at the point z = 1. Now we consider the functions Fk = Fk (z) that are used to get an asymptotic expansion of the function pk (n). For Fk , the two theorems formulated above can be applied, namely, with the following parameter values: ρ=

1 , k

α = Γ(1 +

1 1 )ζ (1 + ), k k

1 β=− , 2

γ=

1 . 24

This leads to the desired asymptotic expansion of the function pk (n). Thus, the Wright function ϕ(ρ, β; z) was introduced as an auxiliary function for investigation of the partition number function pk (n). However, Wright recognized its potential importance for other applications and devoted several papers to this function and its generalizations including [58–61].

4.2 Fractional diffusion-wave equation The Wright function appeared in fractional calculus many years later its first application in the partition theory, namely, by attempts to derive a closed form formula for the fundamental solutions of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation [17, 36–39]: 𝜕α u(x, t) 𝜕2 u(x, t) = , 𝜕t α 𝜕x 2 where the time-fractional derivative 𝜕α u(x, t) ={ 𝜕t α

𝜕n u(x,t) , 𝜕t n t 1 ∫ (t Γ(n−α) 0

𝜕α u(x,t) 𝜕t α

−

0 < α ≤ 2,

(42)

is defined in the Caputo sense:

n τ)n−α−1 𝜕 u(x,τ) 𝜕τn

α = n ∈ ℕ, dτ,

n − 1 < α < n.

(43)

Equation (42) is referred to as the time-fractional diffusion and as the time-fractional diffusion-wave equation in the cases 0 < α ≤ 1 and 1 < α ≤ 2, respectively. The difference between these two cases can be seen in the formula for the Laplace transform of the Caputo fractional derivative of order α [39]: n−1 𝜕α u(x, t) 𝜕k u(x, t) 󵄨󵄨󵄨󵄨 ̃ s) − ∑ sα−1−k , ÷ sα u(x, 󵄨 α 𝜕t 𝜕t k 󵄨󵄨󵄨t=0+ k=0

n − 1 < α ≤ n, n ∈ ℕ.

(44)

258 | Yu. Luchko For equation (42) and 0 < α ≤ 1, we consider the Cauchy problem u(x, 0+) = g(x), −∞ < x < +∞;

u(∓∞, t) = 0, t > 0,

(45)

and the signaling problem u(x, 0+) = 0, x > 0;

u(0+, t) = h(t), u(+∞, t) = 0, t > 0.

(46)

̇ 0+), have to be added If 1 < α ≤ 2, the initial values of the first time-derivative u(x, to the conditions (45) and (46). To ensure the continuous dependence of the solutions on the parameter α in the transition from α = 1− to α = 1+, we agree to assume ̇ 0+) = 0. u(x, The problems mentioned above are well studied in the cases α = 1 and α = 2 so that in the further discussions we restrict ourselves to the case 0 < α < 2, α ≠ 1. For the sake of convenience, we use the abbreviation β=

α , 2

0 < β < 1.

(47)

In what follows, we focus on the fundamental solutions 𝒢c (x, t; β) and 𝒢s (x, t; β) for the Cauchy and signaling problems for equation (42), respectively, that are solutions to these problems with g(x) = δ(x) in (45) and h(t) = δ(t) in (46). To derive the fundamental solutions 𝒢c (x, t; β) and 𝒢s (x, t; β), the technique of the Laplace transform is used (see [36–39] for details). Remarkably, they can be expressed in terms of the function Mν (or the function Fν ) that is a particular case of the Wright function defined by (11): 𝒢c (x, t; β) = 𝒢s (x, t; β) =

r M (r), 2|x| β

βr M (r), t β

t > 0, r = |x|t −β ,

t > 0, x ≥ 0, r = xt −β .

(48) (49)

In the multi-dimensional case, the fundamental solutions to the Cauchy and signaling problems for the fractional diffusion-wave equation of type (42) have more complicated form compared to the one-dimensional case [4, 12, 22, 31] and are not directly connected to the Wright function. In contrary, the fundamental solution to the Cauchy problem for the multidimensional space-fractional diffusion-wave equation can be always written in terms of the generalized Wright function. In the rest of this subsection, we formulate the problem and give a short overview of the obtained results [4, 31]. Let us consider the Cauchy problem for the multi-dimensional space-fractional diffusion-wave equation in the following form: α 𝜕u (x, t) = −(−Δ) 2 u(x, t), 𝜕t

x ∈ ℝn , t > 0, 1 < α ≤ 2,

(50)

The Wright function and its applications | 259

α

where (−Δ) 2 is the fractional Laplacian that for sufficiently well-behaved functions f : ℝn → ℝ is defined as a pseudo-differential operator with the symbol |κ|α [53, 54]: α

(ℱ (−Δ) 2 f )(κ) = |κ|α (ℱ f )(κ),

(51)

where (ℱ f )(κ) is the Fourier transform of a function f at the point κ ∈ ℝn defined by (ℱ f )(κ) = f ̂(κ) = ∫ eiκ⋅x f (x) dx.

(52)

ℝn

For 0 < α < m, m ∈ ℕ and x ∈ ℝn , the fractional Laplacian can also be represented as a hypersingular integral [54]: 1

α

(−Δ) 2 f (x) =

dn,m (α)

∫ ℝn

(Δm h f )(x) dh |h|n+α

(53)

with the suitably defined finite differences operator (Δm h f )(x) and the normalization constant dn,m (α). In [4, 31], the first fundamental solution Gα,n (x, t) to the Cauchy problem for equation (50) was represented in the form of the following Mellin–Barnes integral: γ+i∞

Γ(s)Γ( αn − α2 s) |x| −2s 1 ( 1 ) ds, ∫ n Γ( n2 − s) (4π) 2 2πi 2t α n

Gα,n (x, t) =

2t − α

1 − α/2 < γ < 1.

γ−i∞

The integration contour in the last integral representation can be transformed to the loop L−∞ starting and ending at −∞ and encircling all poles of the function Γ(s) and then the integral can be represented as a series by applying the Jordan lemma and then the Cauchy residue theorem: n 2 2k (−1)k Γ( α + α k) |x| ( ) . n 1 n (4π) 2 k=0 k! Γ( 2 + k) 2t α n

Gα,n (x, t) =

∞

2t − α

∑

The last series can be represented as n

Gα,n (x, t) =

2t − α (4π)

n 2

1 Ψ1 [

( αn , α2 ) ( n2 , 1)

;−

|x|2 2

4t α

]

(54)

in terms of the generalized Wright function 1 Ψ1 defined by (32). It is easy to check that the generalized Wright function from the right-hand side of (54) is a particular case of the function given by (38). As a consequence, it is completely monotone with respect 2 to the variable s = |x|2 and thus the fundamental solution Gα,n is nonnegative (in fact, 4t α

it is a pdf in space evolving in time). The space-fractional diffusion equation (50) was investigated in detail in [44] in the one-dimensional case and discussed in [4, 21, 31] in the multi-dimensional case.

260 | Yu. Luchko

4.3 Scale-invariant solutions of the fractional diffusion-wave equation In this subsection, we consider special solutions to the one-dimensional fractional diffusion-wave equation that are invariant under the scaling transformations Tλ of the (x, t, u)-space in the form x̄ = λa x,

t ̄ = λb t,

ū = λc u,

(55)

where a, b, and c are some constants and λ is a real parameter restricted to an open interval I containing the value λ = 1. It is well known (see, e. g., [50]) that if a second order partial differential equation G(x, t, u, ux , ut , uxx , utt , uxt ) = 0

(56)

is invariant under Tλ given by (55) then the variable transformation u(x, t) = t c/b v(z),

z = xt −a/b ,

(57)

reduces equation (56) to a second order ordinary differential equation of the form g(z, v, v󸀠 , v󸀠󸀠 ) = 0.

(58)

For the fractional partial differential equations, the same method for determination of the scale-invariant solutions can be applied, too [6, 19, 33]. In the case of the onedimensional fractional diffusion-wave equations with both Caputo and Riemann– Liouville fractional derivatives, the scale-invariant solutions can be expressed in terms of the Wright and the generalized Wright functions. Here we present some results from [6, 19, 33]. Let us first consider the fractional diffusion-wave equation (42) with the Caputo fractional derivative given by (43) on the semi-axis (x ≥ 0). Its group of scaling transformations is described in the following theorem. Theorem 10 ([6]). Let Tλ be a one parameter group of scaling transformations for equation (42) of the form Tλ ∘ (x, t, u) = (λx, λb t, λc u). Then the invariants of the group Tλ are given by the expressions η1 (x, t) = xt −1/b = xt −α/2 , with b =

2 α

η2 (x, t, u) = t −c/b u = t −γ u,

(59)

and a real parameter γ = cα/2.

Remark 8. It is worth mentioning that the first scale-invariant η1 of (59) coincides with the similarity variable that is in argument of the fundamental solution to the signaling problem for equation (42). It is a consequence of the fact that equation (42) is invariant under the corresponding group of scaling transformations.

The Wright function and its applications | 261

The theory of the Lie groups and the previous theorem yield that the scaleinvariant solutions of equation (42) have the form u(x, t) = t γ v(y),

y = xt −α/2 ,

(60)

where the function v depends just on one variable and has to be determined. Furthermore, the general theory says that the substitution of u in the form (60) reduces the fractional diffusion-wave equation (42) to an ordinary fraction differential equation with an unknown function v. Theorem 11 ([6]). The scale-invariant solutions of the fractional diffusion-wave equation (42) have the form (60) and are determined by the ordinary fractional differential equation γ−n+1,α

(∗ P2/α

v)(y) = v󸀠󸀠 (y),

y > 0,

(61)

where ∗ Pδτ,α is the Caputo type modification of the left-hand sided Erdélyi–Kober fractional differential operator of order δ (0 < δ, n − 1 < α ≤ n ∈ ℕ) defined as follows: n−1 1 d (∗ Pδτ,α g)(y) := (Kδτ,n−α ∏(τ + j − u )g)(y), δ du j=0

y > 0,

∞

1 ∫ (u − 1)α−1 u−(τ+α) g(yu1/δ ) du, α > 0, (Kδτ,α g)(y) := { Γ(α) 1 g(y), α = 0,

(62) (63)

being the left-hand sided Erdélyi–Kober fractional integral operator. Remark 9. In the case α = n ∈ ℕ, equation (61) is a linear ordinary differential equation of order max{n, 2}. In the case α = 1 (the diffusion equation) (61) takes the form 1 v󸀠󸀠 (z) + yv󸀠 (y) − γv(y) = 0. 2

(64)

In the case α = 2 (the wave equation) equation (61) is reduced to the ordinary differential equation of the second order: (y2 − 1)v󸀠󸀠 (y) − 2(γ − 1)yv󸀠 (y) + γ(γ − 1)v(y) = 0.

(65)

The complete discussion of these cases one can find, e. g, in [50]. The case α = n ∈ ℕ, n > 2 was considered in [6]. In [6], the closed form solutions of equation (61) were determined in terms of the Wright function and the four parameter Wright function. Theorem 12 ([6]). The scale-invariant solutions of the fractional diffusion equation (42) with 0 < α ≤ 1 have the form α u(x, t) = C1 t γ ϕ(− , 1 + γ; −y) 2

(66)

262 | Yu. Luchko in the case −1 < γ, γ ≠ 0, and α u(x, t) = C1 ϕ(− , 1; −y) + C2 2

(67)

α

in the case γ = 0, where y = xt − 2 is the first scale invariant (59) and C1 , C2 are arbitrary constants. Theorem 13 ([6]). The scale-invariant solutions of the fractional diffusion-wave equation (42) with 1 < α < 2 have the form α u(x, t) = C1 t γ ϕ(− , 1 + γ; −y) 2 γ−1 γ−1 α 1 ); y2 )) + C2 t γ ( ϕ(− , 1 + γ; y) − y2+2 α ϕ((−α, 2 − α), (2, 3 + 2 2 2 α in the case 1 − α < γ < 1, γ ≠ 1 − α2 , γ ≠ 0, and α u(x, t) = C1 ϕ(− , 1; −y) 2 2 α 2 1 + C2 ( ϕ(− , 1; y) − y2− α ϕ((−α, 2 − α), (2, 3 − ); y2 )) + C3 2 2 α α

in the case γ = 0, where y = xt − 2 is the first scale invariant (59) and C1 , C2 , C3 are arbitrary constants. For the results regarding the scale-invariant solutions to the fractional diffusionwave equation with the Riemann–Liouville time- and space-fractional derivatives that are expressed in terms of the generalized Wright function we refer the interested reader to [19] and [33].

4.4 Subordination formulas for the fractional diffusion-wave equations Subordination principles are an important tool for investigation of properties of solutions to the Cauchy problems for the fractional partial differential equations based on solutions of equations of the same kind of higher order. In [3] and [2], the subordination formula ∞

Sβ (t)x = ∫ t −γ ϕ(−γ, 1 − γ; −st −γ )Sδ (s)x ds,

t > 0, x ∈ X

(68)

0

was proved for 0 < β < δ ≤ 2, γ = β/δ. In this formula, Sβ (t) denotes a solution operator to the Cauchy problem for the abstract fractional evolution equation in the form Dβ u(t) = Au(t),

u(0) = x,

u(k) (0) = 0,

k = 0, . . . , n − 1, x ∈ X,

(69)

The Wright function and its applications | 263

where Dβ is the Caputo fractional derivative of order β, n − 1 < β ≤ n, n ∈ ℕ and A is a linear closed unbounded operator densely defined in a Banach space X subject to some additional conditions (see [3] and [2] for details). It is worth mentioning that the Wright function ϕ(−γ, 1 − γ; −τ) is non-negative for τ ∈ ℝ+ and can be interpreted as a probability density function (see the previous section). In this subsection, we follow the presentation of [32] and discuss subordination formulas of type (68) for the Cauchy problem for the fractional evolution equation (69) α with the fractional Laplacian A = −(−Δ) 2 : β

α

Dt u(x, t) = −(−Δ) 2 u(x, t),

x ∈ ℝn , t > 0, 0 < α ≤ 2, 0 < β ≤ 2.

(70)

β

In (70), Dt denotes the Caputo time-fractional derivative of order β, β > 0 and the fractional Laplacian is defined as in (51). Because the problem under consideration is linear, its solution can be represented in the form u(x, t) = ∫ Gα,β,n (ζ , t)φ(x − ζ ) dζ ,

(71)

ℝn

where Gα,β,n is the so-called first fundamental solution to the fractional diffusionwave equation (70) that corresponds to the initial condition in the form of the Dirac δ-function. Thus the behavior of solutions to the problem under consideration is determined by the fundamental solution Gα,β,n and in this subsection we focus on subordination formulas for the fundamental solution Gα,β,n in the form ∞

Gα,β,n (x, t) = ∫ Φ(s, t)Gα,̂ β,n ̂ (x, s) ds,

(72)

0

where the kernel function Φ = Φ(s, t) can be interpreted as a probability density function in s, s ∈ ℝ+ for each value of t, t > 0. The subordination formulas for Gα,β,n can be deduced based on its Mellin–Barnes representations; they were derived in [29, 30] in the case β = α, in [5] in the case β = α/2, and in [4, 31] in the general case: βn

γ+i∞

1 t− α 1 Gα,β,n (x, t) = ∫ Kα,β,n (s)z −s ds, n α (4π) 2 2πi Kα,β,n (s) =

γ−i∞ s Γ( 2 )Γ( αn − αs )Γ(1 − αn + αs ) . β β Γ(1 − α n + α s)Γ( n2 − 2s )

z=

|x| β

2t α

,

(73) (74)

264 | Yu. Luchko Representing the kernel function (74) as a product of two factors and employing the Mellin convolution theorem lead to various integral representations of the fundamental solution Gα,β,n . In the case that one of the factors of this product corresponds to the fundamental solution with different parameters, we arrive at a subordination formula. Thus, to obtain the subordination formula of type (68), we compare the kernel functions Kα,β,n (s) and Kα,δ,n (s) defined by formula (74) with 0 < β < δ ≤ 2. Evidently, we can represent Kα,β,n (s) as a product of two factors: Kα,β,n (s) = Kα,δ,n (s) × Φ∗α,β,n (s),

Φ∗α,β,n (s) =

Γ(1 − Γ(1 −

δn α βn α

+ αδ s) β

+ α s)

.

(75)

The function Φα,β,n (τ) can be determined as the inverse Mellin integral transform of Φ∗α,β,n (s) and then represented as a series α α α (−1)k 1 (τ) δ k+ δ −n , β β δ k! Γ(1 − − k) k=0 δ δ

∞

Φα,β,n (τ) = ∑

(76)

which can be recognized to be a special case of the Wright function: Φα,β,n (τ) =

α β β α αδ −n τ ϕ(− , 1 − ; −τ δ ). δ δ δ

(77)

Applying now the Mellin convolution theorem and after some simplifications, we arrive at the subordination formula of type (68). The method described above can be applied to connect fundamental solutions with different orders of the space- and time-fractional derivatives. Theorem 14 ([32]). For the fundamental solution Gα,β,n (x, t) to the Cauchy problem for the multi-dimensional space-time-fractional diffusion-wave equation (70) with 0 < β ≤ 1, 0 < α ≤ 2, and 2β + α < 4 the following subordination formula is valid: ∞

2β

2β

Gα,β,n (x, t) = ∫ t − α Φα,β (st − α )G2,1,n (x, s) ds,

(78)

0

where the fundamental solution to the conventional diffusion equation is given by G2,1,n (x, t) =

1 |x|2 ) exp(− 4t (√4πt)n

and the kernel function Φα,β is a probability density function that is defined by formula (37) in terms of the four parameter Wright function. Remark 10. For the time-fractional diffusion equation (α = 2, 0 < β ≤ 1 in equation (70)) the subordination formula (78) with the kernel function Φα,β given by the 1st

The Wright function and its applications | 265

line of (37) is valid. In this case, the four parameter Wright function is reduced to the conventional Wright function and we arrive at the well-known formula ∞

G2,β,n (x, t) = ∫ t −β ϕ(−β, 1 − β; −st −β )G2,1,n (x, s) ds,

0 < β < 1.

(79)

0

For the space-fractional diffusion equation (β = 1, 0 < α ≤ 2 in equation (50)) the subordination formula (78) has to be applied with the kernel function Φα,β given by the 2nd line of (37). The four parameter Wright function from (37) is reduced to the conventional Wright function and we arrive at the subordination formula ∞

α α Gα,1,n (x, t) = ∫ s−1 ϕ(− , 0; −s− 2 t)G2,1,n (x, s) ds, 2

0 < α < 2.

(80)

0

For other subordination formulas, we refer to [44] and [45] in the case of the one-dimensional fractional diffusion-wave equations, to [32] in the case of the multidimensional fractional diffusion-wave equations, and to [1] in the case of the vectorvalued subordination principles for an abstract two-parametric family of fractional evolution equations. In all cases, the Wright type functions play a decisive role in formulation of the subordination formulas.

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Rudolf Gorenflo, Francesco Mainardi, and Sergei Rogosin

Mittag-Leffler function: properties and applications

Abstract: The chapter presents a survey of results on the properties and applications of the Mittag-Leffler function and its generalizations. Keywords: Mittag-Leffler function, integral transforms, fractional calculus, integral and differential equations of fractional order, fractional modelling MSC 2010: 33E12, 26A33, 34A08, 45K05, 44Axx, 60G22

1 Introduction In this chapter we present the basic properties of the two-parametric Mittag-Leffler function (or simply Mittag-Leffler function) zn , Γ(αn + β) n=0 ∞

Eα,β (z) := ∑

Re(α) > 0, β ∈ ℂ,

(1)

where Γ(p) is the Euler gamma-function, where an integral representation of Γ(p) is ∞ 1 = 0, k = 0, 1, 2, . . .. Γ(p) := ∫0 up−1 e−u du, p ∈ ℂ \ {0, −1, −2, . . .}, Γ(−k) The function (1) is the most straightforward generalization of the classical MittagLeffler function introduced by Magnus Gustaf (Gösta) Mittag-Leffler zn , Γ(αn + 1) n=0 ∞

Eα (z) := ∑

Re(α) > 0,

(2)

which in turn is a simple generalization of the exponential function, ∞ zn zn = ∑ . n! n=0 Γ(n + 1) n=0 ∞

exp(z) := ∑

Note: Rudolf Gorenflo (31.07.1930–20.10.2017). Acknowledgement: The authors appreciate constructive remarks and suggestions of the referees that helped to improve the manuscript. The work by FM has been carried out in the framework of the activities of the National Group of Mathematical Physics (INdAM-GNFM). The work by SR is supported by Belarusian Fund for Fundamental Scientific Research (grant F17MS-002). Rudolf Gorenflo, Department of Mathematics, Free University Berlin, Berlin, Germany Francesco Mainardi, Department of Physics and Astronomy, University of Bologna, Via Irnerio 46, I-40126 Bologna, Italy, e-mail: [email protected] Sergei Rogosin, Department of Economics, Belarusian State University, Nezavisimosti ave 4, BY-220030 Minsk, Belarus, e-mail: [email protected] https://doi.org/10.1515/9783110571622-011

270 | R. Gorenflo et al. The functions (1) and (2) have numerous generalizations mostly related to the fractional calculus and fractional modeling. Partly, the material of the chapter is based on the results from our monograph [16]. Due to the importance of the Mittag-Leffler function and its applications, the second edition of [16] is now in preparation. The (classical) Mittag-Leffler function has been introduced to give an answer to a classical question of complex analysis, namely, to describe the procedure of analytic continuation of power series outside the disc of their convergence. Much later, theoretical applications at the study of integral equations and more practical applications to the modeling of “non-standard” processes have been found for the Mittag-Leffler function. The importance of the Mittag-Leffler function was re-discovered when its connection to fractional calculus was fully understood. Different aspects of the distinguished role of this function in fractional theory and applications have been described in several monographs and surveys on fractional calculus (see, e. g., [7, 8, 14, 15, 17, 21, 34, 40]) and fractional modeling (see, e. g., [39, 4, 5, 18, 25, 3, 43–45]). In this chapter the main attention is paid to the results which can be useful for researchers of different subjects. Many other specific results can be found in [16], as well as in the survey papers [17, 28, 38], and in other sources where this function is helpful in solving specific problems.

2 History of the Mittag-Leffler function At the end of the nineteenth century Mittag-Leffler started to work on the problem of analytic continuation of monogenic functions of one complex variable. To solve this problem he had used the Laplace–Abel type integral +∞

∫ e−ω F(ωz)dω,

(3)

0 kν ν 1 ν where F(z) = ∑∞ ν=0 ν! z , lim supν→∞ √|kν | = r . Here r is the radius of convergence of the function whose analytic continuation was studied by Mittag-Leffler using his method. Mittag-Leffler proposed the following generalization of the Laplace–Abel integral: +∞

∫ e−ω Eα (ωα z)dω,

(4)

0

with Eα being the function (2). The properties of the latter were studied by him in the series of five notes published in 1901–1905 (see, e. g., [16, Ch. 2]). Nowadays, the function Eα is known as the Mittag-Leffler function (the results on it refer to item 33E12 in the 2000 Mathematics Subject Classification).

Mittag-Leffler function: properties and applications | 271

Practically at the same time several other functions were introduced related to the problem studied by Mittag-Leffler. Among them are the functions introduced by Le Roy [23] zn , (n!)p n=0 ∞

p > 0,

∑

(5)

by Lindelöf [24] zn , nαn n=0 ∞

∑

∞

∑(

n=0

α > 0, n

z ) , log(n + 1/α)

0 < α < 1,

(6) (7)

and by Malmquist [27] z n−2 , n n=2 Γ[1 + (log n)α ] ∞

∑

0 < α < 1.

(8)

The direct generalization of the Mittag-Leffler, namely the function (1), was proposed by Wiman in his work [46] on zeros of the function (2). Later this function was re-discovered and intensively studied by Agarval and Humbert (see, e. g. [16, Ch. 4]). With β = 1 this function coincides with the classical Mittag-Leffler function (2), i. e. Eα,1 = Eα . The next period in the development of the theory of the Mittag-Leffler function is connected with increasing of the number of parameters. Thus, a three-parametric Mittag-Leffler type function was introduced by Prabhakar [37], γ

(γ)n z n , n!Γ(αn + β) n=0 ∞

Eα,β (z) := ∑

Re(α) > 0, Re(β) > 0, γ > 0,

(9)

where (a)n = a(a + 1) ⋅ ⋅ ⋅ (a + n − 1), n ∈ ℕ, (a)0 = 1, a ≠ 0 is the so-called Pochhammer , whenever Γ(a) is defined. symbol. (a)n can be represented in the form (a)n = Γ(a+n) Γ(a) When γ = 1 this function coincides with the two-parametric Mittag-Leffler function (1) and with β = γ = 1 it coincides with the classical Mittag-Leffler function (2), i. e. 1 1 Eα,β = Eα,β , Eα,1 = Eα . Later, in relation to the solution of certain type fractional differential equations, Kilbas and Saigo introduced [19] another kind of three-parametric Mittag-Leffler type function, ∞

Eα,m,l (z) := ∑ cn z n , n=0

Re(α) > 0, m > 0, l ∈ ℂ,

(10)

Γ(α[jm+l]+1) with c0 = 1, cn = ∏n−1 j=0 Γ(α[jm+l+1]+1) , and parameters α, m, l such that α[jm + l] ≠ −1, −2, −3, . . . for any j = 0, 1, 2, . . .. With m = 1 this function reduces to the twoparametric Mittag-Leffler function (1), namely, Eα,1,l (z) = Γ(αl + 1)Eα,αl+1 (z).

272 | R. Gorenflo et al. Both the above functions are essentially used at an explicit representation of solutions to fractional integral and differential equations (see [21, 29]). All the above-mentioned Mittag-Leffler type functions are related to the general class of special function, the so-called Fox H-functions (see [20, 30]), 󵄨󵄨 (a , α ), . . . , (a , α ) 1 󵄨 m,n m,n m,n p p ]= Hp,q (z) = Hp,q (z) [z 󵄨󵄨󵄨 1 1 (s)z s ds, ∫ Hp,q 󵄨󵄨 (b1 , β1 ), . . . , (bq , βq ) 2πi

(11)

L

where L is a suitable path on the complex plane ℂ, z s is a suitably chosen singlevalued branch of the corresponding multi-valued function and m,n (s) = Hp,q m

A(s) = ∏ Γ(bj − βj s), j=1

A(s)B(s) , C(s)D(s) n

B(s) = ∏ Γ(1 − aj + αj s), j=1

q

C(s) = ∏ Γ(1 − bj + βj s), j=m+1

p

D(s) = ∏ Γ(aj − αj s). j=n+1

The relation of the Mittag-Leffler type functions to H-functions is due to the so-called Mellin–Barnes integrals (see [32] or [16, Appendix D]), the integrals I(z) =

1 ∫ f (s)z −s ds 2πi

(12)

L

with suitably chosen contour L and the function f being a ratio of the products of the Γ-functions. Further generalizations of the Mittag-Leffler function (multi-parametric MittagLeffler functions) were proposed along the same lines (see the detailed description in [16, Ch. 6]). Among them we have to point out the 2m-parametric function [1] (Luchko– Kilbas–Kiryakova function; see also [22]) ∞

E((α, β)m ; z) = ∑

zn

m n=0 ∏j=1 Γ(αj n

+ βj )

.

(13)

In particular, for m = 1 this definition coincides with definition of the two-parametric function (1), but the four-parametric function zn Γ(α1 n + β1 )Γ(α2 n + β2 ) n=0 ∞

Eα1 ,β1 ;α2 ,β2 (z) = ∑

(14)

is more close by its properties to the Wright function [16, Appendix F.2], zn . n!Γ(αn + β) n=0 ∞

ϕ(α, β; z) = ∑

(15)

Mittag-Leffler function: properties and applications | 273

A generalization of Prabhakar function (9) was given by Shukla and Prajapati [42]: (γ)κn z n , n!Γ(αn + β) n=0 ∞

γ,κ

Eα,β (z) := ∑

Re(α) > 0, Re(β) > 0, κ > 0, γ > 0.

(16)

This definition was combined with (13) by Saxena and Nishimoto [41] in the following definition of the generalized 2m + 2-parametric function: (γκ)n z n . m n=0 n! ∏j=1 Γ(αj n + βj ) ∞

γ,κ

E(α,β) (z) = ∑ m

(17)

Paneva-Konovska proposed another type of multi-parametric function (the 3m-parametric Mittag-Leffler function) [31] (γ1 )n . . . (γm )n z n . m n! n=0 ∏j=1 Γ(αj n + βj ) ∞

(γ ),m

E(αj ),(β ) (z) = ∑ j

j

(18)

Recently an interest in the Le Roy type function (5) rose again, namely, the function (called also the Gerhold–Garra–Polito function) (γ)

zn , (Γ(αn + β))γ n=0 ∞

Fα,β (z) = ∑

α, β, γ > 0,

(19)

has been introduced independently by Gerhold [13] and jointly by Garra and Polito [10] (see [12]). The interest in this function is due to certain applications. If γ = m is (m) a positive integer, then Fα,β is a special case of the 2m-parametric Mittag-Leffler function (13). There are some other generalizations of the Mittag-Leffler function mentioned, e. g., in [16, Ch. 6].

3 Analytic properties 3.1 Relations to elementary and simple special functions For particular values of parameters the Mittag-Leffler function (1) coincides with some elementary functions and simple special functions. In particular, ez − 1 1 , |z| < 1; E1,1 (z) = E1 (z) = exp z; E1,2 (z) = , 1−z z E2,1 (z 2 ) = E2 (z 2 ) = cosh z; E2,1 (−z 2 ) = E2 (−z 2 ) = cos z,

E0 (z) = E2,2 (z 2 ) =

sinh z ; z

E2,2 (−z 2 ) =

sin z ; z

1 E4 (z) = [cos z 1/4 + cosh z 1/4 ], 2

(20) (21) (22)

274 | R. Gorenflo et al. 1 1/3 1/3 √3 1 E3 (z) = [ez + 2e− 2 z cos( z 1/3 )], 2 2

E1/2 (±z 1/2 ) = ez [1 + erf(±z 1/2 )] = ez erfc(∓z 1/2 ),

(23) (24)

where the error function erf(z) and the complementary error function erfc(z) are defined by the formulas z

2 2 ∫ e−t dt, √π

erf(z) =

erfc(z) = 1 − erf(z).

0

There exist also more general formulas giving representations of the Mittag-Leffler function (2) with rational parameters: q−1

γ(1 − k/q, z) ], Γ(1 − k/q) k=1

Ep/q (z p/q ) = ez [1 + ∑ E1/q (z 1/q ) =

p−1

1 ∑ E (z 1/p e2πim/p ), p m=0 1/q

q = 2, 3, . . . ,

p = 1, 2, . . . , q = 2, 3, . . . .

(25) (26)

z

Here γ(p, z) is incomplete Gamma-function γ(p, z) = ∫0 up−1 e−u du, Re(p) > 0.

3.2 Recurrence, differential and integral relations Recurrence relations The following recurrence relations are helpful in analysis with the Mittag-Leffler function Eα,β (z): Eα,β+α (z) = zEα,β (z) − r−1

1 , Γ(β)

zn , Γ(β + nα) n=0

Eα,β+rα (z) = z r Eα,β (z) − ∑

Re(α) > 0, β ∈ ̸ −ℕ0 ,

Re(α) > 0, r ∈ ℕ, (β + nα) ∈ ̸ −ℕ0 .

(27) (28)

We have to mention also duplication formulas for Eα,β (z) Eα,β (z) + Eα,β (−z) = 2E2α,β (z 2 ),

Eα,β (z) − Eα,β (−z) = 2zE2α,α+β (z 2 ).

(29)

γ

For the Prabhakar function Eα,β (z) the recurrence relation has the form γ

γ

γ−1

zEα,α+β (z) = Eα,β (z) − Eα,β (z),

Re(α) > 0, Re(β) > 0, Re(α + β) > 0.

(30)

From this formula one can deduce the relations between special cases of the Prabhakar function and the Mittag-Leffler function, 2 αEα,β (z) = Eα,β−1 (z) + (1 + α − β)Eα,β (z),

Re(α) > 0, Re(β) > 1;

(31)

Mittag-Leffler function: properties and applications | 275

3 zEα,β (z) =

1 [E (z) − (2β − 3α − 3)Eα,β−α−1 (z) 2α2 α,β−α−2 + (2α2 + β2 − 3αβ + 3α − 2β + 1)Eα,β−α (z)],

(32)

where Re α > 0, Re(β) > 2 + Re(α) > 0, and 3 Eα,β (z) =

1 [E (z) − (2β − 3α − 3)Eα,β−1 (z) 2α2 α,β−2 + (2α2 + β2 − 3αβ + 3α − 2β + 1)Eα,β (z)],

(33)

where Re(α) > 0, Re(β) > 2. Recurrence relations for the Kilbas–Saigo function Eα,m,l (z) are valid if the parameters of this function satisfy the conditions α > 0;

m, l ∈ ℝ;

zEα,m,l+m (z) =

m > 0, α(jm + l) ≠ −1, −2, . . . , −n (∀j ∈ ℕ0 ).

(34)

Γ(αl + α + 1) [Eα,m,l (z) − 1], Γ(αl + 1)

n−1

z n Eα,m,l+m (z) = ∏ j=0

(35)

Γ(α(jm + l + 1) + 1) Γ(α(jm + l) + 1) n−1

k−1

k=0

j=0

× [Eα,m,l (z) − 1 − ∑ (∏

Γ(α(jm + l) + 1) )z k ], Γ(α(jm + l + 1) + 1)

n = 2, 3, . . . . (36)

Differential relations Differential properties of the Mittag-Leffler function are used in the study of fractional differential equations. They are obtained directly from a series representation of the Mittag-Leffler function and have a predicable form. We have n

(

d ) [z β−1 Eα,β (λz α )] = z β−n−1 Eα,β−n (λz α ), Re β > n, n ∈ ℕ, dz n

( n

(

d γ γ ) [z β−1 Eα,β (λz α )] = z β−n−1 Eα,β−n (λz α ), Re β > n, n ∈ ℕ, dz

n d ) [z n(l−m+1) En,m,l (λz nm )] = ∏[n(l − m) + j]z n(l−m) + λz nl En,m,l (λz nm ). dz j=1

(37) (38) (39)

The last relation is valid if the following conditions are satisfied: α = n ∈ ℕ;

m, l ∈ ℝ;

m > 0, n(jm + l) ≠ −1, −2, . . . , −n (∀j ∈ ℕ0 ).

(40)

276 | R. Gorenflo et al.

Integral relations By the direct integration of the series representation of the Mittag-Leffler function one can easily verify that the following integral relation holds: z

∫ Eα,β (λt α )t β−1 dt = z β Eα,β+1 (λz α )

(Re(α) > 0, Re(β) > 0);

(41)

0

furthermore, we have the more general relation z

1 ∫(z − t)μ−1 Eα,β (λt α )t β−1 dt = z μ+β−1 Eα,μ+β (λz α ) Γ(α)

(42)

0

(μ > 0, Re(α) > 0, Re(β) > 0), with integration along the straight line connecting the points 0 and z. Analogous to (41) the following formula is valid for the Prabhakar function: z

γ

γ

∫ Eα,β (λt α )t β−1 dt = z β Eα,β+1 (λz α )

(Re(α) > 0, Re(β) > 0, γ ∈ ℂ).

(43)

0

3.3 The Mittag-Leffler function as an entire function The Mittag-Leffler function (or the two-parametric Mittag-Leffler function) zn Γ(αn + β) n=0 ∞

Eα,β (z) = ∑

is an entire function of the variable z for any α, β ∈ ℂ, Re(α) > 0; the power series in its definition converges for each finite z and thus its sum is an analytic function in the whole complex plane ℂ since its radius of convergence R = lim

n→+∞

|an | =∞ |an+1 |

due to the asymptotics of the Gamma-function (Stirling formula) Γ(az + b) ≈ √2πe−az (az)az+b−1/2 ,

z → ∞.

By using the Stirling formula it is possible to calculate the order ρ and the type σ of the Mittag-Leffler function: ρ = lim sup n→+∞

n log n

log

1 |an |

,

σ=

1 lim sup(n|an |ρ/n ). eρ n→+∞

Mittag-Leffler function: properties and applications | 277

For the Mittag-Leffler function Eα,β (z) we have for any α, β (Re(α) > 0, β ∈ ℂ) ρ=

1 , Re(α)

σ = 1.

(44)

It means that for any ε > 0 there exist rε > 0 and a sequence zk = rk eiθk , rk → +∞, (θk ∈ [0, 2π) \ A, where A is relatively small exceptional set (see [16]), such that 󵄨󵄨 (σ+ε)r ρ iθ 󵄨 , ∀r > rε , θ ∈ [0, 2π), 󵄨󵄨Eα,β (re )󵄨󵄨󵄨 ≤ e ρ 󵄨󵄨 (σ−ε)rk iθ 󵄨 . 󵄨󵄨Eα,β (rk e k )󵄨󵄨󵄨 ≥ e

(45) (46)

γ

The Prabhakar function Eα,β (z) is an entire function for any α, β, γ ∈ ℂ, Re(α) > 0, and has the order and type ρ=

1 , Re(α)

σ = 1.

(47)

The Kilbas–Saigo function Eα,m,l (z) is an entire function, entailing the conditions (34), and it has the order and type ρ=

1 , α

σ=

1 . m

(48)

Zeros distribution Since the Mittag-Leffler function Eα,β (z) is an entire function it can have zeros (and it does have zeros in all cases, except α = β = 1, when E1,1 (z) = exp z). Let us formulate some results on the distribution of the zeros following [36]. Let us introduce the following constants: cβ = cβ =

α , Γ(β − α)

α , Γ(β − 2α)

dβ =

dβ =

α , Γ(β − 2α)

α , Γ(β − 3α)

τβ = 1 +

τβ = 2 +

1−β , α

1−β , α

β ≠ α − l, l ∈ ℤ+ ,

β = α − l, l ∈ ℤ+ , α ∈ ̸ ℕ.

(49) (50)

Let the values of parameters α, β satisfy one of the following relations: 1) 0 < α < 2, β ∈ ℂ, and β ≠ 0, −1, −2, . . . when α = 1; 2) α = 2, Re(β) > 3. Then all zeros zn of the Mittag-Leffler function Eα,β (z) with sufficiently large modulus are simple and the asymptotic relation holds for n → ±∞ (zn )1/α = 2πin − τβ α log 2πin +

dβ /cβ

(2πin)α

+ (τβ α)2

log cβ log 2πin − (τβ α)2 + αn , 2πin 2πin

(51)

278 | R. Gorenflo et al. where for α < 2 αn = O(

log |n| 1 log2 |n| ) + O( ) + O( ), |n|1+α |n|2α |n|2

(52)

but for α = 2 αn =

log |n| log2 |n| 1 e±iπβ ). ) + O( ) + O( + O( |n|2 cβ2 (2πn)−4τβ |n|1−4τβ |n|−8τβ

(53)

The situation with zeros of the Mittag-Leffler function for large values of the parameter α is slightly different. Let α > 2. Then all zeros zn of the Mittag-Leffler function Eα,β with sufficiently large modulus are simple and the following asymptotic formula holds: zn = (

α

π 1 β−1 (n − − ) + αn ) , sin π/α 2 α

(54)

where the sequence αn is defined as described below: 1) For all pairs (α, β) which are not mentioned in (49), (50) the remainder αn in (54) π 3π π satisfies the relation αn = O(e−πn(cos α −cos α )/ sin α ). π 2) If 2 < α < 4, then αn = O(n−α Re(τβ ) e−πn cot α ). π 3π π 3) If α ≥ 4, then αn = e−πn cot α (O(eπn cos α / sin α ) + O(n−α Re(τβ ) )). If β is real then all zeros zn with sufficiently large modulus are real too.

Asymptotic behavior of the Mittag-Leffler function We also mention asymptotic results for the Mittag-Leffler function Eα,β (z) which are basically a refinement of results by Dzhrbashyan (see [8]). For all 0 < α < 2, β ∈ ℂ, m ∈ ℕ the following asymptotic formulas hold: If |arg z| < min{π, πα}, then Eα,β (z) =

1 (1−β)/α z 1/α m z −k e −∑ z + O(|z|−m−1 ), α Γ(β − kα) k=1

|z| → ∞.

(55)

If 0 < α < 1, πα < |arg z| < π, then m

z −k + O(|z|−m−1 ), Γ(β − kα) k=1

Eα,β (z) = − ∑

|z| → ∞.

(56)

For all α ≥ 2, β ∈ ℂ, m ∈ ℕ the following asymptotic formula is valid: Eα,β (z) =

1 α

∑

|arg z+2πn|< 3πα 4 m

(z 1/α e2πin/α )

1−β z 1/α e2πin/α

z −k + O(|z|−m−1 ), Γ(β − kα) k=1

−∑

e

|z| → ∞.

(57)

Mittag-Leffler function: properties and applications | 279

For asymptotic formulas for other functions of Mittag-Leffler type we refer to our book [16].

3.4 Relations to other special functions First of all we mention the relation of the Mittag-Leffler function of half-integer parameter with the (generalized) hypergeometric function, Em/2 (z) = 0 Fm−1 ( ; +

1 2 m−1 , ,..., ; z) m m m

3m z 2 m+2 m+3 2(m+1)/2 z , ,..., ; ), 1 F2m−1 (1; 2m 2m 2m mm n!√π

m ∈ ℕ,

(58)

where the generalized hypergeometric function p Fq is defined by the formula p Fq (a1 , a2 , . . . , ap ; b1 , b2 , . . . , bq ; z)

(a1 )n (a2 )n ⋅ ⋅ ⋅ (ap )n z n (b1 )n (b2 )n ⋅ ⋅ ⋅ (bq )n n! n=0 ∞

= ∑

(59)

and (c)n is the Pochhammer symbol. As already mentioned in Section 2, the Mittag-Leffler function can be considered as a special case of general family of special functions, namely the H-functions (see (11)). The corresponding relation gives the following formula: 󵄨󵄨 (0, 1) 󵄨 1,1 ]. Eα,β (z) = H1,2 (z) [−z 󵄨󵄨󵄨 󵄨󵄨 (0, 1), (1 − β, α)

(60)

Such a relation can be rewritten in terms of generalized Wright function (or Wright– Fox function) p Wq (z), p, q ∈ ℕ0 , 󵄨󵄨 (1, 1) 󵄨 ], Eα,β (z) = 1 W1 [z 󵄨󵄨󵄨 󵄨󵄨 (β, α)

(61)

where the Wright–Fox function is defined by the following series: p Wq (z)

∞ ∏p Γ(a + α n) n 󵄨󵄨 (a , α ), . . . , (a , α ) z l l 󵄨 p p . ] = ∑ l=1 = p Wq [z 󵄨󵄨󵄨 1 1 q 󵄨󵄨 (b1 , β1 ), . . . , (bq , βq ) n! n=0 ∏j=1 Γ(bj + βj n)

(62)

γ

For the Prabhakar function Eα,β (z) the following relations with special functions are known: γ

β+m−1 z β β+1 1 ,..., ; m ), m ∈ ℕ, F (γ; , Γ(β) 1 m m m m m 󵄨 󵄨󵄨 (1 − γ, 1) 1 γ H 1,1 (z) [−z 󵄨󵄨󵄨 ], Eα,β (z) = 󵄨󵄨 (0, 1), (1 − β, α) Γ(γ) 1,2

Em,β (z) =

(63) (64)

280 | R. Gorenflo et al.

γ

Eα,β (z) =

(γ, 1) 󵄨󵄨󵄨󵄨 1 󵄨 z] . 1 W1 [ (β, α) 󵄨󵄨󵄨 Γ(γ)

(65)

If the conditions (40) are satisfied, then the Kilbas–Saigo function Eα,m,l (z) is related to generalized hypergeometric function En,m,l (z) = 1 Fn (1;

nl + 1 nl + 2 nl + n z , ,..., ; ). nm nm nm (nm)n

(66)

3.5 The Mittag-Leffler function of real variable Recall that a smooth function f on (0, +∞) is called completely monotonic if its successive derivatives have an alternating sign starting from positive (−1)k f (k) (x) ≥ 0,

∀x ∈ (0, +∞); k = 0, 1, 2, . . .

By the celebrated Bernstein theorem, the function f is completely monotonic if it is the Laplace transform ∞

f (x) = ∫ e−xt μ(dt) 0

for a positive σ-finite measure μ. The following results are known for the Mittag-Leffler function: the classical Mittag-Leffler function Eα (−x) with α ∈ [0, 1] and the two-parametric Mittag-Leffler function Eα,β (−x) with α ∈ [0, 1], β ≥ α, are completely monotonic for x ∈ (0, +∞). Feller (see [9]) discusses completely monotonic functions through their relationship with infinitely divisible measures, which are fundamental in defining Lévy processes, which are stochastic process with independent, stationary increments. He showed [9, p. 450] that the function ω is the Laplace transform of an infinitely divisible probability distribution iff ω = e−ψ , where ψ has a completely monotone derivative and ψ(0) = 0. In past several decades, Lévy processes and Lévy stable distributions have gained popularity in financial modeling, as well as in biology and physics; this is probably a reason for the increased interest in completely monotonic functions, too.

Integral transforms of the Mittag-Leffler function Here the results of application of classical integral transforms to the Mittag-Leffler function are presented (see, e. g., [6]), namely, the Laplace transform ∞

Lφ (t) = ∫ e−xt φ(x)dx, 0

Mittag-Leffler function: properties and applications | 281

the Mellin transform ∞

Mψ (s) = ∫ ψ(τ)τs−1 dτ, 0

and the Fourier transform +∞

Fϕ (κ) = ∫ eitκ ϕ(t)dt. −∞

The one most used in applications is the following form of the Laplace transform of the two-parameter Mittag-Leffler function: ∞

∫ e−at Eα,β (λt α )t β−1 dt = 0

aα−β aα − λ

󵄨󵄨 λ 󵄨󵄨 󵄨 󵄨 (Re(α) > 0, Re(β) > 0, Re(a) > 0, 󵄨󵄨󵄨 α 󵄨󵄨󵄨 < 1). 󵄨󵄨 a 󵄨󵄨

(67)

Similar to the Laplace transform formula we have for the Prabhakar function ∞

γ

∫ e−at Eα,β (λt α )t β−1 dt = 0

aαγ−β (aα − λ)γ

󵄨󵄨 λ 󵄨󵄨 󵄨 󵄨 (Re(α) > 0, Re(β) > 0, Re(a) > 0, 󵄨󵄨󵄨 α 󵄨󵄨󵄨 < 1). (68) 󵄨󵄨 a 󵄨󵄨

Laplace transforms of the Mittag-Leffler function can be represented via the Volterra function (see [2, Ch. 6] and [11]), ∞

1 t u+α uβ μ(t, β, α) = du ∫ Γ(β + 1) Γ(u + α + 1)

(Re(β) > −1).

0

From the Mellin–Barnes integral representation of the two-parametric MittagLeffler function (see, e. g., [16]) we arrive at the following formula for the Mellin transform of this function: ∞

M{Eα,β (−t); s} = ∫ Eα,β (−t)t s−1 dt = 0

Γ(s)Γ(1 − s) Γ(β − αs)

(0 < Re(s) < 1).

(69)

For the Prabhakar function the analogous formula has the form γ

∞

M{Eα,β (−t); s} = ∫ Eα,β (−t)t s−1 dt = 0

Γ(s)Γ(γ − s) Γ(γ)Γ(β − αs)

(0 < Re(s) < γ).

(70)

To conclude this subsection, we consider the Fourier transform of the twoparameter Mittag-Leffler function Eα,β (|t|) with α > 1: F{|t|Eα,α+β (|t|); κ} = −

2 1 W [− κ2 2 1 κ2

󵄨󵄨 (2, 2), (1, 1) 󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 (α + β, 2α)

(α > 1, β ∈ ℂ),

where 2 W1 is a Wright function (62) with parameters p = 2, q = 1.

(71)

282 | R. Gorenflo et al.

3.6 Relations to fractional calculus Here we present a few formulas related to the values of the fractional integrals and derivatives of the two-parameter Mittag-Leffler function. Let us start with the left-sided Riemann–Liouville integral: α (I0+ f (τ))(t) :=

t

1 f (τ)dτ ∫ Γ(α) (t − τ)1−α

(t > 0).

0

Let Re(α) > 0, Re(β) > 0, Re(γ) > 0, λ ∈ ℝ. Then by using the series representation and the left-sided Riemann–Liouville integral of the power function we get α γ−1 τ Eβ,γ (λτβ ))(t) = t α+γ−1 (Eβ,α+γ (λt β )), (I0+

(72)

and, in particular, if λ ≠ 0, then α γ−1 τ Eα,γ (λτα ))(t) = (I0+

t γ−1 1 (Eα,γ (λt α ) − ). λ Γ(γ)

(73)

Analogously, one can calculate the right-sided fractional Riemann–Liouville (or Liouville) integral (I−α f (τ))(t)

+∞

f (τ)dτ 1 := ∫ Γ(α) (τ − t)1−α t

of the two-parameter Mittag-Leffler function in the case Re(α) > 0, Re(β) > 0, λ ∈ ℝ, λ ≠ 0; we have (I−α τ−α−γ Eβ,γ (λτ−β ))(t) = t −γ (Eβ,α+γ (λt −β )).

(74)

If we suppose additionally that Re(α + γ) > Re(β), then this formula can be rewritten as (I−α τ−α−γ Eβ,γ (λτ−β ))(t) =

t β−γ 1 (Eβ,α+γ−β (λt −β ) − ), λ Γ(α + γ − β)

(75)

1 t α−β (Eα,β (λt −α ) − ). λ Γ(β)

(76)

and, in particular, (I−α τ−α−β Eα,β (λτ−α ))(t) =

Fractional differentiation of the two-parameter Mittag-Leffler function is presented here by the formulas only for the Riemann–Liouville derivative of order α, Re(α) > 0 with non-integer real part m − 1 < Re(α) < m (more relations can be found in [16]). First we calculate the left-sided Riemann–Liouville derivative: (Dα0+ f (τ))(t)

t

1 dm f (τ)dτ . := ∫ Γ(m − α) dt m (t − τ)α+1−m 0

Mittag-Leffler function: properties and applications | 283

(Dα0+ τγ−1 Eβ,γ (λτβ ))(t) = t γ−α−1 (Eβ,γ−α (λt β )),

Re(α) > 0, Re(β) > 0, λ ∈ ℝ.

(77)

If we assume an extra condition on the parameters, namely Re(γ) > Re(β), Re(γ) > Re(α + β), λ ≠ 0, then the following relations hold: (Dα0+ τγ−1 Eβ,γ (λτβ ))(t) =

t γ−α−1 + λt γ−α+β−1 Eβ,γ−α+β (λt β ). Γ(γ − α)

(78)

In particular (see [19]), for Re(α) > 0, Re(β) > Re(α) + 1, one can prove (Dα0+ τβ−1 Eα,β (λτα ))(t) =

t β−α−1 + λt β−1 (Eα,β (λt α )). Γ(β − α)

(79)

Finally, the right-sided (Liouville) fractional derivative (Dα− f (τ))(t)

+∞

(−1)m dm f (τ)dτ := ∫ Γ(m − α) dt m (τ − t)α+1−m t

of the two-parameter Mittag-Leffler function satisfies the relation (see, e. g., [21, p. 86]) (Dα− τα−β Eα,β (λτ−α ))(t) =

t −β + λt −α−β (Eα,β (λt −α )), Γ(β − α)

(80)

valid for all Re(α) > 0, Re(β) > Re(α) + 1.

4 Application to fractional order equations 4.1 Solution to integral equations Among the applications of the Mittag-Leffler function to the solution of integral equations we have to single out the Abel equation of the second kind (see [14]) t

u(t) +

λ u(τ) dτ = f (t), ∫ Γ(α) (t − τ)1−α

α > 0, λ ∈ ℂ.

(81)

0

In terms of the fractional integral operator such an equation reads α ((1 + λI0+ )u(τ))(t) = f (t).

(82)

The Abel integral equations occur in many situations where physical measurements are to be evaluated. In many of these, the independent variable is the radius of a circle or a sphere and only after a change of variables the integral operator has the form I α , usually with α = 1/2, and the equation is of the first kind. For instance, there

284 | R. Gorenflo et al. are applications in the evaluation of spectroscopic measurements of cylindrical gas discharges, the study of the solar or a planetary atmosphere, the investigation of star densities in a globular cluster, the inversion of travel times of seismic waves for the determination of terrestrial sub-surface structure, spherical stereology. Descriptions and analysis of several problems of this kind can be found in the books by Gorenflo and Vessella [15]. Another field in which the Abel integral equations or integral equations with more general weakly singular kernels are important is the study of inverse boundary value problems in partial differential equations, in particular parabolic ones, in which naturally the independent variable has the meaning of time. The Abel integral equation (82) can be formally solved as follows: ∞

α αn ) f (τ))(t) = ((1 + ∑ (−λ)n I0+ u(t) = ((1 + λI0+ )f (τ))(t). −1

(83)

n=1

The formula is obtained by using standard technique of the successive approximation method. Convergence of the Neumann series simply follows for any function f ∈ αn C[0, a] (cf., e. g., [15, p. 130]). Note that (I0+ f (τ))(t) = Φαn (t) ∗ f (t) = ∞ 1 ∫ (t Γ(αn) 0

τ)αn−1 f (τ)dτ, +

− solution reads

t+αn−1 Γ(αn)

∗ f (t) =

where v+ = v if v > 0, v+ = 0 if v < 0. Thus the formal ∞

u(t) = f (t) + ( ∑ (−λ)n n=1

t+αn−1 ) ∗ f (t). Γ(αn)

(84)

We recall n

(−λt α ) , Γ(αn + 1) n=0 ∞

eα (t; λ) := Eα (−λt α ) = ∑ ∞

∑ (−λ)n

n=1

t > 0, α > 0, λ ∈ ℂ,

t+αn−1 d = E (−λt α ) = eα󸀠 (t; λ), Γ(αn) dt α

t > 0.

(85) (86)

Finally, the (formal) solution can be written as u(t) = f (t) + eα󸀠 (t; λ) ∗ f (t).

(87)

Observing that, because of the rapid growth of the gamma function, the infinite series in (84) and (86) are uniformly convergent in every bounded interval of the variable t, we conclude that the term-wise integration and differentiation are well defined. Following [14] one can use the alternative technique of the Laplace transform, which will allow us to obtain the solution in different forms, including the result (87). Applying the Laplace transform to (81) we obtain [1 +

λ sα ̃ sα−1 ̃ ̃ (s) = ̃f (s) 󳨐⇒ u ̃ (s) = α ]u f (s) = s α f (s). α s s +λ s +λ

(88)

Mittag-Leffler function: properties and applications | 285

By proceeding with the inverse Laplace transform using a Laplace transform pair, eα (t; λ) := Eα (−λt α ) ÷

sα−1 󸀠 sα−1 , e (t; λ) ÷ s , sα + λ α sα + λ

(89)

we obtain the following representation of the solution to (81): t

u(t) = f (t) + eα󸀠 (t; λ) ∗ f (t) = f (t) + ∫ f (t − τ)eα󸀠 (τ; λ)dτ.

(90)

0

Formally, one can apply integration by parts and rewrite (90) (note that this formula is more restrictive with respect to condition on the given function f ): t

u(t) = f (t) + ∫ f 󸀠 (t − τ)eα (τ; λ)dτ + f (0+)eα (t; λ).

(91)

0

If the function f is continuous on the interval [0, a] then formula (87) (or, what is the same, (90)) gives the unique continuous solution to the Abel integral equation of the second kind (81) for any real λ. The existence of the integral in (87) follows for any absolutely integrable function f . A number of integral equations which are reduced to the Abel integral equation of the first or the second kind are presented in the Handbook of Integral Equations [35]. Among them we can single out the following equations: t 1o (92) ∫(1 + b√t − τ)y(τ)dτ = f (t), b = const, 0

2o

which is reduced to the Abel integral equation of the second kind by differentiating in t; t

∫(b + 0

1 )y(τ)dτ = f (t), √t − τ

b = const,

(93)

which can be solved as a combination of the Abel integral equation of the first and the second kind.

4.2 Solution to fractional ordinary differential equations One of the leading solution methods for linear fractional ordinary differential equations is their (equivalent) reduction to certain Volterra integral equations in proper functional spaces. An extended technique based on this method is presented in the monograph [21] (see also [16, Ch. 7]). Other methods discussed, e. g., in [21] are the compositional method, the operational method, and the integral transforms method.

286 | R. Gorenflo et al. It was noted, in particular, that different types of fractional derivatives involved in the equations lead to different kinds of initial conditions. For instance, if a differential equation contains the Riemann–Liouville fractional derivative, then the natural initial conditions are the so-called Cauchy type conditions: β

Da+k (a+) = bk ; but in the case of the Caputo derivatives it is natural to pose the standard Cauchy conditions. We give below two examples of the solution to initial problems for fractional ordinary differential equations involving the Mittag-Leffler function. A lot of other fractional ordinary differential equations possessing a closed form solution are presented in the monograph [21]. First consider a simple linear ordinary differential equation with one Riemann–Liouville fractional derivative, (Dαa+ y(τ))(t) − λy(t) = f (t)

(a < t ≤ b; α > 0; λ ∈ ℝ).

(94)

Standard initial conditions for such an equation are so-called Cauchy type initial conditions, (Dα−k a+ y(τ))(a+) = bk

(bk ∈ ℝ, k = 1, . . . , n = −[−α]),

(95)

where [⋅] means the integral part of a number. If we suppose that the right-hand side in (94) is Hölder-continuous, i. e. f ∈ Cγ [a, b], 0 ≤ γ < 1, γ < α, then (see [21, p. 172]) the Cauchy type problem (94)–(95) is equivalent in the space Cn−α [a, b] to the Volterra integral equation n

y(t) = ∑ j=1

bj (t − a)α−j

Γ(α − j + 1)

+

t

t

a

a

1 λ y(τ) f (τ) dτ + dτ. ∫ ∫ Γ(α) (t − τ)1−α Γ(α) (t − τ)1−α

(96)

Here and in what follows C0 [a, b] means simply the set of continuous functions on [a, b], i. e. C[a, b]. One can solve equation (94) by the method of successive approximations (for justification of this method in the case considered, see, e. g. [21, pp. 172, 222]). If we set n

y0 (t) = ∑ j=1

bj

Γ(α − j + 1)

(t − a)α−j ,

(97)

then we get the recurrent relation t

t

a

a

y (τ) f (τ) 1 λ dτ. ym (t) = y0 (t) + ∫ m−1 1−α dτ + ∫ Γ(α) (t − τ) Γ(α) (t − τ)1−α

(98)

Mittag-Leffler function: properties and applications | 287

It can be rewritten in terms of the Riemann–Liouville fractional integrals, α α ym (t) = y0 (t) + λ(Ia+ ym−1 (τ))(t) + (Ia+ f (τ))(t).

(99)

Performing successive substitutions one can obtained from (99) the following formula for the mth approximation ym to the solution of (96): n

m+1

j=1

k=1

ym (t) = ∑ bj ∑

t

m λk−1 (t − a)αk−j λk−1 + ∫[ ∑ (t − τ)αk−1 ]f (τ)dτ. Γ(αk − j + 1) Γ(αk) k=1

(100)

a

Taking the limit as m → ∞ we get the solution of the integral equation (96) (and thus of the Cauchy type problem (94)–(95)) n

t

∞ λk−1 λk−1 (t − a)αk−j + ∫[ ∑ (t − τ)αk−1 ]f (τ)dτ, y(t) = ∑ bj ∑ Γ(αk − j + 1) Γ(αk) j=1 k=1 k=1 ∞

(101)

a

or n

t

∞ λk (t − a)αk+α−j λk + ∫[ ∑ (t − τ)αk+α−1 ]f (τ)dτ. Γ(αk + α − j + 1) Γ(αk + α) k=0 k=0 ∞

y(t) = ∑ bj ∑ j=1

(102)

a

The latter yields the following representation of the solution to (94)–(95) in terms of the Mittag-Leffler function: n

α−j

y(t) = ∑ bj (t − a) j=1

α

t

Eα,α−j+1 [λ(t − a) ] + ∫(t − τ)α−1 Eα,α [λ(t − τ)α ]f (τ)dτ.

(103)

a

Another important problem for linear ordinary differential equation is the Cauchy problem for FDE with one fractional derivative of Caputo type, (C Dαa+ y(τ))(t) − λy(t) = f (t) (k)

y (a) = bk

(a ≤ t ≤ b; n − 1 ≤ α < n; n ∈ ℕ; λ ∈ ℝ), (bk ∈ ℝ, k = 1, . . . , n − 1),

(104) (105)

where the Caputo fractional derivative C Dαa+ is defined by the following formula: C

(

Dαa+ y(τ))(t)

t

1 y(n) (τ)dτ := ∫ Γ(n − α) (t − τ)α+1−n

(n − 1 < α < n).

0

In this case it is possible to pose initial conditions in the same form as that for ordinary differential equations (i. e. by involving usual derivatives) due to the properties of the Caputo derivative (see the corresponding discussion in [16, Appendix E]).

288 | R. Gorenflo et al. Under the assumption f ∈ Cγ [a, b], 0 ≤ γ < 1, γ < α, the Cauchy problem (104)– (105) is equivalent (see, e. g., [21, pp. 172, 230]) to the Volterra integral equation, n−1

bj

t

t

a

a

1 λ y(τ) f (τ) y(t) = ∑ (t − a) + dτ + dτ. ∫ ∫ 1−α j! Γ(α) Γ(α) (t − τ) (t − τ)1−α j=0 j

(106)

By applying the method of successive approximations with the initial approximation bj j y0 (t) = ∑n−1 j=0 j! (t − a) we get the solution to the Volterra equation (106) (and thus to the Cauchy problem (104)–(105)) in the form n−1

t

∞ λk−1 λk (t − a)αk+j + ∫[ ∑ (t − τ)αk−1 ]f (τ)dτ. y(t) = ∑ bj ∑ Γ(αk + j + 1) Γ(αk) j=0 k=1 k=0 ∞

(107)

a

The latter yields the following representation of the solution to (104)–(105) in terms of the Mittag-Leffler function: n

t

j=1

a

y(t) = ∑ bj (t − a)j Eα,j+1 [λ(t − a)α ] + ∫(t − τ)α−1 Eα,α [λ(t − τ)α ]f (τ)dτ.

(108)

4.3 Solution to fractional partial differential equations (main types) Classification of linear and non-linear partial differential equations of fractional order is still far from complete. Several results for partial differential equations are described in [21]. Anyway this area is rapidly growing, since most of the results are related to different types of applications. It is impossible to describe all existing results. Partly, they are presented in [16]. We also note that many authors have applied methods of fractional integro-differentiation to constructing solutions of ordinary and partial differential equations, to investigating integro-differential equations, and to obtaining a unified theory of special functions. Recently a number of books presenting results in this area were published (e. g., [4, 7, 25, 43, 44]). One can find in these references different aspects of the theory and applications of partial fractional differential equation. Anyway, we have to note that this branch of analysis is far from complete. The simplest partial differential equation with a Riemann–Liouville fractional derivative is the so-called fractional diffusion equation, (Dα0+,t u)(x, t) = λ2

𝜕2 u 𝜕x 2

(x ∈ ℝ; t > 0; λ > 0; α > 0).

(109)

Here Dα0+,t is the Riemann–Liouville fractional derivative of order α with respect to time t.

Mittag-Leffler function: properties and applications | 289

Partial differential equations with the Riemann–Liouville fractional derivative in time are supplied by initial conditions which are known as Cauchy type conditions. Let us consider equation (109) for 0 < α < 2. The Cauchy type initial conditions then have the form (Dα−k 0+,t u)(x, 0+) = fk (x),

x ∈ ℝ,

(110)

where k = 1 if 0 < α < 1, and two conditions with k = 1, 2 if 1 < α < 2. The problem (109)–(110) is solved usually by the method of integral transforms. Let us apply to equation (109) successively the Laplace transform with respect to time ∞ variable t, Lt {u(x, t); s} = ∫0 u(x, t)e−st dt (x ∈ ℝ; (s) > 0), and the Fourier transform +∞

with respect to spatial variable x, Fx {u(x, t); κ} = ∫−∞ u(x, t)eixκ dx (σ ∈ ℝ; t > 0). Then l

α−j

α j−1 Lt {Dα−k 0+,t u(x, t); s} = s Lt {u(x, t); s} − ∑ s (D0+,t u)(x, 0+) j=1

(x ∈ ℝ),

(111)

where l − 1 < α ≤ l, l ∈ ℕ. In the considered case (0 < α < 2) we take into account the initial conditions (110) and we get from (109) k

sα Lt {u(x, t); s} = ∑ sj−1 fj (x) + λ2 ( j=1

𝜕2 L {u(x, t); s}). 𝜕x 2 t

(112)

Here either k = 1 or k = 2. For the Fourier transform the following relation is well known: Fx {[

𝜕2 u (x, t)]; κ} = −|κ|2 Fx {u(x, t); κ}. 𝜕x 2

(113)

Thus applying the Fourier transform to (112) we get for k = 1 or k = 2, k

Fx Lt {u(x, t); (κ, s)} = ∑ j=1

sj−1 sα + λ2 |κ|2

(κ ∈ ℝ; s > 0).

(114)

In order to obtain an explicit solution to problem (109)–(110) we use the inverse Fourier and the inverse Laplace transform and the corresponding tables of these transforms. The final result reads (see, e. g., [21, Thm. 6.1]): Let 0 < α < 2 and λ > 0. Then the formal solution of the Cauchy type problem (109)–(110) is represented in the form k +∞

u(x, t) = ∑ ∫ Gjα (x − τ, t)fj (τ)dτ,

(115)

j=1 −∞

where k = 1 if 0 < α < 1, and k = 2 if 1 < α < 2, Gjα (x, t) =

α α |x| 1 α/2−j t ϕ(− , − j + 1; − t −α/2 ) 2λ 2 2 λ

(j = 1, 2)

(116)

290 | R. Gorenflo et al. ϕ(α, β; z) is the classical Wright function (15). The formal solution (115) becomes the exact one if the integrals in the right-hand side of (115) converge. Another example is the solution to the Cauchy problem for the partial fractional differential equation with the Caputo fractional derivative, (C Dα0+,t u)(x, t) = λ2

𝜕2 u 𝜕x2

(x ∈ ℝ; t > 0; λ > 0; 0 < α < 2).

(117)

This equation is a particular case of the fractional diffusion-wave equation (C Dα0+,t u)(x, t) = λ2 Δx u(x, t) (x ∈ ℝn ; t > 0; λ > 0; 0 < α < 2).

(118)

Equation (117) is supplied by the Cauchy condition(s) 𝜕k u (x, 0) = fk (x) 𝜕xk

(119)

(x ∈ ℝ),

where k = 0, if 0 < α < 1, and two conditions with k = 0, 1, if 1 < α < 2; the 0th order derivative means the value of the solution u at points (x, 0). To solve the Cauchy problem (119) for the fractional differential equation (117) we use the same method as in Section 4.2. We apply first the Laplace integral transform with respect to time variable t using the relation k−1

(Lt CDα0+,t u)(x, s) = sα (Lt u)(x, s) − ∑ sα−j−1 j=0

𝜕j u (x, 0), 𝜕t j

(120)

and then the Fourier transform with respect to spatial variable x. Then in view of the Cauchy initial condition(s) we obtain from equation (117) k−1

(Fx Lt u)(σ, s) = ∑

j=0

sα

sα−j−1 (F f )(σ). + λ2 |σ|2 x k

(121)

By using the inverse Fourier and the inverse Laplace transform and the corresponding tables of these transforms we get the final result (see, e. g., [21, Thm. 6.3]): Let 0 < α < 2 and λ > 0. Then the formal solution of the Cauchy problem (117), (119) is represented in the form k−1 +∞

u(x, t) = ∑ ∫ Gjα (x − τ, t)fj (τ)dτ,

(122)

j=0 −∞

where k = 1 if 0 < α < 1, and k = 2 if 1 < α < 2, Gjα (x, t) =

α α |x| 1 j−α/2 t ϕ(− , j + 1 − ; − t −α/2 ) 2λ 2 2 λ

(j = 0, 1)

(123)

ϕ(α, β; z) is the classical Wright function (15). The formal solution (122) becomes the exact one if the integrals in the right-hand side of (122) converge.

Mittag-Leffler function: properties and applications | 291

5 Fractional modeling with the Mittag-Leffler function 5.1 Deterministic models The following differential equation of fractional order α > 0, with the Caputo fractional derivative Dα∗ (in the notation of [14]) m−1 k

Dα∗ u(t) := Dα0+ (u(t) − ∑

k=0

t (k) + u (0 )) = −u(t) + q(t), k!

t > 0, m − 1 < α ≤ m,

(124)

providing the initial conditions u(k) (0+ ) = ck ,

k = 0, 1, . . . , m − 1,

(125)

is called the simple fractional relaxation equation and the simple fractional oscillation equation, whenever 0 < α < 1 and 1 < α < 2, respectively. The application of the Laplace transform to (124)–(125) yields m−1

̃ = ∑ ck u(s) k=0

sα−k−1 1 ̃ q(s). + sα + 1 sα + 1

(126)

By using the notation sα−1 , sα + 1

(127)

k = 0, 1, . . . , m − 1,

(128)

eα (t) ≡ eα (t; 1) := Eα (−t α ) ÷ uk (t) := J k eα (t) ÷

sα−k−1 , sα + 1

t

k where J k := I0+ is a k-times repeated integral, J 1 u(t) = ∫0 u(τ)dτ. Inversion of the Laplace transform gives the representation of the solution to (124)–(125) m−1

t

m−1

t

0

k=0

0

u(t) = ∑ ck uk (t) − ∫q(t − τ)u󸀠0 (τ)dτ = ∑ ck uk (t) − ∫q(t − τ)eα󸀠 (τ)dτ. k=0

(129)

Here we use the relation sα

1 sα−1 = −(s α − 1) ÷ −u󸀠0 (t) = −eα󸀠 (t), u0 (0+ ) = eα (0+ ) = 1. +1 s +1

Besides, the m functions uk (t) = J k eα (t) with k = 0, 1, . . . , m − 1 represent those particular solutions of the homogeneous equation (124) which satisfy the initial conditions (0+ ) = δk,h , h, k = 0, 1, . . . , m − 1, and thus they are fundamental solutions to this u(h) k equation.

292 | R. Gorenflo et al. An analysis of the solution (129) is given, e. g., in [16] (see also [25, 26]). We have to mention also that the Mittag-Leffler function is used in the study of fractional models of linear visco-elasticity, in particular, in the representation of the creep compliance J(t) and the relaxation modulus G(t). These models are studied by the Laplace transform method e. g. in [25]. The most general form of the corresponding constitutive equation is the fractional operator equation for stress σ = σ(t) and strain ϵ = ϵ(t), p

[1 + ∑ ak k=1

q

dνk d νk ]σ(t) = [m + ∑ bk ν ]ϵ(t), ν dt k dt k k=1

(130)

with νk = k + ν − 1, and tν ν { J {1 − E [−(t/τ ) ]} + J J(t) = J + , ∑ n ν ϵ,n + g { { Γ(1 + ν) n { t −ν { {G(t) = G + ∑ G E [−(t/τ )ν ] + G , n ν σ,n − e Γ(1 − ν) { n where all the coefficients are non-negative.

5.2 Stochastic models Taking into account the complete monotonicity of the Mittag-Leffler function Pillai [33] has introduced the probability distribution (which he called the Mittag-Leffler distribution). The theory of a process that has been devised by Pillai [33] as an increasing Lévy process on the spatial half-line x ≥ 0 occurring in natural time t ≥ 0. Let us consider the probability distribution function, Fα (x) = 1 − Eα (−x α ),

x ≥ 0, 0 < α ≤ 1,

(131)

d E (−xα ), dx α

x ≥ 0, 0 < α ≤ 1.

(132)

and its density fα (x) = −

Their Laplace transforms (denoting by ξ the Laplace parameter corresponding to x) are ξ α−1 1 1 , = F̃α (ξ ) = − ξ 1 + ξ α ξ (1 + ξ α )

̃f (ξ ) = α

1 . 1 + ξα

(133)

According to Feller [9], the distribution Fα (x) is infinite divisible if its density can be written as ̃fα (ξ ) = exp(−gα (ξ )), ξ ≥ 0, where gα (ξ ) is (in a more modern terminology) a Bernstein function, meaning that gα (ξ ) is non-negative and has a completely

Mittag-Leffler function: properties and applications | 293

monotone derivative. Here we have gα (ξ ) = ln(1 + ξ α ) ≥ 0 so gα󸀠 (ξ ) = gα󸀠 (ξ )

αξ α−1 . 1+ξ α

As a con-

sequence the derivative turns out to be a completely monotone function being a product of two completely monotone functions and thus it follows that gα (ξ ) is a Bernstein function. We can define a stochastic process x = x(t) in the half-line x ≥ 0 occurring in time t ≥ 0 by its density fα (x, t) (density in x evolving in t) taking ̃f (ξ , t) = α

1 −t t = (1 + ξ α ) = (̃fα (ξ )) . (1 + ξ α )t

(134)

For a Laplace inversion of ̃fα (ξ , t) we write for ξ > 1 ∞ ̃f (ξ , t) = ξ −αt (1 + ξ −α )−t = ∑ (−t )ξ −α(t+k) . α k k=0

(135)

Then, using the correspondence xα(t+k)−1 ÷ ξ −α(t+k) , Γ(α(t + k))

(136)

we obtain finally the distribution function of the considered process, ∞

Fα (x, t) = ∑ (−1)k k=0

Γ(t + k) xα(t+k) , k!Γ(t)Γ(α(t + k) + 1)

(137)

Γ(t + k) xα(t+k)−1 . k!Γ(t)Γ(α(t + k))

(138)

and its density, ∞

fα (x, t) = ∑ (−1)k k=0

Another stochastic model, where the Mittag-Leffler function is used, is the renewal model of the Mittag-Leffler type. The standard Poisson process is generalized by replacing the exponential function (as a waiting time density) by the Mittag-Leffler function (taking into account its power law behavior at infinity). The corresponding renewal process can be called the fractional Poisson process or the Mittag-Leffler waiting time process. By taking the diffusion limit of the Mittag-Leffler renewal process one can arrive at the space-time fractional diffusion equation. More details of these models can be found in [16, Ch. 9].

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[26] F. Mainardi and R. Gorenflo, On Mittag-Leffler functions in fractional evolution processes, J. Comput. Appl. Math., 118 (2000), 283–299. [27] J. Malmquist, Sur le calcul des integrales d’un système d’équations différentielles par la methodé de Cauchy–Lipschitz, Ark. Mat. Astron. Fys., 1 1903, 149–156. [28] A. M. Mathai, Mittag-Leffler function, Ch. 2, in A. M. Mathai and H. J. Haubold (eds.), An Introduction to Fractional Calculus, pp. 19–46, Nova Science Publishers, 2017. [29] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. [30] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function. Theory and Applications, Springer, Dordrecht, 2010. [31] J. Paneva-Konovska, From Bessel to Multi-Index Mittag-Leffler Functions Enumerable Families. Series in them and Convergence, World Scientific, Singapore, 2016. [32] R. B. Paris and D. Kaminski, Asymptotic and Mellin–Barnes Integrals, Cambridge Univ. Press, Cambridge, 2001. [33] R. N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Stat. Math., 42(1) (1990), 157–161. [34] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [35] A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, 2nd ed., CRC Press, Boca Raton, FL, 2008. [36] A. Yu. Popov and A. M. Sedletskii, Zeros distribution of Mittag-Leffler functions, Contemp. Math., Fundam. Dir., 40 (2011), 3–171 (Russian); translation in J. Math. Sci., 190 (2013), 209–409. [37] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15. [38] S. Rogosin, The role of the Mittag-Leffler function in fractional modeling, Mathematics, 3 (2015), 368–381. [39] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer Verlag, Berlin etc., 2007. [40] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, New York and London, Gordon and Breach Science Publishers, 1993 (Extended edition of the Russian original, Nauka i Tekhnika, Minsk, 1987). [41] R. K. Saxena and K. Nishimoto, N-Fractional Calculus of Generalized Mittag-Leffler functions, J. Fract. Calc., 37 (2010), 43–52. [42] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811. [43] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, vol. I. Background and Theory, Springer & Higher Education Press, Berlin & Beijing, 2013. [44] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, vol. II. Applications, Springer & Higher Education Press, Berlin & Beijing, 2013. [45] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer Verlag, New York, 2003. [46] A. Wiman, Über die Nullstellen der Funktionen Eα (x), Acta Math., 29 (1905), 217–234.

Richard B. Paris

Asymptotics of the special functions of fractional calculus Abstract: A summary of the large-variable asymptotic theory of integral functions of hypergeometric type is presented. This theory provides a basis for the asymptotic description of several special functions that find application in the field of fractional calculus. These include the one-, two- and three-parameter Mittag-Leffler functions, the generalised Bessel function, and the Wright function. Attention is given to the determination of the coefficients that appear in the exponential expansions of these functions. An algorithm for the calculation of these coefficients is described in the Appendix. The chapter concludes with a discussion of the main asymptotic properties of the Fox H-function. Keywords: Integral functions of hypergeometric type, asymptotic expansions, MittagLeffler functions, generalised Bessel function, Wright function, Fox H-function MSC 2010: 30E15, 33C70, 33E20, 34E05, 41A60

1 Introduction In this chapter we consider the asymptotic expansion of several special functions that find application in the field of Fractional Calculus. These functions include the MittagLeffler function and the Wright function, together with its generalisations, and the Fox H-function. The common approach we shall use throughout to discuss the first group of functions is to employ the asymptotic theory of integral functions of hypergeometric type. This theory has a long history and was developed in the first half of the last century by Ford [4], Newsom [10], Hughes [6], Wright [28] for a more general class of integral function, and there appeared a long and detailed study by Braaksma [2] for integral functions of hypergeometric type; for details, see [20, § 2.3]. It is with this last-mentioned development of the theory that we shall be concerned here. We first present a summary of the asymptotic theory of integral functions of hypergeometric type p Ψq (z) (sometimes referred to as a generalised Wright function). We then apply these results to each of the specialised cases of relevance to fractional calculus. We conclude the chapter with a discussion of the main asymptotics of the Fox H-function.

Richard B. Paris, Abertay University, Dundee, DD1 1HG, UK, e-mail: [email protected] https://doi.org/10.1515/9783110571622-012

298 | R. B. Paris

2 The asymptotic theory of p Ψq (z) for large |z| The integral function of hypergeometric type p Ψq (z) is defined by ∞ (α1 , a1 ), . . . , (αp , ap ) 󵄨󵄨󵄨󵄨 zn Ψ (z) ≡ Ψ ( g(n) z) = , ∑ 󵄨 p q p q (β1 , b1 ), . . . , (βq , bq ) 󵄨󵄨󵄨 n! n=0

g(n) =

∏pr=1 Γ(αr n + ar ) ∏qr=1 Γ(βr n + br )

(1) (2)

,

where p and q are nonnegative integers. The parameters αr and βr are real and positive and ar and br are arbitrary complex numbers. We also assume that the αr and ar are subject to the restriction αr n + ar ≠ 0, −1, −2, . . .

(n = 0, 1, 2, . . . ; 1 ≤ r ≤ p)

(3)

so that no gamma function in the numerator in (1) is singular. In the special case αr = βr = 1, the function p Ψq (z) reduces to a multiple of the ordinary hypergeometric function p Ψq (z) =

∏pr=1 Γ(ar )

p Fq ∏qr=1 Γ(br )

a , . . . , ap ( 1 ; z) ; b1 , . . . , bq

see, for example, [22, p. 40]. We introduce the parameters associated1 with g(n) in (2) given by p

q

κ = 1 + ∑ βr − ∑ αr , p

r=1

q

r=1

p

q

r=1

r=1

h = ∏ αrαr ∏ βr−βr ,

1 ϑ = ∑ ar − ∑ br + (q − p), 2 r=1 r=1

ϑ󸀠 = 1 − ϑ.

(4)

If it is supposed that αr and βr are such that κ > 0 then p Ψq (z) is uniformly and absolutely convergent for all finite z. If κ = 0, the sum in (1) has a finite radius of convergence equal to h−1 , whereas for κ < 0 the sum is divergent for all nonzero values of z. The parameter κ will be found to play a critical role in the asymptotic theory of p Ψq (z) by determining the sectors in the z-plane in which its behaviour is either exponentially large, algebraic or exponentially small in character as |z| → ∞. In this section we state the standard asymptotic expansions of the integral function p Ψq (z) as |z| → ∞ for κ > 0 and finite values of the parameters given in [2] and [28]; see also [18, § 2.3]. 1 Empty sums and products are to be interpreted as zero and unity, respectively.

Asymptotics of the special functions of fractional calculus | 299

We first introduce the exponential expansion ℰp,q (z) and the algebraic expansion Hp,q (z) associated with p Ψq (z). The expansion ℰp,q (z) can be obtained from the Ford– Newsom theorem [4, 10], for which a simpler derivation in the case p Ψq (z) based on the Abel–Plana form of the well-known Euler–Maclaurin summation formula is given in [20, pp. 42–50]. We have the formal asymptotic sum ϑ Z

∞

Z = κ(hz)1/κ ,

−j

ℰp,q (z) := Z e ∑ Aj Z , j=0

(5)

where the coefficients Aj are those appearing in the inverse factorial expansion of g(s)/s! given by M−1 Aj ρM (s) g(s) s = κ(hκκ ) { ∑ + }. 󸀠 + j) Γ(1 + s) Γ(κs + ϑ Γ(κs + ϑ󸀠 + M) j=0

(6)

Here g(s) is defined in (2) with n replaced by s, M is a positive integer and ρM (s) = O(1) for |s| → ∞ in | arg s| < π. The leading coefficient A0 is specified by 1

1

p

a − 21

A0 = (2π) 2 (p−q) κ− 2 −ϑ ∏ αr r r=1

q

1

∏ βr2 r=1

−br

.

(7)

The coefficients Aj are independent of s and depend only on the parameters p, q, αr , βr , ar and br . An algorithm for their evaluation when κ > 0 is described in the Appendix. The algebraic expansion Hp,q (z) follows from the Mellin–Barnes integral representation [18, § 2.4] ∞i

1 −s ∫ Γ(s)g(−s)(ze∓πi ) ds, p Ψq (z) = 2πi −∞i

1 󵄨 󵄨󵄨 󵄨󵄨arg(−z)󵄨󵄨󵄨 < π(1 − κ), 2

(8)

where the path of integration is indented near s = 0 to separate2 the poles of Γ(s) from those of g(−s) situated at s = (ar + k)/αr ,

k = 0, 1, 2, . . . (1 ≤ r ≤ p).

(9)

In general there will be p such sequences of simple poles though, depending on the values of αr and ar , some of these poles could be multiple poles or even ordinary points if any of the Γ(βr s + br ) are singular there. Displacement of the contour to the right over the poles of g(−s) then yields the algebraic expansion of p Ψq (z) valid in the sector in (8). 2 This is always possible when the condition (3) is satisfied.

300 | R. B. Paris If it is assumed that the parameters are such that the poles in (9) are all simple we obtain the algebraic expansion given by p

−1 −am /αm Hp,q (z) := ∑ αm z Sp,q (z; m) m=1

(10)

and Sp,q (z; m) denotes the formal asymptotic sum 󸀠p (−)k k + am ∏r=1 Γ(ar − αr (k + am )/αm ) −k/αm z , ) q Γ( k! αm ∏r=1 Γ(br − βr (k + am )/αm ) k=0 ∞

Sp,q (z; m) := ∑

(11)

with the prime indicating the omission of the term corresponding to r = m in the product. This expression in (10) consists of (at most) p expansions each with the leading behaviour z −am /αm (1 ≤ m ≤ p). When the parameters αr and ar are such that some of the poles are of higher order, the expansion (11) is invalid and the residues must then be evaluated according to the multiplicity of the poles concerned; this will lead to terms involving log z in the algebraic expansion. The three main expansion theorems are as follows. Throughout we let ϵ denote an arbitrarily small positive quantity. Theorem 1. If 0 < κ < 2, then ℰp,q (z) + Hp,q (ze∓πi ) Ψ (z) ∼ { p q Hp,q (ze∓πi )

(| arg z| ≤ 21 πκ),

( 21 πκ + ϵ ≤ | arg z| ≤ π),

(12)

as |z| → ∞. The upper or lower sign is chosen according as arg z > 0 or arg z < 0, respectively. It is seen that the z-plane is divided into two sectors, with a common vertex at z = 0, by the rays arg z = ± 21 πκ. In the sector | arg z| < 21 πκ, the asymptotic character of 1 p Ψq (z) is exponentially large whereas in the complementary sector 2 πκ < | arg z| ≤ π, the dominant expansion of p Ψq (z) is algebraic in character. On the rays arg z = ± 21 πκ the expansion ℰp,q (z) is oscillatory and is of comparable significance to the algebraic expansion. Theorem 2. If κ = 2 then p Ψq (z)

∼ ℰp,q (z) + ℰp,q (ze∓2πi ) + Hp,q (ze∓πi )

(13)

as |z| → ∞ in the sector | arg z| ≤ π. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. The rays arg z = ± 21 πκ now coincide with the negative real axis. It follows that p Ψq (z) is exponentially large in character as |z| → ∞ except in the neighbourhood of the negative real axis, where the algebraic expansion becomes asymptotically significant.

Asymptotics of the special functions of fractional calculus | 301

Figure 1: The exponentially large and algebraic sectors associated with p Ψq (z) in the complex z-plane with θ = arg z when 0 < κ < 1. The Stokes and anti-Stokes lines are indicated.

Theorem 3. When κ > 2 we have3 p Ψq (z)

N

∼ ∑ ℰp,q (ze2πin ) + Hp,q (ze∓πi ) n=−N

(14)

as |z| → ∞ in the sector | arg z| ≤ π. The integer N is chosen to be the smallest integer satisfying 2N + 1 > 21 κ and the upper or lower sign is chosen according as arg z > 0 or arg z < 0, respectively. In this case the asymptotic behaviour of p Ψq (z) is exponentially large for all values of arg z and, consequently, the algebraic expansion may be neglected. The sums ℰp,q (ze2πin ) are exponentially large (or oscillatory) as |z| → ∞ for values of arg z satisfying | arg z + 2πn| ≤ 21 πκ. The division of the z-plane into regions where p Ψq (z) possesses exponentially large or algebraic behaviour for large |z| is illustrated in Figure 1. When 0 < κ < 2, the exponential expansion ℰp,q (z) is still present in the sectors 21 πκ < | arg z| < min{π, πκ}, where it is subdominant. The rays arg z = ±πκ (0 < κ < 1), where ℰp,q (z) is maximally subdominant with respect to Hp,q (ze∓πi ), are called Stokes lines.4 As these rays are crossed (in the sense of increasing | arg z|) the exponential expansion switches off according to Berry’s now familiar error-function smoothing law [1]; see [13] for details. The rays arg z = ± 21 πκ, where ℰp,q (z) is oscillatory and comparable to Hp,q (ze∓πi ), are called anti-Stokes lines. 3 In [27], the expansion was given in terms of the two dominant expansions only, viz. ℰp,q (z) and ℰp,q (ze∓2πi ), corresponding to n = 0 and n = ±1 in (14). 4 The positive real axis arg z = 0 is also a Stokes line where the algebraic expansion is maximally subdominant.

302 | R. B. Paris In view of the above interpretation of the Stokes phenomenon a more precise version of Theorem 1 is as follows. Theorem 4. When 0 < κ ≤ 2, then ℰp,q (z) + Hp,q (ze∓πi ) (| arg z| ≤ min{π − ϵ, πκ − ϵ}), { { { ∓πi ) (min{πκ + ϵ, π} ≤ | arg z| ≤ π; 0 < κ ≤ 1), p Ψq (z) ∼ {Hp,q (ze { { ∓2πi ∓πi {ℰp,q (z) + ℰp,q (ze ) + Hp,q (ze ) (| arg z| ≤ π; 1 < κ ≤ 2),

(15)

as |z| → ∞. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. We omit the expansion on the Stokes lines arg z = ±πκ when 0 < κ < 1 and the exponentially small contribution on the Stokes lines arg z = ±π when κ = 1; the details in the case p = 1, q ≥ 0 are discussed in [15, Section 4]. The expansions5 in (15a) and (12a) were given by Wright [27, 28] in the sector | arg z| ≤ min{π, 32 πκ − ϵ} as he did not take into account the Stokes phenomenon. Since ℰp,q (z) is exponentially small in 21 πκ < | arg z| ≤ π, then in the sense of Poincaré, the expansion ℰp,q (z) can be neglected and there is no inconsistency between Theorems 1 and 4. Similarly, ℰp,q (ze−2πi ) is exponentially small compared to ℰp,q (z) in 0 ≤ arg z < π and there is no inconsistency between the expansions in (12a) and (15c) when 1 < κ < 2. However, in the vicinity of arg z = π, these last two expansions are of comparable magnitude and, for real parameters, they combine to generate a real result on this ray. A similar remark applies to ℰp,q (ze2πi ) in −π < arg z ≤ 0. The following theorem was given by Braaksma [2, p. 331]. Theorem 5. If p = 0, so that g(s) has no poles and κ > 1, then H0,q (z) ≡ 0. When 1 < κ ≤ 2, we have the expansion 0 Ψq (z)

∼ ℰ0,q (z) + ℰ0,q (ze∓2πi )

(16)

as |z| → ∞ in the sector | arg z| ≤ π The upper or lower sign is chosen according as arg z > 0 or arg z < 0, respectively. The dominant expansion 0 Ψq (z) ∼ ℰ0,q (z) holds in the reduced sector | arg z| ≤ π − ϵ. It can be seen that (16) agrees with (15c) when Hp,q (z) ≡ 0. Braaksma gave the result (16) valid in a sector straddling the negative real axis given by π − δ ≤ arg z ≤ π + δ, where 0 < δ < 21 π(1 − 21 κ). The expansion ℰ0,q (z) in (16) is exponentially large in the sector | arg z| < 21 πκ. The other expansion ℰ0,q (ze−2πi ) is subdominant in the upper half-plane but combines with ℰ0,q (z) on the negative real axis to produce (for real parameters) a real expansion. 5 In displayed formulae consisting of several lines, such as (15), we indicate the equation referred to by the addition of the letters a, b, c in an obvious order.

Asymptotics of the special functions of fractional calculus | 303

Since the exponential factors associated with ℰ0,q (z) and ℰ0,q (ze−2πi ) are exp[|Z|eiθ/κ ] and exp[|Z|ei(θ−2π)/κ ], respectively, where θ = arg z and we recall that Z is defined in (5), the greatest difference between these factors occurs when θ θ − 2π sin( ) = sin( ); κ κ that is, when θ = π(1 − 21 κ). Consequently, as arg z decreases in the upper half-plane, the expansion ℰ0,q (ze−2πi ) in (16) switches off across the Stokes line arg z = π(1 − 21 κ). Thus, when 1 < κ < 2, the expansion of 0 Ψq (z) on the positive real axis is described by ℰ0,q (z). Similar considerations apply to ℰ0,q (ze2πi ) and the Stokes line arg z = −π(1− 21 κ) in the lower half-plane; see [16] for details.

3 The Mittag-Leffler function Eα,β (z) We now apply the above general theory summarised in Section 2 to the first of the special functions encountered in fractional calculus. The (two-parameter) Mittag-Leffler function Eα,β (z) is defined by zn , Γ(αn + β) n=0 ∞

(17)

Eα,β (z) = ∑

where we consider α > 0. Special cases of this function are E1,2 (z) = z −1 (ez − 1),

E2,2 (z) = z −1/2 sinh z 1/2 ,

and E1,β (z) = z 1−β ez P(β − 1, z),

(18)

where P(a, z) = γ(a, z)/Γ(a) denotes the (lower) normalised incomplete gamma function [11, (8.2.4)]. The series in (17) corresponds to a case of 1 Ψ1 ((1, 1); (α, β); z) with g(s) in (2) given by g(s) = Γ(1 + s)/Γ(αs + β) and the parameters κ = α, h = α−α , ϑ = 1 − β. Then from (5) and (7), we have Z = z 1/α , A0 = 1/α and, from (6), it is seen that Aj = 0 for j ≥ 1. The exponential and algebraic expansions are from (5), (10) and (11) given by ℰ1,1 (z) =

1 (1−β)/α z exp[z 1/α ], α

∞

z −k . Γ(β − αk) k=1

H1,1 (ze∓πi ) = − ∑

Then, from Theorems 2, 3 and 4, we obtain the following asymptotic expansions as |z| → ∞:

304 | R. B. Paris (i) When 0 < α < 1 { 1 z (1−β)/α exp[z 1/α ] − ∑∞ k=1 Eα,β (z) ∼ { α ∞ z −k − ∑k=1 Γ(β−αk) {

z −k Γ(β−αk)

(| arg z| ≤ πα − ϵ),

(πα + ϵ ≤ arg z ≤ π);

(19)

(ii) when 1 < α < 2 1 (1−β)/α z z exp[z 1/α ] − ∑∞ (| arg z| ≤ π − ϵ), { k=1 Γ(β−αk) α { { Eα,β (z) ∼ { α1 z (1−β)/α exp[z 1/α ] + α1 (ze∓2πi )(1−β)/α exp[(ze∓2πi )1/α ] { { ∞ z −k { − ∑k=1 Γ(β−αk) (| arg z| ≤ π); −k

(20)

(iii) when α = 2 Eα,β (z) ∼

1 1 (1−β)/α (1−β)/α 1/α z exp[z 1/α ] + (ze∓2πi ) exp[(ze∓2πi ) ] α α ∞ z −k −∑ (| arg z| ≤ π); Γ(β − αk) k=1

(21)

(iv) when α > 2 Eα,β (z) ∼

∞ z −k 1 N (1−β)/α exp[z 1/α e2πin/α ] − ∑ ∑ (ze2πin ) α n=−N Γ(β − αk) k=1

(22)

(| arg z| ≤ π), where N is the smallest integer6 satisfying 2N + 1 > 21 α. The upper or lower signs are taken according as arg z > 0 or arg z < 0, respectively. In the particular case α = 1, it follows from the expansion of the incomplete gamma function [11, (8.2.5), (8.11.2)] that the expansion of E1,β (z) in (18) is given by (19a) as |z| → ∞ in | arg z| ≤ π. On the negative real axis we put z = −x, with x > 0. From (20b), we note that when 1 < α < 2 the two exponential terms combine to yield 1 1 (1−β)/α (1−β)/α exp[x1/α eπi/α ] + (xe−πi ) (xeπi ) exp[x1/α e−πi/α ] α α (1 − β) π 2 π ]. = x(1−β)/α exp[x 1/α cos ] cos[x1/α sin + π α α α α

Fα,β (x) :=

6 The more refined treatment of Eα,1 (z) discussed in [18, Section 5.1.4] has the integer N satisfying N < 21 α < N + 1. The additional exponential expansions present in (22) with this choice of N are, however, exponentially small for | arg z| ≤ π.

Asymptotics of the special functions of fractional calculus | 305

Then, when 1 < α < 2, we obtain ∞

(−x)−k Γ(β − αk) k=1

Eα,β (−x) ∼ Fα,β (x) − ∑

(23)

as x → +∞. The presence of the additional exponential expansion ℰ1,1 (ze∓2πi ) in (20b) is seen to be essential in order to obtain a real result (when β is real) on the negative z-axis. We remark that the result (23) can also be deduced by use of the identity Eα,β (−x) = 2E2α,β (x2 ) − Eα,β (x). Insertion of the expansions for Eα,β (x) and E2α,β (x2 ) as x → +∞ from (20a) and (22) (with N = 1), followed by some routine algebra, then yields the result in (23). When 0 < α < 1, it is established in [12] when β = 1 (see also [25] for the case β ≠ 1) that the exponential term α−1 z (1−β)/α exp[z 1/α ] in (19a) is multiplied by the approximate factor involving the error function (πα ∓ θ) √ |z| 1 1 + erf[ ] 2 2 α 2 as |z| → ∞ in the neighbourhood of the Stokes lines θ = arg z = ±πα, respectively. This shows that this exponential term indeed switches off in the familiar manner [1] as one crosses the Stokes lines in the sense of increasing |θ| and that consequently the expansion in (19a) is valid in | arg z| ≤ πα − ϵ. The (one-parameter) Mittag-Leffler function Eα (z) is defined by zn . Γ(αn + 1) n=0 ∞

Eα (z) = ∑

When α is an integer we have the exact result Eα (z) =

M+ξ

1 ∑ exp[z 1/α e2πin/α ], α n=−M

0 ξ ={ 1

(α = 2M + 1),

(α = 2M + 2).

(24)

This produces the evaluations E1 (z) = ez ,

E2 (z) = cosh z 1/2 ,

1 z 1/3 2 − 21 z 1/3 √3 e + e cos( z 1/3 ), 3 3 2 1 1 E4 (z) = cosh z 1/4 + cos z 1/4 . 2 2 We remark that (i) the algebraic expansion in (22) vanishes when β = 1 and α is an integer and (ii) that the additional exponential terms in (24) not present on the righthand side of (22) are all exponentially small. These additional exponentially small contributions are not accounted for in the theory presented in (14). The asymptotic expansion of Eα (z) for large |z| in | arg z| ≤ π then immediately follows from (19)–(22) upon substitution of the value β = 1. E3 (z) =

306 | R. B. Paris

4 The generalised Bessel function 0 Ψ1 (z) The generalised Bessel function is defined by ∞ −− 󵄨󵄨󵄨󵄨 zn Ψ (z) ≡ Ψ ( z) = ∑ 󵄨 0 1 0 1 (a, b) 󵄨󵄨󵄨 Γ(an + b)n! n=0

(25)

which converges for all finite z provided a > −1. This function was first studied by Wright [26, 29] and more recently in [24]. When a = 1, 0 Ψ1 (z) reduces to the modified Bessel function z (1−b)/2 Ib−1 (2√z). The case b = 0 finds application in probability theory and is discussed in [19]. The asymptotic expansion of 0 Ψ1 (z) as |z| → ∞ separates into two distinct cases according as a > 0 and −1 < a < 0.

4.1 The case a > 0 The function 0 Ψ1 (z) has g(s) in (2) given by g(s) = 1/Γ(as + b) and, from (4), the parameters κ = 1 + a, h = a−a , ϑ = 21 − b. It follows that when a > 0 we have κ > 1. Since p = 0, the algebraic expansion H0,1 (z) ≡ 0 and the exponential expansion is ϑ Z

∞

−j

ℰ0,1 (z) = Z e ∑ Aj Z , j=0

Z = κ(hz)1/κ ,

where, from (7), A0 = (a/κ)ϑ /√2πκ. The normalised coefficients cj ≡ Aj /A0 can be obtained by the algorithm described in the Appendix to yield the first few coefficients given by c0 = 1, c2 =

c1 = −

1 {(2 + a)(1 + 2a) − 12b(1 + a − b)}, 24a

1 {(2 + a)(1 + 2a)(2 − 19a + 2a2 ) + 24b(1 + a)(2 + 7a − 6a2 ) 1152a2 − 24b2 (4 − 5a − 20a2 ) − 96b3 (1 + 5a) + 144b4 }.

(26)

An example of the coefficients cj for the case a = 21 , b = 45 is presented in Table 1. Then from Theorems 2, 4 and 5 we have the expansions ℰ0,1 (z) + ℰ0,1 (ze∓2πi )

0 Ψ1 (z) ∼ {

∑Nn=−N ℰ0,1 (ze2πin )

(1 < κ ≤ 2),

(κ > 2),

(27)

for |z| → ∞ in | arg z| ≤ π. The upper or lower sign is chosen according as arg z > 0 or arg z < 0, respectively and N is the smallest integer satisfying 2N + 1 > 21 (1 + a). We remark that when κ = 2 (a = 1) the expansion of 0 Ψ1 (z) can be obtained from that of the modified Bessel function.

Asymptotics of the special functions of fractional calculus | 307 Table 1: The normalised coefficients cj = Aj /A0 for 0 Ψ1 (z) in (25) when a = j 1 3 5 7 9

cj

j

5 − 48 85085 − 663552 − 1511535025 6115295232 + 215144256952625 84537841287168 + 1304836837479714163625 14023813415445725184

2 4 6 8 10

1 2

and b =

5 . 4

cj 455 − 4608

24079055 − 127401984

26957055125 + 1761205026816

+ 570314645402376875 32462531054272512

+ 560395062780446967448375 1346286087882789617664

From the discussion after Theorem 5, the expansions ℰ0,1 (ze∓2πi ) are subdominant in the upper (lower) half-plane and switch on (in the sense of increasing | arg z|) across the Stokes lines arg z = ±π(1 − 21 κ). Consequently, when 1 < κ < 2, these expansions are not present on the positive real axis, but make a significant contribution on the negative real axis. Thus, on the real axis we have, as x → +∞, with X = κ(hx)1/κ , 0 Ψ1 (x)

∞

∼ ℰ0,1 (x) = X ϑ eX ∑ Aj X −j j=0

(1 < κ < 2)

(28)

and 0 Ψ1 (−x)

∼ ℰ0,1 (xeπi ) + ℰ0,1 (xe−πi ) (1 < κ ≤ 2) ∞

= 2X ϑ eX cos π/κ ∑ Aj X −j cos [X sin j=0

π π + (ϑ − j)]. κ κ

(29)

An example when the parameter κ > 2 is furnished by the following. When κ = 4, so that N = 1, we have as x → +∞ 0 Ψ1 (x)

∼ ℰ0,1 (x) + ℰ0,1 (xe2πi ) + ℰ0,1 (xe−2πi ) ∞

∞

j=0

j=0

= X ϑ eX ∑ Aj X −j + 2X ϑ ∑ Aj X −j cos [X +

π (ϑ − j)] 2

and, upon neglecting an exponentially small contribution, 0 Ψ1 (−x)

∼ ℰ0,1 (xeπi ) + ℰ0,1 (xe−πi ) ∞

= 2X ϑ eX cos π/4 ∑ Aj X −j cos [X sin j=0

π π + (ϑ − j)]. 4 4

308 | R. B. Paris

4.2 The case −1 < a < 0 Let a = −σ, with 0 < σ < 1. Then use of the reflection formula for the gamma function, together with the fact that ϑ = 21 − b, yields 0 Ψ1 (z) =

=

1 ∞ ∑ Γ(σn+1−b)z n sin π(−σn + b) π n=0

1 πiϑ {e 1 Ψ0 (zeπiσ ) + e−πiϑ 1 Ψ0 (ze−πiσ )}, 2π

(30)

where 1 Ψ0 (z) ≡ 1 Ψ0 ((σ, 1 − b); −−; z). The expansion of 0 Ψ1 (z) with −1 < a < 0 can then be constructed from knowledge of the expansion of the associated functions ±πiσ ). The details of the expansion of 0 Ψ1 (z) for large complex z are given in 1 Ψ0 (ze [29]; see also [24] for a summary. To keep the presentation as clear as possible, we restrict our attention here to the most commonly occurring case of real z in (30). We first deal with the expansion of the associated function7 1 Ψ0 (z)

∞

Γ(σn + δ) n z n! n=0

= ∑

(0 < σ < 1; δ = 1 − b);

(31)

the case δ = 0 and z → −∞ has been given in a probabilistic context in [7, Thm. 2.4.6, p. 62]. The parameters associated with (31) are κ = 1 − σ, h = σ σ and ϑ = δ − 21 . From (5) and (10) the algebraic and exponential expansions are H1,0 (z) =

1 ∞ (−)k k + δ −(k+δ)/σ Γ( )z , ∑ σ k=0 k! σ

ϑ Z

∞

−j

ℰ1,0 (z) = Z e ∑ Aj Z , j=0

(32)

with Z defined in (5) and A0 = (2π/κ)1/2 (σ/κ)ϑ . Then, since 0 < κ < 1, we obtain from Theorem 4 the large-z expansion 1 Ψ0 (z)

ℰ1,0 (z) + H1,0 (ze∓πi )

∼{

H1,0 (ze

∓πi

)

(| arg z| ≤ πκ − ϵ)

(πκ + ϵ ≤ | arg z| ≤ π),

where the upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. The first few normalised coefficients cj = Aj /A0 are c0 = 1, c2 =

c1 =

1 {2 + 7σ + 2σ 2 − 12δ(1 + σ) + 12δ2 }, 24σ

1 {4 + 172σ + 417σ 2 + 172σ 3 + 4σ 4 − 24δ(1 + 41σ + 41σ 2 + σ 3 ) 1152σ 2 + 120δ2 (4 + 11σ + 4σ 2 ) − 480δ3 (1 + σ) + 144δ4 }, . . . .

(33)

7 This requires Γ(σn+δ) to be regular for n = 0, 1, 2, . . . ; the expansion (35) is still valid if this condition is not met; see [29, § 4].

Asymptotics of the special functions of fractional calculus | 309

If we denote the coefficients in (26) by cj ≡ cj (a, b), then it can be shown that the coefficients in (33) when δ = 1 − b are given by cj (−σ, b). The expansion on the Stokes lines arg z = ±πκ is of a more recondite nature and is discussed in [15, § 5]; see also [17, § 4]. It is found that8 1 Ψ0 (xe

±πiκ

)∼

k+δ m −1 e±πiδ o Γ( σ ) −(k+δ)/σ x ∑ σ k=0 k!

∞ iBj 1 ϑ )(−X)−j + (Xe±πi ) e−X ∑ ( Aj ± √ 2 2πX j=0

(34)

as x → +∞, where X = κ(hx)1/κ and mo ∼ X denotes the optimal truncation index for the algebraic expansion. The coefficients Bj (which we do not specify here) depend on the coefficients Aj ; see [17, § 4] for details. We now return to consideration of the expansion of the generalised Bessel function 0 Ψ1 (x) in (30) as x → +∞. The algebraic component of this expansion is, from (30) and (32) with δ = 1 − b, 1 πiϑ Ĥ 0,1 (x) := {e H1,0 (xeπiσ . e−πi ) + e−πiϑ H1,0 (xe−πiσ . eπi )} 2π 1 ∞ x−(k+1−b)/σ . = ∑ σ k=0 k!Γ(1 − k+1−b )

(35)

σ

The exponential component is, with X± := Xe±πiσ/κ and X as defined above, ̂ (x) := ℰ0,1 =

∞ ∞ 1 {eπiϑ X+ϑ eX+ ∑ Aj X+−j + e−πiϑ X−ϑ eX− ∑ Aj X−−j } 2π j=0 j=0

1 πσ π X ϑ X cos πσ/κ ∞ j e + (ϑ − j)] (σ < ). ∑ (−) Aj X −j cos [X sin π κ κ 2 j=0

Let us denote the points xe±πiσ in the z-plane that appear in the arguments in (30) by P± . Then when 0 < σ < 31 , P± lie in the exponentially large sector | arg z| < 21 πκ in ̂ (x) is exponentially large as x → +∞. When σ = 1 , Figure 2(a) and consequently ℰ0,1 3 ̂ (x) is P± lie on the anti-Stokes lines arg z = ± 1 πκ; on these rays cos πσ/κ = 0 and ℰ0,1 2

oscillatory with an algebraically controlled amplitude. When 31 < σ < 21 , P± lie in the ̂ (x) is exponentially small. When exponentially small and algebraic sectors so that ℰ0,1 1 σ = 2 , P± lie on the Stokes lines arg z = ±πκ, where the subdominant exponential expansions are in the process of switching off. Finally, when 21 < σ < 1, P± are situated in the algebraic sectors, where the expansions are purely algebraic. 8 There is a misprint in [15, (4.24)]: the sign of i should be reversed in both instances.

310 | R. B. Paris

Figure 2: The Stokes and anti-Stokes lines when 0 < κ < 1 and the location of the arguments P± (indicated by heavy dots): (a) P± = xe±πiσ and (b) P± = xe±πiκ .

Then we obtain the expansion when −1 < a < 0 ̂ (x) + Ĥ 0,1 (x) (0 < σ < 1 ), ℰ0,1 2 Ψ (x) ∼ { 0 1 Ĥ 0,1 (x) ( 21 ≤ σ < 1),

(36)

as x → +∞. It should be remarked that in (36b) we have omitted the exponentially small contribution when σ = 21 ; for details of this case, see [17]. When z = −x, x → +∞ we find from (30) and (31), upon replacing z by xe∓πi and using the result 1 Ψ0 (ze2πi ) = 1 Ψ0 (z), that 0 Ψ1 (−x)

=

1 πiϑ {e 1 Ψ0 (xe−πiκ ) + e−πiϑ 1 Ψ0 (xeπiκ )}. 2π

(37)

It is now apparent that the arguments of the 1 Ψ0 functions are situated on the Stokes lines for all values of σ in the range 0 < σ < 1, where the exponential expansions will be in the process of switching off; see Figure 2(b). The algebraic component of this expansion vanishes since, from (34), it is given by k+δ m −1 cos π(δ−ϑ) o Γ( σ ) −(k+δ)/σ x ≡ 0, ∑ πσ k! k=0

upon recalling that δ = 21 +ϑ. In the combination in (37), the exponentially small series involving the coefficients Bj in (34) cancel. This therefore yields the final result 0 Ψ1 (−x) ∼

X ϑ −X ∞ j e ∑ (−) Aj X −j 2π j=0

(0 < σ < 1; x → +∞).

(38)

Asymptotics of the special functions of fractional calculus | 311

5 The Wright function 1 Ψ1 (z) The Wright function is defined by 1 Ψ1 (z)

∞ (α, a) 󵄨󵄨󵄨󵄨 Γ(αn + a) z n ≡ 1 Ψ1 ( , 󵄨󵄨 z) = ∑ (β, b) 󵄨󵄨 Γ(βn + b) n! n=0

(39)

where we consider α, β > 0. Then when β > α−1 the series converges for all finite values of z. A special case of this function corresponding to the three-parameter Mittag-Leffler function, or Prabhakar function, is discussed in Section 5.1. The function 1 Ψ1 (z) is associated with g(s) in (2) given by g(s) = Γ(αs + a)/Γ(βs + b) and, from (4), the parameters κ = 1 + β − α, h = αα β−β and ϑ = a − b. The algebraic and exponential expansions are given by H1,1 (z) =

Γ( k+a ) 1 ∞ (−)k α z −(k+a)/α , ∑ (k+a) α k=0 k! Γ(b − β ) α ϑ Z

∞

−j

ℰ1,1 (z) = Z e ∑ Aj Z , j=0

Z = κ(hz)1/κ ,

where, from (7), 1

1

1

A0 = κ−ϑ− 2 αa− 2 β 2 −b . From (A.4), the normalised coefficient c1 = A1 /A0 is c1 =

1 {α(α − 1)(1 − 6b + 6b2 ) + β(1 + β)(1 − 6a + 6a2 ) + αβ[α − β − 2(1 + 6ab − 6b)]. 12αβ

The complexity of the coefficient c2 for general parameters prevents its presentation. Then, from Theorems 3 and 4, we obtain the expansions as |z| → ∞: ℰ1,1 (z) + H1,1 (ze∓πi ) (| arg z| ≤ min{π − ϵ, πκ − ϵ}), { { { { { {H1,1 (ze∓πi ) (πκ + ϵ ≤ | arg z| ≤ π; 0 < κ < 1), Ψ (z) ∼ 1 1 { ∓2πi {ℰ1,1 (z) + ℰ1,1 (ze ) { { { { ∓πi (| arg z| ≤ π; 1 < κ ≤ 2), { + H1,1 (ze )

(40)

and 1 Ψ1 (z)

N

∼ ∑ ℰ1,1 (ze2πin ) n=−N

(| arg z| ≤ π; κ > 2),

(41)

where N is the least integer satisfying 2N + 1 > 21 κ. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. As an example using the algorithm described in the Appendix, we present in Table 2 the normalised coefficients cj when α = 21 , β = 1 (κ = 32 ) and a = 41 , b = 43 . Other numerical examples of 1 Ψ1 (z) are considered in [16].

312 | R. B. Paris Table 2: The normalised coefficients cj = Aj /A0 for 1 Ψ1 (z) in (39) when α = 21 , β = 1, a = j

cj

j

cj

1

61 192 22783285 42467328 30375638199305 6262062317568

2

23161 73728 44604509425 32614907904 162721816250787605 7213895789838336

3 5

4 6

1 4

and b =

3 . 4

5.1 The three-parameter Mittag-Leffler (or Prabhakar) function (ρ)

The three-parameter Mittag-Leffler function Eα,β (z) (also known as the Prabhakar function) is obtained as a special case of the Wright function 1 Ψ1 (z) by setting the numerator gamma function in (39) equal to Γ(n + ρ). Thus we have (ρ)

(ρ)n (1, ρ) 󵄨󵄨󵄨󵄨 zn 1 = 󵄨 z) . 1 Ψ1 ( (α, β) 󵄨󵄨󵄨 Γ(αn + β) n! Γ(ρ) n=0 ∞

Eα,β (z) = ∑

(42)

Then, with κ = α, h = α−α , ϑ = ρ − β and Z = z 1/α , we obtain the algebraic and exponential expansions (we add a tilde to account for the additional factor 1/Γ(ρ) in their definitions) z −ρ ∞ (−)k Γ(k + ρ) z −k , H̃ 1,1 (z) = ∑ Γ(ρ) k=0 k! Γ(β − α(k + ρ))

1/α

ϑ/α z ̃ (z) = z e ℰ1,1 Γ(ρ)

∞

∑ Aj z −j/α ,

j=0

(43)

where A0 = α−ρ . The normalised coefficients cj = Aj /A0 simplify considerably in this case to yield c0 = 1,

1 c1 = (ρ − 1)(ρ(1 + α) − 2β), 2

1 (ρ − 1)(ρ − 2) [3ρ2 (1 + α)2 + 12β(1 + β) − ρ(1 + α)(5 + α + 12β)], 24 1 (ρ − 1)(ρ − 2)(ρ − 3) [ρ3 (1 + α)3 − ρ2 (1 + α)2 (5 + α + 6β) − 8β(1 + β)(2 + β) c3 = 48 + 2ρ(1 + α)(3 + α(1 + β) + 11β + 6β2 )], . . . . c2 =

(ρ)

It is seen when ρ = 1 (so that Eα,β (z) reduces to the ordinary Mittag-Leffler function in Section 3) that the above coefficients c1 = c2 = c3 = 0 in accordance with the exponential expansion in (19) and (20) consisting of a single term. From (40) and (41) we then obtain the expansions as |z| → ∞ (an equivalent discussion has been given recently in [5, § 4]) ̃ (z) + H̃ 1,1 (ze∓πi ) (| arg z| ≤ min{π − ϵ, πα − ϵ}) ℰ1,1 { { { { { {H̃ 1,1 (ze∓πi ) (πα + ϵ ≤ | arg z| ≤ π; 0 < α < 1) (ρ) Eα,β (z) ∼ { ∓2πi {ℰ1,1 ̃ (z) + ℰ1,1 ̃ (ze ) { { { { ∓πi ̃ (| arg z| ≤ π; 1 < α ≤ 2) { + H1,1 (ze )

(44)

Asymptotics of the special functions of fractional calculus | 313

and N

̃ (ze2πin ) Eα,β (z) ∼ ∑ ℰ1,1 (ρ)

n=−N

(| arg z| ≤ π; α > 2),

(45)

where N is the least integer satisfying 2N + 1 > 21 α. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively. If we introduce the notation for r = 1, 2, . . . ̃ (xeπir ) + ℰ1,1 ̃ (xe−πir ) Sr (x) := ℰ1,1 =

πr ∞ πr πr 2x ϑ/α exp [x 1/α cos ] ∑ Aj x−j/α cos [x1/α sin + (ϑ − j)], Γ(ρ) α j=0 α α

then on the positive axis (45) can be rewritten as N

(ρ)

̃ (x) + ∑ S2n (x) Eα,β (x) ∼ ℰ1,1 n=1

(α > 2)

as x → +∞. On the negative axis we put z = −x = xeπi , where x > 0. Then from (44) we obtain the expansions H̃ 1,1 (x) (0 < α ≤ 1) (ρ) Eα,β (−x) ∼ { ̃ H1,1 (x) + S1 (x) (1 < α ≤ 2)

(46)

as x → +∞. It is easily verified that S1 (x) is exponentially small as x → +∞ for 1 < α < 2, becoming of algebraic order when α = 2. In addition, when α = 1 we have (ρ) E1,β (−x) = 1 F1 (ρ; β; −x)/Γ(β), where 1 F1 is the confluent hypergeometric function. Use of the standard asymptotics9 of this function shows that the expansion H̃ 1,1 (x) in (46a)

holds when α = 1. When α > 2, we obtain from (45) (ρ)

N−1

Eα,β (−x) ∼ ∑ S2n+1 (x) n=0

(α > 2),

(47)

where N is the least integer satisfying 2N + 1 > 21 α. We have neglected the series ̃ (xe(2N+1)πi ) in (47) since it is exponentially small as x → +∞; this is easily seen ℰ1,1 to be the case because cos((2N + 1)π/α) < 0. This neglected series has a complex value (ρ) and it might be questioned how this may be reconciled with the fact that Eα,β (−x) must 9 The treatment of the maximally subdominant exponentially small contribution in the expansion of 1 F1 (a; b; −x) as x → +∞ is considered in [14].

314 | R. B. Paris be real-valued when the parameters are real. As with the ordinary Mittag-Leffler function in Section 3, there are additional exponentially small contributions not accounted for in the expansion (47); these would combine to generate a real, exponentially small (ρ) contribution to Eα,β (−x), in addition to the real (exponentially large) contribution from (47), when the parameters are real.

6 The function 1 Ψ2 (z) The function defined by 1 Ψ2 (z)

∞ 󵄨󵄨 (1, 1) zn 󵄨󵄨 ≡ 1 Ψ2 ( 󵄨󵄨 z) = ∑ (β1 , b1 ), (β2 , b2 ) 󵄨󵄨 Γ(β1 n + b1 )Γ(β2 n + b2 ) n=0

(48)

generalises the function in Section 4 through the replacement of n! by Γ(β2 n + b2 ). We consider β1 , β2 > 0 so that the series converges for all finite values of z. If one of the β parameters is negative (say β1 = −σ), then the above series can be expressed in terms of the Wright function 1 Ψ1 (z) discussed in Section 5, since application of the reflection formula for the gamma function shows that (when β2 > σ) 1 ∞ Γ(σn + 1 − b1 ) n zn =− ∑ z sin π(σn − b1 ) Γ(−σn + b1 )Γ(β2 n + b2 ) π n=0 Γ(β2 n + b2 ) n=0 ∞

∑

1 πib1 {e 1 Ψ1 (ze−πiσ ) − e−πib1 1 Ψ1 (zeπiσ )}, 2πi

=

where 1 Ψ1 (z) ≡ 1 Ψ1 ((σ, 1 − b1 ); (β2 , b2 ); z). The function 1 Ψ2 (z) is associated with the function in (2) given by g(s) = Γ(1 + s)/(Γ(β1 s + b1 )Γ(β2 s + b2 )) and, from (4), the parameters −β

−β

h = β1 1 β2 2 ,

κ = β1 + β2 ,

ϑ=

3 − b1 − b2 . 2

Since p = 1 (with α1 = a1 = 1) the algebraic expansion is, from (10) and (11), (−)k z −k−1 . Γ(b1 −β1 −β1 k)Γ(b2 −β2 −β2 k) k=0 ∞

H1,2 (z) = ∑ The exponential expansion is

ϑ Z

∞

−j

ℰ1,2 (z) = Z e ∑ Aj Z , j=0

Z = κ(hz)1/κ ,

where, from (7), 1

A0 =

κ− 2 −ϑ 21 −b1 21 −b2 β β2 . √2π 1

Asymptotics of the special functions of fractional calculus | 315

The normalised coefficients cj = Aj /A0 are obtained from the algorithm in the Appendix to yield c0 = 1, c2 =

c1 = −

1 {2β12 P11 (b2 ) − β1 β2 P12 (b1 , b2 ) + 2β22 P11 (b1 )}, 24β1 β2

1 {4β14 P21 (b2 ) + 4β13 β2 P22 (b1 , b2 ) + 3β12 β22 P23 (b1 , b2 ) 1152(β1 β2 )2

+ 4β1 β23 P22 (b2 , b1 ) + 4β24 P21 (b1 )}, where P11 (b) = 1 − 6b + 6b2 ,

P12 (b1 , b2 ) = 7 − 12(b1 − 2b1 b2 + b2 ),

and P21 (b) = 1 + 12b(1 − 2b − 2b2 + 3b3 ),

P22 (b1 , b2 ) = 5 + 6b2 (1 − 7b2 + 4b22 ) − 12b1 (1 − 4b2 − 6b22 + 12b32 ),

P23 (b1 , b2 ) = 11 − 8(b1 + b2 + 4b21 + 4b22 ) + 16b1 b2 (7 − 6(b1 + b2 ) + 18b1 b2 ). It will be observed that c1 and c2 are symmetrical functions of β1 , β2 and b1 , b2 . In the special case β1 = β2 the coefficients simplify to yield c0 = 1, c2 =

c1 =

1 {1 − 4b21 + 8b1 b2 − 4b22 }, 8

1 {9 + 8b21 (2b21 − 5) + 8b1 b2 (10 − 8b21 + 9b1 b2 − 8b22 ) + 8b22 (2b22 − 5)}, 128

which are independent of β1 (or β2 ). Then from Theorems 3 and 4 we have the expansions as |z| → ∞ ℰ1,2 (z) + H1,2 (ze∓πi ) (| arg z| ≤ min{π − ϵ, πκ − ϵ}), { { { ∓πi ) (min{πκ + ϵ, π} ≤ | arg z| ≤ π; 0 < κ ≤ 1), 1 Ψ2 (z) ∼ {H1,2 (ze { { ∓2πi ∓πi {ℰ1,2 (z) + ℰ1,2 (ze ) + H1,2 (ze ) (| arg z| ≤ π; 1 < κ ≤ 2),

(49)

when 0 < κ ≤ 2, and 1 Ψ2 (z)

N

∼ ∑ ℰ1,2 (ze2πin ) + H1,2 (ze∓πi ) n=−N

(| arg z| ≤ π),

(50)

when κ > 2. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively and the integer N is chosen to be the smallest integer satisfying 2N + 1 > 1 κ. 2

316 | R. B. Paris

7 The Fox H-function The Fox H-function is defined by the integral 󵄨 m,n m,n (a1 , α1 ), (a2 , α2 ), . . . , (ap , αp ) 󵄨󵄨󵄨 Hp,q (z) = Hp,q ( 󵄨 z) (b1 , β1 ), (b2 , β2 ), . . . , (bq , βq ) 󵄨󵄨󵄨 =

∏m Γ(br + βr s) ∏nr=1 Γ(1 − ar − αr s) 1 z −s ds, ∫ p r=1 2πi ∏r=n+1 Γ(ar + αr s) ∏qr=m+1 Γ(1 − br − βr s)

(51)

ℒ

where the parameters αr and βr are real and positive, ar are br are arbitrary complex numbers and the integers m, n, p, q satisfy 0 ≤ m ≤ q and 0 ≤ n ≤ p. There are two sequences of poles of the numerator gamma functions. The poles of Γ(1 − ar − αr s) are ∗ situated at s = σrk (1 ≤ r ≤ n) and those of Γ(br + βr s) are situated at s = σrk (1 ≤ r ≤ m), where σrk =

1 − ar + k , αr

∗ σrk =−

br + k βr

(k = 0, 1, 2, . . .).

(52)

It will be assumed throughout this section that all these poles are simple; see Section 7.2 for an example where this condition is not satisfied. The integration contour ℒ is taken to be a path in the complex s-plane that separates these two sequences of poles. We define the following parameters10 associated with (51): m

p

r=1 m

r=1 p

n

p

m

q

r=1

r=n+1

r=1

r=m+1

κm = ∑ βr − ∑ αr ,

p

q

r=1

r=1

h = ∏ αrαr ∏ βr−βr ,

κ = κq ,

1 ϑm = ∑ br − ∑ ar + (p − m + 1), 2 r=1 r=1 μ = ∑ αr − ∑ αr + ∑ βr − ∑ βr .

ϑ = ϑq ,

ϑ󸀠 = 1 − ϑ, (53)

The contour ℒ can be taken to be a left-hand loop described in the positive sense, ∗ m,n with endpoints at −∞, that encloses only the poles σrk . Such a path defines Hp,q (z) for κ > 0, z ≠ 0, or κ = 0, |z| < 1/h. If ℒ is taken to be a right-hand loop described in the negative sense, with endpoints at +∞, that encloses only the poles σrk , then m,n this defines Hp,q (z) for κ < 0, z ≠ 0, or κ = 0, |z| > 1/h. If ℒ is chosen to be the path (−∞i, ∞i) (which may be suitably indented if necessary) to lie to the left of all the ∗ m,n poles σrk but to the right of σrk , then the integral in (51) defines Hp,q (z) in the sector 1 | arg z| < 2 πμ. 10 We employ an analogous notation to that in Section 2. In terms of the notation employed in [8, § 1.1], we have μ = α∗ , κ = Δ, h = 1/δ, ϑ = μ + 21 and κm = a∗1 (when n = 0) and in [9, § 1.2] we have μ = α, κ = μ, h = 1/β, ϑ = δ + 21 .

Asymptotics of the special functions of fractional calculus | 317

With the integration path chosen to be either the right-hand loop (when κ < 0) or the left-hand loop (when κ > 0) we have the power series expansions n

∞

m,n Hp,q (z) = ∑ ∑ hrk z (ar −1−k)/αr r=1 k=0 m ∞

m,n Hp,q (z) = ∑ ∑ h∗rk z (br +k)/βr r=1 k=0

(κ < 0, z ≠ 0; κ = 0, |z| > 1/h),

(54)

(κ > 0, z ≠ 0; κ = 0, 0 < |z| < 1/h),

(55)

where hrk =

m 󸀠n (−)k ∏j=1 Γ(bj + βj σrk ) ∏ j=1 Γ(1 − aj − αj σrk ) k!αr ∏pj=n+1 Γ(aj + αj σrk ) ∏qj=m+1 Γ(1 − bj − βj σrk )

(56)

and h∗rk

n 󸀠m ∗ ∗ (−)k ∏ j=1 Γ(bj + βj σrk ) ∏j=1 Γ(1 − aj − αj σrk ) . = ∗) ∗ ) ∏q k!βr ∏pj=n+1 Γ(aj + αj σrk Γ(1 − bj − βj σrk j=m+1

(57)

The primes in the second numerator product in hrk and the first numerator product in h∗rk denote the omission of the gamma function corresponding to j = r. The representations in (54) and (55) assume that all the poles in (52) are simple. If some of these poles are of higher order then the residues must be calculated accordingly and will result in terms involving log z; see [8, § 1.4] for details. A detailed account of the properties of the H-function can also be found in [9, Chapter 1]. m,n (z) for large |z| and finite parameters To present the asymptotic expansion of Hp,q we introduce the following quantities. The constants C0 and D0 are given by p

m

r=n+1

r=1

C0 = (2πi)m+n−p−1 exp[πi( ∑ ar − ∑ br )], p

m

r=n+1

r=1

D0 = (−2πi)m+n−p−1 exp[−πi( ∑ ar − ∑ br )].

(58) (59)

The exponential expansion is ϑ Z

∞

Z = κ(hz)1/κ ,

−j

ℰ (z) = Z e ∑ Aj Z , j=1

(60)

where 1

p

1

A0 = (2π)(p−q+1)/2 κ − 2 −ϑ ∏ αr2 r=1

−r

q

b − 21

∏ βr r r=1

.

The coefficients Aj are defined by ∏pj=1 Γ(1 − aj + αj s) ∏qj=1 Γ(1

− bj + βj s)

M−1

= κ(hκ)s { ∑

j=0

Aj

Γ(κs + ϑ󸀠 + j)

+

ρM } Γ(κs + ϑ󸀠 + M)

318 | R. B. Paris for positive integer M. The quantity ρM = O(1) for |s| → ∞ in | arg s| < π and ϑ󸀠 = 1 − ϑ, where ϑ is defined in (53). The coefficients Aj can be determined by means of the algorithm described in the Appendix. We have retained the same form for the exponential expansion as used in Section 2; this has required a small modification to the equivalent expression (and the constants C0 and D0 ) used in [8].

7.1 Statement of the expansion theorems For large z and finite values of the parameters, we have the following expansions [2, 8]. Theorem 6. When κ > 0, μ > 0 we have the algebraic expansion (obtained by displacement of the integration path (−∞i, ∞i) in (51) to the right over the poles σjk in (52) assumed to be all simple) n ∞

m,n (z) ∼ ∑ ∑ hjk z (aj −1−k)/αj Hp,q j=1 k=0

(61)

as |z| → ∞ in the sector | arg z| ≤ 21 πμ − ϵ, where hjk is defined in (56) when all the poles σjk are simple. If some of these poles are coincident then the residues of the integrand in (51) must be evaluated accordingly. m,n If κ ≤ 0, the expansion of Hp,q (z) is given by the convergent series in (54). Theorem 7. When κ > 0, μ = 0 and the poles σjk in (52) are all simple, we have the expansion n ∞

m,n (z) ∼ ∑ ∑ hjk z (aj −1−k)/αj + C0 ℰ (zeπiκ/2 ) + D0 ℰ (ze−πiκ/2 ) Hp,q j=1 k=0

(62)

as |z| → ∞ in the sector | arg z| ≤ δ, where 1 0 < δ < π min{αi , βj } 2

(n + 1 ≤ i ≤ p, 1 ≤ j ≤ m).

and the constants C0 , D0 and the exponential expansion ℰ (z) are defined in (58), (59) and (60). Theorem 8. Let n = 0 and the parameters κ, κm , C0 and D0 be given in (53), (58) and (59). Then the following exponential expansions hold. (i) When m = q, μ > 0 we have q,0 Hp,q (z) ∼ C0 ℰ (zeπiκ )

(63)

as |z| → ∞ in the sector | arg z| ≤ 21 πκ − ϵ. When m = q, n = 0 the constant C0 = (2π)q−p−1 e−πiϑ , where ϑ is defined in (53).

Asymptotics of the special functions of fractional calculus | 319

(ii) When m < q, μ ≥ 0 we have m,0 Hp,q (z) ∼ C0 ℰ (zeπiκm ) + D0 ℰ (ze−πiκm )

(64)

as |z| → ∞ in the sector | arg z| ≤ ω − ϵ, where 1

ω = { 41 2

πμ π min{αi , βj }

(1 ≤ i ≤ p, 1 ≤ j ≤ m)

(μ > 0),

(μ = 0),

and the exponential expansion ℰ (z) is defined in (60). When z = x, x > 0 and with X = κ(hx)1/κ , we have from Theorem 8 the leading asymptotic forms q,0 (x) ∼ (2π)q−p−1 A0 X ϑ e−X Hp,q

(65)

and, when m < q and ϑ is real, m,0 (x) ∼ 2(2π)m−p−1 A0 X ϑ eX cos πκm /κ cos[X sin Hp,q

κ πκm + π( m − ϑm )] κ κ

(66)

as x → +∞. From Theorems 6 and 7 we have, when all the poles σjk are simple, the leading asymptotic forms O(xρ ) (μ > 0), m,n (x) = { Hp,q 1 ρ ϑ O(x ) + O(X ) cos [X − 2 πξ ] (μ = 0),

(67)

as x → +∞, where ρ = max{(ℜ(ar ) − 1)/αr } (1 ≤ r ≤ n) and q

p

m n 1 ξ = ∑ br − ∑ br + ∑ ar − ∑ ar − m − n + (p + q + 1). 2 r=1 r=m+1 r=1 r=n+1

7.2 An example arising in fractional diffusion ∗ As an example, which involves the poles σrk possessing an infinite sequence of double poles, we consider the function arising in the context of fractional diffusion problems (see [3]) 2,0 H1,2 (z)

≡

(1, α) 2,0 H1,2 ( 1 ( 2 n, 1), (1, 1)

∞i 󵄨󵄨 Γ( 21 n + s)Γ(1 + s) −s 1 󵄨󵄨 z ds ∫ 󵄨󵄨 z) = 󵄨󵄨 2πi Γ(1 + αs) −∞i

320 | R. B. Paris valid for | arg z| < π(1 − 21 α) with 0 < α < 2, where n is a positive integer. The case α = 1 can be eliminated from our consideration since, by the Cahen–Mellin integral [18, § 3.3.1], [23, p. 280], 2,0 H1,2 (z) = z n/2 e−z

(α = 1).

∗ For odd integer n the sequence σrk consists of simple poles at s = −1 − k and 1 s = − 2 n−k, k = 0, 1, 2, . . . . Then we have from (55) and (57) the power series expansion11 ∞

2,0 H1,2 (z) = z ∑

(−z)k Γ( 21 n − 1 − k) k!Γ(1 − α − αk)

k=0

∞

+ z n/2 ∑

k=0

(−z)k Γ(1 − 21 n − k) k!Γ(1 − 21 αn − αk)

.

(68)

When α = 1, the first sum vanishes and the second sum correctly reduces to z n/2 e−z . ∗ When n is an even integer, let n = 2m, m = 1, 2, . . . . The sequence σrk now consists of simple poles at s = −1, −2, . . . , −m + 1 and an infinite sequence of double poles at s = −m − k, k = 0, 1, 2, . . . . To evaluate the residues at the double poles we set s = −m − k + ϵ and let ϵ → 0 making use of the results Γ(−k + ϵ) = (−)k /(ϵk!) + O(1) and Γ(a+ϵ) = Γ(a){1+ϵψ(a)+O(ϵ2 )}, where ψ(a) is the logarithmic derivative of the gamma function. Then the residues are given by (−)m z m+k (log z + Ψk ) , k!Γ(m + k)Γ(1−αm−αk)

Ψk := αψ(1−αm−αk) − ψ(1 + k) − ψ(m + k)

to yield the power-logarithmic series expansion valid when α ≠ 1 m−2

2,0 H1,2 (z) = z ∑

k=0

∞ z k (log z + Ψk ) (−z)k Γ(m − 1 − k) + (−z)m ∑ . k!Γ(1 − α − αk) k!Γ(m + k)Γ(1−αm−αk) k=0

(69)

2,0 The expansion of H1,2 (z) for large z is exponentially small in | arg z| < π(1 − 21 α) and is given by Theorem 8. With the parameters p = 1, q = 2, κ = 2 − α, h = αα , ϑ = 21 n, A0 = κ−(n+1)/2 α−1/2 and C0 = e−πiϑ , we obtain the leading behaviour ϑ

2,0 H1,2 (z) ∼ C0 A0 (Zeπi ) e−Z ,

=

Z = κ(hz)1/κ

(αα z)n/(2(2−α)) 1/(2−α) exp [−(2 − α)(αα z) ] {α(2 − α)}1/2

as |z| → ∞ in the sector | arg z| < π(1 − 21 α). 11 It is worth noting that (68) holds for arbitrary n > 0, provided n ≠ 2, 4, . . . .

(70)

Asymptotics of the special functions of fractional calculus | 321

Appendix. An algorithm for the computation of the coefficients cj = Aj /A0 We describe an algorithm for the computation of the normalised coefficients cj = Aj /A0 appearing in the exponential expansion ℰp,q (z) in (5). Methods of computing these coefficients by recursion in the case αr = βr = 1 have been given by Riney [21] and Wright [30]; see [18, Section 2.2.2] for details. Here we describe an algebraic method for arbitrary αr > 0 and βr > 0. The inverse factorial expansion (6) can be re-written as M−1 cj g(s)Γ(κs + ϑ󸀠 ) O(1) s + } = κA0 (hκκ ) { ∑ 󸀠 Γ(1 + s) (κs + ϑ )j (κs + ϑ󸀠 )M j=0

(A.1)

for |s| → ∞ uniformly in | arg s| ≤ π − ϵ, where g(s) is defined in (2) and we recall that 1 1 ϑ󸀠 = 1 − ϑ. Introduction of the scaled gamma function Γ∗ (z) = Γ(z)(2π)− 2 ez z 2 −z leads to the representation 1

1

Γ(αs + a) = (2π) 2 e−αs (αs)αs+a− 2 e(αs; a)Γ∗ (αs + a), where αs+a− 21

a e(αs; a) := e (1 + ) αs −a

1 a = exp [(αs + a − ) log (1 + ) − a]. 2 αs

Then, after some routine algebra, we find that (A.1) becomes g(s)Γ(κs + ϑ󸀠 ) s = κA0 (hκκ ) Rp,q (s) ϒp,q (s), Γ(1 + s)

(A.2)

where ϒp,q (s) := Rp,q (s) :=

∏pr=1 Γ∗ (αr s + ar ) Γ∗ (κs + ϑ󸀠 ) , ∏qr=1 Γ∗ (βr s + br ) Γ∗ (1 + s) ∏pr=1 e(αr s; ar ) e(κs; ϑ󸀠 ) . ∏qr=1 e(βr s; br ) e(s; 1)

Substitution of (A.2) in (A.1) then yields the inverse factorial expansion (6) in the alternative form M−1

Rp,q (s) ϒp,q (s) = ∑

j=0

as |s| → ∞ in | arg s| ≤ π − ϵ.

cj

(κs + ϑ󸀠 )j

+

O(1) (κs + ϑ󸀠 )M

(A.3)

322 | R. B. Paris We now expand Rp,q (s) and ϒp,q (s) for s → +∞ making use of the well-known expansion (see, for example, [18, p. 71]) ∞

Γ∗ (z) ∼ ∑ (−)k γk z −k k=0

(|z| → ∞; | arg z| ≤ π − ϵ),

where γk are the Stirling coefficients, with γ0 = 1,

γ1 = −

1 , 12

γ2 =

1 , 288

γ3 =

139 , 51840

γ4 = −

571 , 2488320

....

Then we find Γ∗ (αs + a) = 1 −

γ1 + O(s−2 ), αs

e(αs; a) = 1 +

a(a − 1) + O(s−2 ), 2αs

whence Rp,q (s) = 1 +

𝒜

2s

+ O(s−2 ),

ϒp,q (s) = 1 +

ℬ

12s

+ O(s−2 ),

where we have defined the quantities 𝒜 and ℬ by p

p

q

ar (ar − 1) b (b − 1) ϑ −∑ r r − (1 − ϑ), α βr κ r r=1 r=1

𝒜=∑

q

1 1 1 −∑ + − 1. α β κ r=1 r r=1 r

ℬ=∑

Upon equating coefficients of s−1 in (A.3) we then obtain 1 1 c1 = κ(𝒜 + ℬ). 2 6

(A.4)

The higher coefficients are obtained by continuation of this expansion process in inverse powers of s. We write the product on the left-hand side of (A.3) as an expansion in inverse powers of κs in the form M−1

Rp,q (s)ϒp,q (s) = 1 + ∑ j=1

Cj

(κs)j

+ O(s−M )

(A.5)

as s → +∞, where the coefficients Cj are determined with the aid of Mathematica. From the expansion of the ratio of two gamma functions in [11, (5.11.13)] we obtain M−1 (−)k (j)k (1−j) 󸀠 1 1 (ϑ ) + O(s−M )}, B = { ∑ (κs + ϑ󸀠 )j (κs)j j=0 (κs)k k! k

where B(s) (x) are the generalised Bernoulli polynomials defined by k (

s ∞ B(s) (x) t xt k tk ) e = ∑ k! et − 1 k=0

Here we have s = 1 − j ≤ 0 and B(s) 0 (x) = 1.

(|t| < 2π).

Asymptotics of the special functions of fractional calculus | 323

Then the right-hand side of (A.3) as s → +∞ becomes M−1

1+ ∑ j=1

cj

(κs + ϑ󸀠 )j M−1

=1+ ∑

(κs)j

j=1

M−1

=1+ ∑ j=1

cj

Dj

(κs)j

+ O(s−M ) M−1

∑

k=0

(−)k (j)k (1−j) 󸀠 (ϑ ) + O(s−M ) B (κs)k k! k

+ O(s−M )

(A.6)

with j−1

j−1 (k−j+1) 󸀠 (ϑ ), Dj = ∑ (−)k ( ) cj−k Bk k k=0 where we have made the change in index j + k → j and used ‘triangular’ summation (see [22, p. 58]). Substituting (A.5) and (A.6) into (A.3) and equating the coefficients of like powers of κs, we then find Cj = Dj for 1 ≤ j ≤ M − 1, whence j−1

j−1 (k−j+1) 󸀠 (ϑ ). ) cj−k Bk cj = Cj − ∑ (−)k ( k k=1

(A.7)

Thus we find c 1 = C1 ,

󸀠 c2 = C2 − c1 B(0) 1 (ϑ ),

󸀠 (0) 󸀠 c3 = C3 − 2c2 B(−1) 1 (ϑ ) + c1 B2 (ϑ ),

...,

and so on, from which the coefficients cj can be obtained recursively. With the aid of Mathematica this procedure is found to work well in specific cases when the various parameters have numerical values, where up to a maximum of 100 coefficients have been so calculated.

Bibliography [1] [2] [3] [4]

M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. R. Soc. Lond., A422 (1989), 7–21. B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compos. Math., 15 (1963), 239–341. S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion problems, J. Differ. Equ., 199 (2004), 211–255. W. B. Ford, The Asymptotic Developments of Functions Defined by Maclaurin Series, University of Michigan, Science Series, vol. II, 1936.

324 | R. B. Paris

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R. Garra and R. Garrappa, The Prabhakar or three parameter Mittag-Leffler function, Comm. Nonlinear Sci. Numer. Simulat. 56 (2018), 314–329. H. K. Hughes, On the asymptotic expansions of entire functions defined by Maclaurin series, Bull. Am. Math. Soc., 50 (1944), 425–430. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Walters-Noordhoff, Groningen, 1971. A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications, Chapman & Hall/CRC Press, Boca Raton, 2004. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010. C. V. Newsom, On the character of certain entire functions in distant portions of the plane, Am. J. Math., 60 (1938), 561–572. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. R. B. Paris, Exponential asymptotics of the Mittag-Leffler function, Proc. R. Soc. Lond., 458A (2002), 3041–3052. R. B. Paris, Exponentially small expansions in the asymptotics of the Wright function, J. Comput. Appl. Math., 234 (2010), 488–504. R. B. Paris, Exponentially small expansions of the confluent hypergeometric functions, Appl. Math. Sci., 7 (2013), 6601–6609. R. B. Paris, Exponentially small expansions of the Wright function on the Stokes lines, Lith. Math. J., 54 (2014), 82–105. R. B. Paris, Some remarks on the theorems of Wright and Braaksma on the Wright function p Ψq (z), arXiv:1708.04824 (2017). R. B. Paris, The asymptotics of the generalised Bessel function, Math. Aeterna, 7 (2017), 381–406, arXiv:1711.03006. R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2001. R. B. Paris and V. Vinogradov, Asymptotic and structural properties of the Wright function arising in probability theory, Lith. Math. J., 56 (2016), 377–409. R. B. Paris and A. D. Wood, Asymptotics of High Order Differential Equations, Pitman Research Notes in Mathematics, vol. 129, Longman Scientific and Technical, Harlow, 1986. T. D. Riney, On the coefficients in asymptotic factorial expansions, Proc. Am. Math. Soc., 7 (1956), 245–249. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. E. T. Whittaker and G. N. Watson, Modern Analysis, Cambridge University Press, Cambridge, 1966. R. Wong and Y.-Q. Zhao, Smoothing of Stokes’s discontinuity for the generalized Bessel function I, II, Proc. R. Soc. Lond., A455 (1999), 1381–1400 and 3065–3084. R. Wong and Y.-Q. Zhao, Exponential asymptotics of the Mittag-Leffler function, Constr. Approx., 18 (2002), 355–385. E. M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. Lond. Math. Soc., 38 (1934), 257–270. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. Lond. Math. Soc. (Ser. 2), 10 (1935), 286–293. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. Lond. Math. Soc. (Ser. 2), 46 (1940), 389–408.

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[29] E. M. Wright, The generalized Bessel function of order greater than one, Qu. J. Math., 11 (1940), 36–48. [30] E. M. Wright, A recursion formula for the coefficients in an asymptotic expansion, Proc. Glasgow Math. Assoc., 4 (1958), 38–41.

Moulay Rchid Sidi Ammi and Delfim F. M. Torres

Analysis of fractional integro-differential equations of thermistor type Abstract: We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results as regards fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: problems concerned with Riemann–Liouville, Caputo, and time-scale fractional operators. The existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence. Keywords: Integral and differential equations, fractional operators, nonlocal thermistor problems, time scales, dynamic equations, positive solutions, local and global existence, fixed-point theorems MSC 2010: 26A33, 26E70, 35A01, 35B09, 45M20, 58J20

1 Introduction Fractional calculus covers a wide range of classical fields in mathematics and its applications, such as Abel’s integral equation, viscoelasticity, analysis of feedback amplifiers, capacitor theory, fractances, electric conductance, mathematical biology and optimal control [4]. In particular, Abel integral equations have been well studied, with many publications devoted to its applications in different fields. One can say that Abel’s integral equations, of the first and second kind, are the most celebrated integral equations of fractional order [45]. The former investigations on such equations are due to Niels Henrik Abel himself, for the first kind [67], and to Hille and Tamarkin for the second kind [42]. Abel was led to his equations studying the tautochrone problem (from the Greek tauto, meaning same, and chrono, meaning time), that is, the problem of determining the shape of a curve for which the time taken by an object sliding withAcknowledgement: The authors are grateful to the R&D unit UID/MAT/04106/2019 (CIDMA) and to two anonymous referees for constructive comments. Moulay Rchid Sidi Ammi, Group of Mathematical and Numerical Analysis of PDEs and Applications (AMNEA), Department of Mathematics, Moulay Ismail University, Faculty of Science and Technology, B.P. 509, Errachidia, Morocco, e-mail: [email protected], http://orcid.org/0000-0002-4488-9070 Delfim F. M. Torres, Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal, e-mail: [email protected], http://orcid.org/0000-0001-8641-2505 https://doi.org/10.1515/9783110571622-013

328 | M. R. Sidi Ammi and D. F. M. Torres out friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, simultaneously the tautochrone and the brachistochrone curve, which brings together the subjects of fractional calculus and the calculus of variations [45, 58]. Fractional integral equations occur in many situations where physical measurements are to be evaluated, e. g., in evaluation of spectroscopic measurements of cylindrical gas discharges, the study of the solar or a planetary atmosphere, the investigation of star densities in a globular cluster, the inversion of travel times of seismic waves for determination of terrestrial sub-surface structure, and spherical stereology [33, 34]. For detailed descriptions and analysis of such integral equations of fractional order, we refer the reader to the books of Gorenflo and Vessella [36] and Craig and Brown [22]. See also [32]. Another field, in which fractional integral equations with general weakly singular kernels appear naturally, is inverse boundary value problems in partial differential equations, in particular parabolic ones [35]. Here we are mainly interested in questions involving existence and uniqueness of the solutions of fractional integral equations. The existence and uniqueness of the solutions for FDEs have been intensely studied by many mathematicians [41, 47, 52, 60]. However, most available results were concerned with the existence–uniqueness of the solutions for FDEs on a finite interval. Since continuation theorems for FDEs are not well developed, results about global existence-uniqueness of the solution of FDEs on the half axis [0, +∞), by using directly the results from local existence, have only recently flourished [12, 54]. Here we address such issues. To motivate our study, we can mention two types of electrical circuits, which are related with fractional calculus. Circuits of the first type consist of capacitors and resistors, which are described by conventional (integer-order) models, but for which the circuit itself, as a whole, may have non-integer-order properties, becoming a fractance device, that is, an electrical element that exhibits fractionalorder impedance properties. Circuits of the second type may consist of resistors and capacitors, both modeled in the classical sense, and fractances. In particular, we can consider thermistor-type problems, which are highly nonlinear and mathematically challenging [72]. Inspired by modern developments of the study of thermistors, where fractional partial differential equations have a crucial role to play, we consider here mathematical models and tools that serve as prototypes for other integral problems. It turns out that the available computational methods for such mathematical problems are not theoretically sound, in the sense they rely on results of local existence. Here we review the recent theory of global existence for nonlocal fractional problems of thermistor type [70, 74, 76, 77]. We begin, in Section 2, with an historical account of the theory of FDEs dealing with Cauchy-type problems and their reduction to Volterra integral equations. Section 3 contains some of the main tools in the area: we recall a necessary and sufficient condition for a subspace of continuous functions to be precompact; Schauder’s fixed-point theorem; and a useful generalization of Gron-

Analysis of fractional integro-differential equations of thermistor type | 329

wall’s inequality. In Section 4, our main concern is existence and uniqueness of the solution to a fractional-order nonlocal Riemann–Liouville thermistor problem of the form D2α u(t) =

T (∫0 β

λf (u(t)) f (u(x)) dx)2

I u(t)|t=0 = 0,

+ h(t) ,

t ∈ (0, T) ,

(1)

∀β ∈ (0, 1],

under suitable conditions on f and h (see Theorem 1). We also establish the boundedness of u (Theorem 2). Here the constant λ is a positive dimensionless real parameter. The unknown function u may be interpreted as the temperature generated by an electric current flowing through a conductor [51, 83]. We assume that T is a fixed positive real and α > 0 a parameter describing the order of the FD. In the case α = 1 and h ≡ 0, (1) becomes the one-dimensional nonlocal steady state thermistor problem; the values of 0 < α < 21 correspond to intermediate processes. In Section 5, a more general Caputo thermistor problem (8) on the half axis [0, +∞) is considered, instead of the bounded interval [0, T) of (1). One of the main difficulties lies in handling the nonlocal term t (∫0

λf (t, u(t)) f (x, u(x)) dx)2

of problem (8), where, in contrast with problem (1), function f depends on both time and the unknown function u. We are concerned with local existence on a finite interval (Theorem 3), as well as results of continuation (Theorem 4) and global existence (Theorems 5 and 6) via Schauder’s fixed-point theorem. Finally, in Section 6, we consider fractional integral and differential equations on time scales, that is, on arbitrary nonempty closed subsets of the real numbers. The investigation of dynamic equations on time scales allows one to unify and extend the theories of difference and differential equations into a single theory [13]. A time scale is a model of time, and the theory has found important applications in several contexts that require simultaneous modeling of discrete and continuous data. Its usefulness appears in many different areas, and the reader interested in applications is refereed to [3, 5, 20, 59, 63] and the references therein. The idea to join the two subjects of FC and the calculus on time scales, on a single theory, was born with the work in [14–16] and is subject to strong current research since 2011: see, e. g., [17, 19, 62, 63, 77]. Using Schauder’s fixed-point theorem, we obtain existence and uniqueness results of positive solutions for a fractional Riemann– Liouville nonlocal thermistor problem on arbitrary nonempty closed subsets of the real numbers (Theorems 8 and 9). We end with Section 7, presenting our conclusions.

330 | M. R. Sidi Ammi and D. F. M. Torres

2 Historical account Abel’s integral equations of first and second kinds can be formulated, respectively, as x

f (x) = ∫ 0

k(x, s)g(s) ds, (x − s)α

0 < α < 1,

0 ≤ x ≤ b,

(2)

and x

f (x) = ∫ 0

k(x, s)g(s) ds + g(x), (x − s)α

0 < α < 1,

0 ≤ x ≤ b,

(3)

where g is the unknown function to be found, f is a well behaved function, and k is the kernel. These celebrated equations appear frequently in many physical and engineering problems, like semi-conductors, heat conduction, metallurgy and chemical reactions [32, 36]. The special case α = 1/2 arises often. If the kernel is given by 1 , then (2) is a fractional integral equation of order 1 − α [45]. This probk(x, s) = Γ(1−α) lem is a generalization of the tautochrone problem, and it is related with the birth of the fractional calculus of variations [10]. For solving integral equations (2)–(3) of Abel type, several approaches are possible, e. g., using transformation techniques [38], orthogonal polynomials [61], integral operators [81], fractional calculus [30, 80], Bessel functions [78], wavelets [11], methods based on semigroups [43, 44], as well as many other techniques [55, 65]. FDEs, in which an unknown function is contained under the operation of a FD, have a long history, enriched by the intensive development of the theory of fractional calculus and their applications in the last decades. Fractional ordinary differential equations have the following form: F(x, y(x), Dα1 y(x), Dα2 y(x), . . . , Dαn y(x)) = g(x), where F(x, y1 , y2 , . . . , yn ) and g(x) are given functions, Dαk are fractional differentiation operators of real order αk > 0, k = 1, 2, . . . , n [68]. For example, one can consider nonlinear differential equations of the form Dα y(x) = f (x, y(x))

(4)

with real α > 0 or complex α, Re(α) > 0. Similarly to the investigation of ordinary differential equations, the methods for FDEs are essentially based on the study of equivalent Volterra integral equations. Equations (2)–(3) are examples of Volterra integral equations of the first and second kinds, respectively. Several authors have developed methods to deal with fractional integro-differential equations and construct solutions for ordinary and partial differential equations, with the general goal of obtaining a unified theory of special functions [49, 68]. In [35], analytical solutions to some linear

Analysis of fractional integro-differential equations of thermistor type | 331

operators of fractional integration and fractional differentiation are obtained, using Laplace transforms. The 1918 note of O’Shaughnessy and Post [64], is one of the first references to develop a method for solving the equation 1

(D 2 y)(x) =

y x

with the Riemann–Liouville derivative. Fifteen years later, in 1933, Fujiwara considered the FDE α

α (Dα+ y)(x) = ( ) y(x) x using the Hadamard FD of order α > 0 [28]. Provided f (x, y) is bounded in a special domain and satisfies a Lipschitz condition with respect to y, Pitcher and Sewell have shown in 1938 how the nonlinear FDE (Dαa+ y)(x) = f (x, y(x)),

0 < α < 1,

a ∈ ℝ,

(5)

in the sense of Riemann–Liouville, can be reduced to a Volterra integral equation, proving existence and uniqueness of a continuous solution to (5) [66]. In 1965, AlBassam considered the nonlinear Cauchy-type problem of fractional order (Dαa+ y)(x) = f (x, y(x)),

(Dα−1 a+ y)(x)/(x

= a) =

0 < α < 1,

1−α y)(x)/(x (Ia+

a ∈ ℝ,

= a) = b1 ,

b1 ∈ ℝ.

(6)

Similarly as before, he applied the method of successive approximations to the equivalent reduced Volterra integral equation and the contraction mapping method, establishing existence of a unique solution [9]. In 1978, Al-Abedeen and Arora considered the problem (Dαa+ y)(x) = f (x, y(x)),

1−α (Ia+ y)(c) = y0 ,

a < c < b,

y0 ∈ ℝ,

with 0 < α ≤ 1, and they proved an existence and uniqueness result for the corresponding Volterra nonlinear integral equation [8]. On the basis of Picard method and Schauder’s fixed-point theorem, Tazali obtained in 1982 two local existence results of a continuous solution to (6) [82]. Interestingly, more general existence and uniqueness results than the ones of [82], also obtained using a fixed-point theorem and equivalent nonlinear integral formulations, have been published in 1977 by Leskovskiĭ [53]. Similar results, on the basis of fixed-point theorems and integral equations, were derived by Semenchuk in 1982 [69]. In 1988 [27], El-Sayed examined problem (4) on a finite interval where the FD is considered in the sense of Gelfand and Shilov [29]. Hadid, in 1995, used a fixed-point theorem to prove existence of a solution to the corresponding integral equation of (6) [37]. Using the contraction mapping method on a complete

332 | M. R. Sidi Ammi and D. F. M. Torres metric space, Hayek et al. established in 1999 existence and uniqueness of a continuous solution to the Cauchy-type problem described by the following system of FDEs: (Dα0+ y)(x) = f (x, y(x)),

y(a) = b,

0 < α ≤ 1,

a > 0,

b ∈ ℝn ;

see [39]. In 2000, Kilbas, Bonilla and Trujillo studied the Cauchy-type problem (Dαa+ y)(x) = f (x, y(x)), α−k) y)(x)/(x (Da+

= a) = bk ,

n − 1 < α ≤ n, bk ∈ ℝ,

n = −[−α],

k = 1, 2, . . . , n,

with complex α, α ∈ ℂ with Re(α) > 0, on a finite interval [a, b] [46]. By using the method of successive expansions and Avery–Henderson and Leggett–Williams multiple fixed-point theorems on cones, they proved existence of multiple positive solutions for the corresponding nonlinear Volterra integral equation [46]. See also [24, 48]. It should be noted, however, that in some function spaces the proof of equivalence between solutions of Cauchy-type problems for FDEs and corresponding reduced Volterra integral equations constitute a major difficulty [21]. For more recent results, we refer the reader to [2]. The techniques are, however, similar to the ones already mentioned; details are below.

3 Fundamental results Let C([0, T]) be the space of all continuous functions on [0, T]. The following three auxiliary lemmas are particularly useful for our purposes. Lemma 1 (See [54]). Let M be a subset of C([0, T]). Then M is precompact if and only if the following conditions hold: 1. {u(t) : u ∈ M} is uniformly bounded, 2. {u(t) : u ∈ M} is equicontinuous on [0, T]. Lemma 2 (Schauder’s fixed-point theorem [23]). Let U be a closed bounded convex subset of a Banach space X. If T : U → U is completely continuous, then T has a fixed point in U. We also recall the following version of Gronwall’s lemma. Lemma 3 (Generalized Gronwall’s inequality [40, 84]). Let v : [0, b] → [0, +∞) be a real function and w(⋅) be a nonnegative, locally integrable function on [0, b]. Suppose that there exist a > 0 and 0 < α < 1 such that t

v(t) ≤ w(t) + a ∫ 0

v(s) ds. (t − s)α

Analysis of fractional integro-differential equations of thermistor type | 333

Then there exists a constant k = k(α) such that t

v(t) ≤ w(t) + ka ∫ 0

w(s) ds (t − s)α

for t ∈ [0, b].

4 Nonlocal Riemann–Liouville problem Let 0 < α
0 we can prove that the map F : X → X is a contraction and it has a fixed point u = Fu. Hence, there exists a unique u ∈ X that is the solution to the integral equation (7). The result follows from Lemma 4.

4.2 Boundedness We now show that the assumption that electrical conductivity f (u) is bounded (hypothesis (H1 )) allows one to assert the boundedness of u. Theorem 2. Under hypotheses (H1 ) and (H2 ) and λ > 0, if u is the solution of (7), then ‖u‖ ≤

( (c λT)2 f (0) + h∞ ) 1

N 2α

λLf

e (c1 TN α )2 .

For more details on the subject see [74].

5 Nonlocal Caputo thermistor problem Now, our main goal consists to prove global existence of the solutions for a fractional Caputo nonlocal thermistor problem. Precisely, we consider the following fractionalorder initial value problem: C

D2α 0+ u(t) =

t (∫0

λf (t, u(t)) f (x, u(x)) dx)2

,

t ∈ (0, ∞) ,

(8)

u(t)|t=0 = u0 ,

1 where C D2α 0+ is the fractional Caputo derivative operator of order 2α with 0 < α < 2 a real parameter. We shall assume the following hypotheses: (H1 ) f : ℝ+ × ℝ+ → ℝ+ is a Lipschitz continuous function with Lipschitz constant Lf with respect to the second variable such that c1 ≤ f (s, u) ≤ c2 with c1 and c2 two positive constants; (H2 ) there exists a positive constant M such that f (s, u) ≤ Ms2 ; (H3 ) |f (s, u) − f (s, v)| ≤ s2 |u − v| or, in a more general manner, there exists a constant ω ≥ 2 such that |f (s, u) − f (s, v)| ≤ sω |u − v|.

Analysis of fractional integro-differential equations of thermistor type | 335

5.1 Local existence theorem In this subsection, a local existence theorem of the solutions for (8) is obtained by applying Schauder’s fixed-point theorem. In order to transform (8) into a fixed-point problem, we give in the following lemma an equivalent integral form of (8). Lemma 5. Suppose that (H1 )–(H3 ) holds. Then the initial value problem (8) is equivalent to t

λ f (s, u(s)) u(t) = u0 + ds. ∫(t − s)2α−1 t Γ(2α) (∫ f (x, u) dx)2 0

(9)

0

Proof. It is a simple exercise to see that u is a solution of the integral equation (9) if and only if it is also a solution of the IVP (8). Theorem 3. Suppose that conditions (H1 )–(H3 ) are verified. Then (8) has at least one solution u ∈ C[0, h] for some T ≥ h > 0. Proof. Let E = {u ∈ C[0, T] : ‖u − u0 ‖C[0,T] = sup |u − u0 | ≤ b}, 0≤t≤T

where b is a positive constant. Further, put Dh = {u : u ∈ C[0, h], ‖u − u0 ‖C[0,h] ≤ b}, where −1

1

2α λM , T} ) ) h = min{(b( Γ(2α + 1)c12

and 0 < α < 21 . It is clear that h ≤ T. Note also that Dh is a nonempty, bounded, closed, and convex subset of C[0, h]. In order to apply Schauder’s fixed-point theorem, we define the following operator A: t

(Au)(t) = u0 +

f (s, u(s)) λ ds, ∫(t − s)2α−1 t Γ(2α) (∫ f (x, u) dx)2 0

t ∈ [0, h].

(10)

0

It is clear that all solutions of (8) are fixed points of (10). Then, by assumptions (H1 ) and (H2 ), it follows that ADh ⊂ Dh . Our next step, in order to prove Theorem 3, consists to use the following two technical lemmas. Lemma 6. The operator A is continuous. Lemma 7. The operator ADh is continuous.

336 | M. R. Sidi Ammi and D. F. M. Torres One can prove that {(Au)(t) : u ∈ Dh } is equicontinuous. Taking into account that ADh ⊂ Dh , we infer that ADh is precompact. This implies that A is completely continuous. As a consequence of Schauder’s fixed-point theorem and Lemma 5, we conclude that problem (8) has a local solution. This completes the proof of Theorem 3.

5.2 Continuation results Now we give a continuation theorem for the fractional Caputo nonlocal thermistor problem (8). First, we present the definition of a noncontinuable solution. ̃ on (0, β)̃ be both solutions of (8). If Definition 1 (See [50]). Let u(t) on (0, β) and u(t) ̃ ̃ ̃ for t ∈ (0, β), then we say that u(t) ̃ can be continued to (0, β). β < β and u(t) = u(t) A solution u(t) is noncontinuable if it has no continuation. The existing interval of the noncontinuable solution u(t) is called the maximum existing interval of u(t). Theorem 4. Assume that conditions (H1 )–(H3 ) are satisfied. Then u = u(t), t ∈ (0, β), β is noncontinuable if and if only for some η ∈ (0, 2 ) and any bounded closed subset ∗ S ⊂ [η, +∞) × ℝ there exists a t ∈ [η, β) such that (t ∗ , u(t ∗ )) ∉ S. Proof. Suppose that there exists a compact subset S ⊂ [η, +∞) × ℝ such that {(t, u(t)) : t ∈ [η, β)} ⊂ S. Compactness of S implies β < +∞. The proof follows from Lemmas 8 and 9. Lemma 8. The limit limt→β− u(t) exists. Proof. The proof is based on the Cauchy convergence criterion. The second step of the proof of Theorem 4 consists to show the following result. Lemma 9. The function u(t) is continuable. Proof. As S is a closed subset, we can say that (β, u∗ ) ∈ S. Define u(β) = u∗ . Hence, u(t) ∈ C[0, β]. Then we define the operator K by t

λ f (s, v(s)) (Kv)(t) = u1 + ds, ∫(t − s)2α−1 t Γ(2α) (∫ f (x, v) dx)2 0

β

where β

f (s, v(s)) λ u1 = u0 + ds, ∫(t − s)2α−1 t Γ(2α) (∫ f (x, v) dx)2 0

0

v ∈ C([β, β + 1]), t ∈ [β, β + 1]. Set 󵄨 󵄨 Eb = {(t, v) : β ≤ t ≤ β + 1, |v| ≤ max 󵄨󵄨󵄨u1 (t)󵄨󵄨󵄨 + b} β≤t≤β+1

Analysis of fractional integro-differential equations of thermistor type | 337

and 󵄨 󵄨 Eh = {v ∈ C[β, β + 1] : max 󵄨󵄨󵄨v(t) − u1 (t)󵄨󵄨󵄨 ≤ b, v(β) = u1 (β)}, t∈[β,β+h] 1

λM −1 2α where h = min{(b( Γ(2α+1)c ) , 1}. Similarly to the proof of Theorem 3, we show that 2) 1

K is completely continuous on Eb , which shows that the operator K is continuous. We show that KEh is equicontinuous. Consequently, K is completely continuous. Then Schauder’s fixed-point theorem can be applied to see that the operator K has a fixed ̃ ∈ Eh . On other words, we have point u(t) t

̃ f (s, u(s)) λ ds ∫(t − s)2α−1 t Γ(2α) ̃ dx)2 (∫ f (x, u(x))

̃ = u1 + u(t)

β

0

t

= u0 +

̃ λ f (s, u(s)) ds, ∫(t − s)2α−1 t Γ(2α) ̃ dx)2 (∫ f (x, u(x)) 0

0

t ∈ [β, β + h], where u(t), t ∈ (0, β], ̃ ={ u(t) ̃ u(t), t ∈ [β, β + h]. ̃ ∈ C[0, β + h] and It follows that u(t) t

̃ = u0 + u(t)

̃ f (s, u(s)) λ ds. ∫(t − s)2α−1 t Γ(2α) ̃ dx)2 (∫ f (x, u(x)) 0

0

̃ is a solution of (8) on (0, β + h]. This is absurd Therefore, according to Lemma 5, u(t) because u(t) is noncontinuable. This completes the proof of Lemma 9. Similarly to Theorem 1, uniqueness of the solution to problem (8) is derived from the proof of Theorem 4 for a well chosen λ.

5.3 Global existence of the solutions Now we provide two sets of sufficient conditions for the existence of a global solution for (8) (Theorems 5 and 6). We begin with an auxiliary lemma. Lemma 10. Suppose that conditions (H1 )–(H3 ) hold. Let u(t) be a solution of (8) on (0, β). If u(t) is bounded on [τ, β) for some τ > 0, then β = +∞. Proof. The proof follows immediately from the results of Subsection 5.2.

338 | M. R. Sidi Ammi and D. F. M. Torres Theorem 5. Suppose that conditions (H1 )–(H3 ) hold. Then (8) has a solution in C([0, +∞)). Proof. The existence of a local solution u(t) of (8) is ensured thanks to Theorem 3. We already know, by Lemma 5, that u(t) is also a solution to the integral equation t

u(t) = u0 +

λ f (s, u(s)) ds. ∫(t − s)2α−1 t Γ(2α) (∫ f (x, u(x)) dx)2 0

0

Suppose that the existing interval of the noncontinuable solution u(t) is (0, β), β < +∞. Then t 󵄨󵄨 󵄨󵄨 󵄨󵄨 f (s, u(s)) λ 󵄨 󵄨󵄨 󵄨󵄨 ds󵄨󵄨󵄨 ∫(t − s)2α−1 t 󵄨󵄨u(t)󵄨󵄨󵄨 = 󵄨󵄨󵄨u0 + Γ(2α) 󵄨󵄨󵄨 (∫0 f (x, u(x)) dx)2 󵄨󵄨󵄨 0 t

λ 1 󵄨 󵄨 ≤ |u0 | + ∫(t − s)2α−1 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds Γ(2α) (c1 t)2 t

≤ |u0 | +

0

λ 1 |u(s)| ds. ∫ Γ(2α) c12 (t − s)1−2α 0

By Lemma 5, there exists a constant k(α) such that, for t ∈ (0, β), we have t

λ 1 󵄨 󵄨󵄨 ∫(t − s)2α−1 ds, 󵄨󵄨u(t)󵄨󵄨󵄨 ≤ |u0 | + k|u0 | Γ(2α) c2 1 0

which is bounded on (0, β). Thus, by Lemma 10, problem (8) has a solution u(t) on (0, +∞). Next we give another sufficient condition ensuring global existence for (8). Theorem 6. Suppose that there exist positive constants c3 , c4 and c5 such that c3 ≤ |f (s, x)| ≤ c4 |x| + c5 . Then (8) has a solution in C([0, +∞)). Proof. Suppose that the maximum existing interval of u(t) is (0, β), β < +∞. We claim that u(t) is bounded on [τ, β) for any τ ∈ (0, β). Indeed, we have t 󵄨󵄨 󵄨󵄨 󵄨󵄨 λ f (s, u(s)) 󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 ds󵄨󵄨󵄨 ∫(t − s)2α−1 t 󵄨󵄨u(t)󵄨󵄨 = 󵄨󵄨u0 + 󵄨󵄨 Γ(2α) (∫0 f (x, u(x)) dx)2 󵄨󵄨󵄨 󵄨 0

≤ |u0 | +

t

t

0

0

c3 c2 |u(s)| λ λ ds. ∫(t − s)2α−1 ds + ∫ Γ(2α) (c1 τ)2 Γ(2α) (c1 τ)2 (t − s)1−2α

Analysis of fractional integro-differential equations of thermistor type | 339

If we take t

c3 λ w(t) = |u0 | + ∫(t − s)2α−1 ds, Γ(2α) (c1 τ)2 0

which is bounded, and a=

λc2 1 , Γ(2α) (c1 β)2

it follows, according with Lemma 3, that v(t) = |u(t)| is bounded. Thus, by Lemma 10, problem (8) has a solution u(t) on (0, +∞). For more details on the subject, see [70, 71, 74, 75, 79].

6 Fractional problems on arbitrary time scales Throughout the remainder of this chapter, we denote by 𝕋 a time scale, which is a nonempty closed subset of ℝ with its inherited topology. For convenience, we make the blanket assumption that t0 and T are points in 𝕋. Our main concern is to prove existence and uniqueness of the solution to a fractional-order nonlocal thermistor problem of the form λf (u(t)) 𝕋 2α , t ∈ (t0 , T) , Dt0+ u(t) = T (∫t f (u(x)) △ x)2 (11) 0 𝕋 β It0+ u(t0 )

= 0,

∀ β ∈ (0, 1),

under suitable conditions on f as described below. We assume that α ∈ (0, 1) is a parameter describing the order of the FD; 𝕋 D2α t0+ is the left Riemann–Liouville FD operator β

of order 2α on 𝕋; 𝕋 It0+ is the left Riemann–Liouville FI operator of order β defined on 𝕋 by [19] (see Section 6.1, where these definitions and main properties of the fractional operators on time scales are recalled). As before, u may be interpreted as the temperature inside the conductor and f (u) the electrical conductivity of the material. In the literature, many existence results for dynamic equations on time scales are available [25, 26]. In recent years, there has also been significant interest in the use of FDEs in mathematical modeling [6, 56, 85]. However, much of the work published to date has been concerned with it separately by the time-scale community and by the fractional one. Results on fractional dynamic equations on time scales are scarce [7]. Here we give existence and uniqueness results for the fractional-order nonlocal thermistor problem on time scales (11), putting together time-scale and fractional domains. According with [57, 62, 63], this is quite appropriate from the point of view of practical applications. Our main aim is to prove existence of the solutions for (11) using a fixed-point theorem and, consequently, uniqueness (see Theorems 8 and 9). For more details see [77].

340 | M. R. Sidi Ammi and D. F. M. Torres

6.1 Fractional calculus on time scales We deal with the notions of Riemann–Liouville FIs and FDs on time scales, the so called BHT fractional calculus on time scales [62]. For local approaches, we refer the reader to [17, 18]. Here we are interested in nonlocal operators, which are the ones who make sense with respect to thermistor-type problems [72, 73]. Although we restrict ourselves to the delta approach on time scales, similar results are trivially obtained for the nabla fractional case [31]. Definition 2 (Riemann–Liouville FI on time scales [19]). Let 𝕋 be a time scale and [a, b] an interval of 𝕋. Then the left fractional integral on time scales of order 0 < α < 1 of a function g : 𝕋 → ℝ is defined by 𝕋 α Ia+ g(t)

t

=∫ a

(t − s)α−1 g(s) △ s, Γ(α)

where Γ is the Euler gamma function. The left Riemann–Liouville FD operator of order α on time scales is then defined using Definition 2 of FI. Definition 3 (Riemann–Liouville FD on time scales [19]). Let 𝕋 be a time scale, [a, b] an interval of 𝕋, and α ∈ (0, 1). Then the left Riemann–Liouville FD on time scales of order α of a function g : 𝕋 → ℝ is defined by 𝕋 α Da+ g(t)

t

△

(t − s)−α g(s) △ s) . = (∫ Γ(1 − α) a

Remark 1. If 𝕋 = ℝ, then we obtain from Definitions 2 and 3, respectively, the usual left Rieman–Liouville FI and FD. Proposition 1 (See [19]). Let 𝕋 be a time scale, g : 𝕋 → ℝ and 0 < α < 1. Then 𝕋 α Da+ g

1−α

= △ ∘ 𝕋 I a+ g.

Proposition 2 (See [19]). If α > 0 and g ∈ C([a, b]), then 𝕋 α Da+

α

∘ 𝕋 I a+ g = g. 1−α

Proposition 3 (See [19]). Let g ∈ C([a, b]), 0 < α < 1. If 𝕋 I a+ u(a) = 0, then 𝕋 α Ia+

α

∘ 𝕋 Da+ g = g.

α Theorem 7 (See [19]). Let g ∈ C([a, b]), α > 0, and 𝕋 a It ([a, b]) be the space of functions that can be represented by the Riemann–Liouville △-integral of order α of some C([a, b])-function. Then

g∈

𝕋 α a It ([a, b])

Analysis of fractional integro-differential equations of thermistor type

| 341

if and only if 𝕋 1−α Ia+ g

∈ C 1 ([a, b])

and 𝕋 1−α Ia+ g(a)

= 0.

The following result of the calculus on time scales is also useful. Proposition 4 (See [7]). Let 𝕋 be a time scale and g an increasing continuous function on the time-scale interval [a, b]. If G is the extension of g to the real interval [a, b] defined by g(s) G(s) := { g(t)

if s ∈ 𝕋, if s ∈ (t, σ(t)) ∉ 𝕋,

then b

b

∫ g(t) △ t ≤ ∫ G(t)dt, a

a

where σ : 𝕋 → 𝕋 is the forward jump operator of 𝕋 defined by σ(t) := inf{s ∈ 𝕋 : s > t}.

6.2 Existence We begin by giving an integral representation to our problem (11). Note that the case 0 < α < 21 is coherent with our fractional operators with 2α − 1 < 0. Lemma 11. Let 0 < α < 21 . Problem (11) is equivalent to t

f (u(s)) λ u(t) = △ s. ∫(t − s)2α−1 T Γ(2α) (∫t f (u) △ x)2 t 0

(12)

0

Proof. We have 𝕋 2α Dt0+ u(t)

t

=

λ f (u(s)) △ s) (∫(t − s)2α−1 T Γ(2α) (∫ f (u) △ x)2

=

△ (𝕋 It1−2α u(t)) 0+

t0

t0

= (△ ∘ 𝕋 It1−2α )u(t). 0+

The result follows from Proposition 3: 𝕋 It2α ∘ (𝕋 D2α t0+ (u)) = u. 0+

△

342 | M. R. Sidi Ammi and D. F. M. Torres For the sake of simplicity, we take t0 = 0. It is easy to see that (11) has a solution u = u(t) if and only if u is a fixed point of the operator K : X → X defined by t

Ku(t) =

λ f (u(s)) △ s. ∫(t − s)2α−1 T Γ(2α) (∫ f (u) △ x)2 0

(13)

0

To prove existence of the solution, we begin by showing that the operator K defined by (13) verifies the conditions of Schauder’s fixed-point theorem. Lemma 12. The operator K is continuous. Lemma 13. The operator K sends bounded sets into bounded sets on ℂ([0, T], ℝ). Lemma 14. The operator K sends bounded sets into equicontinuous sets of ℂ(I, ℝ). It follows by Schauder’s fixed-point theorem that (11) has a solution on I. We have just proved Theorem 8. Theorem 8 (Existence of the solution). Let 0 < α < 21 and f satisfies hypothesis (H1 ). Then there exists a solution u ∈ X of (11) for all λ > 0.

6.3 Uniqueness We now derive uniqueness of the solution to problem (11). Theorem 9 (Uniqueness of the solution). Let 0 < α < the hypothesis (H1 ). If 0 a. If 𝒥 achieves a minimum value at x ⋆ , and if the maps t 󳨃→ DαA− (𝜕3 L[x⋆ ]α (t)), for t ∈ [a, A], and t 󳨃→ Dαb− (𝜕3 L[x⋆ ]α (t)), for t ∈ [a, b], exist and are continuous, then α

α

Dαb− (𝜕3 L[x⋆ ] (t)) − DαA− (𝜕3 L[x⋆ ] (t)) = 0,

t ∈ [a, A],

and α

α

𝜕2 L[x⋆ ] (t) + Dαb− (𝜕3 L[x⋆ ] (t)) = 0,

t ∈ [A, b].

Moreover, the following transversality conditions are fulfilled: 1−α I 1−α (𝜕3 L[x⋆ ]α (t)) − IA− (𝜕3 L[x⋆ ]α (t)) = 0 at t = a, if x(a) is free; { { { b− 1−α IA− (𝜕3 L[x⋆ ]α (t)) = 0 at t = A, if x(A) is free; { { { 1−α ⋆ α at t = b, if x(b) is free. {Ib− (𝜕3 L[x ] (t)) = 0

A survey on fractional variational calculus | 353

For our next result, besides the boundary conditions, an integral constraint is imposed on the set of admissible functions. Such type of problems are known as isoperimetric problems, and the first example goes back to Dido, Queen of Cartage in Africa. She was interested if finding the shape of a curve with fixed perimeter, of maximum possible area. As is well known, the solution is given by a circle. The calculus of variations presents a solution to such problem, with an integral constraint of type b

2

∫ √1 + (x󸀠 (t)) dt = constant. a

In our case, we consider that the new constraint depends also on a fractional derivative, and it is of form b

G(x) := ∫ M(t, x(t), C Dαa+ x(t)) dt = K,

K ∈ ℝ,

(10)

a

where M : [a, b] × ℝ2 → ℝ is a constant, such that the functions 𝜕2 M and 𝜕3 M exist and are continuous. Theorem 4. Suppose that 𝒥 (3), subject to the constraints (4) and (10), attains a minimum value at x⋆ . If x⋆ is not a solution for 𝜕2 M[x]α (t) + Dαb− (𝜕3 M[x]α (t)) = 0,

∀t ∈ [a, b],

(11)

and if there exist and are continuous the functions t 󳨃→ Dαb− (𝜕3 L[x⋆ ]α (t)) and t 󳨃→ Dαb− (𝜕3 M[x⋆ ]α (t)) on [a, b], then there exists λ ∈ ℝ such that x⋆ satisfies 𝜕2 F[x]α (t) + Dαb− (𝜕3 F[x]α (t)) = 0,

∀t ∈ [a, b],

where we define the function F as F := L + λM. The case when x⋆ is a solution of (11) can easily be included. Theorem 5. Suppose that 𝒥 (3), subject to the constraints (4) and (10), attains a minimum value at x ⋆ . If there exist and are continuous the functions t 󳨃→ Dαb− (𝜕3 L[x⋆ ]α (t)) and t 󳨃→ Dαb− (𝜕3 M[x⋆ ]α (t)) on [a, b], then there exist λ0 , λ ∈ ℝ, not both zero, such that x⋆ satisfies the equation 𝜕2 F[x]α (t) + Dαb− (𝜕3 F[x]α (t)) = 0,

∀t ∈ [a, b],

where we define the function F as F := λ0 L + λM. Next, we present another constrained type problem, but now the restriction is given by g(t, x(t)) = 0,

∀t ∈ [a, b],

(12)

354 | R. Almeida and D. F. M. Torres where x = (x1 , x2 ) is a vector and g : [a, b] × ℝ2 → ℝ is differentiable with respect to x1 and x2 . Also, we have the boundary conditions x(a) = xa

and x(b) = xb ,

xa , xb ∈ ℝ2 .

(13)

Theorem 6. Consider the functional b

α

𝒥 (x) = ∫ L[x]2 (t) dt, a

where [x]α2 (t) := (t, x1 (t), x2 (t), C Dαa+ x1 (t), C Dαa+ x2 (t)), defined on C 1 [a, b] × C 1 [a, b], subject to the constraints (12) and (13). If 𝒥 attains an extremum at x ⋆ = (x1⋆ , x2⋆ ), the maps t 󳨃→ Dαb− (𝜕i+3 L[x ⋆ ]α2 (t)), i = 1, 2, are continuous, and 𝜕3 g(t, x(t)) ≠ 0 for all t ∈ [a, b], then there exists a continuous function λ : [a, b] → ℝ such that α

α

𝜕i+1 L[x⋆ ]2 (t) + Dαb− (𝜕i+3 L[x⋆ ]2 (t)) + λ𝜕i+1 g(t, x(t)) = 0 for all t ∈ [a, b] and i = 1, 2. Infinite horizon problems are an important field of research that deals with phenomena that spread in time [1, 13, 20]. In this case, the cost functional is evaluated in an infinite interval [a, ∞[: ∞

α

𝒥 (x) := ∫ L[x] (t) dt,

(14)

a

defined on a set of functions with fixed initial conditions: x(a) = xa . Since we are dealing with improper integrals, some attention should be paid to the question what a minimizer is. We say that x⋆ is a local minimizer for 𝒥 as in (14) if there exists some ϵ > 0 such that, for all x, if ‖x⋆ − x‖ < ϵ, then b

α

lim inf ∫[L[x⋆ ] (t) − L[x]α (t)] dt ≤ 0.

T→∞ b≥T

a

Also, let us define the functions b

A(ϵ, b) := ∫ a

L[x ⋆ + ϵv]α (t) − L[x⋆ ]α (t) dt; ϵ b

α

α

V(ϵ, T) := inf ∫[L[x⋆ + ϵv] (t) − L[x⋆ ] (t)]dt; b≥T

a

A survey on fractional variational calculus | 355

W(ϵ) := lim V(ϵ, T), T→∞

where v ∈ C 1 [a, ∞[ is a function and ϵ a real. Theorem 7. Let x⋆ be a local minimizer for 𝒥 as in (14). Suppose that: 1. limϵ→0 V(ϵ,T) exists for all T; ϵ 2. limT→∞ V(ϵ,T) exists uniformly for all ϵ; ϵ 3. for every T > a and ϵ ≠ 0, there exists a sequence (A(ϵ, bn ))n∈ℕ such that lim A(ϵ, bn ) = inf A(ϵ, b)

n→∞

b≥T

uniformly for ϵ. If there exists and is continuous the function t 󳨃→ Dαb− (𝜕3 L[x⋆ ]α (t)) on [a, b], for all b > a, then α

α

𝜕2 L[x⋆ ] (t) + Dαb− (𝜕3 L[x⋆ ] (t)) = 0, for all b > a. Also, we have α

1−α (𝜕3 L[x⋆ ] (t)) = 0 at t = b. lim inf Ib−

T→∞ b≥T

So far, we considered variational problems with order α ∈ ]0, 1[. Now we proceed by extending them to functionals depending on higher-order derivatives. With this purpose, consider the functional b

C

α

C

α

𝒥 (x) := ∫ L(t, x(t), Da+1 x(t), . . . , Da+m x(t)) dt,

(15)

a

defined on C m [a, b], such that x(i) (a) and x(i) (b) are fixed reals for i ∈ {0, 1, . . . , m − 1}. Here, m is a positive integer, αi ∈ ]i − 1, i[, for all i ∈ {1, . . . , m}, and L : [a, b]×ℝm+1 → ℝ is differentiable with respect to the ith variable, for i ∈ {2, 3, . . . , m + 1}. Theorem 8. Let x ⋆ be a minimizer of 𝒥 (15), and suppose that, for all i ∈ {1, . . . , m}, the α functions t 󳨃→ Db−i (𝜕i+2 L[x⋆ ]αm (t)) exist and are continuous on [a, b]. Then α

m

α

α

𝜕2 L[x⋆ ]m (t) + ∑ Db−i (𝜕i+2 L[x⋆ ]m (t)) = 0 i=1

α

α

for all t ∈ [a, b], where [x ⋆ ]αm (t) := (t, x(t), C Da+1 x⋆ (t), . . . , C Da+m x⋆ (t)). After solving the presented Euler–Lagrange equations, we still need to verify if we are in the presence of a minimizer of the functional or not. We recall that the Euler– Lagrange equation is just a necessary optimality condition of first order, and thus its

356 | R. Almeida and D. F. M. Torres solutions may not be a solution for the problem. For the question of the existence of solutions, we refer to [8, 9]. One possible way to check if we have a candidate for a minimizer or a maximizer is to apply the Legendre condition, which is a second-order necessary optimality condition [3, 17]. Under suitable convexity of the Lagrangian, sufficient conditions for a global minimizer hold [6]. For our next result, we recall that a function L : [a, b] × ℝ2 → ℝ is convex with respect to the second and third variables if L(t, x + v, y + w) − L(t, x, y) ≥ 𝜕2 L(t, x, y)v + 𝜕3 L(t, x, y)w for all t ∈ [a, b] and x, y, v, w ∈ ℝ. Theorem 9. If L is convex and if x ⋆ satisfies the Euler–Lagrange equation (9), then x ⋆ minimizes 𝒥 (3) when restricted to the boundary conditions (4).

4 Direct methods There are several different numerical approaches and methods to solve fractional differential equations [2, 10, 12]. Different discretizations of fractional derivatives are possible, but many of them do not preserve the fundamental properties of the systems, such as stability [15, 29]. For numerical calculations, a simple and powerful method preserving stability is obtained from Grünwald–Letnikov discretizations. For a detailed numerical treatment of fractional differential equations, based on Grünwald–Letnikov fractional derivatives, we refer the interested reader to [24, 28]. Here we just mention that both explicit and implicit methods are possible: Theorem 5.1 of [28] shows that the explicit and implicit Grünwald–Letnikov methods are asymptotically stable; while Theorem 5.2 of [28] gives conditions on the step size for the explicit method to be absolute stable, asserting that the implicit method is always absolutely stable, without any step size restriction. It is also worth to underline the good convergence properties and error behavior of the Grünwald–Letnikov methods [28]. We start this section by recalling the (left) Grünwald–Letnikov fractional derivative of a function x. Let α > 0 be a real. The Grünwald–Letnikov fractional derivative of order α is defined by GL α a Dx x(t)

:= lim+ h→0

1 ∞ α ∑ (−1)k ( )x(t − kh), hα k=0 k

where (αk ) stands for the generalization of binomial coefficients to real numbers. As usual, we will adopt the notation α (wkα ) := (−1)k ( ). k

A survey on fractional variational calculus | 357

This fractional derivative is particularly useful to approximate the Caputo derivative. The method is now briefly explained. Given an interval [a, b] and a fixed integer N, let tj := a + jh, j = 0, 1, . . . , N and h > 0, be a partition of the interval [a, b]. Then C

Dαa+ x(tj ) =

j

1 x(a) (t − a)−α + O(h). ∑ (wkα )x(tj−k ) − α h k=0 Γ(1 − α) j

Thus, truncated Grünwald–Letnikov fractional derivatives are first-order approximations of the Caputo fractional derivatives. The approximation used for the Caputo derivative is then C

Dαa+ x(tj ) ≈

j

x(a) 1 ̃ j ). (t − a)−α := Dx(t ∑ (wα )x(tj−k ) − hα k=0 k Γ(1 − α) j

The next step is to discretize the functional (3). For simplicity, let h = (b − a)/N and let us consider the grid tj = a + jh, j = 0, 1, . . . , N. Then tk

N

C

α

𝒥 (x) = ∑ ∫ L(t, x(t), Da+ x(t)) dt k=1 t

k−1

N

≈ ∑ hL(tk , x(tk ), C Dαa+ x(tk ))

(16)

k=1 N

̃ k )) dt. ≈ ∑ hL(tk , x(tk ), Dx(t k=1

The right hand side of (16) can be regarded as a function Ψ of N − 1 unknowns: N

̃ k )). Ψ(x1 , x2 , . . . , xN−1 ) = ∑ hL(tk , x(tk ), Dx(t k=1

To find an extremum for Ψ, one has to solve the following system of algebraic equations: 𝜕Ψ = 0, 𝜕xi

i = 1, . . . , N − 1.

Suppose that h goes to zero and the solution obtained by this method converges to a function x⋆ . Then x⋆ is a solution of the Euler–Lagrange equation (9) (for details, see [4, Theorem 8.1]). Example 1. Consider the functional 10

C

0.5

𝒥 (x) = ∫( D0+ x(t) − 0

2

2 t 1.5 ) dt Γ(3/2)

358 | R. Almeida and D. F. M. Torres Table 1: Errors for the numerical solution of problem of Example 1. n Error

10

50

100

200

0.425189

0.107622

0.056370

0.029215

Figure 1: Plot of the numerical solution of problem of Example 2. 2 subject to the boundary conditions x(0) = 0 and x(10) = 100. Since C D0.5 0+ t =

2 t 1.5 , Γ(3/2)

and 𝒥 is nonnegative, we conclude that the function x(t) = t 2 is a minimizer of the functional. If we discretize the functional, for different values of n, we obtain several numerical approximations of x. In Table 1 we present the error of each n, where the error is given by the maximum of the absolute value of the difference between x and the numerical approximation.

Example 2. For our second example, we consider a problem where we do not know its exact solution. Let 1

C

0.5

2

2

𝒥 (x) = ∫(x(t)( D0+ x(t)) − sin(x(t))) dt 0

subject to the boundary conditions x(0) = 0 and x(1) = 1. In Figure 1, we present the result for n = 100.

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R. Almeida, General necessary conditions for infinite horizon fractional variational problems, Appl. Math. Lett., 26(7) (2013), 787–793. R. Almeida and N. R. O. Bastos, A numerical method to solve higher-order fractional differential equations, Mediterr. J. Math., 13(3) (2016), 1339–1352.

A survey on fractional variational calculus | 359

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Teodor M. Atanacković, Sanja Konjik, and Stevan Pilipović

Variational principles with fractional derivatives

Abstract: We consider several generalizations of the Hamilton principle and study optimality conditions involving Euler–Lagrange equations. We present the concept of the fractional complementary variational principle. Moreover, we propose an expansion formula for the fractional derivative as a tool for solving the Euler–Lagrange equations. Keywords: Fractional derivatives, fractional variational principles, Euler–Lagrange equations, expansion formula MSC 2010: 26A33, 49K05, 34A08

1 Introduction The use of derivatives and integrals with real or even complex number order has opened possibilities for novel theoretical and applied research, whose expansion is particularly evident in the last several decades. Applications of fractional operators can be found in such diverse fields as physics, mechanics, engineering, control theory, modeling, probability, finance, economics, biology, chemistry, medicine, pharmacology, etc. We refer here to several extensive monographs and reference books that cover the basics of the theory of fractional calculus along with comprehensive studies on advanced topics: [15, 16, 28, 33, 36, 42, 45, 46, 50, 39]. This paper is devoted to the application of the fractional calculus to the calculus of variations. We focus on variational problems whose Lagrangians involve fractional derivatives, the Euler–Lagrange equations, the optimality conditions for the action integrals and approximations. Research in this direction has appeared with the paper of Riewe [48, 49], who investigated non-conservative Lagrangian and Hamiltonian mechanics and formulated a version of the Euler–Lagrange equations. Jumarie [30–32] applied fractional variational calculus to the analysis of fractional Brownian motion, fractional stochastic mechanics and stochastic optimal control, while Agrawal [1–3] Acknowledgement: This work is supported by Projects 174005 and 174024 of the Serbian Ministry of Science and Project 142-451-2384 of the Provincial Secretariat for Higher Education and Scientific Research. Teodor M. Atanacković, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21000 Novi Sad, Serbia, e-mail: [email protected] Sanja Konjik, Stevan Pilipović, Faculty of Sciences, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110571622-015

362 | T. M. Atanacković et al. considered different types of variational problems involving Riemann–Liouville, Caputo and Riesz fractional derivatives, and derived the corresponding Euler–Lagrange equations. Further significant contributions to the theory include the work of research groups of Torres [5, 4, 25, 39–41], Baleanu [21, 22, 19, 20, 24, 23, 43], Atanackovic [14, 13, 18, 11, 12, 9, 10], and Klimek [34–36]. The content of the paper is organized as follows. After preliminaries of Section 2, our main results are presented in the next three sections. We present in Subsection 3.1 the optimality of the fractional variational problem allowing for the case that the interval, which defines the action integral, is different from the interval over which fractional derivatives are taken. Then we study in Subsection 3.2 the optimality conditions taking the order of the real fractional derivative as a variable with respect to which optimization is performed. In Subsection 3.3 we consider the action integral with the fractional derivatives with complex orders and discuss its optimality. Section 4 contains a short presentation (without proofs) of a complementary variational principle that involves fractional derivatives. Motivated by limitations that occur due to the presence of both left and right fractional derivatives in the Euler–Lagrange equations, we turn our attention to approximations and numerical procedures in order to obtain solutions explicitly. This is done in Section 5, where we propose the so-called expansion formula, which expresses fractional derivatives in terms of a function and its moments and show its use in the approximation procedure for solving the Euler–Lagrange equations. Results presented in this article are based on investigations presented in [14, 13, 18, 11, 9, 10, 17, 37]

2 Notations The aim of this section is a brief introduction of the necessary notation and mathematical preliminaries that will be used throughout this paper. Let AC([a, b]) be the set of continuous functions having the first derivative almost everywhere in [a, b] being integrable, that is, it is in L1 ([a, b]). Then, for k = 2, 3, . . ., ACk ([a, b]) = {f ∈ Ck−1 ([a, b]) | f (k−1) ∈ AC([a, b])}, where Ck ([a, b]) is the set of k-times continuously differentiable functions. Fractional operators of complex order are introduced as follows (see [38, 50]): Let η ∈ ℂ with 0 < Re η < 1. Then the definition of the left Riemann–Liouville (RL) fractional integral of an absolutely continuous function on [a, b], b > a ≥ 0, is given by η a It u(t)

t

1 := ∫(t − τ)η−1 u(τ) dτ, Γ(η) a

t ∈ [a, b].

If η = iθ, θ ∈ ℝ, the latter integral diverges, and hence one introduces the fractional integration of imaginary order as

Variational principles with fractional derivatives | 363

iθ a It u(t)

t

d d 1 := a It1+iθ u(t) = ∫(t − τ)iθ u(τ) dτ, dt Γ(1 + iθ) dt

t ∈ [a, b].

a

(1)

However, in both cases the left RL fractional derivative of order η ∈ ℂ with 0 ≤ Re η < 1 is given by η a Dt u(t)

t

1 d d 1−η u(τ) dτ, := a It u(t) = ∫ dt Γ(1 − η) dt (t − τ)η a

t ∈ [a, b],

while the right RL fractional derivative is given by η t Db u(t)

b

1 u(τ) d d 1−η dτ, := t Ib u(t) = ∫ dt Γ(1 − η) dt (τ − t)η t

t ∈ [a, b].

If η = α ∈ (0, 1), then we have the left and right RL fractional derivatives of real order α, a Dαt u. η When applying operator 0 Dt with η ∈ ℂ to a real valued function, one obtains, in general, a complex valued function. Therefore, in order to have a real valued function, one should take the real part of the fractional derivative (cf. [17]): η a Dt u(t)

1 η̄ η := (a Dt u(t) + a Dt u(t)), 2

η ∈ ℂ,

where η̄ denotes the complex conjugate of η. The next assertion, called the fundamental lemma, is a crucial ingredient in the derivation of the Euler–Lagrange equations (cf. [26, 27, 29]): Let u ∈ C([a, b]). If b

∫ u(t)φ(t) dt = 0 a

for all continuous functions φ on [a, b] that vanish at t = a and t = b, then u ≡ 0 on [a, b]. Except for Subsection 3.3, we will consider real RL fractional derivatives throughout. If y ∈ C1 ([a, b]) then limα→1− a Dαt y = y󸀠 ; see, e. g., [44, 50]. Recall that a sufficient condition for the existence of the left, resp. right RL fract u(τ) b u(τ) tional derivative of order α, 0 ≤ α < 1, is that ∫a (t−τ) α dτ, resp. ∫ (τ−t)α dτ are absot lutely continuous functions in [a, b]. Also, one can calculate fractional derivatives for certain less regular functions, e. g., α a Dt (t

− a)−μ =

Γ(1 − μ) 1 , Γ(1 − μ − α) (t − a)μ+α

μ < 1.

(2)

364 | T. M. Atanacković et al. Notice here that with μ := 1 − α in (2), one obtains a Dαt (t − a)α−1 = 0, while for μ = 0 we have a Dαt c ≠ 0, for any constant c ≠ 0. These facts are of crucial importance in the analysis of fractional conserved quantities and the fractional Noether theorem. The definition of complex RL fractional derivatives can be extended for arbitrary η = α + iβ, β ∈ ℝ, k − 1 < α < k, k ∈ ℕ, and u ∈ ACk ([a, b]), by η

a Dt u :=

k

dk k−α+iβ d k−α+iβ u, resp. t Dαb u := (− ) t Ib u, t ∈ [a, b]. aI dt dt k t

If α = k − 1, we have to use the modification according to (1). The fractional integration by parts formula holds for absolutely continuous functions f , g, and α ∈ ℂ (cf., e. g., [37, Prop. 6.4.3]): b

b

a

a

∫ f (t)a Dαt g(t) dt = ∫ g(t)t Dαb f (t) dt.

(3)

Let α ∈ [k − 1, k), k ∈ ℕ. Then (cf. [50, Th. 2.4]) α α a Dt a It u

n−1

(t − a)α−k−1 dn−k−1 󵄨󵄨󵄨󵄨 I n−α u. 󵄨 Γ(α − k) dt 󵄨󵄨󵄨t=a a t k=0

= u and a Itα a Dαt u = u − ∑

This also holds for η = α + iβ, α ∈ (k − 1, k), k ∈ ℕ, β ∈ ℝ. Again, if α = k − 1, we have to use the modification according to (1). Fractional derivatives are linear operators, but they do not obey the Leibniz rule. Let u ∈ C 1 [a, b], α ∈ [0, 1). We recall the definition of the left Caputo fractional derivative of order α: c α a Dt u(t)

t

=

u󸀠 (τ) 1 dτ, ∫ Γ(1 − α) (t − τ)α a

t ∈ [a, b].

The right Caputo fractional derivative and the complex extensions can be done as in the case of RL fractional derivatives. Let u ∈ C 1 ([a, b]). The relation between the RL and Caputo definitions reads α a Dt u(t)

= ca Dαt u(t) +

1 u(a) , Γ(1 − α) (t − a)α

(4)

from which it follows that they coincide provided that u(a) = 0; an analog holds for the corresponding right derivatives under the assumption u(b) = 0. The same condition shows that the first derivative and fractional derivative commute, i. e., (a Dαt u)󸀠 = α 󸀠 a Dt (u ). Contrary to the RL case, the Caputo fractional derivative has the property c α c α a Dt c = t Db c = 0, c ∈ ℝ.

Variational principles with fractional derivatives | 365

3 Variational problems with fractional derivatives The simplest one-dimensional fractional variational problem consists of finding extreme values—minima or maxima—of a functional L given by b

L[u] = ∫ L(t, u(t), a Dαt u) dt, a

α ∈ (0, 1).

(5)

It was proved in [1] that if one wants to minimize (5) among all functions u which have a continuous left αth RL fractional derivative and which satisfy the Dirichlet boundary conditions u(a) = a0 and u(b) = b0 , for some real constant values a0 and b0 , then a minimizer should be sought among all solutions of the Euler–Lagrange equation, 𝜕L 𝜕L + Dα ( ) = 0. 𝜕u t b 𝜕a Dαt u

(6)

In this section, we shall be concerned with the variational principles that involve fractional derivatives in three different situations: 1. Find minimal (or maximal) values of the functional B

L[u] = ∫ L(t, u(t), a Dαt u(t)) dt

(7)

A

2.

with respect to u, where [A, B] ⊂ (a, b) ⊂ ℝ and 0 < α < 1. Find minimal (or maximal) values of the functional b

L[u, α] = ∫ L(t, u(t), a Dαt u(t), α) dt

(8)

a

3.

with respect to u and α ∈ (0, 1). Find minimal (or maximal) values of the functional b

γ

L[u] = ∫ L(t, u(t), a Dαt u(t), a Dt u(t)) dt

(9)

a

with respect to u, where α ∈ (0, 1), Re γ ∈ (0, 1). In all three cases one has to specify regularity properties, both on the Lagrangian L, and on the class of admissible functions u among which extremal values are sought. In addition, certain constraints, e. g., boundary conditions, will be imposed on the set of admissible functions among which the maximum/minimum is sought.

366 | T. M. Atanacković et al.

3.1 Hamilton’s principle on [A, B] ⊂ (a, b) In this subsection α ∈ (0, 1). Consider (7) under the assumption that the Lagrangian L has the properties L ∈ C1 ([a, b] × ℝ × ℝ) } } and } } t→ 󳨃 𝜕3 L(t, u(t), a Dαt u(t)) ∈ AC([a, b]), ∀ u ∈ AC([a, b]).}

(10)

Extrema of (7) will be sought among all absolutely continuous functions in [a, b], which in addition satisfy condition u(a) = a0 , for a fixed a0 ∈ ℝ. Functional in (7) is a generalization of fractional variational problems previously studied in the literature (cf., e. g., [1, 2, 21, 22, 25]) in the sense that the lower bound a in the fractional derivative does not coincide with the lower bound A of the integral that is minimized (see also [20, 24]). The reason for that comes from their different physical meanings: While the interval [A, B] defines the Hamilton action, the value a is related to the memory of the system. The functional in (7) can be generalized in different ways. For instance, one can take the Lagrangian L to depend also on the right RL fractional derivatives of u, or take an arbitrary (finite) number of left and right RL fractional derivatives of u in L, whose order can be ≥ 1. Also, u can be a function from [a, b] into ℝd , d > 1. All these generalizations do not bring about essential novelties in the theory. The special case A = a was considered in [1], where it was proved that the extreme values of (5), with continuous left αth Riemann–Liouville fractional derivative and satisfying the Dirichlet boundary conditions u(a) = a0 and u(b) = b0 , a0 , b0 ∈ ℝ, should be sought among all solutions of the Euler–Lagrange equation (6). The following result shows the Euler–Lagrange equation for (7). Theorem 3.1 ([9]). Let u∗ ∈ AC([a, b]) be an extremum of the functional L in (7), whose Lagrangian L satisfies (10). Then u∗ satisfies the following Euler–Lagrange equations: 𝜕L c α 1 1 𝜕L 𝜕L 󵄨󵄨󵄨󵄨 + t DB ( = 0, in (A, B), ) + 󵄨󵄨 α α 󵄨 𝜕u 𝜕a Dt u 𝜕a Dt u 󵄨t=B Γ(1 − α) (B − t)α α t DB (

𝜕L 𝜕L ) − t DαA ( ) = 0, in (a, A). α 𝜕a Dt u 𝜕a Dαt u

We present a sketch of the proof. A necessary condition for a solution u∗ of the fractional variational problem defined by (7) is that the first variation of L is zero at the solution u∗ , i. e., B

0 = δL[u] = ∫[ A

𝜕L 𝜕L δu(t) + Dα δu(t)] dt, 𝜕u 𝜕a Dαt u a t

(11)

Variational principles with fractional derivatives | 367

where δu is the Lagrangian variation of u, i. e., δu(a) = 0, δu(B) = 0. An integration by parts formula (3) gives B

∫ a

B

𝜕L 𝜕L ) dt. Dα δu(t) dt = ∫ δu(t)t DαB ( 𝜕a Dαt u a t 𝜕a Dαt u a

Thus, we obtain B

∫ a

B

A

A

a

𝜕L 𝜕L 𝜕L )dt + ∫ δu(t)t DαB ( )dt. Dα δu(t)dt = ∫ δu(t)t DαB ( 𝜕a Dαt u a t 𝜕a Dαt u 𝜕a Dαt u

From the last equality, we conclude that B

∫ A

𝜕L Dα δu(t)dt 𝜕a Dαt u a t B

= ∫ δu(t)t DαB ( A

A

𝜕L 𝜕L 𝜕L )dt + ∫[t DαB ( ) − t DαA ( )]δu(t)dt. 𝜕a Dαt u 𝜕a Dαt u 𝜕a Dαt u a

If we insert this into (11), we obtain 𝜕L 𝜕L + Dα = 0, 𝜕u t B 𝜕a Dαt u

t ∈ (A, B),

and α t DB (

𝜕L 𝜕L ) − t DαA ( ), 𝜕a Dαt u 𝜕a Dαt u

t ∈ (a, A).

By (4), the claim now follows if we replace the right Riemann–Liouville by the right Caputo fractional derivative in the first equation. We note that the last equation defines the optimal history of the process u(t), t ∈ (a, A) that minimizes the action integral.

3.2 Generalized Hamilton’s principle We continue to assume that α ∈ (0, 1). We will analyze stationary points for a generalized Hamilton action integral, also considering the order α of the fractional derivative as a variable. The method we are going to present here offers a way for choosing α precisely, so that it minimizes L. We consider (8) where u ∈ U, α ∈ A := [0, α0 ], α0 ≤ 1. (We already have defined the left fractional derivative for α = 1.) The problem is to find min

(u,α)∈U×A

L[u, α],

(12)

368 | T. M. Atanacković et al. or min(min L[u, α]). α∈A

(13)

u∈U

Analysis of the third possibility minu∈U (minα∈A L[u, α]) is more complicated, and less natural in applications. Therefore, we are looking for stationary conditions for (12) with respect to the set of admissible functions and the order of fractional derivative, and for conditions which guarantee the equivalence of (12) and (13). When studying (12) there are two cases for choosing a set of admissible variables: α0 < 1 and α0 = 1. In the first case, Ul := {u ∈ L1 ([a, b]) | a Dαt u ∈ L1 ([a, b])}, while in the second case one takes u ∈ Ul with the additional assumption that a D1t u exists and a D1t u = u󸀠 is an integrable function. For instance, C 1 ([a, b]) ⊂ Ul . The set of all admissible functions Ul which satisfy the specified boundary conditions u(a) = a0 , u(b) = a1 will be denoted by U. Assumptions on the Lagrangian density L are similar to those in (10): L ∈ C1 ([a, b] × ℝ × ℝ × [0, 1]) } } and } } t 󳨃→ 𝜕3 L(t, u(t), a Dαt u(t), α) ∈ Ur , ∀ u ∈ U,}

(14)

where Ur := {u ∈ L1 ([a, b]) | t Dαb u ∈ L1 ([a, b])}. In this set up, one can notice that the fractional variational problem (7) is a special case of (12) obtained by taking A = {α}, α ∈ [0, 1]. In the following theorem we give a necessary condition for the existence of solutions to (12). Again, we give only the main steps of the proof. Theorem 3.2 ([11]). Let L satisfy (14). Then a necessary condition for the functional (8) to have an extreme point at (u∗ , α∗ ) ∈ U × A is that (u∗ , α∗ ) is a solution of the system of equations

b

∫( a

𝜕L 𝜕L + t Dαb ( ) = 0, 𝜕u 𝜕a Dαt u

(15)

𝜕L 𝜕L ) dt = 0, G(u(t), α) + 𝜕a Dαt u(t) 𝜕α

(16)

where 𝜕a Dαt u d = (f1 ∗t u)(t, α), f1 (t, α) 𝜕α dt 1 [ψ(1 − α) − ln t], t > 0, = α t Γ(1 − α)

G(u(t), α) =

with the Euler function ψ(z) =

d dz

t

ln Γ(z), and (f1 ∗t u)(t, α) = ∫a f1 (τ, α)u(t − τ) dτ.

Variational principles with fractional derivatives | 369

Proof. Let (u∗ , α∗ ) be an element of U × A for which L[u, α] has an extreme value. Let u(t) = u∗ (t) + ε1 f (t), α = α∗ + ε2 , ε1 , ε2 ∈ ℝ, with f ∈ Ul , and the boundary conditions on f are specified so that the varied path u∗ + ε1 f is an element of U. Then L[u, α] = L[u∗ + ε1 f , α∗ + ε2 ] =: L(ε1 , ε2 ). A necessary condition for an extreme value of L[u, α] is 𝜕L(ε1 , ε2 ) 󵄨󵄨󵄨󵄨 = 0, 󵄨 𝜕ε1 󵄨󵄨󵄨ε1 =0,ε2 =0

𝜕L(ε1 , ε2 ) 󵄨󵄨󵄨󵄨 = 0. 󵄨 𝜕ε2 󵄨󵄨󵄨ε1 =0,ε2 =0

This leads to b

∫( a

𝜕L 𝜕L + Dα ( ))f (t) dt = 0. 𝜕u t b 𝜕a Dαt u

From this equation, using the fundamental lemma of the calculus of variations,α we 𝜕 D u conclude that condition (15) holds for the optimal values u∗ and α∗ . The term a𝜕αt in (16) is transformed by the use of the expression t

𝜕a Dαt u 1 d ln(t − τ)u(τ) = ψ(1 − α) a Dαt u − dτ ∫ 𝜕α Γ(1 − α) dt (t − τ)α d = (f1 ∗t u)(t, α)G(u, α), dt

a

(y, α) ∈ U × A

(cf. [8, p. 592]). In this way we obtain (16). Under additional assumptions, problem (12) becomes equivalent to problem (13), which may, in some instances, be easier to solve. Theorem 3.3 ([11]). Let L satisfy (14). Assume that for every α ∈ [0, 1] there is a unique u∗ (t, α) ∈ U, the solution to (15), and that the mapping α 󳨃→ u∗ (t, α) is differentiable as a mapping from [0, 1] to U. Then the problem min(u,α)∈U×A L[u, α] is equivalent to the problem minα∈A (minu∈U L[u, α]). We present a sketch of the proof. We first solve (15) with the corresponding boundary conditions to obtain u∗ = u∗ (t, α). According to the assumption, the solution u∗ is unique. Then we insert u∗ in (16) to obtain α∗ . In this case, the functional L[u, α] becomes a functional depending only on α, α 󳨃→ L[u∗ (t, α), α] = L[α]. Therefore, (16) transforms to the total derivative of L[α] since b 𝜕 𝜕L 𝜕L dL[α] dL[α + ε] 󵄨󵄨󵄨󵄨 = ( = ) dt, Dα )u + ( ∫ 󵄨󵄨 󵄨󵄨ε=0 dα dε 𝜕a Dαt u 𝜕α a t 𝜕α a

where we used fractional integration by parts formula (3) in the third, and equation (15) in the last equality. This proves the claim. The following simple assertion is of particular interest.

370 | T. M. Atanacković et al. Proposition 3.4 ([11]). Let L satisfy (10). Assume that for every α ∈ [0, 1] there exists a unique uα ∈ U, the solution to the fractional variational problem (12), and that L[uα , α] is the corresponding minimal value of the functional L. Assume additionally that 󵄨󵄨 dL > 0, (u, α)󵄨󵄨󵄨 󵄨u=uα dα

∀uα ∈ U.

Then the minimal, resp. maximal value of the functional L[u, α] is attained at α = 0, resp. at α = 1. We illustrate the obtained results by the following example. Example 3.5 ([11]). Let a = 0, b = 1, where L is of the form 1 L(t, u(t), 0 Dαt u(t), α) := Γ(1 − α)0 Dαt u(t) − cu2 (t), 2

t ∈ [0, 1], c > 0, c ≠ 1,

U := {u ∈ Ul | u(0) = c1 }, α ∈ [0, α0 ], α0 < 21 , and consider the problem of finding stationary points for the functional (8). Recently, Lagrangians linear in the velocity have been treated in [21, 43]. Equations (15) and (16) become Γ(1 − α)t Dα0 1 − cu = u

(17)

and 1

∫(Γ(1 − α) 0

𝜕0 Dαt u 𝜕Γ(1 − α) + ) dt = 0. 𝜕α 𝜕α

Equation (17) has a unique solution u∗ = L[u∗ , α] =

1 , c(1−t)α

t ∈ [0, 1], α ∈ [0, α0 ]. This implies

1

1 1 dt. ∫ 2c (1 − t)2α 0

Since α0 < 1/2, we see that L[u , α] exists and that it is an increasing function with respect to α. Hence, L[u∗ , α] attains its minimal value at α = 0, and it equals 1/(2c), while the maximal value of L[u∗ , α] is attained at α0 . ∗

3.3 Hamilton’s principle with complex derivatives As we already stated, we consider complex exponents (9). Assume that L satisfies L ∈ C1 ([a, b] × ℝ × ℝ × ℝ) } } } } and } } γ α α 1 t 󳨃→ t Db (𝜕3 L(t, u(t), a Dt u(t), a Dt u(t))) ∈ C([a, b]), ∀u ∈ C ([a, b]) } } } } and } } γ γ α 1 t 󳨃→ a Dt (𝜕4 L(t, u(t), a Dt u(t), a Dt u(t))) ∈ C([a, b]), ∀u ∈ C ([a, b]).}

(18)

Variational principles with fractional derivatives | 371

Let U = {u ∈ C1 ([a, b]) | u(a) = 0 ∧ u(b) = 0} be the set of admissible functions y. We consider the following problem: Find u∗ ∈ U such that b

γ

min L[u] = min ∫ L(t, u(t), a Dαt u(t), a Dt u(t)) dt u∈U

u∈U

a

b

γ

= ∫ L(t, u∗ (t), a Dαt u∗ (t), a Dt u∗ (t)) dt.

(19)

a

The Euler–Lagrange equations for (9) are derived in the following theorem. Theorem 3.6 ([18]). Let L satisfy (18). Then the minimizer u∗ of the functional (9) satisfies the Euler–Lagrange equation 𝜕L 𝜕L 𝜕L γ + t Dαb ( ) + t Db ( γ ) = 0, α 𝜕u 𝜕a Dt u 𝜕a Dt u

t ∈ [a, b].

(20)

4 Complementary variational principles with fractional derivatives The aim of this short section is to present, without proofs, the main ideas of the complementary variational principles given in our paper [13]. Let u =∈ AC1 ([a, b]), 1 2 L(t, u, a Dαt u) = (a Dαt u) + Π(t, u), 2 where L ∈ C2 ([a, b] × ℝ × ℝ) and Π ∈ C2 ([a, b] × ℝ). We consider the primal functional J[u] ≡

L[u, a Dαt u]

b

1 2 = ∫[ (a Dαt u) + Π(t, u)] dt. 2

(21)

a

We refer to the principle of least action as the primal variational principle. Requiring that (21) attains its minimum, we obtain the Euler–Lagrange equations α α t Db (a Dt u)

+

𝜕Π = 0, 𝜕u

(22)

where u belongs to a space of admissible functions U, defined as U = {u ∈ AC1 ([a, b]) | “u satisfies prescribed boundary conditions”}.

372 | T. M. Atanacković et al. Equivalently, the Euler–Lagrange equations can be written as a system of two equations with independent functions u and pα . These equations are called the canonical equations (or the Hamilton equations). As in the classical case, we define the generalized momentum and the corresponding Hamiltonian as pα :=

𝜕L = Dα u, 𝜕 a Dαt u a t

(23)

1 H(t, u, pα ) := pα a Dαt u − L(t, u, a Dαt u) = p2α − Π(t, u). 2

(24)

Equation (22) now becomes α t Db pα

=−

𝜕Π . 𝜕u

(25)

Thus, the canonical equations (23), (25) are obtained by minimizing (21). Next we write (21) in the form b

α α ̄ L[u, a Dt u, pα ] = ∫[pα a Dt u − H(t, u, pα )] dt, a

b

1 = ∫[pα a Dαt u − p2α + Π(t, u)] dt. 2 a

The integration by parts formula gives b

L[u, pα ] = ∫[(t Dαb pα ) ⋅ u − H(t, u, pα )] dt, a

b

1 = ∫[(t Dαb pα ) ⋅ u − p2α + Π(t, u)] dt. 2

(26)

a

Since (26) is a functional with independent functions u and pα , we assume additionally that pα belongs to the space P := {pα ∈ AC1 ([a, b])}. Requiring that the functional L attains a minimum, by arbitrary independent variation of pα and u we obtain the Hamilton equations α a Dt u

=

𝜕H = pα , 𝜕pα

α t Db pα

=

𝜕H 𝜕Π =− . 𝜕u 𝜕u

(27)

System (27) is identical to (23), (25). With this notation, we have the following. : ℝ → D be a bijection of ℝ onto an open subset D of ℝ, Proposition 4.1 ([13]). Let 𝜕Π(t,u) 𝜕u 2

for every t ∈ [a, b]. Let | 𝜕𝜕uΠ2 | ≠ 0, for every t ∈ [a, b]. Then there exists Φ : [a, b] × D → ℝ of class C2 such that u(t) = Φ(t, t Dαb pα ), t ∈ [a, b], and α t Db pα

≡−

𝜕Π(t, Φ(t, t Dαb pα )) , 𝜕Φ

t ∈ [a, b].

(28)

Variational principles with fractional derivatives | 373

Now (26) implies pα ∈ P, G[pα ] := L[Φ(t, t Dαb pα ], pα ) b

1 = ∫[(t Dαb pα ) ⋅ Φ(t, t Dαb pα ) − p2α + Π(t, Φ(t, t Dαb pα ))] dt. 2

(29)

a

Requiring that the functional G attains its maximum, we obtain equations equivalent to the Euler–Lagrange equations (22). We refer to this property of the functional G as the dual variational principle. Primal and dual variational principles together form a complementary principle. This means that if the functional J given by (21) attains a minimum at a function u, then the functional G given by (29) attains a maximum at a function pα , where the connection between u and pα is given by (23). In the sequel we shall refer to J as the primal functional and to G as the dual functional, while to J and G together we refer as the complementary functionals. Our main result is the following theorem. Theorem 4.2 ([13]). Let the Hamiltonian be of the form (24). Additionally to (28) assume 𝜕2 Π(t,x) > 0, t ∈ [a, b], x ∈ ℝ. Then the complementary principle for the functionals (21) 𝜕x 2 and (29) holds. Moreover, J[u] ≤ J[U],

G[pα ] ≥ G[Pα ],

where U = u + δu and Pα = pα + δpα , for u, U ∈ U and pα , Pα ∈ P, and there exists a constant k(α) > 0, such that ‖U − u‖L2 ≤ √

J[U] − G[Pα ] . k(α)

The proof of this theorem can be found in [13].

5 Approximations and the expansion formula In the previous section we derived the Euler–Lagrange equations for several classes of fractional variational principles. The mutual property of all those equations is that they contain both left and right fractional derivatives, except for the case when Lagrangian is linear with respect to the fractional derivative of a function. This implies that finding analytical solutions to the Euler–Lagrange equations could be quite difficult on one hand, and on the other hand, it is very important to calculate solutions explicitly for further applications. Motivated by all these facts, the idea is to use approximations of fractional derivatives, and in that way reduce the problem to the classical one. For instance, one can use the approximation of the RL fractional derivative

374 | T. M. Atanacković et al. by integer order ones (with [a, b] = [0, T], for simplicity): α 0 Dt y(t)

∞ α t n−α = ∑( ) y(n) (t), n Γ(n + 1 − α) n=0

n−1

αΓ(n−α) (cf. [50, p. 278]). However, it requires function y to be anawhere (αn ) = (−1) Γ(1−α)Γ(n+1) lytic and derivatives of y satisfy specified boundary conditions, which are very strong conditions. Here we propose an expansion formula that holds true for continuously differentiable functions, without any additional restrictions on initial conditions. It is important to underline that the expansion in (31) below depends only on the function itself and its moments, but does not depend on derivatives of that function. The pth moment of the function y, p ∈ ℕ, denoted by Vp (y)(t), is defined by t

Vp (y)(t) := ∫ τp y(τ) dτ,

p ∈ ℕ,

0

and it has the following properties: Vp(1) (y)(t) = t p y(t)

and

Vp (y)(0) = 0,

t ∈ [0, T], p ∈ ℕ.

(30)

Theorem 5.1 ([14]). Let y ∈ C1 ([0, T]) and 0 < α < 1. Then α 0 Dt y(t)

=

N Vp−1 (y)(t) y(t) Cp−1 (α) A(N, α) − + QN+1 (y)(t), ∑ α t t p+α p=1

t ∈ (0, T],

(31)

where A(N, α) :=

N 1 Γ(p + α) +∑ Γ(1 − α) p=1 Γ(1 − α)Γ(α)p!

Γ(N + 1 + α) α Γ(1 − α)Γ(α)N! Γ(p + α) , Cp−1 (α) := Γ(1 − α)Γ(α)(p − 1)!

(32)

=

(33)

and the remaining term QN+1 (y)(t) satisfies the following estimate: |QN+1 (y)(t)| ≤

C ⋅ Mt t 1−α ⋅ α , Γ(1 − α)Γ(α) N 1

t ∈ [0, T],

with 0 < α1 < 1 − α, Mt := max0≤τ≤t |y(1) (τ)| and certain constant C > 0. Thus, lim QN+1 (y)(t) = 0,

N→∞

t ∈ [0, T].

(34)

Variational principles with fractional derivatives | 375

The derivation of the expansion formula (31) goes as follows. Let y ∈ C1 ([0, T]). By definition, we have α 0 Dt y(t)

t

−α

1 τ = α (∫ y(1) (τ)(1 − ) dτ + y(0)), t Γ(1 − α) t

t ∈ [0, T].

0

p

(−1) Γ(q+1) p We use the binomial expansion formula (1 − z)q = 1 + ∑∞ p=1 Γ(q+1−p)p! z , which converges for |z| < 1, and uniformly on each closed subinterval of (−1, 1), and the identity (−1)p Γ(q+1) Γ(p−q) = Γ(−q)p! . With this we obtain (with q = −α) Γ(q+1−p)p! α 0 Dt y(t)

t

p

∞ Γ(p + α) τ y(0) 1 ( ) ) dτ + α = α ∫ y(1) (τ)(1 + ∑ t Γ(1 − α) Γ(α)p! t t Γ(1 − α) p=1 0

=

t

N Γ(p + α) 1 p 1 y(t) + α (t y(t) − p ⋅ ∫ τp−1 y(τ) dτ) ∑ α t Γ(1 − α) t Γ(1 − α) p=1 Γ(α)p! t p 0

t

+

t α Γ(1

Γ(p + α) τ 1 ( ) dτ ∫ y(1) (t) ∑ − α)Γ(α) p! t p=N+1 N

=

p

∞

0

Vp−1 (y)(t) y(t) A(N, α) − ∑ Cp−1 (α) + QN+1 (y)(t), tα t p+α p=1

t ∈ [0, T],

with A(N, α) and Cp−1 (α) being as in (32) and (33), respectively. This proves the expansion formula (31). Proving estimate (34), we will have the convergence of the expansion formula (31) and improve the corresponding result given in [6]. Recall the Stirling formula for the Gamma function: 1

Γ(z + α) ≈ √2π z z+α− 2 e−z ,

|z| → ∞, α ∈ ℝ, |α| < ∞.

Together with Γ(n + 1) = n!, n ∈ ℕ, we have Γ(n + α) 1 ≈ 2−α , (n + 1)! n

n ∈ ℕ, n → ∞, α ∈ ℝ, |α| < ∞.

Thus, for t ∈ [0, T], we have t

|QN+1 (y)(t)| ≤

0

t

∞ ε ⋅ Mt p + 1 τp dτ ≤ α ∫ ∑ t Γ(1 − α)Γ(α) p=N+1 p2−α t p 0 1−α

≤

p

∞ 1 Γ(p + α) τ ( ) dτ ∫ |y(1) (τ)| ∑ α t Γ(1 − α)Γ(α) p! t p=N+1

C ⋅ Mt t ⋅ , Γ(1 − α)Γ(α) N α1

376 | T. M. Atanacković et al. where α1 is chosen in such a way that α1 < 1 − α. This implies that ∑∞ p=N+1 1 ε⋅∑∞ p=N+1 p2−α−α1

1 p2−α−α1

is a

convergent Dirichlet series; ε is a constant, and C := . Now, for t ∈ [0, T], letting N → ∞, we obtain QN+1 (y)(t) → 0, which proves the claim. Remark 5.2. In the case when y ∈ C2 ([0, T]), one obtains the following expansion formula (t ∈ (0, T]): α 0 Dt y(t)

≈

N Vp−2 (y)(t) y(t) (1) 1−α A(N, α) + B(N, α)y (t)t + Cp−2 (α) p−1+α , ∑ tα t p=2

(35)

where A(N, α) =

N 1 Γ(p − 1 + α) 1 − , ∑ Γ(1 − α) Γ(2 − α) p=2 Γ(α − 1)(p − 1)!

B(N, α) =

N Γ(p − 1 + α) 1 [1 + ∑ ], Γ(2 − α) p=1 Γ(α − 1)p!

Cp−2 (α) =

Γ(p − 1 + α) . Γ(2 − α)Γ(α − 1)(p − 2)!

Numerical experiments show that the expansion (35) gives a better approximation of a fractional derivative than (31), i. e., the same accuracy is obtained with a smaller number of moments; cf. [6, 47]. Note that B(N, α) tends to zero as N tends to infinity (see [7, Eq. (16)]). We now want to apply the expansion formula (31) in solving variational problems that involve fractional derivatives. For that purpose, let T

L[u] = ∫ L(t, u(t), 0 Dαt u(t)) dt,

u ∈ U,

(36)

0

and L ∈ C1 ([0, T] × ℝ × ℝ), } } and } } t→ 󳨃 𝜕3 L(t, u(t), a Dαt u(t)) ∈ AC([0, T]), ∀ u ∈ C1 ([0, T]).}

The variational problem consists of minimizing functional L over the set of admissible functions U = {u ∈ C1 ([0, T]) | u(0) = a0 , u(T) = b0 , a0 , b0 ∈ ℝ}. As shown above, the corresponding Euler–Lagrange equation takes the form 𝜕L 𝜕L + Dα ( ) = 0. 𝜕u t T 𝜕0 Dαt u

(37)

Our idea here is to insert (31) into the Lagrangian L in (36), and derive the corresponding Euler–Lagrange equations. In that way we shall reduce the variational problem (36) to the classical one (meaning that the Lagrangian does not depend on fractional derivatives of functions), which will allow us to apply the classical theory. Then

Variational principles with fractional derivatives | 377

we relate the results obtained through the approximation procedure with solutions to (37). Suppose that u ∈ C1 ([0, T]), and write 0 Dαt u as α 0 Dt u(t)

= N0̂ Dαt u(t) + QN+1 (u)(t),

t ∈ [0, T],

where ̂α N 0 Dt u(t)

=

N Vp−1 (u)(t) u(t) A(N, α) − , Cp−1 (α) ∑ tα t p+α p=1

t ∈ [0, T],

(38)

and QN+1 (u) is the remainder term, and insert it into (36). Thus, we consider the problem of minimizing the following functional: T

L[u] = ∫ L(t, u(t), N0̂ Dαt u(t) + QN+1 (u)(t)) dt. 0

Applying the mean value theorem on L, one obtains T

L[u] =

Dαt u(t)) dt ∫ L(t, u(t), N0̂ 0

T

+ ∫ QN+1 (u)(t)𝜕3 L(t, u(t), ξ (t)) dt, 0

where ξ (t) lies between N0̂ Dαt u(t) and N0̂ Dαt u(t) + QN+1 (u)(t), t ∈ (0, T]. Set T

LN [u] := ∫ L(t, u(t), N0̂ Dαt u(t)) dt. 0

Then T

L = LN + ∫ QN+1 (u)(t)𝜕3 L(t, u(t), ξ (t)) dt.

(39)

0

The relation between the functionals L and LN is established as follows (cf. [14]). Assume that there exist u∗ ∈ U and u∗N ∈ U which minimize the functionals L and LN , respectively. We will show that L[u∗ ] = lim LN [u∗N ]. N→∞

Let u∗ ∈ U be a minimizer of (36), i. e., T

min L[u] = L[u ] = ∫ L(t, u∗ (t), 0 Dαt u∗ (t)) dt, ∗

u∈U

0

(40)

378 | T. M. Atanacković et al. and let u∗N ∈ U be a minimizer of LN , i. e., min LN [u] = u∈U

LN [u∗N ]

T

= ∫ L(t, u∗N (t), N0̂ Dαt u∗N (t)) dt. 0

Consider (39). The assumptions on L and u, and the estimate (34) imply that for any T ε > 0 there is N0 ∈ ℕ such that | ∫0 QN+1 (u)(t)𝜕3 L(t, u(t), ξ (t)) dt| < ε, for all N ≥ N0 . Thus, L[u∗ ] = LN [u∗ ] − δ ≥ LN [u∗N ] − δ and L[u∗ ] ≤ L[u∗N ] = LN [u∗N ] + δ, for arbitrarily small δ ∈ ℝ. Writing this together, we obtain LN [u∗N ] − δ ≤ L[u∗ ] ≤ LN [u∗N ] + δ, which implies (40). Moreover, if U (with the usual Hausdorff topology of C1 ([0, T])) is compact, and ∗ if u and u∗N are unique solutions of the minimization problems L → min and LN → min, respectively, then u∗N → u∗ , as N → ∞. This follows from the uniqueness of a minimizer and the fact that every subsequence of {u∗N }N∈ℕ has a convergent subsequence. Hence, we are now looking for minimizers of LN , i. e., T

Dαt u(t)) dt, min LN [u] = min ∫ L(t, u(t), N0̂ u∈U

u∈U

(41)

0

where N0̂ Dαt u(t) is given by (38). We shall derive necessary conditions for minimizers of (41). Substituting (38) into (41), we obtain T

LN [u] = ∫ L(t, u(t), 0

N Vp−1 (u)(t) u(t) A(N, α) − ∑ Cp−1 (α) )dt, α t t p+α p=1

where Vp−1 (u), p = 1, . . . , N, satisfy (30). Set LN (t, u, V0 , . . . , VN−1 ) := L(t, u(t),

N Vp−1 (u)(t) u(t) A(N, α) − ). Cp−1 (α) ∑ tα t p+α p=1

Then the new Lagrangian LN is a function depending on the independent variable t, and N dependent variables (functions of t): u, V0 , . . . , VN−1 . Following the well-known procedure for deriving the Euler–Lagrange equations for a variational problem subject

Variational principles with fractional derivatives | 379

to constraints, we construct a modified functional N

(1) (u)(t) − t p−1 u(t)) Φ(t, u, V0 , . . . , VN−1 ) := LN (t, u, V0 , . . . , VN−2 ) + ∑ λp−1 (t)(Vp−1 p=1

and obtain 𝜕2 L + 𝜕3 L

A(N, α) N − ∑ λp−1 (t)t p−1 = 0, tα p=1

(1) (t) = −𝜕3 L λp−1

Cp−1 (α)

t p+α p−1 (1) Vp−1 (u)(t) = t u(t),

(42)

p = 1, 2, . . . , N, λp−1 (T) = 0,

,

Vp−1 (u)(0) = 0,

p = 1, 2, . . . , N.

(43) (44)

Here, as in (10), we use 𝜕2 L and 𝜕3 L in order to denote partial derivatives of L with respect to the second and the third variable, respectively. The question now arises how the optimality conditions (42)–(44) are related to the necessary condition for the optimality (37). More precisely, we want to show that the solutions to (42)–(44) converge to the solution of (37), as N → ∞, in a weak sense. We shall now explain the precise meaning of the weak convergence in this context. Set A = {φ : φ ∈ C1 ([0, T])}. Every function f ∈ C([0, T]) defines an element of the dual space A󸀠 via T

φ 󳨃→ ⟨f , φ⟩ = ∫ f (t)φ(t) dt,

φ ∈ A.

0

As usual, we say that f and g from A are equal in the weak sense if 󸀠

⟨f , φ⟩ = ⟨g, φ⟩,

∀ φ ∈ A.

Therefore, the weak form of (37) reads ⟨𝜕2 L + t DαT (𝜕3 L), φ⟩ = 0,

∀ φ ∈ A.

Theorem 5.3 ([14]). Let L be given by (36) with the Lagrangian L ∈ C1 ([0, T] × ℝ × ℝ). Assume that t 󳨃→ 𝜕3 L(t, u(t), 0 Dαt u(t)) ∈ AC([0, T]). Denote by (Θ) the left hand side of the fractional Euler–Lagrange equation (37), and by (ΘN ) the left hand side of (42), together with the constraints (43)–(44), which correspond to the variational problem (41)–(30) in which the left RL fractional derivative of y ∈ U is approximated by (38). Then in the weak sense, as N → ∞.

(ΘN ) → (Θ)

After the explanations given in the previous part, we give only some steps of the proof. We have ⟨t DαT (𝜕3 L), φ⟩

T

=

∫ t DαT (𝜕3 L) φ(t) dt 0

T

= ∫ 𝜕3 L 0 Dαt φ(t) dt. 0

(45)

380 | T. M. Atanacković et al. Now, the expansion formula (31) for 0 Dαt φ(t) yields α 0 Dt φ(t)

= lim [ N→∞

N Vp−1 (φ)(t) φ(t) A(N, α) − ∑ Cp−1 (α) ]. α t t p+α p=1

Substituting this into (45), we obtain T

T

∫ 𝜕3 L ⋅ 0 Dαt φ(t)dt = lim ∫[𝜕3 L N→∞

0

0

N Vp−1 (φ)(t) φ(t) A(N, α) − ]dt. 𝜕3 L Cp−1 (α) ∑ α t t p+α p=1

(46)

Inserting (43) into the last term in (46) one has T N

∫ ∑ 𝜕3 L Cp−1 (α)

Vp−1 (φ)(t) t p+α

0 p=1

N T

dt ∑ ∫ λp−1 (t)t p−1 φ(t)dt, p=1 0

where we used (30). Collecting all the above results, we obtain ⟨𝜕2 L +

α t DT (𝜕3 L), φ⟩

T

= lim ∫[𝜕2 L + N→∞

0

N 𝜕3 L A(N, α) − λp−1 (t)t p−1 ]φ(t) dt, ∑ tα p=1

which proves the claim. An example of the application of the expansion formula in the calculus of variations can be found in [14, Sec. 4]. Remark 5.4. Variational calculus within the fractional calculus is still a wide area of investigations which gives possibilities for generalizations and applications. In order to suggest possibilities for further investigations, we have given in references an extended list of papers and books which can serve for a new insight into variational fractional calculus. Moreover, we quote some possible directions of investigations: Calculation of the best constants in various norm inequalities of the form “Find a minimum of ‖L(f )‖V ” under the condition ‖u‖U = 1, where L : U → V is a continuous linear mapping and L is a differential operator of a special type involving fractional derivatives, for example a version of a fractional gradient. Complementary variational problems with more general conditions on Lagrangian and the formulation of a weak form of the minimum of an action integral. Pseudo-convexity or any other generalization of the Lagrangian. Especially the weak formulation of the variational problem involving various types of fractional derivatives. In all the previous observations the main idea is the extension of the use of such a generalized fractional variational calculus in various problems of mechanics or other fields of the natural or engineering sciences.

Variational principles with fractional derivatives | 381

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Mark Meerschaert and Hans Peter Scheffler

Continuous time random walks and space-time fractional differential equations Abstract: The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of this model is governed by a fractional diffusion equation. If the step length of the random walk follows a power law, we get a space fractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives. Keywords: Continuous time random walks, governing equations, stable laws, extended central limit theorem MSC 2010: 60G52, 60J50, 26A33, 60J22

1 Introduction The continuous time random walk (CTRW) is a model from statistical physics, introduced by Montroll and Weiss [34] and developed further by Scher and Lax [38], Klafter and Silbey [14], and Hilfer and Anton [12]. Start with a random walk S(n) = Y1 + ⋅ ⋅ ⋅ + Yn where the independent and identically distributed (iid) random variables {Yn } represents the particle jumps. Now assume a sequence of iid positive random variables {Jn }, and suppose that the waiting time Jn separates the n−1st and the nth jumps. Then T(n) = J1 + ⋅ ⋅ ⋅ + Jn is the time of the nth jump. The number of jumps by time t ≥ 0 is N(t) = max{n ≥ 0 : T(n) ≤ t}, and the CTRW X(t) = S(N(t)) gives the particle location at time t ≥ 0. If the waiting times Jn and the jumps Yn are independent, this is called an uncoupled CTRW. Metzler and Klafter [31, 32] survey a wide variety of CTRW applications in biology, geophysics, geomorphology, finance, material science, particle physics, and turbulence. Berkowitz et al. [6] review CTRW models in hydrology. Scalas [36] reviews Acknowledgement: Mark M. Meerschaert was partially supported by ARO MURI grant W911NF-15-10562 and USA National Science Foundation grants DMS-1462156 and EAR-1344280. Mark Meerschaert, Department of Statistics and Probability, Michigan State University, East Lansing, Michigan, USA, e-mail: [email protected] Hans Peter Scheffler, Department of Mathematics, University of Siegen, Siegen, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110571622-016

386 | M. Meerschaert and H. P. Scheffler applications of the CTRW model in finance. Sokolov [44] reviews the physical assumptions behind applications of the CTRW. Zaburdaev, Denisov and Klafter [49] review the “Lévy walk” model, a strongly coupled CTRW where the step length is proportional to the waiting time. Metzler et al. [33] review applications of the CTRW model to single particle tracking. Scher and Montroll [39] apply the CTRW model to transient photocurrent in amorphous materials. Uchaikin and Sibatov [47] develop CTRW theory for fractional kinetics in solids. A CTRW model for the migration of cancer cells was presented in Fedotov and Iomin [9]. Schumer and Jerolmack [41] develop an interesting CTRW model for sediment deposition in the geological record. Ganti et al. [10] propose a CTRW model for gravel bed load transport in rivers. Benson and Meerschaert [4] outline a CTRW model for contaminant transport that segregates the mobile and immobile phases. Schulz et al. [40] apply the CTRW model to cell movements. Meerschaert et al. [30] propose a CTRW model for sound transmission in complex media.

2 Uncoupled CTRW Now suppose that the iid jumps have a probability density function (pdf) f (x), and the waiting times have a pdf ψ(t). Montroll and Weiss [34] compute the exact pdf of the uncoupled CTRW using transforms. Using the Fourier transform (FT) ∞

f ̂(k) = 𝔼[eikYn ] = ∫ eikx f (x) dx

(1)

−∞

and the Laplace transform (LT) ∞

̃ ψ(s) = 𝔼[e−sJn ] = ∫ e−st ψ(t) dt,

(2)

0

we apply a simple conditioning argument. First note that ∞

p(x, t) = ℙ[S(N(t)) = x] = ∑ ℙ[S(N(t)) = x|N(t) = n]ℙ[N(t) = n]. n=0

If N(t) = n, then S(N(t)) = S(n) is the sum of n iid random variables, so its FT is 𝔼[eik(Y1 +⋅⋅⋅+Yn ) ] = 𝔼[eikY1 ] ⋅ ⋅ ⋅ 𝔼[eikYn ] = f ̂(k)n . Clearly t

q(0, t) = ℙ[N(t) = 0] = ℙ[J1 > t] = 1 − ℙ[J1 ≤ t] = 1 − ∫ ψ(u) du, 0

(3)

Continuous time random walks and space-time fractional differential equations | 387

t ̃ ̃ is the LT of ∫ g(u) du. More ̃ s) = s−1 (1 − ψ(s)), and hence q(0, using the fact that s−1 g(s) 0 generally, for N(t) = n > 0 we require that T(n) = u ≤ t and Jn+1 > t − u, and hence t

q(n, t) = ℙ[N(t) = n] = ∫ ψn∗ (u)Ψ(t − u) du, 0

where ψn∗ is the n-fold convolution, and Ψ(t) = ℙ[Jn+1 > t]. Taking LT we see that ̃ ̃ n s−1 (1 − ψ(s)) ̃ s) = ψ(s) for all n ≥ 0. Taking FT and LT in (3) leads to the Montroll– q(n, Weiss formula ∞

∞

−∞

0

̄ s) = ∫ eikx ∫ e−st p(x, t) dt dx p(k, ̃ 1 1 − ψ(s) ̃ n s−1 (1 − ψ(s)) ̃ = = ∑ f ̂(k)n ψ(s) , ̃ s 1 − f ̂(k)ψ(s) n=0 ∞

(4)

which gives the exact Fourier–Laplace transform (FLT) for the pdf of the uncoupled CTRW. Rewrite (4) in the form 1 ̃ ̃ p(k, ̄ s) = f ̂(k)ψ(s) ̄ s) + (1 − ψ(s)), p(k, s invert the FT, and then invert the LT to obtain the master equation from Klafter and Silbey [14]: t

∞

∞

p(x, t) = ∫ ψ(t − u) ∫ f (x − y)p(y, u) dy du + δ(x) ∫ ψ(u) du. 0

(5)

t

−∞

̃ If ψ(t) = λe−λt for t > 0, then ψ(s) = λ/(λ + s), and the Montroll–Weiss equation reduces to ̄ s) = p(k,

1

s + λ(1 − f ̂(k))

.

Inverting the LT yields ̂

̂ t) = e−λt(1−f (k)) , p(k, which is the well-known formula for the compound Poisson pdf with jump pdf f (x) (e. g., see [26, Example 3.3]). The compound Poisson is a special case of the CTRW with exponential waiting times. Because the exponential distribution has no memory, this CTRW is a Markov process: Once the value X(t) = S(N(t)) is known, the pdf of X(t + s) has no further dependence on the past history of X(u) for 0 ≤ u < t. In fact, it

388 | M. Meerschaert and H. P. Scheffler is even a Lévy process: The pdf of X(s) is the same as that of X(t + s) − X(t) (stationary increments), and the random variables X(t) and X(t + s) − X(t) are independent (independent increments). However, a CTRW without exponential waiting times is not a Lévy process, or even a Markov process. The influence of the memory can be seen in the master equation (5). Next we give a heuristic explanation of the connection between CTRW and fractional calculus (see, e. g., Scalas, Gorenflo and Mainardi [37]). Suppose that ℙ[Xn > x] = Ax−α for some A > 0 and some 1 < α < 2. Then μ = 𝔼[Xn ] exists, and we can take Yn = Xn − μ. Suppose also that ℙ[Jn > t] = Bt −β for some B > 0 and some 0 < β < 1. A calculation [26, Proposition 1.7] shows that f ̂(k) = 1 + D(−ik)α + O(k 2 ) where D = ̃ AΓ(2 − α)/(α − 1). A similar calculation [26, Theorem 3.37] shows that ψ(s) = 1 − sβ + O(s) when B = 1/Γ(1 − β). Now in order to obtain a limit pdf, replace Yn by c−1/α Yn and Jn by c−1/β Jn . Then the particle jumps have FT f ̂(c−1/α k) = 1 + Dc−1 (−ik)α + O(c−2/α k 2 ) and the ̃ −1/β s) = 1 − c−1 sβ + O(c−1/β s). Plug into the Montroll–Weiss waiting times have LT ψ(c formula, multiply by c on top and bottom, and let c → ∞ to get the CTRW scaling limit, p̄ c (k, s) = →

1 c−1 sβ + O(c−1/β s) −1 s c sβ − Dc−1 (−ik)α + ⋅ ⋅ ⋅ sβ−1 = p̄ ∞ (k, s). sβ − D(−ik)α

(6)

Rewrite as sβ p̄ ∞ (k, s) = D(−ik)α p̄ ∞ (k, s) + sβ−1 , and invert the FT and LT to get the space-time fractional diffusion equation β

𝜕t p∞ (x, t) = D𝜕xα p∞ (x, t) + δ(x)

t −β Γ(1 − β)

(7)

that governs the pdf of the long-time CTRW scaling limit in terms of Riemann–Liouville β ̃ 𝜕xα f (x) has FT fractional derivatives. Here we use the fact that 𝜕t g(t) has LT sβ g(s), α ̂ −β β−1 (−ik) f (k), and t /Γ(1 − β) has LT s [26, Example 2.9]. This statistical physics argument illustrates how the fractional derivative in space codes long particle jumps, and the fractional derivative in time represents long waiting times. The argument is not completely rigorous because the LT inversion requires more assumptions; see, e. g., the proof of [25, Theorem 3.1]. Remark 2.1. Mainardi [17] computes a solution to the time fractional diffusion equation (7) with α = 2 using Wright functions. Mainardi and Gorenflo [18] solve time fractional differential equations using the Mittag-Leffler function. Hilfer and Anton [12] and Mainardi, Luchko and Pagnini [19] use Mittag-Leffler functions to solve the general equation (7). The Mittag-Leffler function also has a special role in CTRW modeling: Mainardi, Gorenflo and Scalas [20] note that, for Mittag-Leffler waiting times ℙ[Jn > t] = Eβ (−λt β ), the time process is already in its asymptotic form; see also Meerschaert, Nane, and Vellaisamy [29].

Continuous time random walks and space-time fractional differential equations | 389

Another way to derive the CTRW scaling limit uses the extended central limit theorem. Since T(n) has LT ̃ n 𝔼[e−sT(n) ] = 𝔼[e−s(J1 +⋅⋅⋅+Jn ) ] = 𝔼[e−sJ1 ] ⋅ ⋅ ⋅ 𝔼[e−sJn ] = ψ(s)

(8)

the rescaled sum c−1/β T([ct]) has LT 𝔼[e−sc

−1/β

T([ct])

̃ −1/β ) ] = ψ(sc

[ct] [ct]

= (1 −

sβ + o(c−1 )) c

β

→ e−ts

(9)

as c → ∞, using the fact that (1 + a/c + o(1/c))c → ea . The limit is the LT of a stable Lévy process D(t) with index β, and the continuity theorem for the LT implies that c−1/β T([ct]) ⇒ D(t) in distribution. A similar argument [26, Section 1.2] shows that c−1/α S([ct]) ⇒ A(t), an α-stable Lévy process with 𝔼[eikA(t) ] = exp(Dt(−ik)α ). Since the renewal process N(t) is the inverse of the random walk T(n), it has an inverse limit [24, Theorem 3.2]: Observe that {N(t) ≥ u} = {T(⌈u⌉) ≤ t}, where ⌈u⌉ is the smallest integer n ≥ u. Define the inverse time process E(t) = inf{u > 0 : D(u) > t}, and note that {E(t) ≤ u} = {D(u) ≥ t}. Then ℙ[c−β N(ct) ≤ u] = ℙ[N(ct) ≤ cβ u]

= ℙ[T(⌈cβ u⌉) ≥ ct]

= ℙ[c−1 T(⌈cβ u⌉) ≥ t]

= ℙ[(cβ )

−1/β

T(⌈cβ u⌉) ≥ t] → ℙ[D(u) ≥ t] = ℙ[E(t) ≤ u]

so that c−β N(ct) ⇒ E(t). Then an argument [24, Theorem 4.2] using the continuous mapping theorem (e. g., see Billingsley [8] or Whitt [48]) shows that c−β/α S(N(ct)) = (cβ )

−1/α

S(cβ c−β N(ct)) ≈ (cβ )

−1/α

S(cβ E(t)) ⇒ A(E(t)).

Remark 2.2. Technically, the continuous mapping argument in [24, Theorem 4.2] requires process convergence: Not only does c−1/α S([ct]) ⇒ A(t) for a single t > 0, but also (c−1/α S([ct1 ]), . . . , c−1/α S([ctn ])) ⇒ (A(t1 ), . . . , A(tn )) for any 0 ≤ t1 < ⋅ ⋅ ⋅ < tn as random vectors. This is called convergence of finite dimensional distributions. To extend to all t ≥ 0, one considers {A(t) : t ≥ 0} as a random element of the space of right-continuous functions from [0, ∞) to the real line. Then [24, Theorem 4.1] establishes the convergence c−1/α S([ct]) ⇒ A(t) for all t ≥ 0 in the Skorokhod J1 topology on that space of functions. In this setting, [24, Theorem 4.2] establishes CTRW process convergence in the Skorokhod M1 topology. Straka and Henry [46, Theorem 3.6] establish CTRW process convergence in the stronger J1 topology. For more details, see [26, Chapter 4].

390 | M. Meerschaert and H. P. Scheffler β

̃ u) = e−us by (9), and it follows easily that The pdf g(t, u) of t = D(u) has LT g(s, 1/β D(u) has the same pdf as u D(1). Hence g(t, u) = u−1/β gβ (u−1/β t) where gβ (t) = g(t, 1) is the density of D(1). Then ℙ[E(t) ≤ u] = ℙ[D(u) ≥ t] 1/β

= ℙ[u

−1/β

D(1) ≥ t] = ℙ[D(1) ≥ tu

tu−1/β

] = 1 − ∫ gβ (u) du.

(10)

0

Differentiate (10) to see that u = E(t) has density h(u, t) =

t −1−1/β u gβ (tu−1/β ). β

(11)

Next we compute the LT of this pdf. Since ∞

ℙ[E(t) ≤ u] = ℙ[D(u) ≥ t] = ∫ g(w, u) dw t

the inner process E(t) has density t

h(u, t) =

d d ℙ[E(t) ≤ u] = [1 − ∫ g(w, u) dw] du du 0

with LT β β d ̃ s) = − d [s−1 g(s, ̃ u)] = − [s−1 e−us ] = sβ−1 e−us h(u, du du

using the fact that integration corresponds to multiplication by s−1 in LT space. Now a simple conditioning argument, similar to (3), shows that the CTRW limit A(E(t)) has pdf ∞

p∞ (x, t) = ∫ q(x, u)h(u, t) du ≈ ∑ ℙ(A(u) = x|E(t) = u)ℙ(E(t) = u), u

0

α

̂ u) = eDt(−ik) , the CTRW limit density has where q(x, u) is the pdf of x = A(u). Since q(k, FLT ∞

α

β

p̄ ∞ (k, s) = ∫ eDu(−ik) sβ−1 e−us du = 0

sβ−1 , sβ − D(−ik)α

which agrees with (6). Hence the CTRW limit pdf p∞ (x, t) solves the space-time fractional diffusion equation (7); see Becker-Kern et al. [3, Example 5.1] for more details. This probabilistic argument provides a rigorous connection between the CTRW and the space-time fractional diffusion equation, as well as a stochastic model for the long-time CTRW limit. The method extends naturally to vector particle jumps; see [26, Chapter 6].

Continuous time random walks and space-time fractional differential equations | 391

Remark 2.3. Since the CTRW scaling limit X(t) = A(E(t)) is not a Markov process, its transition density p(x, t) does not completely characterize the process. Meerschaert and Straka [27, 28] develop a method for computing the joint pdf of X(t) at multiple times. This method is based on a semi-Markov representation of the CTRW limit, where the memory is explicitly included; see also Germano et al. [11]. Krüsemann, Schwarz and Metzler [16] demonstrate how the non-Markovian nature (ageing, or memory) can be observed in Scher–Montroll experiments on transient photocurrent in amorphous materials. Barkai and Cheng [2] develop a theory of ageing CTRW. Remark 2.4. A closely related model called the continuous time random maximum (CTRM) describes the biggest jump, rather than the sum. Using the same setup as before, let M(n) = max(Y1 , . . . , Yn ) and consider the CTRM M(N(t)) that describes the x biggest jump by time t ≥ 0. Letting F(x) = ∫−∞ f (y) dy denote the cumulative distribution function (cdf) of the jumps, note that ℙ[M(n) ≤ x] = ℙ[Y1 ≤ x, . . . , Yn ≤ x] = ℙ[Y1 ≤ x] ⋅ ⋅ ⋅ ℙ[Yn ≤ x] = F(x)n and argue in exactly the same way as before that the CTRM has cdf ∞

P(x, t) = ℙ[M(N(t)) ≤ x] = ∑ ℙ[M(N(t)) ≤ x|N(t) = n]ℙ[N(t) = n]

(12)

∞ ̃ 1 1 − ψ(s) ̃ n s−1 (1 − ψ(s)) ̃ ̃ s) = ∑ F(x)n ψ(s) = P(x, . ̃ s 1 − F(x)ψ(s) n=0

(13)

n=0

with LT

If ℙ[Yn > x] = Dx−α for some α > 0, then c−1/α M([ct]) has cdf F [ct] (c1/α x) = (1 −

[ct]

Dx−α ) c

→ e−Dtx

−α

as c → ∞. It follows that c−1/α M([ct]) ⇒ Z(t), a max-stable process with cdf G(x, t) = −α e−Dtx for all x > 0. Then c−β/α M(N(ct)) ⇒ Z(E(t)) as c → ∞. The CTRM limit has cdf ∞

P∞ (x, t) = ∫ G(x, u)h(u, t) du 0

with LT ∞

β

P̃ ∞ (x, s) = ∫ e−Dux sβ−1 e−us du = 0

−α

sβ

sβ−1 + Dx−α

β

for all s > 0. Rewrite as s P̃ ∞ (x, s) = −Dx−α P̃ ∞ (x, s) + sβ−1 and invert the LT to see that the cdf of the CTRW limit solves the time fractional ordinary differential equation β

𝜕t P∞ (x, t) = −Dx −α P∞ (x, t) + δ(x)

t −β . Γ(1 − β)

See Benson et al. [5] for more details, and an application to rainfall data.

(14)

392 | M. Meerschaert and H. P. Scheffler

3 Coupled CTRW In Section 2, we considered the uncoupled CTRW, where the waiting times are independent of the particle jumps. Now we consider the more general coupled CTRW, where the length of the particle jump can depend on the waiting time. This model extension is useful to bound particle velocity, the ratio of jump length over waiting time. Let (Yi , Ji ) be iid with (Y, J) on ℝ × ℝ+ , where Yi models the ith jump of a walker and Ji is the waiting time before or after the ith jump. Set T(n) = J1 + ⋅ ⋅ ⋅ + Jn

and S(n) = Y1 + ⋅ ⋅ ⋅ + Yn ,

so that (S(n), T(n)) is a space-time random walk on ℝ × ℝ+ . Let N(t) = max{n ≥ 0 : T(n) ≤ t}

(15)

denote the number of jumps by time t ≥ 0. For t ≥ 0 we define the continuous time random walk (CTRW) S(N(t)) = Y1 + ⋅ ⋅ ⋅ + YN(t)

(16)

and the overshooting continuous time random walk (OCTRW) S(N(t) + 1) = Y1 + ⋅ ⋅ ⋅ + YN(t) + YN(t)+1 ,

(17)

which involves one additional jump. Observe that the CTRW corresponds to the “first wait, then jump” scenario, whereas the OCTRW corresponds to the “first jump, then wait” picture. That is, in the CTRW we begin with a waiting time, then jump, then repeat. In the OCTRW we begin with a jump, then wait, then repeat. See Figures 1 and 2 for an illustration.

Figure 1: The CTRW model (16). Each random waiting time Ji is followed by a random jump Yi . In the coupled CTRW, the pdf of the particle jump Yi can depend on the previous waiting time Ji .

Continuous time random walks and space-time fractional differential equations | 393

Figure 2: The OCTRW model (17). Each random waiting time Ji follows a random jump Yi . In the coupled OCTRW, the pdf of the waiting time Ji can depend on the previous particle jump Yi .

Since Yi and Ji can be dependent, S(n) and N(t) can be dependent, which makes the analysis of the long-time limiting behavior of the coupled CTRW process in (16) and the coupled OCTRW process in (17) more involved than the uncoupled case. Hence the analysis of the limit process and the governing equation is more delicate than the special case discussed in Section 2. In order to prove limit theorems for these processes, we need to make an assumption on the joint distribution of Y and J, i. e., the distribution of the random vector (Y, J). In order to make this exposition as simple as possible, we assume that for some 0 < α < 2 and 0 < β < 1 we have (n−1/α S(n), n−1/β T(n)) ⇒ (A, D)

(18)

as n → ∞, where A and D are nondegenerate. It follows from (18) by projecting on either coordinate that A has a strictly α-stable distribution and D has a β-stable distribution. It follows [3, Eq. (2.18)] that for any t > 0 we have (c−1/α S(ct), c−1/β T(ct)) ⇒ (A(t), D(t))

(19)

as c → ∞, where {(A(t), D(t))}t≥0 is a Lévy process on ℝ × ℝ+ with (A(1), D(1)) = (A, D). Observe again that A(t) and D(t) can and in general will be dependent. The characteristic function of (A(u), D(u)) for u > 0 is characterized by a variant of the well-known Lévy–Khintchine formula. Namely, in the present case, under assumption (18), we have [3, Lemma 2.1] 𝔼[e−sD(u)+ikA(u) ] = exp(−uψ(k, s)),

(20)

for all (k, s) ∈ ℝ × ℝ+ , where the symbol is given by ψ(k, s) = iak +

∫ ℝ×ℝ+ \{(0,0)}

(1 − eikx e−st +

ikx )ϕ(dx, dt) 1 + x2

(21)

394 | M. Meerschaert and H. P. Scheffler for some a ∈ ℝ. The so-called Lévy measure ϕ(dx, dt) is finite outside every neighborhood of the origin and satisfies (x 2 + t)ϕ(dx, dt) < ∞.

∫ 0 0. Furthermore, observe [3, Corollary 2.3] that A and D and hence the Lévy processes {A(u)}u≥0 and {D(u)}u≥0 are independent, so that the CTRW is uncoupled, if and only if ϕ(dx, dt) = δ0 (dx)ϕD (dt) + ϕA (dx)δ0 (dt)

(24)

where δ0 denotes the point mass at zero. Note that {D(u)}u≥0 is a β-stable subordinator and hence the sample paths of D(u) are cádlág, strictly increasing and D(u) → ∞ as u → ∞. Define the first passage time process by E(t) = inf{u ≥ 0 : D(u) > t}

(25)

for t ≥ 0. Finally, observe that the symbol ψ(k, s) in (21) induces a pseudo-differential operator ψ(i𝜕x , 𝜕t ) which for suitable functions f : ℝ × ℝ+ → ℝ has the representation

Continuous time random walks and space-time fractional differential equations | 395

ψ(i𝜕x , 𝜕t )f (x, t) = −a𝜕x f (x, t) − ∫ (H(t − u)f (x − y, t − u) − f (x, t) + ℝ×ℝ+

y𝜕x f (x, t) )ϕ(dy, du) 1 + y2

(26)

where H(t) = I(t ≥ 0) denotes the Heaviside step function. In fact, if we denote by L1ω (ℝ × ℝ+ ) the Banach space of measurable functions for which the norm ‖f ‖ω := ∫ e−ωt |f (t, x)| dx dt ℝ×ℝ+

exists, then (26) is valid for all functions in L1ω (ℝ × ℝ+ ) whose weak first and second order spatial derivatives as well as weak first order time derivatives belong to L1ω (ℝ × ℝ+ ); see Baeumer et al. [1, Theorem 3.2].

4 Limit theorems and governing equations In this section we derive the long-time scaling limit of the coupled CTRW and OCTRW processes. Moreover, the governing pseudo-differential equations for the densities of the limit processes are obtained. Recall the definition of the first passage time from (25) above. The following result is due to Jurlewicz et al. [13, Theorem 3.1]. Theorem 4.1. Suppose that (Yi , Ji ) are iid random vectors on ℝ×ℝ+ such that (18) holds. (a) For the CTRW in (16) we have for any t > 0 c−β/α S(N(ct)) ⇒ A(E(t)−)

(27)

as c → ∞. (b) For the OCTRW in (17) we have for any t > 0 c−β/α S(N(ct) + 1) ⇒ A(E(t)) as c → ∞. Sketch of the proof. By projecting on the second coordinate in (19) we see that c−1/β T(ct) ⇒ D(t) as c → ∞. Using (15) and (25) this implies that c−β N(ct) ⇒ E(t) as c → ∞. In fact, we even get from (19) that (c−β/α S(cβ t), c−β N(ct)) ⇒ (A(t), E(t))

(28)

396 | M. Meerschaert and H. P. Scheffler as c → ∞. Now write for the CTRW c−β/α S(N(ct)) = c−β/α S(cβ c−β N(ct)) and use the continuity of the composition mapping to see that (27) holds true. The proof of (28) is similar, using a result from Silvestrov [43] on randomly stopped processes.

Figure 3: Illustration of the difference between the CTRW limit process A(E(t)−) and the OCTRW limit process A(E(t)) in the special case where A(u) = D(u).

Example 4.2. Figure 3 illustrates the difference between CTRW limit process A(E(t)−) and the OCTRW limit process A(E(t)) in the special case where A(u) = D(u). This occurs when the jumps equal the waiting times, i. e., Yn = Jn for all n; see Example 5.2 for more details. At u = E(t), since D(u−) < D(u) at a jump, we also have D(E(t)−) < D(E(t)). In fact D(E(t)−) is the value of the subordinator A = D just before the jump, and D(E(t)) is the value of the subordinator after the jump. Since D(E(t)−) < t and D(E(t)) > t with probability one for any t > 0 (e. g., see Bertoin [7, III, Theorem 4]), the situation in Figure 3 is typical. Recall that a stochastic process {X(t)}t≥0 is called self-similar with index H if for d

d

any scale c > 0 we have X(ct) = cH X(t) for all t ≥ 0, where = denotes equality in distribution.

Corollary 4.3 ([13, Corollary 3.3]). The limit processes A(E(t)−) and A(E(t)) in Theorem 4.1 are both self-similar with index β/α. Proof. This follows easily since the scaling factor in both (27) and (28) is c−β/α . We now present the governing pseudo-differential equations of the CTRW limit process A(E(t)−) and the OCTRW limit process A(E(t)) obtained in Theorem 4.1. We show that the governing equations of the CTRW and OCTRW only differ in their initial/boundary conditions. While this may seem like a minor difference, the result can

Continuous time random walks and space-time fractional differential equations | 397

be quite dramatic, as we shall see in the examples in Section 5. Recall the representation of the pseudo-differential operator ψ(i𝜕x , 𝜕t ) from (26). Also observe that for t > 0 the set ℝ × (t, ∞) is bounded away from (0, 0) and hence ϕ(dx, (t, ∞)) is a finite measure. Theorem 4.4 ([13, Theorem 4.1]). (a) The density c(x, t) of the CTRW limit A(E(t)−) is a solution to the governing equation ψ(i𝜕x , 𝜕t )c(x, t) = δ0 (dx)ϕD (t, ∞).

(29)

(b) The density a(x, t) of the OCTRW limit A(E(t)) is a solution to the governing equation ψ(i𝜕x , 𝜕t )a(x, t) = ϕ(dx, (t, ∞)).

(30)

Remark 4.5. In the uncoupled case, where A and D are independent we get using (24) that ϕ(dx, (t, ∞)) = δ0 (dx)ϕD (t, ∞) so that in the uncoupled case the CTRW limit A(E(t)−) and the OCTRW limit A(E(t)) are identical. Remark 4.6. The densities in (29) and (30) are the point source solutions to those equations, that is c(x, 0) = δ0 (x) and a(x, 0) = δ(x). If one has a (smooth) initial condition p(y) the CTRW and OCTRW governing equations read ψ(i𝜕x , 𝜕t )c(x, t) = p(x)ϕD (t, ∞) and ∞

ψ(i𝜕x , 𝜕t )a(x, t) = ∫ p(x − y)ϕ(dy, (t, ∞)), −∞

respectively. Solving the governing equations (29) and (30) for the CTRW and OCTRW limit processes relies heavily on Fourier–Laplace transform (FLT) techniques. Let {X(t)}t≥0 be a stochastic process and let m(x, t) denote the density of X(t). Then the FLT of m(x, t) is defined as ∞

̄ s) = ∫ ∫ eikx e−st m(x, t) dx dt m(k,

(31)

0 ℝ

for k ∈ ℝ and s > 0. The following result gives the FLT of the densities of the CTRW and OCTRW limit. Recall the definition of the symbols of the Lévy processes {(A(t), D(t))}t≥0 , {A(t)}t≥0 and {D(t)}t≥0 from (21), (22) and (23) above.

398 | M. Meerschaert and H. P. Scheffler Theorem 4.7 ([13, Proposition 4.2]). (a) The density c(x, t) of the CTRW limit A(E(t)−) has FLT ̄ s) = c(k,

1 ψD (s) s ψ(k, s)

(32)

for k ∈ ℝ, s > 0. (b) The density a(x, t) of the OCTRW limit A(E(t)) has FLT ̄ s) = a(k,

1 ψ(k, s) − ψA (k) s ψ(k, s)

(33)

for k ∈ ℝ, s > 0.

5 Examples In this section we will present several concrete examples of coupled CTRW and OCTRW limits, and solve the corresponding governing equations. In the coupled case, these equations involve coupled space-time fractional derivative operators. Example 5.1 (uncoupled case). Here we revisit the uncoupled case from Section 2, to show how the same results follow from the more general coupled CTRW limit theory. If Yn and Jn are independent, then so are A(t) and D(t). Then the FL-symbol is ψ(k, s) = ψA (k) + ψD (s) and in view of Remark 4.5 we have ϕ(dx, (t, ∞)) = δ0 (dx)ϕD (t, ∞). Suppose that the stable Lévy motion {A(t)}t≥0 is totally positively skewed with Fouriersymbol ψA (k) = −b(−ik)α for some 0 < α ≤ 2, α ≠ 0. Suppose further the {D(t)}t≥0 is a standard β-stable subordinator with Laplace symbol ∞

ψD (s) = sβ = ∫ (1 − e−su )ϕD (du).

(34)

0

Then in view of [23, Theorem 7.3.7] we have ϕD (t, ∞) =

t −β . Γ(1 − β)

(35)

Since in the uncoupled case the CTRW limit A(E(t)−) and the OCTRW limit A(E(t)) are identical, the governing equations (29) and (30) read β

𝜕t c1 (x, t) = b𝜕xα c1 (x, t) + δ0 (x)

t −β Γ(1 − β)

(36)

where b < 0 if 0 < α < 1 and b > 0 for 1 < α ≤ 2. The β-stable random variable D has a smooth density gβ (u) supported on u > 0 and the stable Lévy motion A(t) has a

Continuous time random walks and space-time fractional differential equations | 399

Figure 4: Solution c1 (x, t) to the uncoupled OCTRW limit equation (37) with t = 1.0, α = 2, and b = 1 in the case β = 0.6 (solid line), compared with the solution to (37) with t = 1.0, α = 2, and b = 1 in the traditional diffusion case β = 1 (dashed line). In the uncoupled case, the CTRW and OCTRW are governed by the same equation.

smooth density p(x, t). It follows from a simple conditioning argument, as in Section 2, that A(E(t)) = A(E(t)−) has the density ∞

c1 (x, t) = ∫ p(x, (t/s)β )gβ (s) ds = 0

∞

t ∫ p(x, u)gβ (tu−1/β )u−1/β−1 dx, β

(37)

0

that solves the governing equation (36). See [21, 24] for details. Equation (36) is called a space-time fractional diffusion equation. Figure 4 plots the solution for α = 2, so that A(t) is a traditional Brownian motion, and (36) reduces to the time fractional diffusion equation. The plot compares the case of heavy tailed waiting times β = 0.6 with the case of light tailed waiting times β = 1. In the light tailed case, (36) reduces to the traditional diffusion equation. The introduction of a time fractional derivative produces a sharper peak at the origin, and heavier tails. Both are the consequence of long waiting times between jumps. The plot was drawn in the open source programming language R [35] using the stabledist package. Codes for all the figures in this paper are available from the authors upon request. The remaining examples are coupled. Suppose that Jn are iid with D, a standard β-stable random variable with Laplace symbol (34) and Lévy measure (35). For any probability measure ω on ℝ and any p > β/2, suppose that the conditional distribution of Yn given Jn = t is ω(t −p dx). Then [3, Theorem 2.2] shows that (18) holds, and that the Lévy measure of (A, D) is given by ϕ(dx, dt) = (t p ω)(dy)ϕD (dt). In this case, A is stable with index α = β/p.

(38)

400 | M. Meerschaert and H. P. Scheffler Example 5.2 (Lévy walk). Suppose that Yn = Jn as in Kotulski [15]. Take Jn iid with D, a standard β-stable random variable. From (38) with p = 1 and ω = ε1 (the point mass in one) we see that the Lévy measure of (A, D) is given by ϕ(dy, dt) = εt (dy)ϕD (dt)

(39)

which is concentrated on the line y = t. It follows that ψA (k) = ψD (−ik) and ψ(k, s) = (s − ik)β . Since A = D, the joint distribution of (A(s), D(s)) is given by P(A(s),D(s)) (dx, dt) = εt (dx)PD(s) (dt). Theorem 4.7 shows that the CTRW limit A(E(t)−) = D(E(t)−) in (27) has FLT c̄2 (k, s) =

1 ψD (s) sβ−1 . = s ψ(k, s) (s − ik)β

(40)

As in [3, Example 5.4] one can invert the FLT in (40) to get c2 (x, t) =

xβ−1 (t − x)−β Γ(β)Γ(1 − β)

0 < x < t.

(41)

It solves the coupled governing equation (29) which can be written as (𝜕t + 𝜕x )β c2 (x, t) = δ0 (x)

t −β . Γ(1 − β)

(42)

It follows from (33) that the OCTRW limit A(E(t)) = D(E(t)) in (28) has FLT ā 2 (k, s) =

1 ψ(k, s) − ψA (k) 1 (s − ik)β − (−ik)β = . s ψ(k, s) s (s − ik)β

(43)

Inverting the FLT as in [3, Example 5.4] yields a2 (x, t) =

β

t x−1 ( ) Γ(β)Γ(1 − β) x − t

x>t

(44)

is the density of the OCTRW limit D(E(t)). It solves the governing equation ∞

1 (𝜕t + 𝜕x ) a2 (x, t) = ∫ δ0 (x − u)βu−β−1 du Γ(1 − β) β

(45)

t

Both governing equations (42) and (45) involve the fractional material derivative (𝜕t + 𝜕x )β considered by Sokolov and Metzler [45]. Figure 5 compares the CTRW and OCTRW limit pdf in the case where both the waiting times and the jumps are heavy tailed with β = 0.45. Note the striking difference between the CTRW and OCTRW limit pdf. The CTRW limit density c2 (x, t) in (41) is

Continuous time random walks and space-time fractional differential equations | 401

Figure 5: Solution a2 (x, t) to the coupled OCTRW limit equation (45) at t = 1.0 in the case β = 0.45 (solid line), compared with the solution c2 (x, t) to the coupled CTRW limit equation (42) with t = 1.0 and β = 0.45 (dashed line).

supported on 0 < x < t and has moments of all orders. The OCTRW limit a2 (x, t) in (44) falls of like x−1−β as x → ∞ and hence its moments of order > β diverge. Recall that in this model both the jumps and the waiting times are positive random variables. The coupled CTRW S(N(t)) lies between 0 and t because the jumps and the waiting times are equal, but at any time t > 0 that is not a jump time T(n), the particle has experienced a portion of the waiting time Jn+1 , but not the jump Yn+1 . The coupled OCTRW S(N(t) + 1) lies between t and ∞ because at any time t > 0 that is not a jump time T(n), the particle has already experienced the jump Yn+1 , but only a portion of the waiting time Jn+1 . Hence the limit CTRW pdf is concentrated on 0 < x < t, and the limit OCTRW pdf is supported on x > t. It may seem strange that the difference of a single jump can have such a profound effect on the limit pdf. However, in the case of heavy tails, we explained in Example 2.4 that c−1/α M([ct]) ⇒ Z(t), where M(n) = max(Y1 , . . . , Yn ). Since we also have c−1/α S([ct]) ⇒ A(t), the largest jump is the same order of magnitude as the entire sum. Hence a single jump Yi can be comparable to the entire sum of jumps S(n), and likewise for the waiting times. Example 5.3 (Gaussian mixture). Suppose that D is a β-stable random variable with β 𝔼(e−sD ) = e−s and the conditional distribution of Y given D = t is normal with mean zero and variance 2t, as in Shlesinger, Klafter, and Wong [42]. Then Y is symmetric stable with index α = 2β, since 2

2β

𝔼[eikY ] = 𝔼[e−Dk ] = e−|k| ,

402 | M. Meerschaert and H. P. Scheffler 2

using the fact that a normal with mean zero and variance 2t has FT e−tk . If we take (Yn , Jn ) iid with (Y, J) then (18) holds and it follows from (38) that the Lévy measure of (A, D) is given by ϕ(dx, dt) = N0,2t (dx)ϕD (dt) where N0,2t is a normal distribution with mean zero and variance 2t. Then the Lévy symbol of (A, D) equals β

ψ(k, s) = (s + k 2 ) .

(46)

By (32) the CTRW limit A(E(t)−) has FLT c̄3 (k, s) =

sβ−1 . (s + k 2 )β

(47)

Inverting the FLT [3, Example 5.2] shows that the CTRW limit has the density t

c3 (x, t) = ∫ 0

1 x2 exp(− )c2 (u, t) du √4πu 4u

(48)

with c2 (u, t) as in (41). This density solves the governing equation β

(𝜕t − 𝜕x2 ) c3 (x, t) = δ0 (x)

t −β . Γ(1 − β)

(49)

By (33) the OCTRW limit A(E(t)) has FLT ā 3 (k, s) =

(s + k 2 )β − |k|2β (s + k 2 )β

(50)

and has the density [13, Example 5.3] ∞

a3 (x, t) = ∫ t

x2 1 exp(− )a2 (u, t) du √4πu 4u

(51)

with a2 (u, t) as in (44). In view of (30) it solves the governing equation β

(𝜕t − 𝜕x2 ) a3 (x, t) =

∞

1 x2 1 exp(− )βu−β−1 du ∫ Γ(1 − β) √4πu 4u

(52)

t

Figure 6 plots the pdf of the OCTRW limit and the CTRW limit in the case β = 0.8 at time t = 1. The difference is striking. The CTRW limit pdf has a sharp peak, and a much lighter tail than the OCTRW limit pdf. See Meerschaert and Scalas [22] for an application to finance.

Continuous time random walks and space-time fractional differential equations | 403

Figure 6: Solution a3 (x, t) to the coupled OCTRW limit equation (52) with t = 1.0 and β = 0.8 (solid line), and solution c3 (x, t) to the corresponding CTRW limit equation (49) with t = 1.0 and β = 0.8 (dashed line).

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404 | M. Meerschaert and H. P. Scheffler

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Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy

Inverse subordinators and time fractional equations Abstract: The inverse stable subordinator is the first passage time of a standard stable subordinator with index 0 < β < 1. The probability density of the inverse stable subordinator can be used to solve time fractional Cauchy problems, where the usual first derivative in time is replaced by a Caputo fractional derivative of order β. If the Cauchy problem governs a Markov process, then the fractional Cauchy problem governs a time-changed process, where the time parameter is replaced by the inverse stable subordinator. Applications include delayed Brownian motion, and the fractional Poisson process. Keywords: Inverse subordinator, Cauchy problem, delayed Brownian motion, fractional Poisson process MSC 2010: 60K05, 33E12, 26A33

1 Introduction Zaslavsky [52] introduced the time fractional differential equation β

𝜕t m(x, t) = D𝜕x2 m(x, t)

(1)

with 0 < β < 1 and D > 0 as a model for Hamiltonian chaos; see also Nigmatullin [40]. Zaslavsky called the stochastic process governed by this equation a “fractal Brownian motion.” Meerschaert and Scheffler [27] showed that equation (1) governs a timechanged Brownian motion B(Et ) where B(t) is a Brownian motion, and Et is an independent inverse stable subordinator of index β. A typical plot of this “delayed Brownian motion” is shown in Figure 1. The effect of the time change is to introduce delays in the particle motion.

Acknowledgement: M. M. Meerschaert was partially supported by ARO MURI grant W911NF-15-1-0562. Mark M. Meerschaert, Department of Statistics and Probability, Michigan State University, East Lansing, Michigan, USA, e-mail: [email protected] Erkan Nane, Department of Mathematics and Statistics, 221 Parker Hall, Auburn University, Auburn AL 36849, USA, e-mail: [email protected] P. Vellaisamy, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India, e-mail: [email protected] https://doi.org/10.1515/9783110571622-017

408 | M. M. Meerschaert et al.

Figure 1: Typical sample path of delayed Brownian motion Xt = B(Et ) with D = 0.5 and β = 0.8, from [39].

Equation (1) is an example of a fractional Cauchy problem, where the usual first derivative in time is replaced by a Caputo fractional derivative β 𝜕t f (t)

t

1 f 󸀠 (s) ds = ∫ Γ(1 − β) (t − s)β

(2)

0

for some 0 < β < 1. Baeumer and Meerschaert [5] showed that the solution to a fractional Cauchy problem can be written in terms of the probability density function (pdf) of the inverse stable subordinator Et . First note that p(x, t) =

2 1 e−x /(4Dt) √4πDt

solves the traditional diffusion equation 𝜕t p(x, t) = D𝜕x2 p(x, t),

(3)

a special case of (1) with β = 1. Then [5, Theorem 3.1] shows that the solution to the time fractional diffusion equation (1) is given by ∞

m(x, t) = ∫ p(x, u)h(u, t) du,

(4)

0

where h(u, t) is the pdf of the inverse stable subordinator Et . Since the Brownian motion B(u) has pdf p(x, u), and the independent inverse stable subordinator u = Et has pdf h(u, t), a simple conditioning argument shows that (4) is also the pdf of the timechanged process x = B(Et ). Figure 2 plots the solution to (1) to show the behavior over time. Note the sharper peak and heavier tails, compared to a bell-shaped normal pdf. Since the solutions spread at the rate t β/2 , slower than the usual t 1/2 spreading for a classical diffusion, (1) models subdiffusion.

Inverse subordinators and time fractional equations | 409

Figure 2: Solution (4) to the time fractional diffusion equation (1) at times t = 0.1 (solid line), t = 0.3 (dotted line), and t = 0.8 (dashed line) with β = 0.75 and D = 1.0.

2 The inverse stable subordinator A subordinator Dt is a nondecreasing Lévy process. A Lévy process is a stochastic process with stationary, independent increments [4, 45]. The distribution of Dt is strictly stable if Dct ≃ c1/β Dt (same distribution) for some 0 < β < 1. The pdf g(x, t) of the stable subordinator Dt cannot generally be written in closed form, but computer codes to compute g(x, t) are widely available [30, Chapter 5]. Due to the distributional scaling relation, one can also write g(x, t) = t −1/β g(xt −1/β , 1). The inverse stable subordinator Et = inf{u : D(u) > t}

(5)

is the first passage time of the stable subordinator above the level t ≥ 0. Properties of the inverse stable subordinator are detailed in [27, Section 3] and Meerschaert and Straka [31]. An explicit formula for the moments of Et was given by Piryatinska, Saichev and Woyczynski [41]. It follows from the definition (5) that ∞

ℙ[Et ≤ u] = ℙ[Du ≥ t] = ∫ g(w, u) dw,

(6)

t

hence the inverse stable subordinator has pdf t

d d h(u, t) = ℙ[Et ≤ u] = [1 − ∫ g(w, u) dw]. du du

(7)

0

Now a simple calculation [27, Corollary 3.1] shows that the pdf of the inverse stable subordinator Et is h(x, t) =

t −1−1/β x g(tx−1/β , 1) β

(8)

410 | M. M. Meerschaert et al.

Figure 3: Inverse stable density h(x, t) with β = 0.6 and t = 1, from [31].

for all x > 0 and t > 0. This formula together with (4) was used to plot Figure 2. Figure 3 plots a typical inverse stable density h(x, t). The density is supported on the positive half-line, and is discontinuous at the origin. Using asymptotic properties of stable densities, it is not hard to show (e. g., see [31, Section 4]) that h(0+, t) = lim h(x, t) = x↓0

t −β Γ(1 − β)

(9)

for all t > 0. The Laplace transform (LT) of g(x, t) is (e. g., see [30, Proposition 3.10 and p. 114]) ∞

̃ t) = ∫ e−sx g(x, t) dx = e−tcs g(s,

β

(10)

0

and we assume c = 1 to get the standard stable subordinator. Using the fact that integration corresponds to multiplication of the Laplace transform by s−1 , it follows from (7) that the (standard) inverse stable subordinator pdf has Laplace transform ̃ s) = − d [s−1 g(s, ̃ u)] h(u, du β β d = − [s−1 e−us ] = sβ−1 e−us du

(11)

for all s > 0. The processes t = Du and u = Et are inverses. The graph of the inverse stable subordinator u = Et is just the graph of the stable subordinator t = Du with the axes swapped, i. e., a reflection through the diagonal line t = u. See Figures 4 and 5 for an illustration. Here the stable subordinator and its inverse were simulated using freely available codes [30, Chapter 5]. The stable subordinator is a strictly increasing pure jump process. Therefore, the inverse stable subordinator is continuous, and its graph has flat periods (resting times), which correspond to the jumps in the stable subordinator. The lengths of those resting periods follow a power law distribution, since they

Inverse subordinators and time fractional equations | 411

Figure 4: A typical sample path of the stable subordinator t = Du with index β = 0.8.

Figure 5: The inverse stable subordinator u = Et with index β = 0.8, using the same sample path as in Figure 4. The graphs are the same, with the axes swapped.

are the same as the jump distribution of the stable subordinator: Jumps larger than any given cutoff ε > 0 follow a Pareto distribution, where the probability of a jump length exceeding x > ε is proportional to x−β , e. g., see [30, Section 3.4]. Since the resting periods of the inverse stable subordinator have the same distribution, they are not exponentially distributed, and hence Et is not a Markov process.

3 Fractional Cauchy problems The fractional diffusion equation (1) is an example of a fractional Cauchy problem. A Cauchy problem is an abstract differential equation of the form 𝜕t p(x, t) = Lx p(x, t);

p(x, 0) = p0 (x)

(12)

where Lx is some spatial operator. A Banach space is a Cauchy complete, normed vector space. A familiar example is L1 (ℝ), the space of real-valued functions of one real variable, with the norm ‖f ‖1 = ∫ |f (x)| dx. A semigroup is a family of linear operators {Tt } on that space, with the property that T0 is the identity operator, and Tt+s = Tt Ts .

412 | M. M. Meerschaert et al. A C0 semigroup is bounded and continuous in the Banach space norm. Then the generator Lx f (x) = lim

t→0

Tt f (x) − T0 f (x) , t−0

(13)

where the limit is taken in the Banach space norm, is defined on a dense subset of that space. The generator can contain ordinary derivatives as in (1), fractional derivatives in space, variable coefficients, and boundary conditions. The fractional Cauchy problem β

𝜕t m(x, t) = Lx m(x, t);

m(x, 0) = p0 (x)

(14)

uses a Caputo fractional derivative (2) of order 0 < β < 1. The mathematical study of fractional Cauchy problems was initiated by Kochubei [20, 21] and Schneider and Wyss [46]. Later Baeumer and Meerschaert [5, Theorem 3.1] showed that if p(x, t) solves the Cauchy problem (12), then (4) solves the corresponding fractional Cauchy problem, where the function h(x, t) is given by the formula (8). A few years afterwards, Meerschaert and Scheffler [27, Corollary 3.1] identified this function as the pdf of the inverse β-stable subordinator. Hence if Lx is the generator of some Markov process B(t), it follows that the fractional Cauchy problem (14) governs the non-Markovian process B(Et ). As illustrated in Figure 1, the time fractional derivative models long resting times between motions of the original Markov process. In [33, Theorem 3.6] this idea is used to show that, under some mild technical conditions, the fractional Cauchy problem (14) with d

Lx f = ∑

𝜕(aij (x)(𝜕f /𝜕xi )) 𝜕xj

i,j=1

(15)

and aij (x) = aji (x) on a bounded domain with zero Dirichlet boundary conditions has a unique solution ∞

m(x, t) = ∑ f ̄(n)Eβ (−λn t β )ψn (x) n=1

(16)

where {ψn } is a complete orthonormal basis of eigenfunctions for Lx with Lx ψn = λn ψn , f ̄(n) = ∫D f (x)ψn (x) dx, and the Mittag-Leffler function zk Γ(1 + βk) k=0 ∞

Eβ (z) = ∑

(17)

for any complex z. For β = 1, Eβ (−λn t β ) = e−λn t and we recover the well-known solution to the Cauchy problem (12) obtained using separation of variables. In short, equation ∞ (16) comes from (4), using the fact that Eβ (−λt β ) = ∫0 e−λu h(u, t) du. Chen et al. [16,

Inverse subordinators and time fractional equations | 413

Theorem 5.1] show that the same formula (16) yields pointwise solutions to the spacetime fractional Cauchy problem (14) on a bounded domain with Lx = −(−Δ)α , the fractional Laplacian. In both cases, the Cauchy problem (12) governs the probability densities of a killed Markov process B(t), and the fractional Cauchy problem (14) governs the time-changed process B(Et ). Note that fractional Cauchy problems have exactly the same boundary conditions as the original Cauchy problem, since these boundary conditions are part of the specification of the generator and the Banach space. A very special case of the fractional Cauchy problem gives the governing equation of the pdf h(x, t) of the inverse stable subordinator itself. Take Lx = −𝜕x , the generator of the shift semigroup Tt f (x) = f (x − t) corresponding to the non-random process B(t) = t (e. g., see [30, Example 3.21]). Then the pdf of B(Et ) = Et solves the fractional Cauchy problem β

𝜕t h(x, t) = −𝜕x h(x, t)

(18)

with the point source initial condition h(x, 0) = δ(x) written in terms of the Dirac delta function, reflecting the fact that E0 = 0 with probability one. Meerschaert and Straka [31] review several equivalent forms of the governing equation (14). One form uses the Riemann–Liouville fractional derivative β

𝔻t f (t) =

t

1 f (s) ds d , ∫ Γ(1 − β) dt (t − s)β

(19)

0

which differs from the Caputo form in that the first derivative is placed outside the integral. Since integration and differentiation do not commute in general, these two forms are not equal. In fact, we have β

β

𝜕t f (t) = 𝔻t f (t) − f (0)

t −β Γ(1 − β)

(20)

when 0 < β < 1; e. g., see [30, Eq. (2.33)]. Then we can also write the fractional Cauchy problem (14) in the form β

𝔻t m(x, t) = Lx m(x, t) + p0 (x)

t −β Γ(1 − β)

(21)

used in the original work of [5, 27, 52]. Then the pdf h(x, t) of the inverse stable subordinator also solves the fractional equation β

𝔻t h(x, t) = −𝜕x h(x, t) + δ(x) for x > 0 and t > 0.

t −β Γ(1 − β)

(22)

414 | M. M. Meerschaert et al.

4 The fractional Poisson process One nice application of the inverse stable subordinator is to define a fractional Poisson process N(Et ), where N(t) is the traditional Poisson process [44]. The probability mass function (pmf) of the traditional Poisson process p(n, t) = e−λt

(λt)n ; n!

n = 0, 1, 2, 3, . . . ,

(23)

gives the probability that N(t) = n. It solves the Cauchy problem 𝜕t p(n, t) = −λp(n, t) + λp(n − 1, t),

(24)

which says that particles make the transition from state n − 1 to state n at rate λ > 0. Then the fractional Cauchy problem β

𝜕t m(n, t) = −λm(n, t) + λm(n − 1, t)

(25)

governs the pmf m(n, t) of the fractional Poisson process. Now the pmf m(n, t) of the fractional Poisson process is given by the formula (4) where x = n is a nonnegative integer, p(n, u) is given by (23), and h(u, t) is the pdf (8) of the inverse stable subordinator. Laskin [24] defines the fractional Poisson process as the counting process whose pmf solves (25). Repin and Saichev [42] define the fractional Poisson process as the counting process with Mittag-Leffler waiting times between state transitions, so that the probability of waiting longer than some time t > 0 before the next jump equals Eβ (−λt β ). It was shown in [35] that all these definitions are equivalent. The fractional Poisson process differs from the traditional Poisson process in that very long resting times occur more often.

5 Continuous time random walks The continuous time random walk (CTRW) is a model in statistical physics that explains the meaning of the time fractional derivative. Start with a random walk S(n) = J1 + ⋅ ⋅ ⋅ + Jn where the particle jumps J1 , J2 , J3 , . . . are independent and identically distributed (iid), with mean zero and finite variance σ 2 > 0. Then the central limit theorem implies that n−1/2 S([nt]) ⇒ B(t) in distribution, where B(t) is normal with mean zero and variance σ 2 t [30, Section 1.1]. Now assume that the jumps Jn are separated by an iid sequence of random waiting times Wn , independent of the jumps. Then Tn = W1 + ⋅ ⋅ ⋅ + Wn is the time of the nth jump, Nt = max{n ≥ 0 : Tn ≤ t} is the number of jumps by time t ≥ 0, and S(Nt ) is the particle position at time t ≥ 0. Figure 6 illustrates the model.

Inverse subordinators and time fractional equations | 415

Figure 6: A continuous time random walk, where random jumps Jn are separated by random waiting times Wn , explains the physical meaning of the time fractional diffusion equation (1).

Now suppose that the random waiting times have a heavy power law tail: We assume that ℙ[Wn > t] = Ct −β where C = 1/Γ(1 − β). Then the extended central limit theorem [30, Theorem 3.41] implies that n−1/β T[nt] ⇒ Dt , a standard β-stable subordinator. The random walk Tn of waiting times and the renewal process Nt are inverse processes: {Nt ≥ n} = {Tn ≤ t}.

(26)

Then [27, Theorem 3.2] shows that these inverse processes have inverse limits: n−β Nnt ⇒ Et , the inverse stable subordinator (5). Next an application of the continuous mapping theorem [30, Section 4.4] yields n−β/2 S(Nnt ) = (nβ )

−1/2

S(nβ n−β Nnt ) ≈ (nβ )

−1/2

S(nβ Et ) ⇒ B(Et ),

a delayed Brownian motion whose pdf solves the time fractional diffusion equation (1). Hence the time fractional derivative of order 0 < β < 1 models long waiting times distributed according to a power law with the same index β. Many other fractional diffusion models can be investigated using the CTRW model. Suppose for example that Xn ≥ 0 are iid with ℙ[Xn > x] = Cx −α for some C > 0 and some 1 < α < 2. Then the mean μ = 𝔼[Xn ] exists, and we can take Jn = Xn − μ as the particle jumps. Now n−1/α S([nt]) ⇒ A(t), an α-stable Lévy process with pdf p(x, t) [30, Theorem 3.41]. The governing equation of this pdf solves (12) with p(x, 0) = δ(x), the Dirac delta function, Lx = D𝜕xα using a space-fractional derivative, and D = CΓ(2 − α)/(α − 1) [30, p. 84]. The long-time limit of the CTRW is derived as before: n−β/α S(Nnt ) = (nβ )

−1/α

S(nβ n−β Nnt ) ≈ (nβ )

−1/α

S(nβ Et ) ⇒ A(Et ).

The CTRW limit A(Et ) has a pdf m(x, t) that solves the fractional Cauchy problem (14) with the same generator Lx = D𝜕xα and the same initial condition. Furthermore, the

416 | M. M. Meerschaert et al. exact form of the CTRW limit pdf is given by (4) where h(u, t) is the pdf (8) of the inverse stable subordinator. If ℙ[Xn > x] = pCx−α

and ℙ[Xn < −x] = qCx −α

α we get a two-sided α-stable Lévy process with generator Lx = pD𝜕xα + qD𝜕−x that also involves a negative fractional derivative. If the particle jumps Jn are random vectors, the generator involves vector fractional derivatives. For example, if Jn = Rn Θn where ℙ[Rn > r] = Cr −α are iid and Θn are iid uniformly distributed random unit vectors, independent of Rn , then n−1/α S([nt]) ⇒ A(t), a spherically symmetric α-stable Lévy motion with generator Lx = DΔα , using the fractional Laplacian [30, Example 6.24]. The exact form of the constant D is given in [30, Example 6.24]. The pdf m(x, t) of the time-changed process A(Et ) is given by (4), where p(x, u) is the pdf of the vector stable process x = A(u), and h(u, t) is the pdf of the inverse stable subordinator (8). The pdf m(x, t) solves the space-time fractional diffusion equation (14) with the generator Lx = DΔα and a delta function initial condition. If the particle jumps are correlated, then one can obtain a delayed fractional Brownian motion BH (Et ) in the limit: n−H S([nt]) ⇒ BH (t) and n−βH S(Nnt ) ⇒ BH (Et ), see [34, Theorem 2.4].

6 Fractal properties Blumenthal and Getoor [8] showed that the range of a stable subordinator is a random fractal with dimension β. Hence the same number β describes the power law waiting times, the order of the fractional derivative, and the fractal dimension of the stable subordinator. Fractal properties of the inverse stable subordinator, and time-changed processes B(Et ), are derived in [39]. The graph of a Brownian motion B(t) is a random fractal with dimension 3/2. Meerschaert, Nane and Xiao [39, Proposition 2.3] show that the graph of the delayed Brownian motion B(Et ) is a random fractal with dimension 1 + β/2, which reduces to 3/2 in the limit case β = 1. The graph of a scalar-valued α-stable Lévy process A(t) with index 1 < α ≤ 2 is a random fractal with dimension 2 − 1/α. This reduces to 3/2 in the case α = 2, since an α-stable Lévy process with α = 2 is a Brownian motion. The graph of the CTRW limit A(Et ) is a random fractal with dimension 1 + β(1 − 1/α) [39, Proposition 2.3], which reduces to 2 − 1/α in the limit case β = 1. The graph of a fractional Brownian motion with Hurst index 0 < H < 1 is a random fractal with dimension 2 − H, which reduces to 3/2 in the special case H = 1/2 of a Brownian motion. The graph of a delayed fractional Brownian motion BH (Et ) is a random fractal with dimension β +1−Hβ, which reduces to 2−H in the limit case β = 1. Hence, even though the sample paths of the inverse stable subordinator Et are continuous and nondecreasing, they are sufficiently irregular as to influence the fractal dimension of a time-changed process.

Inverse subordinators and time fractional equations | 417

7 Higher order equations In many cases, fractional partial differential equations can be written in equivalent higher order forms, some of which do not involve any fractional derivatives. The inverse stable subordinator explains the equivalence between these equations. One interesting example involves Brownian subordinators. Given a Brownian motion B(t), let B󸀠t be another independent Brownian motion. Allouba and Zheng [1, 2] consider the time-changed process Xt = B(|B󸀠t |), which they call “Brownian time Brownian motion.” Burdzy [9] considers a closely related process called “iterated Brownian motion” where B(t) is a two-sided Brownian motion on −∞ < t < ∞ and Yt = B(B󸀠t ). Both processes have the same pdf, and hence the same governing equation 𝜕t m(x, t) =

𝜕x2 p0 (x) + 𝜕x4 m(x, t); √πt

m(x, 0) = p0 (x)

(27)

for t > 0 and x real [2]. They also consider a vector equation, but we will focus here on the scalar case. Note that (27) is not a Cauchy problem, due to the presence of the first term on the right-hand side, which depends on t > 0. Baeumer, Meerschaert and Nane [7, Corollary 3.2] show that in fact equation (27) is equivalent to the time fractional diffusion equation (1) with β = 1/2. One way to see this is to apply 𝜕t1/2 to both sides of (1), or to the equivalent form (21) with Lx = −𝜕x2 . Another approach is to note that the absolute value |B󸀠t | has the same pdf as the maximum Mt = max{B󸀠u : 0 ≤ u ≤ t} by the reflection principle. But since the stable subordinator Dt with index β = 1/2 is the first passage time of the Brownian motion B󸀠t , and since the maximum process is the inverse of the first passage time, |B󸀠t | has the same pdf as the inverse stable subordinator Et . Since (1) governs the delayed Brownian motion B(Et ) and (27) governs the Brownian time Brownian motion B(|B󸀠t |), and since both processes have the same pdf, the governing equations must be equivalent. More generally, Allouba and Zheng [2] show that if B(t) is a Markov process in one or more dimensions with generator Lx , then the pdf of B(Et ) with β = 1/2 also solves the higher order equation 𝜕t m(x, t) =

Lx p0 (x) + Lx 2 m(x, t); √πt

m(x, 0) = p0 (x).

(28)

Then the pdf of the Brownian time process B(|B󸀠t |) solves the same equation. The higher order equation (28) is equivalent to the fractional Cauchy problem (14) with β = 1/2. Another iterated equation comes from the theory of medical ultrasound. Kelly et al. [19] propose a time fractional wave equation α02 2β 2α0 β+1 1 2 𝔻 m(x, t) = Δx m(x, t) m(x, t) + 𝔻 𝜕 m(x, t) + c0 b t b2 t c02 t

(29)

to model the variations in pressure m(x, t) for acoustic wave conduction in a complex medium (e. g., human tissue), where c0 is the speed of sound in a homogeneous

418 | M. M. Meerschaert et al. medium, and the constant b = cos(πy/2). This equation models power law attenuation, which is commonly seen in applications: An input sound wave attenuates according to a power law, and in particular, the amplitude decays like e−α(ω)t where the attenuation coefficient α(ω) = α0 |ω|β depends on the frequency ω of the input wave according to a power law with index β. Straka et al. [48] show that the higher order equation (29) in one dimension is equivalent to a lower order time fractional equation that involves an inverse stable subordinator. Start with the stable subordinator t/c0 + (α0 /b)1/β Dt where Dt is the standard stable subordinator. The pdf h0 (x, t) of the corresponding inverse stable subordinator solves the governing equation α β 1 𝜕t h0 (x, t) + 0 𝔻t h0 (x, t) = −𝜕x h0 (x, t) c0 b

(30)

for x > 0 and t > 0. Then [48, Section 3] shows that the function h0 (x, t) also solves the higher order equation (29) in one dimension. The key is to note that the operator on the left-hand side of (29) is the same as the operator on the left-hand side of (30) applied twice. A closely related argument in Meerschaert et al. [37] shows that if p(x, t) solves the traditional wave equation (12) with Lx = Δx , then the function m(x, t) given by (4) with h replaced by h0 solves (29) in three dimensions. The inverse stable subordinator also leads to a useful CTRW model for the time fractional wave equation (29); see [37, Section 5].

8 Subordinators and inverse subordinators The standard β-stable subordinator is one example of a subordinator, i. e., a nondecreasing Lévy process. More generally, we can consider a wide array of subordinators and their inverses, both of which can be useful in applications. Any subordinator Dt with pdf g(x, t) has a Laplace transform ∞

̃ t) = ∫ e−sx g(x, t) dx = e−tψD (s) g(s,

(31)

0

where the Laplace symbol can be written in the form ∞

ψD (s) = as + ∫ (1 − e−sy ) ϕD (dy)

(32)

0

using the Lévy–Khintchine formula [4, 45]. Here ϕD (dy) is the Lévy measure, which governs the jumps of the process [30, p. 51]. For a standard stable process we have a = 0 and ϕD (y, ∞) = y−β /Γ(1 − β), which leads to ψD (s) = sβ ; e. g., see [30, p. 114]. For the rest of this section we assume that a = 0, but note that an example with a > 0 was already discussed in Section 7.

Inverse subordinators and time fractional equations | 419

Meerschaert and Scheffler [29, Theorem 3.1] shows that, under some mild technical conditions, the inverse subordinator Et defined by (5) has a pdf t

h(x, t) = ∫ ϕD (t − u, ∞)g(u, x) du.

(33)

0

The formula (8) can be obtained as a special case. They also show that the inverse sub̃ s) = s−1 ψ (s)e−xψD (s) [29, Eq. (3.130)], which ordinator pdf has Laplace transform h(x, D reduces to (11) in the case of a standard β-stable subordinator. Veillette and Taqqu [51] develop numerical methods for computing the pdf h(x, t) of a general inverse subordinator. Now suppose that A(t) is a Lévy process with pdf p(x, t). Since this process can take both positive and negative values, we apply the Fourier transform (FT) ∞

̂ t) = ∫ e−ikx p(x, t) dx. p(k, −∞

The Lévy–Khintchine formula [30, Theorem 3.4] implies that ̂ t) = etψA (k) p(k,

(34)

where the Fourier symbol can be written in the form ∞

ψA (k) = −ikb − Dk 2 + ∫ (e−iky − 1 + −∞

iky ) ϕA (dy). 1 + y2

(35)

The time-changed process B(Et ) has a pdf m(x, t) given by (4) [27, Corollary 3.8]. Taking LT and FT in this equation, we can see that ∞

∞

̄ s) = ∫ e−ikx ∫ e−st m(x, t) dt dx m(k, −∞ ∞ ∞

= ∫( ∫ e

0 ∞ −ikx

p(x, u) dx) ( ∫ e−st h(u, t) dt) du

0 −∞ ∞ uψA (k) −1

= ∫e

0

s ψD (s)e−uψD (s) du

0

=

s−1 ψD (s) ψD (s) − ψA (k)

(36)

which we can rewrite in the form ̄ s) = ψA (k)m(k, ̄ s) + s−1 ψD (s). ψD (s)m(k,

(37)

420 | M. M. Meerschaert et al. The Laplace and Fourier symbols correspond to pseudo-differential operators (e. g., see Jacob [18]). Using the functional calculus, since (ik)f ̂(k) is the FT of the weak derivative 𝜕x f (x), the FT ψA (k)f ̂(k) inverts to ψA (−i𝜕x )f (x). Similarly, since sf ̃(s) is the LT of the weak derivative 𝜕t f (t), the LT ψD (s)f ̃(s) inverts to ψD (𝜕t )f (t). For example, if A(t) is a Brownian motion then ψA (k) = −Dk 2 = D(ik)2 and ψA (k)f ̂(k) is the FT of D𝜕x2 f (x). If Dt is the standard β-stable subordinator, then ψD (s) = sβ , and ψD (s)f ̃(s) β is the LT of 𝔻t f (t). For a discussion of weak versus strong derivatives, and how this relates to the formula sf ̃(s) − f (0) for the LT of the traditional derivative 𝜕t f (t); see [31, Section 3]. In short, the extra term f (0) comes from the weak derivative of the Heaviside function at t = 0. Now invert the LT and FT in (37) to obtain the governing equation of A(Et ): ψD (𝜕t )m(x, t) = ψA (−i𝜕x )m(x, t) + δ(x)ϕD (t, ∞),

(38)

using the fact [29, Eq. (3.12)] that ϕD (t, ∞) has LT s−1 ψD (s). More generally, we can consider generalized Cauchy problems of the form (38) with ψA (−i𝜕x ) replaced by the generator Lx of some semigroup. Chen [15] develops solution to generalized Cauchy problems, extending [5, Theorem 3.1] to the case of a general time operator. This provides a governing equation for the time-changed process B(Et ) where B(t) is a Markov process with generator Lx , and Et is an independent general inverse subordinator. See Toaldo [49] for some related results. If A(t) is a Brownian motion and Dt is the standard stable subordinator, then (38) reduces to (21) with Lx = D𝜕x2 , since ϕD (t, ∞) = t −β /Γ(1 − β). If A(t) = t and Dt is the ̂ t) = eψA (k)t with ψA (k) = −ik, standard stable subordinator, then p(x, t) = δ(x − t), p(k, and (38) reduces to (22) with Lx = −𝜕x . This generator is also a weak derivative, a fact that has caused some confusion in the literature [31, Section 5]: Weak and traditional derivatives are the same for differentiable functions, but since the inverse stable pdf h(x, t) has a jump discontinuity at x = 0, the weak derivative has an extra term h(0+, t)H 󸀠 (x) = δ(x)t −β /Γ(1−β) at x = 0. This cancels the last term in (38), and then (20) yields the alternative governing equation found in Hahn, Kobayashi and Umarov [17]: β

𝜕t h(x, t) = −𝜕x h(x, t) − δ(x)

t −β Γ(1 − β)

(39)

where now 𝜕x is the traditional derivative, which is only defined on x > 0.

8.1 Tempered stable subordinator One issue with the stable subordinator pdf is that its mean and variance are undefined, since g(x, t) ≈ x−β−1 for x large. A useful idea [10, 43] for handling this is to “temper” the heavy tail of the pdf so that all moments exist. The function e−λx g(x, t) has a light tail for any λ > 0, and if the tempering parameter λ is sufficiently small, then the

Inverse subordinators and time fractional equations | 421

difference will not be noticeable for moderate x. Hence tempering is a mathematical construct to avoid diverging moments. Of course, e−λx g(x, t) is no longer a probability density, but we can apply (10) to see that gλ (x, t) = e−λx g(x, t)etcλ

β

(40)

is a pdf, with LT g̃λ (s, t) = e−tcψλ (s)

(41)

where ψλ (s) = (λ+s)β −λβ . It is not clear from this calculation that gλ (x, t) should be the pdf of a Lévy subordinator, but another calculation using (32) with an exponentially ̃ t) is indeed the tempered Lévy measure ϕλ (t, ∞) = ceλy t −β /Γ(1 − β) shows that g(s, Laplace symbol of a subordinator. Taking c = 1 yields the pdf of the standard tempered stable subordinator, which we will denote by Dλt . Substituting Dλt for Dt in (5) yields the inverse tempered stable subordinator, which we will denote by Etλ . Since the jump intensity (Lévy measure) is an exponentially tempered power law, the effect of the tempering is to “cool” the big jumps. Alrawashdeh et al. [3] give an explicit formula for the inverse tempered stable pdf t

β

t

hλ (x, t) = exλ [e−λt h(x, t) + λ ∫ e−λτ h(x, τ) dτ − λβ ∫ e−λτ g(τ, x) dτ] 0

(42)

0

using the inverse stable pdf (8) and the standard β-stable pdf. Plots of the pdf are quite similar to Figure 3; see [3, Fig. 1]. Some alternative forms for the inverse tempered stable pdf hλ (x, t) are given in Kumar and Vellaisamy [23], along with an explicit formula for hλ (0+, t) in terms of the incomplete gamma function, asymptotic behavior of the moments of Etλ , and higher order governing equations for the case β = 1/n for some integer n. See Stanislavsky et al. [47] and Veillette and Taqqu [50] for additional information. If A(t) is a Lévy process with pdf p(x, t) and Fourier symbol ψA (k), then (4) with h replaced by hλ gives the pdf of B(Dλt ). This pdf solves (38) with ψD replaced by ψλ . β,λ We call 𝔻t = ψλ (𝜕t ) the tempered fractional derivative. A calculation using the shift property of the LT [30, p. 209] shows that β,λ

β

𝔻t f (t) = e−λt 𝔻t [eλt f (t)] − λβ f (t).

(43)

If B(u) is a Brownian motion, then (38) reduces to the tempered fractional diffusion equation β,λ

𝔻t m(x, t) = 𝜕x2 m(x, t) + δ(x)ϕλ (t, ∞).

(44)

This model exhibits transient anomalous diffusion, resembling anomalous subdiffusion (1) at small time scales and traditional diffusive behavior at large time

422 | M. M. Meerschaert et al. β

scales. More generally, (14) with 𝜕t replaced by ψλ (𝜕t ) represents a tempered fractional Cauchy problem. Classical solutions to tempered fractional Cauchy problems on bounded domains, similar to (16), are developed in [38]. It has been suggested [3, 29] that a tempered Caputo fractional derivative can be β,λ β,λ defined as 𝜕t f (t) = 𝔻t f (t) − f (0)ϕλ (t, ∞), to simplify (44) into a form resembling (1). Then the inverse tempered stable pdf solves the tempered fractional Cauchy problem β,λ β (18) with 𝜕t replaced by 𝜕t [3, Proposition 3.1]. Some other forms of the tempered fractional derivative have been suggested [6, 25, 26]. A CTRW model for tempered fractional diffusion was developed in Chakrabarty and Meerschaert [11]: The waiting times are exponentially tempered power laws. Several applications of tempered fractional diffusion to problems in geophysics are outlined in Meerschaert, Zhang and Baeumer [32]. Another interesting application is the tempered fractional Poisson process N(Etλ ); see [3, Section 7]. The pmf m(x, t) of this process is given by (4) where p(n, t) is given by (23) and h is replaced by hλ . It solves β,λ β the tempered fractional Cauchy problem (25) with 𝜕t replaced by 𝜕t [3, Eq. (7.4)].

8.2 Distributed order and ultraslow diffusion Chechkin et al. [12–14] consider a model where the fractional order β of the time derivative in (1) is randomized. They define the distributed order fractional diffusion equation p(β)

𝜕t

m(x, t) = D𝜕x2 m(x, t),

(45)

where the distributed order fractional derivative is defined as a mixture of Caputo derivatives of different order: p(β)

𝜕t

1

β

f (t) = ∫ 𝜕t f (t) p(β)dβ.

(46)

0

If the pdf p(β) gives positive probability to values of β near zero, then (45) models ultraslow diffusion, where a plume of particles spreads at a logarithmic rate. Meerschaert and Scheffler [28, Theorem 3.4] show that the model (45) corresponds to a subordinator Dt with Lévy measure 1

ϕD (y, ∞) = ∫ 0

y−β p(β)dβ. Γ(1 − β)

(47)

Hence the jumps of this subordinator are power law, with an index β governed by the pdf p(β). A calculation using (32) with a = 0 shows that 1

ψD (s) = ∫ sβ p(β)dβ. 0

(48)

Inverse subordinators and time fractional equations | 423

Now let Et be the distributed order inverse subordinator defined by (5). Its pdf h(x, t) is given by (33), and so if B(t) is an independent Brownian motion with pdf p(x, t), then the pdf m(x, t) of B(Et ) solves the distributed order fractional diffusion equation (45). This pdf is given by (4), where p(x, t) is the pdf of B(t), and h(x, t) is the pdf of the distributed order inverse subordinator. If the pdf p(β) gives positive probability to values of β near zero, then the subordinator Dt is ultrafast, and hence its inverse Et is ultraslow: The presence of very large jumps in Dt translates to very long waiting times for Et ; see Kovács and Meerschaert [22] for more details. The distributed order fractional diffusion equation (45) is a special case of (38). To see this, recall that s−1 ψD (s) is the LT of ϕD (y, ∞), and substitute (48) into (37) to get 1

β

2

1

̄ s) p(β)dβ = −Dk m(k, ̄ s) + ∫ sβ−1 p(β)dβ. ∫ s m(k, 0

(49)

0

Rearrange to get 1

̄ s) − sβ−1 ) p(β)dβ = −Dk 2 m(k, ̄ s) ∫(sβ m(k, 0

̂ 0) = 1 since B(E0 ) = 0. Invert the FT and LT to arrive at (45). A CTRW and note that m(k, model for distributed order fractional diffusion is developed in [28]: Take Bn iid with pdf p(β), and conditional on the value of Bn , take waiting times iid with ℙ[Wn > t|Bn = β] = Ct −β for some C > 0. Classical solutions to more general distributed order Cauchy p(β) β problems, defined by (14) with 𝜕t replaced by 𝜕t , were developed in [36].

9 Summary Time fractional equations like (1) model time-changed stochastic processes B(Et ) with Et the inverse of a subordinator Dt . The waiting times between particle jumps follow a power law distribution, and the power law index equals the order of the time fractional derivative. The inverse stable time change produces many interesting stochastic models, including delayed Brownian motion and the fractional Poisson process. The range of the β-stable subordinator is a random fractal of dimension β, the same number as the one that describes the power law jumps and the order of the fractional derivative. The inverse stable time change also changes the fractal dimension of the graph of B(Et ). A continuous time random walk model with power law waiting times gives a physical interpretation to the time fractional derivative. Higher order governing equations for B(Et ) exist, and in some cases, do not involve any fractional derivatives. More general inverse subordinators lead to tempered fractional diffusion, a tempered fractional Poisson process, and ultraslow diffusion, where particles spread at a logarithmic rate.

424 | M. M. Meerschaert et al.

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Mark M. Malamud

Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations Abstract: This survey is concerned with the spectral theory of Volterra operators An = ⨁nj bj J αj , αj > 0, which are direct sums of multiples of fractional order Riemann– Liouville operators J αj . We discuss the lattices of invariant and hyperinvariant subspaces of operators An , as well as their commutants, double commutants, and other operator algebras related to An . We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators J α . The Gohberg–Krein conjecture on equivalence of unicellularity and cyclicity properties of a dissipative Volterra operator is also discussed. The problem of the similarity of the Volterra integral operators to the operators J α is discussed too. Keywords: Volterra operator, invariant subspaces, hyperinvariant subspaces, Gohberg–Krein conjecture, similarity problem for Volterra operators MSC 2010: 47G10, 34A08

1 Introduction The first main object of this review is the classical Riemann–Liouville fractional integration operator f → Jαf =

x

1 ∫(x − t)α−1 f (t) dt, Γ(α)

α = n − ε > 0, n ∈ ℕ, ε ∈ [0, 1),

(1)

0

on L2 [0, l], as well as direct sums of their multiples An := ⨁nk=1 (bk J αk ) with bk ∈ ℂ \ {0}. The important property of each summand bJ α , b ∈ ℂ \ {0}, is the following description of its lattices of invariant subspaces: Lat(bJ α ) = {χ[a,1] L2 [0, 1] : 0 ≤ a ≤ 1} meaning their unicellularity. At the same time the direct sums ⨁Nk=1 J α , N ≤ ∞, (finite or infinite) form models for certain classes of Volterra operators. For instance, each operator of the class Λexp (see Definition 7) is unitarily equivalent to a restriction of ⨁Nk=1 J to an appropriate invariant subspace. Zolotarev [79] has generalized this result to the case Mark M. Malamud, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation, e-mail: [email protected] https://doi.org/10.1515/9783110571622-018

428 | M. M. Malamud of direct sums ⨁Nk=1 (J α ) with α < 1. The corresponding class of Volterra operators is characterized by the growth of their Fredholm resolvents (see Theorem 11). In this paper we discuss the spectral theory of operators An including their unicellularity, cyclicity, a description of the lattices of invariant and hyperinvariant subspaces Lat An and Hyplat An , the set of cyclic vectors. We also discuss commutant {An }󸀠 , double commutant {An }󸀠󸀠 , and the algebra a(An ), the weak closure of {p(An ) : p is a polynomial}. An interesting feature is that all three algebras coincide for cJ α and are independent on α and c, while they do not coincide for An with n > 1. Moreover, we show that the lattices Lat An , Hyplat An , and all mentioned algebras except a(An ) split whenever either all exponents αj are pairwise distinct or α1 = ⋅ ⋅ ⋅ = αn , while the arguments of bk are pairwise distinct, i. e. arg bj ≠ arg bk (mod 2π) for 1 ≤ j ≠ k ≤ n.

(2)

An important step towards solution to this problem is the problem of description of the set of intertwining operators for a pair {J α1 , bJ α2 }. In particular, it is shown that this set is nonempty if and only if α1 = α2 and b ∈ ℝ+ \ {1}. This problem has recently attracted certain attention as part of the following general problem: Given a bounded operator T on a Banach space X. Describe the set of all complex numbers c ∈ ℂ \ {1} such that there exists an intertwining operator S(≠ 0) for a pair {T, cT}, i. e. S satisfying TS = cTS. This problem has recently been investigated in a series of papers (see, e. g., [6, 5, 7, 63, 38, 39, 41], and the references therein), where such constants c and operators S are called the extended eigenvalues and the corresponding eigenvectors of T. Unicellular Volterra operators on a separable Hilbert space H play a crucial role in the general theory of Volterra operators and its applications. It is well known that each unicellular Volterra operator A on H is cyclic. Brodskii and Kiselevskii [8, 9, 43] proved that the converse is also true whenever imaginary part AI of A is a trace class operator. Gohberg and Krein have conjectured [24, Complement, Section 4] that the assumption on AI is superfluous, i. e. that for a simple dissipative Volterra operator the cyclicity of A is equivalent to its unicellularity. In Section 4 we present counterexamples using operators J α1 ⊕ bJ α2 . Moreover, we present an affirmative answer to the Gohberg–Krein problem, assuming that A is sectorial with the opening πα and its Fredholm resolvent A(z) := (I − zA)−1 is an entire operator valued function of the order ρ = α−1 and normal type. One more topic discussed in the paper is the problem of similarity for (weak) perturbations of the fractional integration J α , α ∈ (0, ∞), on L2 [0, l], by Volterra integral operators of the form α

K = J (I + K1 ),

x

where K1 : f → ∫ k1 (x, t)f (t) dt. 0

Clearly, if K of the form (3) is similar to J α , it is unicellular and Lat K = Lat J α .

(3)

Spectral theory of fractional order integration operators | 429

Denoting by k(x, t) the kernel of the perturbed operator K, one easily finds that k1 (x, t) = Dαx k(x, t) where Dαx = Dnx J ε is a fractional derivative of order α. In turn, with Volterra operators of the form (3) with sufficiently smooth kernels k1 (x, t) is closely related the third main object of the paper, special fractional order integro-differential operators of order α = n − ε: n−1

x

j=1

0

ln−ε (D)f := Dn−ε f + ∑ qj (x)Dn−ε−j f + ∫ N(x, t)(J ε f )(t)dt.

(4)

These operators defined on the domain D(ln−ε (D)) = W1n−ε [0, l] can be treated as perturbations of the operator Dn−ε defined on the same domain. 0 n−ε Note that operator ln−ε (D) restricted to the domain D(ln−ε (D)) = W1,0 [0, l] is the −1 0 inverse to the Volterra operator K of the form (3), i. e. K = ln−ε (D). So, we discuss the similarity problems for two classes of operators (3) and (4) simultaneously. 0 Note also that the similarity between the operators ln−ε (D) and Dn−ε := (J n−ε )−1 0 is used for proving existence of triangular transformation operators for the equation ln−ε (D)y(x, λ) = λy(x, λ) (see [51]). In turn, transformation operators are applied in [51] to prove the unique recovery of operator (4) (with N(x, t) = 0) from n spectra of boundary value problems (a generalization of the classical Borg–Marchenko theorem on the unique determination of the Sturm–Liouville operator −d2 /dx 2 + q from two spectra). We finalize this introduction by considering one more uniqueness problem for a canonical system: J

dy = λH(t)y(t), dt

J = −J ∗ = −J −1 ,

y = col(y1 , . . . , yn ),

(5)

on a finite interval [0, l] with n × n Hamiltonian H(⋅) ≥ 0. Denote by W(x, λ) the fundamental n×n matrix solution of equation (5) satisfying the initial condition W(0, λ) = In . The matrix function W(λ) := W(l, λ) is called the monodromy matrix W(λ). A lot of papers are devoted to the problem of unique recovery of the Hamiltonian by monodromy matrix W(λ). In the definite case (J = iIn ) the most complete result was obtained by Kisilevskii [43] (see also [24]). A complete solution to this problem in the indefinite case was obtained by de Brange [12] for n = 2 and iJ = diag(1, −1) for real normed (tr H(t) ≡ 1) Hamiltonian. For n > 2 and J ≠ iIn some partial uniqueness results are also known (see, e. g., [3, 55] and the references therein). The proofs of the results mentioned above are based on a connection between the uniqueness problem and the problem on the unicellularity of a simple part of the model Volterra operator KH : f → H

1/2

t

(t)J ∫ H1/2 (s)f (s) ds 0

430 | M. M. Malamud in L2 ([0, l]; ℂn ) discovered by Brodskii and Livshits (see [24, Complement, Theorem 4.2]). Notations. Through the paper X1 , X2 , and X denote Banach spaces, B(X1 , X2 ) denotes the set of bounded linear operators from X1 to X2 ; B(X) = B(X, X); Lp ([0, 1]; ℂn ) = Lp [0, 1] ⊗ ℂn .

2 Spectral theory of fractional powers of the integration operator 2.1 Invariant subspaces and cyclicity The operator J α of fractional integration (the Riemann–Liouville operator) is given by x

1 (J f )(x) = ∫(x − t)α−1 f (t) dt, Γ(α) α

α > 0.

(6)

0

It is well defined on Lp [0, l] for each p ∈ [1, ∞] and l > 0. Moreover, it is a Volterra operator, i. e. it is a compact operator with zero spectrum, σ(J α ) = {0}. The family J α forms a holomorphic semigroup, J α J β = J α+β . Note in this connection that if A is a bounded accretive operator on a Hilbert space H, i. e. ℜ(Af , f ) ≥ 0 for every f ∈ H, then its fractional powers Aα , α ∈ (0, 1), are defined by Aα =

∞

∞

0

0

sin απ sin απ ∫ λα−1 (A + λI)−1 A dλ = − ∫ λα ((A + λI)−1 − λ−1 I)dλ. π π

(7)

Note also that the real part JR := 2−1 (J + J ∗ ) of the integration operator J on L2 [0, 1] is 1 JR f = 21 ∫0 f (t) dt. Clearly, it is one dimensional and nonnegative, 2

󵄨1 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 ℜ(Jf , f ) = (JR f , f ) = 󵄨󵄨∫ f (t) dt 󵄨󵄨󵄨 ≥ 0, 󵄨 󵄨󵄨 2 󵄨󵄨 󵄨 0

i. e. J is accretive. Simple calculations show that definition (6) of J α (for α ∈ (0, 1)) is consistent with general definition (7), i. e. substituting A = J in (7) one arrives at (6) with α ∈ (0, 1). Moreover, it follows from (7) that the operator J α is sectorial with the semi-angle πα/2, i. e. 󵄨 󵄨󵄨 −1 α 󵄨󵄨arg(J f , f )󵄨󵄨󵄨 ≤ 2 πα,

f ∈ L2 (0, 1), α ∈ (0, 1].

(8)

Spectral theory of fractional order integration operators | 431

Definition 1. A bounded operator T on a Banach space X is called unicellular if its lattice of invariant subspaces Lat T is linearly ordered, i. e. if E1 , E2 ∈ Lat T, then either E1 ⊂ E2 or E2 ⊂ E1 . This following classical statement is well known (see, e. g., [8, 21, 24, 59]). Proposition 1. Each operator cJ α , c ∈ ℂ \ {0}, is unicellular and its lattice of invariant subspaces Lat(cJ α ) in Lp [0, 1] does not depend on α and is given by Lat(cJ α ) = {Eap := χ[a,1] Lp [0, 1] : 0 ≤ a ≤ 1}.

(9)

Clearly, Lat(cJ α ) is anti-isomorphic to the segment [0, 1]. It is well known that the space Lp [0, 1], p ∈ [1, ∞], with the convolution product x

x

(f ∗ g)(x) = ∫ f (x − t)g(t)dt = ∫ g(x − t)f (t)dt = (g ∗ f )(x) 0

(10)

0

forms a commutative Banach algebra without unity. A description of zero divisors in Lp [0, 1] is given by the Titchmarsh convolution theorem [8, 21, 24, 59]: (f ∗ g)(x) = 0,

x ∈ [0, 1] 󳨐⇒ supp f ⊂ [α, 1], supp g ⊂ [β, 1],

α + β ≥ 1.

(11)

In particular, f is not a zero divisor in Lp [0, 1] if and only if ε

∫ |f (x)|p dx > 0

for each ε > 0.

(12)

0

This condition is called ε-condition. The first proof of Proposition 1 was relied on the Titchmarsh convolution theorem (see [8, 24, 59]). Later, Kalisch [34] proved that, in fact, the two results are equivalent (see also [36] for further discussion of this topic). Definition 2. A subspace E of a Banach space X is called a cyclic subspace for an operator T ∈ B(X) if span{T n E : n ∈ ℕ0 } = X. A vector f (∈ X) is called cyclic if span{T n f : n ∈ ℕ0 } = X. The operator T is cyclic if it has a cyclic vector. The set of all cyclic vectors of an operator T is denoted by Cyc(T). Definition 3. Let T ∈ B(X). We set μT := inf{dim E : E is a cyclic subspace of the operator T}. μT is called the spectral multiplicity of an operator T ∈ B(X). Note that μT can be infinite. It is clear that the operator T is cyclic if and only if μT = 1.

432 | M. M. Malamud Proposition 2. The operator J α is cyclic. Moreover, the following equivalence holds: f ∈ Cyc(J α )

f satisfies ε-condition (12).

⇐⇒

In fact, this result is equivalent to the unicellularity of the operator J α (see [8, 24, 59]). Note in this connection that each unicellular (not necessarily Volterra) operator is always cyclic (see [24], [73, Corollary 3.7.7]). The converse is not true even for simple Volterra operators (see Section 4). Proposition 3. Let A be a cyclic accretive operator on H. Then for any α ∈ (0, 1) the operator Aα is also cyclic. The proof is easily deduced from the following formula expressing the resolvent R(Aα ; z) of the operator Aα by means of the resolvent R(A; z) (see [45]): sin πα sα (A + sI)−1 R(A ; z) = ds ∫ 2α π s − 2sα z cos(πα) + z 2 ∞

α

(13)

0

Next we introduce the algebra of operators that leave invariant the lattice Lat T: Alg Lat T := {S ∈ B(X) : SE ⊆ E for every E ∈ Lat T}.

(14)

Proposition 4. The algebra Alg Lat J α in H = L2 [0, l] consists of all bounded triangular operators: x

d Alg Lat J = {K ∈ B(H) : Kf = ∫ k(x, t)f (t) dt, k(x, t) ∈ L2 (Ω)}. dx α

(15)

0

By definition, the commutant {T}󸀠 of an operator T ∈ B(X) consists of all bounded operators that commute with T. p , and let α > 0. Then the commutant {J α }󸀠 of the Proposition 5. Let p ∈ [1, ∞), p󸀠 = p−1 operator J α in Lp [0, 1] consists of all bounded operators of the form x

d Kf = ∫ k(x − t)f (t)dt, dx

k(x) ∈ Lp󸀠 [0, 1].

(16)

0

In particular, the commutant {J α }󸀠 is commutative and does not depend on α. Definition 4. An invariant subspace E of an operator T ∈ B(X) is called hyperinvariant for T if E is invariant for any bounded operator that commutes with T, that is, it is invariant for all the operators from the commutant {T}󸀠 . The lattice of all the hyperinvariant subspaces of an operator A will be denoted by Hyplat A.

Spectral theory of fractional order integration operators | 433

Combining this result with Proposition 1 and noting that Hyplat T ⊆ Lat T for any bounded T, yields the following description of Hyplat(cJ α ). Proposition 6. The lattice of hyperinvariant subspaces Hyplat(cJ α ) of the operator cJ α in Lp [0, 1] does not depend on α and is given by Hyplat(cJ α ) = Lat(cJ α ) = {Eap : 0 ≤ a ≤ 1}.

(17)

Remark 1. (i) In the case p = 2 another description of the commutant {J}󸀠 in L2 [0, 1] was obtained by Sarason [70]. (ii) Dixmier [13] has found that in the space L1 [0, 1] the commutant {J}󸀠 consists precisely of the convolution operators induced by measures on [0, 1). This fact can also be deduced from the representation (16). One can consider an operator K of the above form as a convolution operator: f → f ∗ k 󸀠 with k 󸀠 being a distribution. However, not all the operators in the commutant {J}󸀠 of J on Lp [0, 1] for p > 1 are generated by measures on [0, 1). For instance, k(x) = xi generates a bounded operator of the form (16), although xi is not a function of a bounded variation on [0, 1].

2.2 Intertwining operators for a pair {J α1 , cJ α2 } and extended eigenvalues of J α First we introduce the classical Mittag-Leffler function (see [14, Chapter 3]) zk , Γ(μ + kα) k=0 ∞

E1/α (z; μ) = ∑

μ > 0,

(18)

which plays an important role in the theory of Riemann–Liouville operators J α . It is known [14] that E1/α (z; μ) is an entire function in z of order 1/α and type 1 not depending on μ. Proposition 7 ([14, Chap. 3]). The Fredholm resolvent of the operator cJ α on Lp [0, l] is given by α

α −1

x

R(z; cJ )f := (I − zcJ ) f (x) = f (x) + zc ∫(x − t)α−1 E1/α (zc(x − t)α ; α)f (t)dt.

(19)

0

Moreover, R(z; cJ α ) is an entire operator-valued function in z of order 1/α and type σ = |c|1/α l. Recall that a bounded operator T ∈ B(X1 , X2 ) is called a quasiaffinity if it is injective and has dense range.

434 | M. M. Malamud Definition 5. (i) Let Xj , be a Banach space and let Aj ∈ B(Xj ), j ∈ {1, 2}. It is said that an operator T ∈ B(X2 , X1 ) intertwines the ordered pair of operators {A1 , A2 } if A1 T = TA2 . (ii) It is said that the operator A1 is a quasiaffine transform of A2 if there exists a quasiaffinity S ∈ B(X1 , X2 ) intertwining the pair {A2 , A1 }, i. e. SA1 = A2 S. (iii) It is said the operators A1 and A2 are quasisimilar if there exist quasiaffinities T and S intertwining the pairs {A1 , A2 } and {A2 , A1 }, respectively. Theorem 1 ([51, 54]). Assume that either α1 ≠ α2 or arg c1 ≠ arg c2 . Then the only bounded operator intertwining the operators c1 J α1 and c2 J α2 in Lp [0, 1] for some p ∈ [1, ∞] is the zero operator. Sketch of the proof. (i) First we assume that α1 ≠ α2 . Let T(≠ 0) be a bounded operator intertwining the pair {c1 J α1 , c2 J α2 }, i. e. J α1 T = cTJ α2

where c = c2 c1−1 .

(20)

Assume for simplicity that p = 2. Then this identity is equivalent to ((I − zJ α1 ) Tf , g) = ((I − zcJ α2 ) f , T ∗ g), −1

f , g ∈ L2 [0, 1], z ∈ ℂ.

−1

(21)

We set f1 = Tf , g1 = T ∗ g, and 1

1

h1 (t) := ∫ f1 (x − t)g(x) dx,

h2 (t) := ∫ f (x − t)g1 (x) dx.

t

(22)

t

Using formulas (19) and (18) for the resolvent (I − λcJ α )−1 and changing the order of integrals one rewrites identity (21) as 1

1

ck z k ∫ t kα2 −1 h2 (t)dt =: φ2 (z). Γ(kα ) 2 k=1 ∞

φ1 (z) := z ∫ t α1 −1 E1/α1 (zt α1 ; α1 )h1 (t) dt = ∑ 0

(23)

0

Assuming for definiteness that α1 > α2 we show that for certain f and g the growth of the entire function φ2 (⋅) is 1/α2 (> 1/α1 ). It follows from (20) that ker T ∈ LatJ α2 = LatJ α1 , hence, by Proposition 1, ker T = Ea2 for some a < 1. Since ker T = Eap , one gets ran T ∗ = L2 [0, 1] ⊖ ker T = L2 [0, 1] ⊖ Ea2 = Ẽa2 = χ[0,a] L2 [0, 1]. Choose f ≥ 1 and g0 ∈ C[0, a] satisfying g0 (x) ≥ 1 for x ∈ [0, a]. If g0 ∈ C[0, a] ∩ ran T ∗ , we set g := g0 . Otherwise one finds a sequence gn = T ∗ fn such that limn→∞ gn = g0 in L2 [0, a]. Hence a

a

rn (t); = ∫ f (s − t)gn (s)ds → ∫ f (s − t)g0 (s)ds =: r0 (t) ≥ a − t. t

t

(24)

Spectral theory of fractional order integration operators | 435

One easily checks that hn (⋅) ∈ C[0, a] and the convergence in (24) is in C[0, a]. It follows that there exists N ∈ ℕ such that rn (t) ≥ a/3

for t ∈ [0, a/2], n ≥ N.

(25)

a

Therefore we can set g := gN = T ∗ fN , and h2 (t) = rN (t)(= ∫t f (s − t)g(s)ds). Inequality (25) implies h2 (t) ≥ a/3 for t ∈ [0, a/2], and hence 1

a

ak = ∫ t α2 (k+1)−1 h2 (t) dt = ∫ t α2 k+α2 −1 h2 (t)dt ≥ 0

0

α2 k+α2 +1

a ≥( ) 3

a/2

a ∫ t α2 k+α2 −1 dt 3

(26)

0

⋅

1 , α2 k + α2

k ∈ ℕ.

(27)

In accordance with the Stirling formula, Γ(αk + α) = √2π(kα)kα+α−1/2 exp(−kα)[1 + o(1)] as k → ∞. To compute the order of growth ρ(φ2 ) of the entire function φ2 in the right hand side of (23) we apply the formula expressing ρ(φ2 ) in terms of its Tailor coefficients

ck =

ck ak , Γ(kα2 )

k ∈ ℕ. We derive

log |ck |−1 log(Γ(kα2 + α2 )|c|−k |ak |−1 ) 1 = lim = lim k→∞ ρ(φ2 ) k→∞ k log k k log k (kα2 + α2 − 1/2) log(kα2 ) − kα2 − k log |c| − log |ak | = lim k→∞ k log k log |ak | = α2 − lim k→∞ k log k (kα2 + α2 + 1) log(2−1 a) − log(kα2 + α2 ) = α2 . k→∞ k log k

≤ α2 − lim

(28)

So, ρ(φ2 ) ≥ 1/α2 . Since the order of the Mittag-Leffler function E1/α2 (cz; α2 ) is 1/α2 , the opposite inequality is obvious. Thus, ρ(φ2 ) = 1/α2 . On the other hand, since the order of φ1 (⋅) does not exceed ρ1 = 1/α1 , the order of E1/α1 (cz; α1 ), equality (23) leads to a contradiction. Thus, T = 0. (ii) The case α1 = α2 but arg c1 ≠ arg c2 is more complicated. Now using the technique of indicator diagrams [46] and the asymptotic formulas for Mittag-Leffler functions (see [14, Chapter 3.2]) it is shown that for certain f and g the functions φ1 and φ2 have different growths on certain different directions (see [54]). To describe the set of intertwining operators for the pair {J α , cJ α } with c > 0 we need some notations. Given any a ∈ (0, 1), an isometric operator Lp [0, 1] → Lp [0, a−1 ] is defined by Ua : Lp [0, 1] ∋ f (x) → ̃f (x) = a1/p f (ax) ∈ Lp [0, a−1 ].

(29)

436 | M. M. Malamud As usual for a < 1 we will identify Lp [a, l] and Lp [0, a] with the subspaces Eap and Ẽap of Lp [0, l]: Eap = {f (x) ∈ Lp [0, l] : f (x) = 0, x ∈ [0, a]} = χ[a,l] Lp [0, l], Ẽ p = {f (x) ∈ Lp [0, l] : f (x) = 0, x ∈ [a, l]} = χ[0,a] Lp [0, l], a

(30)

respectively. Further, let Pa be the projection Pa : Lp [0, a−1 ] → Ẽ1p ⋍ Lp [0, 1]

(31)

of Lp [0, a−1 ] onto Ẽ1p in the decomposition Lp [0, a−1 ] = E1p ⊕ Ẽ1p . According to the decomposition Lp [0, 1] = Eap ⊕ Ẽap , a < 1, we introduce the embedding operator ia : Lp [0, a] → Lp [0, 1] given by 0, x ∈ [0, 1 − a], ia : f (x) → g(x) = { f (x − 1 + a), x ∈ [1 − a, 1].

(32)

Theorem 2 ([54]). Let α > 0 and p ∈ [1, ∞). Then the operator equation J α T = cTJ α ,

c ∈ ℂ,

(33)

in Lp [0, 1] has a nontrivial bounded solution T ∈ B(Lp [0, 1]) if and only if c is positive, c ∈ ℝ+ . Moreover, if a = c−1 ∈ ℝ+ , then T is of the form KPa Ua T = { −1 ia Ua K

if a < 1, if a > 1,

(34)

where K ∈ {J α }󸀠 = {J}󸀠 . Corollary 1 ([51]). For any c ∈ ℂ \ {1} the operator cJ α is not a quasiaffine transform of J α in Lp [0, 1], p ∈ [1, ∞). In particular, the operators cJ α and J α are not quasisimilar in each Lp [0, 1]. Proof. It suffices to consider the case c ∈ ℝ+ because, by Theorem 1, there are no nontrivial bounded intertwining operators for the pair {J α , cJ α } provided that c ∈ ℂ \ ℝ+ . If a = c−1 < 1, then ker(Pa Ua ) = χ[a,1] Lp [0, 1] = Eap ≠ {0}. If a > 1, then p . Thus, the operator T is neither injective no has dense ran(ia Ua−1 K) ⊂ ran ia = E1−a range. Note that a weaker statement, namely the absense of similarity (instead of quasimilarity) between operators cJ α and J α was obtained by G. Kalisch [33]. This result has no finite-dimensional analog. Indeed, the multiple cJn (0), c ∈ ℂ \ {0}, of the Jordan cell is unicellular too and is similar to Jn (0). In connection with Theorems 1 and 2 we introduce the following definition.

Spectral theory of fractional order integration operators | 437

Definition 6 ([6]). A complex number λ is called an extended eigenvalue of an operator A if there is a nonzero bounded operator T such that AT = λTA.

(35)

Each bounded solution T of equation (35) is referred to as an extended eigenvector of A corresponding to the extended eigenvalue λ. The set of extended eigenvalues of A is called the extended point spectrum of A and is denoted 𝔼(A). This notion was recently introduced by Biswas, Lambert, and Petrovic [6]. In [6] and subsequent publications [5, 63] they also characterized the set of extended eigenvalues, for operators acting on finite-dimensional spaces, finite rank operators, Jordan blocks, and C0 -contractions. With Definition 6 Theorem 2 presents a description of the extended point spectrum and the corresponding eigenvalues of the operator J α . In particular, it implies the following result. Corollary 2. For any α > 0 the set of extended eigenvalues of J α in Lp [0, 1] is 𝔼(J α ) = (0, ∞). For α = 1 and integer α ∈ ℕ this result has been rediscovered in [6, 38, 39], and [5], respectively, with other proofs. A further description of extended eigenvalues and extended eigenvectors of different classes of operators in different classes of functional Banach spaces can be found in the works of M. Karaev [39], G. Gasier and H. Alkanjo [11], Gurdal, and others (see [41, 29, 30] and the references therein). In [41] the Banach algebra technique is applied to the calculation of spectral multiplicities and extended eigenvectors of certain linear operators. Note also that Shkarin [71] has shown that there exist Volterra operators with no extended eigenvalues except λ = 1.

n

3 Spectral theory of direct sums ⨁j=1 (bj J αj ) 3.1 Lattices of invariant and hyperinvariant subspaces 3.1.1 The case of distinct αj Let a(T) (au (T)) denote the weakly (uniform) closed subalgebra of B(X) generated by T and the identity. So, a(T) (au (T)) is the weak (uniform) closure of {p(T) : p is a polynomial}. Let {T}󸀠󸀠 denote the double commutant of T, i. e. {T}󸀠󸀠 = {{T}󸀠 }󸀠 . Clearly, a(T) ⊆ {T}󸀠󸀠 ⊆ {T}󸀠 .

(36)

438 | M. M. Malamud First we show that for the operator A = ⨁nj=1 bj J αj with pairwise distinct exponents αj the lattices Lat(A) and Hyplat(A) and algebras Alg Lat A, {A}󸀠 , and {A}󸀠󸀠 always split. Theorem 3 ([52]). Let A = ⨁nj=1 bj J αj where 0 < α1 < α2 < ⋅ ⋅ ⋅ < αn , bj ∈ ℂ, and let Eap = χ[a,1] Lp [0, 1], a ∈ [0, 1], and p ∈ [1, ∞). Then: (i) The lattice of invariant subspaces Lat A of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

j=1

1

j=1

Lat(A) = ⨁ Lat(bj J αj ) = ⨁ Lat(J) = {⨁ Eapj : 0 ≤ aj ≤ 1}. (ii) The algebra Alg Lat A splits: n

n

j=1

j=1

Alg Lat A = ⨁ Alg Lat(bj J αj ) = ⨁ Alg Lat(J). (iii) The lattice Hyplat of hyperinvariant subspaces of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

j=1

1

j=1

Hyplat(A) = ⨁ Hyplat(bj J αj ) = ⨁ Hyplat(J αj ) = {⨁ Eapj : 0 ≤ aj ≤ 1}. (iv) The commutant {A}󸀠 of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

n

j=1

j=1

{A}󸀠 = ⨁{bj J αj } = ⨁{J αj } = ⨁{J}󸀠 = ⨁ a(J). 󸀠

j=1

󸀠

j=1

In particular, the commutant {A}󸀠 is commutative and does not depend on the vector {αj }n1 . (v) The double commutant {A}󸀠󸀠 of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

{A}󸀠󸀠 = ⨁{bj J αj } = ⨁{J αj } = ⨁{J}󸀠󸀠 . j=1

󸀠󸀠

󸀠󸀠

j=1

j=1

(vi) The set Cyc A of A in Lp [0, 1] ⊗ ℂn splits, i. e. the following equivalence holds: f = ⊕nj=1 fj ∈ Cyc A ⇐⇒ fj ∈ Cyc J αj

for each j ∈ {1, . . . , n}.

In particular, the multiplicity of spectrum μA = 1. Note that relations (ii) and (iii) are equivalent for arbitrary direct sums. Relation (iv) is immediate from Theorem 1, while (iii), (iv), and (v) are equivalent for arbitrary operators.

Spectral theory of fractional order integration operators | 439

3.1.2 The case of equal αj Next we investigate the lattices Lat A and Hyplat A of A in the case of α1 = α2 = ⋅ ⋅ ⋅ = αn , i. e. of the Volterra operator A = J α ⊗B on Lp [0, 1]⊗ℂn . We indicate an explicit condition on the eigenvalues of B ensuring the splitting of these lattices. Theorem 4 ([54]). Let α ∈ (0, ∞), let B = diag(b1 , . . . , bn ) be a non-singular diagonal matrix, and let A = J α ⊗B. Then for any p ∈ [1, ∞) the following conditions are equivalent: (i) The arguments of bj are pairwise distinct, i. e. arg bj ≠ arg bk

for 1 ≤ j ≠ k ≤ n.

(37)

(ii) The lattice of invariant subspaces Lat A of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

j=1

1

j=1

Lat(J α ⊗ B) = ⨁ Lat(bj J α ) = ⨁ Lat(J α ) = {⨁ Eapj : 0 ≤ aj ≤ 1}. (iii) The algebra Alg Lat A splits: n

n

j=1

j=1

Alg Lat A = ⨁ Alg Lat(bj J α ) = ⨁ Alg Lat(J). (iv) The lattice Hyplat of hyperinvariant subspaces of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

j=1

1

j=1

Hyplat(J α ⊗ B) = ⨁ Hyplat(bj J α ) = ⨁ Hyplat(J α ) = {⨁ Eapj : 0 ≤ aj ≤ 1}. (v) The commutant {A}󸀠 of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

{J α ⊗ B} = ⨁{bj J α } = ⨁{J α } = ⨁ a(J). 󸀠

󸀠

󸀠

j=1

j=1

j=1

In particular, the commutant {J α ⊗ B}󸀠 is commutative. (vi) The double commutant {A}󸀠󸀠 of A in Lp [0, 1] ⊗ ℂn splits, n

n

n

{A}󸀠󸀠 = ⨁{bj J α } = ⨁{J α } = ⨁{J}󸀠󸀠 . j=1

󸀠󸀠

j=1

󸀠󸀠

j=1

(vii) The set Cyc A of A in Lp [0, 1] ⊗ ℂn splits, i. e. the following equivalence holds: f = ⊕nj=1 fj ∈ Cyc A ⇐⇒ fj ∈ Cyc J α

for each j ∈ {1, . . . , n}.

In particular, the multiplicity of spectrum μA = 1, i. e. the operator A is cyclic. Remark 2. For α = 1 the implication (i) 󳨐⇒ (ii) was first proved in [61, 62] by a method different from that of [54].

440 | M. M. Malamud Remark 3. The algebra a(A) does not split, i. e. n

n

j=1

j=1

a(A) = a(⨁ bj J α ) ⫌ ⨁ a(bj J α ). This relation implies a similar relation for au (A). Remark 4. (i) Note that the equivalences (ii) ⇐⇒ (iii) and (iv) ⇐⇒ (v) ⇐⇒ (vi) are of general nature and hold for any pair of bounded operators {T1 , T2 } (see [10]). Moreover, if the lattice Lat(T1 ⊕ T2 ) splits for a pair {T1 , T2 } of bounded operators then the lattice Hyplat(T1 ⊕ T2 ) also splits, i. e. the implication Lat(T1 ⊕ T2 ) = Lat T1 ⊕ Lat T2 󳨐⇒ Hyplat(T1 ⊕ T2 ) = Hyplat T1 ⊕ Hyplat T2

(38)

is general (see [10]). Therefore the equivalence (i) ⇐⇒ (ii) in Theorem 4 yields all other equivalences. Note also that implication (i) 󳨐⇒ (v) is immediate from (in fact is equivalent to) the second part of Theorem 1. (ii) Note also that, for an operator A in a finite-dimensional space H = ℂn , the splitting property of each of the lattices LatA and Hyplat A is equivalent to the coincidence of the minimal and characteristic polynomials of A. Moreover, in this case the two algebras au (A) and a(A) also split. Combining Theorem 4(v) with Proposition 5, one arrives at the following result. Proposition 8. Let X = Lp [0, 1] ⊗ ℂn , p ∈ [1, ∞), let B = diag(b1 , . . . , bn ) be a diagonal non-singular matrix satisfying condition (37), and let A = J α ⊗ B(∈ B(X)) with α > 0. Then the commutant {A}󸀠 of A splits and is given by {A}󸀠 = {K : K = diag(K1 . . . , Kn ),

Kj ∈ B(Lp [0, 1])},

(39)

where Kj are bounded operators of the form x

d Kj f = ∫ kj (x − t)f (t)dt, dx

kj (x) ∈ Lp [0, 1]. 󸀠

(40)

0

In particular, the commutant {A}󸀠 is commutative. If condition (37) is violated, the description of the commutant {A}󸀠 and the lattice Hyplat A become much more complicated. We consider only the case of eigenvalues bj with equal arguments. Theorem 5 ([54]). Let B = diag(b1 , . . . , bn ) be a diagonal non-singular matrix with eigenvalues of equal arguments: bj = b1 sαj ,

1 = s1 ≤ s2 ≤ ⋅ ⋅ ⋅ ≤ sn .

Spectral theory of fractional order integration operators | 441

Then the lattice Hyplat A of the operator A = J α ⊗ B(∈ B(X)) has the form n

Hyplat A = {⨁ Eapj : (a1 , . . . , an ) ∈ P(s1 , . . . , sn )}, j=1

(41)

where P(s1 , . . . , sn ) := {(a1 , . . . , an ) ∈ ◻n = [0, 1]n :

sj aj+1 ≤ sj+1 aj ≤ sj+1 − sj + sj aj+1 , 1 ≤ j ≤ n − 1}.

(42)

In particular, the lattice Hyplat A is anti-isomorphic to the oblique parallelepiped P(s1 , . . . , sn ). Its dimension equals the number of different sj , 1 ≤ j ≤ n. Corollary 3. Let X = Lp [0, 1] ⊗ ℂn , p ∈ [1, ∞), and let A = J α ⊗ I ∈ B(X). Then Hyplat A is of the form n

Hyplat(J α ⊗ I) = {⨁ Eapj : a1 = a2 = ⋅ ⋅ ⋅ = an ∈ [0, 1]}. j=1

(43)

In particular, Hyplat A is anti-isomorphic to the segment [0, 1]. Combining Theorems 5 and 4 yields the following statement. Corollary 4. Let B = diag(b1 , . . . , bn ) be a diagonal non-singular matrix, and let A = J α ⊗ B be defined on Lp [0, 1] ⊗ ℂn Then Hyplat A = Lat A if and only if condition (37) holds. In this case both lattices split. Remark 5. However, in the case bj = sj > 0 and α = 1 the operator iA = iJ ⊗ B is dissipative with finite-dimensional imaginary part. In accordance with the Nagy–Foias result [73] (a generalization of the von Neumann result), a(A) = {A}󸀠󸀠 . Note also that a characterization of direct sums ⊕Nk=1 J of integration operator up to unitary equivalence have been obtained by G. K. Kalisch [35]. 3.1.3 Operator A on Sobolev spaces Here we briefly discuss the spectral properties of the operator J α ⊗ B defined on a Sobolev space. As usual Wpk [0, 1] (p ∈ [1, ∞), k ∈ ℕ), denotes the Sobolev space consisting of absolutely continuous functions f having k − 1 absolutely continuous derivatives and satisfying f (k) ∈ Lp [0, 1]. Let also k Wp,0 [0, 1] := {f ∈ Wpk [0, 1] : f (0) = f 󸀠 (0) = ⋅ ⋅ ⋅ = f (k−1) (0) = 0}.

In this subsection we consider the cases where the operator A = J α ⊗ B is defined on k kj [0, 1] and ⨁nj=1 Wp j [0, 1]. ⨁nj=1 Wp,0

442 | M. M. Malamud k

j [0, 1] is isometriIn accordance with [16, Lemma 3.1], the operator J α acting Wp,0 α p cally equivalent to the operator J acting on L [0, 1]. Therefore all results on A = J α ⊗ B defined on Lp [0, 1] ⊗ ℂn can be retranslated into the results on the operator A defined kj [0, 1]. on ⨁nj=1 Wp,0

Theorem 6. Let p ∈ [1, ∞) and let k1 , . . . , kn ∈ ℕ. Then the statements of Theorems 3, 4, and Proposition 8 remain true for the operator A = J α ⊗ B acting on the Sobolev space kj [0, 1]. ⨁nj=1 Wp,0 The following result is a counterpart of Theorem 4 for A = J α ⊗ B defined on k ⨁nj=1 Wp j [0, 1]. Theorem 7 ([16, Theorems 4.9, 4.12, 4.20]). Let p ∈ [1, ∞), k1 , . . . , kn ∈ ℕ, and let B = diag(b1 , . . . , bn ) be a non-singular diagonal matrix. Let also the operator A = J α ⊗ B k

be defined on ⨁nj=1 Wp j [0, 1]. Then conditions (i), (iv), (v), and (vi) of Theorem 4 are equivalent. In particular, under condition (37) the lattice of hyperinvariant subspaces Hyplat A of A splits. Remark 6. (i) In connection with Theorem 7 we mention that a complete description of the lattices Lat A of the operators A = J α and A = J α ⊗ B defined on Wpk [0, 1] and k

⨁nj=1 Wp j [0, 1], respectively, were obtained in [15, 16]. An interesting feature of this description is that the lattices Lat A contain both a continuous and a discrete part. In particular, the integration operator J is unicellular on Wpk [0, 1], while its powers

are not. The unicellularity of J on Wpk [0, 1] was first discovered by Tsekanovskii [76]. In their investigations the authors of [15, 16] have been influenced by this result. Romaschenko [64] obtained similar results for both Sobolev and Liouville spaces Wps [0, 1] and Lsp [0, 1] with non-integer exponent s. An interesting feature of the description of the lattices Lat J α in Sobolev spaces Wps [0, 1] depends on the exponent

of embedding of Wps [0, 1] into the space C k [0, 1]. We also mention recent publications [74, 75]. (ii) It is worth to mention that condition (37) for the operator A in Theorem 7 does not imply splitting—neither of the lattice Lat A nor the set Cyc A. Thus A gives a simple counterexample for the implication converse to (38). (iii) Finally we breafly discuss integration Volterra operator J on the Frechet space C ∞ [0, 1] of infinitely differentable functions. In 1966 Mikusinski [60] proved unicellularity of J defined on a subspace C0∞ [0, 1] := {f ∈ C ∞ [0, 1] : f (k) (0) = 0, k ∈ ℤ+ } of C ∞ [0, 1]. However, a problem of whether the operator J is unicellular in C ∞ [0, 1] remaind open. It was positively solved by M. Karaev [37] by emploping Duhamel product technique. This technique has recently been applied by him to investigate

Spectral theory of fractional order integration operators | 443

unicellularity of J in certain Banach spaces of analytic functions on the unit disc (see [40]).

3.2 Splitting of the lattice Lat(A+ ⊕ A− ) with dissipative operators ±A± Theorem 8. Let A+ and −A− be simple dissipative compact operators acting on Hilbert spaces H+ and H− , respectively. Then the lattice of invariant subspaces Lat(A+ ⊕ A− ) splits, that is, Lat(A+ ⊕ A− ) = Lat A+ ⊕ Lat A− .

(44)

Proof. Let E ∈ Lat(A+ ⊕ A− ) and let π+ and π− be the orthoprojections in H := H+ ⊕ H− onto H+ ⊕ {0} and {0} ⊕ H− , respectively. We put E ± := span{An± π± E : n ∈ ℤ+ }

(45)

and denote by P± the orthoprojection in H± onto E ± . Clearly, E ⊂ E + ⊕ E − and E ± ∈ Lat(A± ) and E + ⊕ E − ∈ Lat(A+ ⊕ A− ).

(46)

We show that E = E + ⊕E − . Assuming the contrary we find a nonzero vector g = {g1 , g2 } ∈ E + ⊕E − orthogonal to E. Hence for any h = {h1 , h2 } ∈ E and sufficiently large |z| we have ((A+ − z)−1 h1 , g1 ) + ((A− − z)−1 h2 , g2 ) = 0,

|z| > ‖A± ‖.

(47)

Since A+ is dissipative, ℂ− ⊂ ρ(A+ ). Therefore the first summand of (47) is holomorphic in ℂ− and 1 󵄨 󵄨󵄨 −1 , 󵄨󵄨((A+ − z) h1 , g1 )󵄨󵄨󵄨 ≤ |Im z|

z ∈ ℂ− .

(48)

Similarly, the second summand of (47) is holomorphic in ℂ+ and 1 󵄨 󵄨󵄨 −1 , 󵄨󵄨((A− − z) h2 , g2 )󵄨󵄨󵄨 ≤ Im z

z ∈ ℂ+ .

(49)

Hence ((A+ − z)−1 h1 , g1 ) admits a holomorphic continuation to ℂ+ satisfying 1 󵄨 󵄨󵄨 −1 , 󵄨󵄨((A+ − z) h1 , g1 )󵄨󵄨󵄨 ≤ Im z

z ∈ ℂ+ .

(50)

Since A+ is simple, it has no real eigenvalues and ℝ\{0} ⊂ ρ(A+ ). Thus ((A+ −z)−1 h1 , g1 ) is holomorphic in ℂ\{0} with possible singularity at zero. Due to (48) and (50) it admits a representation ((A+ − z)−1 h1 , g1 ) =

a+ + Φ+ (z), z

(51)

444 | M. M. Malamud where Φ+ (⋅) is an entire function. Due to the relation limz→∞ ‖(A+ − z)−1 ‖ = 0, representation (51) yields lim Φ+ (z) = 0.

(52)

z→∞

By the Liouville theorem the entire function Φ+ (⋅) is identically zero, and ((A+ − z)−1 h1 , g1 ) =

a+ . z

Hence ∗ a+ = (h1 , g1 ) and (Ak+ h1 , g1 ) = 0 = (Ak−1 + h1 , A+ g1 ),

k ∈ ℕ.

(53)

Since h1 ∈ (π1 E) is arbitrary, it follows from definition (45) and relations (53) that P+ A∗+ g1 is orthogonal to E + , that is, P+ A∗+ g1 = 0. Since the operator P+ A∗+ is adjoint to the dissipative operator A+ P+ = P+ A+ P+ , we have also A+ g1 = A+ P+ g1 = 0. Hence g1 = 0, since A+ is simple. Similarly we prove that g2 = 0. Thus, g = {g1 , g2 } = 0 and E = E + ⊕ E − , which proves (44). This result has been applied by G. M. Gubreev [26–28] to the investigation of lattices Lat A of unbounded non-dissipative operators A with two-dimensional imaginary parts.

4 Gohberg–Krein conjecture on unicellularity of cyclic dissipative Voltera operators As already mentioned, each unicellular Volterra operator A on a separable Hilbert space H is cyclic. In this section we discuss certain conditions ensuring the converse implication. To this end we briefly recall the main statements of Brodskii–Kiselevskii’s theory [8, 9, 43]. To this end we recall the following definition [8]. Definition 7 ([8]). (i) A dissipative operator A is referred to the class Λexp if it is quasinilpotent and its Fredholm resolvent (I − zA)−1 is an entire function of exponential type σA . (ii) A Volterra operator A is called simple if it does not admit a representation A = A1 ⊕ A2 with A2 = A∗2 = 0. A dissipative operator A is simple if and only if ker A = {0}. Theorem 9. Let A be a simple dissipative Volterra operator with trace class imaginary part AI := (A − A∗ )/2i. Then: (i) Its Fredholm resolvent is of exponential type 0 < σ = σA ≤ 2 tr AI , i. e. A ∈ Λexp .

Spectral theory of fractional order integration operators | 445

(ii) The operator A is unicellular if and only if σA = 2 tr AI . (iii) The operator A is cyclic if and only if it is unicellular. The converse to statement (i) is also well known: if A ∈ Λexp is unicellular, then AI ∈ S1 , hence σA = 2 tr AI [8, Theorem 30.5]. The statement (iii) of the theorem is due to Kisilevskii ([43], see also [8, Theorem 31.4]). Gohberg and Krein [24, p. 352] have conjectured that a simple cyclic dissipative Volterra operator A is always unicellular, i. e. the assumption AI ∈ S1 in Theorem 9(iii) is superfluous. First we apply Theorem 4(iii) to show that simple counterexamples to this conjecture can be found among appropriate Volterra operators J α ⊗B. These counterexamples (54) to the Gohberg–Krein conjecture were constructed by Atzmon [4]. Independently, they were rediscovered by the author in [52, 53]. Example 1. Choose α ∈ (0, 1) and consider in H = L2 [0, l] ⊗ ℂ2 the operator A = J α ⊕ eiπε/2 J α

with ε < 1 − α.

(54)

By Theorem 4(vii), the operator A is cyclic for any ε > 0. On the other hand, Theorem 4(ii) implies that it is not unicellular in L2 [0, 1] ⊗ ℂ2 . At the same time, in accordance with (8) |arg(Af , f )| ≤ π(α + ε)/2 < π/2,

f ∈ H,

i. e. the operator A is sectorial with a semi-angle π(α + ε)/2. In particular, A is an accretive operator. Thus, the dissipative operator iA presents a counterexample to the Gohberg–Krein’s conjecture. Note, however, that in accordance with Proposition 7 the resolvent (I − zA)−1 of A is an entire function of order ρ = 1/α > 1 and finite type σ = l. Despite this and similar counterexamples we present an affirmative solution to this conjecture imposing certain additional assumptions on an operator A. To this end we introduce a new definition and certain useful results. Definition 8. We say that an operator A ∈ B(H) belongs to a class Λρ,σ with σ ∈ (0, ∞) and ρ ∈ [1, ∞) if: (i) arg(Ah, h) ∈ (−2−1 πα, 2−1 πα) for all h ∈ H and α = ρ−1 . (ii) The Fredholm resolvent A(z) := (I − zA)−1 is an entire operator valued function of the order ρ and type not exceeding σ. We also put Λρ := ⋃σ>0 Λρ,σ . Clearly, the class iΛ1 coincides with the class Λexp . In connection with Definition 8 we present the classical Macaev theorem [23, Theorem V.5.2] describing a wide class of Volterra operators satisfying the condition (ii).

446 | M. M. Malamud Theorem 10. Let A be a Volterra operator in H and let MA (r) := max|z|≤r ‖(I − zA)−1 ‖. Assume that for some ρ > 0 the s-numbers sk (A) of A satisfy sk (A) = O(k −1/ρ )

(sk (A) = o(k −1/ρ ))

as k → ∞.

(55)

Then the resolvent (I − zA)−1 is of order ρ and of finite type σA > 0 (σA = 0), i. e. ln MA (r) = O(r ρ )

(ln MA (r) = o(r ρ ))

as r → ∞.

(56)

In particular, the second estimate in (56) holds provided that A ∈ Sp . Note that estimate (56) for operators A ∈ Sp was obtained for the first time by Keldysh. Note also that operators with similar growth of the resolvent are discussed in [72] in connection with existence of hyperinvariant subspaces. Combining Proposition 7 with relation (8) implies the inclusion J α ∈ Λρ,σ with ρ = α−1 ∈ [1, ∞) and σ = l. Clearly, the operator ∞

̂J α := J α ⊗ I 2 = ⨁ J α l k=1

∞

̂ 2 [0, l] := L2 [0, l] ⊗ l2 (ℕ) = ⨁ L2 [0, l] on L k=1

(57)

as well as its restriction to any invariant subspace also belong to the class Λρ,σ with ρ = α−1 and σ ≤ l. It is worth to note that the growth of the resolvent of any restriction of the operator ̂J α to an invariant subspace always equals ρ, while its type can be arbitrary σ ∈ (0, l). The following result shows the universal character of the operator ̂J α for the classes Λρ,σ . Theorem 11 ([78]). Let A be a completely non-selfadjoint operator in H that belongs to ̂ 2 [0, l] the class Λρ,σ . Then there exists an invariant subspace H1 of the operator ̂J α in L with α = 1/ρ ∈ (0, 1] and l = σ such that A is unitarily equivalent to the operator ̂J α ⌈H1 , the restriction of ̂J α to H1 . This result was obtained by Zolotarev [78] (see also [79]). In turn it generalizes the previous result by Brodskii–Kisilevskii [9] (see also [8, Theorem 8.4]) and coincides with it for α = 1. Now we are ready to state our main positive result towards the Gohberg and Krein conjecture. Theorem 12 ([52]). Let A be a Volterra operator in a Hilbert space H, ker A = {0}, and let the following conditions be fulfilled: (i) πα1 ≤ arg(Af , f ) ≤ πα2 , ∀f ∈ H \ {0} and α := α2 − α1 ≤ 1; (ii) the Fredholm resolvent (I −zA)−1 is and entire function of growth ρ = 1/α and normal type. Then the cyclicity of A is equivalent to its unicellularity.

Spectral theory of fractional order integration operators | 447

Sketch of the proof. Without loss of generality we assume that −α1 = α2 = α/2, i. e. A is sectorial with semi-angle α/2. Then conditions of the theorem mean that A ∈ Λρ,σ . Let A be cyclic. By Theorem 11, the operator A is unitary equivalent to a restriction of ̂J α , α = ρ−1 , to a certain invariant subspace H1 , i. e. U ∗ AU = ̂J α ⌈H1 where U isometrically maps H1 onto H. The rest of the proof is divided in three steps. (i) Lat ̂J α = Lat ̂J. This fact is obvious because ̂J α ∈ a(̂J) and ̂J ∈ a(̂J α ). (ii) Operators ̂J α ⌈H1 and ̂J⌈H1 are cyclic only simultaneously. ∞ ̂α Let H1 ∈ Lat ̂J α , h = ⊕∞ 1 hj ∈ Cyc(J ⌈H1 ), and ‖h‖ = 1. Then for any φ = ⊕1 φj ∈ H1 and any sequence εn ↓ 0 as n → ∞, there exists a sequence of polynomials Pn (t) = ∑nk=1 ak,n t k ∈ ℂ[t] (depending on φ) such that ∞

󵄩2 󵄩 󵄩2 󵄩󵄩 ̂α −1 α 󵄩󵄩Pn (J )h − φ󵄩󵄩󵄩L̂2 [0,l] = ∑ 󵄩󵄩󵄩Pn (J )hj − φj 󵄩󵄩󵄩L2 [0,l] < 2 εn , j=1

n ∈ ℕ.

(58)

m p For the sequence {Pn (⋅)}∞ 1 one finds a sequence of polynomials Qm (t) = ∑p=1 bp,m t ∈ ̃ ‖2 1 ̃n − Q ℂ[t] such that ‖P m L [0,l] ≤ εn /2 where

̃ (t) = ∑ b t p−1 /(p − 1)! ̃ n (t) = ∑ ak,n t αk−1 /Γ(αk) and Q P p,m m 1≤k≤m

1≤k≤n

With these notations one gets ∞

∞

󵄩 ̃ 󵄩2 󵄩 ̃ 󵄩󵄩2 2 ∑ 󵄩󵄩󵄩(Pn (J α ) − Qm (J))hj 󵄩󵄩󵄩L2 [0,l] = ∑ 󵄩󵄩󵄩(P n − Qm ) ∗ hj 󵄩 󵄩L [0,l] j=1

j=1

∞

∞

j=1

j=1

−1 2 −1 2 ̃ ‖2 1 ̃n − Q ≤ ∑ ‖P m L [0,l] ⋅ ‖hj ‖L2 [0,l] ≤ 2 εn ∑ ‖hj ‖L2 [0,l] = 2 εn ,

(59)

where ∗ denotes the convolution (see (10)). Combining (59) with (58) implies ‖Qm (̂J)h− φ‖2 ≤ εn . Since vector φ ∈ H1 is arbitrary, it means that the operator ̂J⌈H1 is cyclic and h ∈ Cyc(̂J⌈H1 ). (iii) By Theorem 9(iii), ̂J⌈H1 is unicellular. Since Lat ̂J = Lat ̂J α , the operator ̂J α ⌈H1 is unicellular too. Thus, the operator A = U(̂J α ⌈H1 )U ∗ is unicellular. Theorem 12 generalizes the Kisilevskii result, Theorem 9(iii), and coincides with it for α = 1. Next we apply Theorem 11 to investigate powers of an operator A ∈ Λρ , ρ ≥ 1. Let A be an accretive operator on H and let β = p + ε ∈ ℝ+ , where p = [β] is an integer part of β, and ε ∈ [0, 1). Then fractional powers Aβ of the operator A are defined by Aβ := Ap ⋅ Aε where Aε is given by (7). However, if A ∈ Λρ , then in accordance with Theorem 11, A = U(̂J α ⌈H1 )U ∗ with α = ρ−1 . This representation allows us to determine the fractional powers Aβ as follows: β

Aβ := U(̂J α ⌈H1 ) U ∗ = U(̂J αβ ⌈H1 )U ∗ .

(60)

448 | M. M. Malamud It can be shown that this definition does not depend on a choice of invariant subspace H1 as well as on a unitary map U intertwining the pair {A, J α ⌈H1 }. Moreover, for operators A ∈ Λρ general definition of fractional powers of an accretive operator coincides with a new one given by (60). Proposition 9. Let A ∈ Λρ , and β ∈ ℝ+ with βρ−1 ≤ 1. Then the operator Aβ belongs to the class Λρ1 with ρ1 = ρ/β ≥ 1. In particular, the operator Aβ is sectorial with the semi-angle πβ/2ρ. This result is easily extracted from the fractional calculus given by (60). Remark 7. If A ∈ Λn with n ∈ ℕ, then An ∈ Λ1 , in particular, An is accretive. The later fact looks very surprising. Indeed, it does not hold for sectorial Volterra operators A ∉ Λn even with arbitrary small semi-angle of sectoriality. To demonstrate this effect in finite-dimensional space one should consider matrices with nonzero spectrum because there are no accretive operators with zero spectrum in ℂn . For instance, let A = ( 0a aε ) and √ε2 < a < ε. Then A is sectorial with semi-angle π/4, while A2 is not accretive. Remark 8. Let A be a sectorial Volterra operator with the semi-angle πα/2 and let ρA be the order of its growth. Then it follows from Theorem 10 that α and ρA are related by the inequality ρA ≥ 1/α. Moreover, it follows from Theorem 10 and the Fragmen– Lindelef theorem that the Fredholm resolvent of A ∈ Λρ is of minimal type (i. e. σA = 0) if and only if A = 0.

5 Similarity of Volterra operators 5.1 Sufficient conditions for similarity to the integration operator J Definition 9. Two bounded operators A1 and A2 on a Banach space X are said to be similar if there exists a topological automorphism T in X such that A2 = TA1 T −1 . This concept takes its origin from the linear algebra classification theorem: Each operator A on ℂn is similar to its model, the direct sum of Jordan cells Jnj (λj ), {λj } = σ(A). Clearly, each Jnj (λj ) is unicellular along with any cJnj (λj ), c ∈ ℂ. Here we present sufficient conditions for the similarity of a Volterra operator x

Kf = ∫ k(x, t)f (t)dt 0

to the integration operator J.

(61)

Spectral theory of fractional order integration operators | 449

Theorem 13 ([50, 51]). Suppose that the kernel k(x, t) of the operator K of the form (61) satisfies the following conditions: (i) k(x, x) = 1; (ii) k(x, t) is absolutely continuous with respect to x for almost all t ∈ [0, l]; (iii) the operator K1 of the form (61) with kernel k1 (x, t) := Dx k(x, t) is weakly compact in L1 [0, 1] and k1 (x, x) ∈ L1 [0, l]; (iv) k1 (x, t) is absolutely continuous with respect to t for almost all x ∈ [0, l] and Dt k1 (x, t) = Dt Dx k(x, t) ∈ L1 (Ω), Ω = {0 ≤ t ≤ x ≤ l}. Then the operator K is similar to the integration operator J in Lp [0, l], p ∈ [1, ∞]. Remark 9. (a) Condition (iii) holds trivially if k1 (x, t) ∈ L∞ (Ω). (b) Note that, since k1 (x, t) is absolutely continuous with respect to t, the repeated inx 1 tegral ∫0 dx ∫0 Dt Dx k(x, t) dt exists and is finite. Therefore, if Dt Dx k(x, t) ≥ 0, then by Fubini’s theorem, the condition Dt Dx k(x, t) ∈ L1 (Ω) is satisfied automatically. Corollary 5. Let {φk (⋅)}∞ 1 be a sequence of absolutely continuous functions on [0, 1], i. e., φk ∈ W11 [0, 1] for all k ∈ ℕ and let εk = ±1, k ∈ ℕ. Suppose further that the following conditions hold: ∞ 󸀠 2 2 (i) the series ∑∞ 1 |φk (x)| and ∑1 |φk (x)| converge almost everywhere in [0, 1] to functions u20 (x) and u21 (x) such that negative functions u0 and u1 satisfy u0 ∈ L∞ [0, 1] and u1 ∈ L1 [0, 1]; 2 (ii) k(x, x) = ∑∞ 1 εk |φk (x)| = 1, x ∈ [0, 1]. Then the series ∞

(62)

k(x, t) = ∑ εk φk (x)φk (t) k=1

converges absolutely and uniformly and defines a kernel k(x, t) ∈ C(Ω) such that the operator K of the form (61) is similar to the integration operator J in the spaces Lp [0, l], p ∈ [1, ∞]. Corollary 6. Let K be a Volterra operator with degenerate kernel (62), i. e. n

k(x, t) = ∑ εk φk (x)φk (t), k=1

εk = ±1

for all

k ≤ n.

(63)

Let also the functions φk (⋅) be absolutely continuous on [0, 1] (i. e., φk ∈ W11 [0, 1] for all k ≤ n). If k(x, x) = 1, then the operator K is similar to the integration operator J in Lp [0, l], p ∈ [1, ∞]. In connection with this corollary we present an interesting result of Gubreev [25]. Theorem 14 ([25]). Let K be a Volterra operator of the form (61) with degenerate kernel (63) and let εk = 1 for 1 ≤ k ≤ n, i. e. the operator iK is dissipative in L2 [0, 1] with

450 | M. M. Malamud n-dimensional imaginary part. Let also the functions {φk (t)}n1 be piecewise absolutely continuous with finitely many discontinuities of the first kind {xj }m 1 . Then the operators K and J are similar in L2 [0, 1] provided that n

∑ φk (xj + 0)φk (xj − 0) ≠ 0

k=1

for all j ≤ m.

(64)

Condition (64) is known as Kisilevskii’s condition. It is interesting to compare this theorem with the following result by Kisilevskii. Theorem 15 ([43, 8]). Let K be a simple Volterra operator of the form (61) with degenerate kernel (63) and let εk = 1 for 1 ≤ k ≤ n. Assume also that the functions {φk (t)}n1 are piecewise continuous with finitely many discontinuities of the first kind {xj }m 1 . If condition (64) is satisfied, then the operator K is unicellular in L2 [0, 1]. This result remains valid for a simple dissipative operator iK with trace class imaginary part (see [8]). It is extracted from the criterion of unicellularity contained in Theorem 9(ii). Remark 10. (i) First results on the similarity between the operators K and J were obtained independently and by different methods by Kalisch [32] and Sakhnovich [67]. Instead of conditions (ii)–(iv) it is assumed in [32] and [67] that k(x, t) ∈ C 2 (Ω) and k1 (x, t), Dt k1 (x, t) ∈ L∞ (Ω), respectively. In several subsequent publications the condition Dt k1 (x, t) ∈ L∞ (Ω) has been weakened (see the papers [49, 50, 47, 57] and references therein and in [18]). This problem is also discussed in the framework of Friedrichs’ method in [22] (see also [18] where the Freeman result is presented with a proof). Theorem 13 in a slightly weaker form was proved in [50], its refinement is contained in [51, Theorem A]. It strengthens all previous results on the similarity to the operator J but one contained in Gubreev’s theorem 14 (see [25] and the discussion in [51]). (ii) Note also that the condition Dt Dx k(x, t) ∈ L1 (Ω) in Theorem 13 is sharp and cannot be weakened in certain classes of kernels. For instance, it is necessary in a class of convolution operators (see Theorem 20 below and discussion in [51]). (iii) Note also that sufficient conditions of the similarity between the operators K and J defined on the Sobolev space Wpk [0, 1] were obtained by Romaschenko [17]. In this case condition (iv) of Theorem 13 is replaced by the condition Dt Dx k(x, t) ∈ Wpk (Ω).

5.2 Sufficient conditions for similarity to operators J α Let 0 < α = n − ε, n ∈ ℕ, ε ∈ [0, 1). Alongside fractional integral operators (6) we introduce the operator of fractional derivative of order α (see [14, 69]), f (α) (x) = Dα f (x) = (Dn J ε f )(x) =

dn ε J f. dxn

(65)

Spectral theory of fractional order integration operators | 451

Next we denote by Wpα [0, l] the Sobolev space of functions f ∈ L1 [0, l] having a fractional derivative f (α) ∈ Lp [0, l]. The functions f ∈ Wpα [0, l] are characterized by the integral representation x

n

1 xα−j + f (x) = ∑ cj ∫(x − t)α−1 f1 (t) dt, Γ(α − j + 1) Γ(α) j=1

(66)

0

where cj = f (α−j) (0), j ∈ {1, . . . , n}, and f1 (⋅) = Dα f (⋅) = f (α) (⋅) ∈ Lp [0, l]. α [0, l] the subspace of those f ∈ Wpα [0, l] for which the Let us denote also by Wp,0

polynomial in (66) is absent, that is, cj = f (α−j) (0) = 0, j ∈ {1, . . . , n}. Here we present the main result on the similarity of the Volterra operator (61) to the operator J α with α ≠ 1. Roughly speaking, the conditions for similarity read as follows: (i) k(x, x − t) should be an entire function in x for all t ∈ [0, l); (ii) the behavior of k(x, t) at the diagonal x = t coincides with that of the trivial α−1 kernel (x−t) . Γ(α) Theorem 16 ([48, 51]). Let p ∈ [1, ∞] and let α = n−ε, ε ∈ [0, 1). Suppose that the kernel k(x, t) of the Volterra operator (61) satisfies the following conditions: (i) for all t ∈ [0, l] the derivatives Dj−ε x k(x, t) ∈ C(Ω)

exist,

j ∈ {0, 1, . . . , n};

(67)

(ii) for all x ∈ [0, l] the derivative Dnt k1 (x, t) ∈ C(Ω) exist, where k1 (x, t) = Dn−ε x k(x, t) ∈ C(Ω); j (iii) Dt k1 (x, x − t0 ) is an entire function in x for all j ∈ {1, . . . , n}, and for all t0 ∈ [0, l]; n−1−ε (iv) [Dj−ε k(x, t)]|t=x = 1. x k(x, t)]|t=x = 0, j ∈ {0, n − 2}, and [Dx Then K is similar in Lp [0, l] to the operator of fractional integration J α . Moreover, if Dn−ε s k(s, t)|t=s = 0, then: (a) there exists a Volterra operator R of the form (61) with a kernel R(x, t) and such that I + R intertwines the operators K and J α , i. e. K(I + R) = (I + R)J α ; ̃ t0 ) := R(x, x − t0 ) admits a holomorphic continu(b) for each t0 ∈ [0, 1) the function R(x, ation to an entire function in x. Moreover, R(x, t) admits a holomorphic continuation to an entire function in both variables whenever k(x, t) does. Denote by Dα0 the restriction of the operator Dα (see (65)) to the domain D(Dα0 ) = Clearly, (J α )−1 = Dα0 . Besides, one shows that the inverse to the Volterra operator K is a fractional order integro-differential operator of the form n−ε Wp,0 [0, l].

K f = −1

0 ln−ε,1 (D)f

n−ε

=D

n−1

f + ∑ qj (x)D j=1

n−ε−j

x

f + ∫ N(x, t)(J ε f )(t)dt 0

(68)

452 | M. M. Malamud 0 n−ε with domain D(ln−ε,1 (D)) = W1,0 [0, l]. Thus, the similarity problem is reduced to the similarity between the operators (68) and Dα0 .

Remark 11. If n = [D−ε x k(x, t)]|t=x = 1.

1, the conditions (iv) are reduced to the single condition

Corollary 7. Let k(x, t) =

(x − t)α−1 P(x, t) and Γ(α)

P(x, x) = 1,

(69)

and suppose that the derivatives Dnx P(x, t) := Q(x, t) ∈ C(Ω),

j

Dt Q(x, t) := Qj (x, t) ∈ C(Ω),

j ∈ {0, . . . , n},

̃ (x, t ) := Q (x, x − exist. Assume, in addition, that for each t0 ∈ [0, 1) the functions Q j 0 j t0 ), (0 ≤ j ≤ n), admit holomorphic continuations to entire functions in x. Then the operator K of the form (61) is similar in Lp [0, l], p ∈ [1, ∞] to the operator of fractional integration J α . Example 2. Let the kernel k(x, t) have the form k(x, t) =

m (x − t)α−1 [1 + ∑ φj (x)ψj (x − t)], Γ(α) j=1

(70)

2n where ∑m j=1 φj (x)ψj (0) = 0, ψj (t) ∈ C [0, l], j ∈ {0, . . . , m}, and φj (x) is an entire function in x for all j ∈ {1, . . . , m}. Then k(x, t) satisfies the conditions of Theorem 16, and hence the Volterra operator K of the form (61) is similar to the operator J α .

Corollary 8. Assume that k(x, t) is a kernel of the form (69) with P(x, t) admitting a holomorphic continuation to entire function in both variables. Then for any n ∈ ℕ there exists a Volterra operator Kn with a kernel kn (x, t) of the form kn (x, t) =

(x − t)α/n−1 Pn (x, t) and Γ(α/n)

Pn (x, x) = 1,

(71)

and such that: (i) Pn (x, t) admits a holomorphic continuation to entire function in both variables; (ii) Kn is an nth root of K, i. e. Knn = K. Next contribution to the subject was done by Ignat’ev [31]. Assuming that α > 2 he simplified the original proof of Theorem 16 by generalization of the reasoning of Hachatryan [42]. The proposed method allows him to weaken the constraints imposed in Theorem 16 on the analyticity domain of the kernel k(x, t). Following [31] we assume that α > 2, l = 1, and denote by Da the quadrilateral in the complex plane with vertices {0, a, a(1 − w)−1 , a(1 − w−1 ) }, −1

w = exp(2πi/α),

(72)

Spectral theory of fractional order integration operators | 453

and put V := {(x, ξ ) : x ∈ [0, 1], ξ ∈ D1−x }. as

A Volterra integral operator is said to be an operator of class A if it can be expressed x

(Nf )(x) = ∫ N(x − t, t)f (t)dt

(73)

0

where N(x, ξ ) is a continuous function defined on V and, for any fixed x ∈ [0, 1], is analytic with respect to ξ on the domain D1−x . Theorem 17 ([31]). Suppose that α > 2 and K = J α (I +JN), where N is a Volterra operator of class A. Then there exists a Volterra operator R of class A such that K = (I+R)J α (I+R)−1 in the spaces Lp [0, l] for p ∈ [1, ∞]. Remark 12. (i) The problem of the similarity between the operators K and J α has a long history going back to the work by Kalisch [32] and Sakhnovich [66, 67], who treated the case of integers α = n ∈ ℕ. Theorem 16 strengthens their result from [32, 66, 67], where the kernel k(x, t) was required to be analytic in both variables instead of being finitely smooth in one of them, as required in Theorem 16. In the case of a non-integer α, Theorem 16 was proved in [51], while a weaker result was announced in [48]. (ii) Note also that similarity result (Theorem 16) has been applied to investigate completeness property of certain boundary value problems for frational order differential equations in [56] and [1, 2]. (iii) Corollary 8 generalizes the classical result of Volterra and Peres [77] and coincides with it for α = n. It was substantially employed by Kalisch [32] in his proof of the similarity of K and J n .

5.3 Convolution Volterra operators 5.3.1 Sufficient conditions of similarity The most complete results on the similarity of the operators J α are known for convolution Volterra operators, x

Kf = ∫ k(x − t)f (t)dt.

(74)

0

We start with a simple corollary of Theorem 13. Corollary 9. Let k(⋅) ∈ W12 [0, l] and k(0) = 1. Then the operator K is similar to the integration operator J in each space Lp [0, l], p ∈ [1, ∞].

454 | M. M. Malamud Next we present our main result on the similarity between the operators J α + K and J α . α Theorem 18 ([48]). Let 0 < α = n − ε, ε ∈ [0, 1), and let k(⋅) ∈ W1α+1 [0, l] ∩ W1,0 [0, l]. α α p Then the operator J + K is similar to the operator J in each space L [0, l], p ∈ [1, ∞].

Sketch of the proof. We shall not give the proof but merely demonstrate how the result α can be deduced from Corollary 9. In fact, the inclusion k(⋅) ∈ W1α+1 [0, l] ∩ W1,0 [0, l] means that representation (66) of k(⋅) is reduced to x

k(x) =

xα−1 1 + ∫(x − t)α−1 k1 (t) dt, Γ(α) Γ(α)

(75)

0

where k1 := k (α) ∈ W11 [0, l]. Therefore the operator K admits a representation K = x J α (I + K1 ), where K1 f = ∫0 k1 (x − t)f (t) dt and k1 ∈ W11 [0, l]. Since the spectrum of I + K1 does not contain zero, it follows from Riesz–Danford’s calculus that the operator K2 = (I + K1 )1/α − I

(76)

is well defined and is also a convolution operator of the form (74) with kernel k2 ∈ W11 [0, l]. Setting K3 = J(I + K2 ), we see that K3 is a convolution operator satisfying K3n = K. The kernel k3 of the operator K3 satisfies k3 ∈ W12 [0, l] and k3 (0) = 1. Therefore the kernel k3 already satisfies conditions of Corollary 9 and, consequently, K3 is similar to J. Using this fact, it is shown that K3α = K is similar to J α . This fact is obvious for integer α = n. For α ≠ n it is based on a special functional calculus (see [49, 51, 58]). [α]+1 [0, l]. Corollary 10. Let K be a convolution operator (74) with k ∈ W1[α]+2 [0, l] ∩ W1,0 α α p Then the operator J + K is similar to the operator J in L [0, l], p ∈ [1, ∞].

Corollary 11. Let α = n − ε and ε > 0, i. e. α ≠ [α]. Assume that k(⋅) admits a representation k(x) = xα−1 k1 (x) where k1 ∈ W n+1 [0, l] and k1 (0) = 0. Then the operator J α +K, where K is the convolution operator of the form (74), is similar to J α in each Lp [0, l], p ∈ [1, ∞]. α Proof. To verify the condition k ∈ W1α+1 [0, l] ∩ W1,0 [0, l] of Theorem 18 it suffices to check that n J ε k ∈ W1n+1 [0, l] ∩ W1,0 [0, l].

(77)

One easily gets x

1

0

0

1 xn−1 (J k)(x) = ∫(x − t)ε−1 t α−1 k1 (t)dt = ∫(1 − u)ε−1 uα−1 k1 (ux) du. Γ(ε) Γ(ε) ε

(78)

Spectral theory of fractional order integration operators | 455

Since k1 ∈ W [α]+2 [0, l], it follows that J ε k ∈ W n+1 [0, l]. It remains to check the conditions at zero. Conditions (Djx J ε k)(0) = 0 for j ≤ n − 2 are immediate from (78). Further, applying the operator Dn−1 = dn−1 /dxn−1 , one gets x 1

n−1 dn−1 n − 1 n−1−j (n−1−j) (Γ(ε)J ε k)(x) = ∑ ( )x (ux)du. ∫(1 − u)ε−1 uα−2−j k1 n−1 j dx j=0

(79)

0

ε ε n Since k1 (0) = 0, it follows that (Dn−1 x J k)(0) = 0. Summing up, J k ∈ W1,0 [0, l] and the kernel k(⋅) meets the conditions of Theorem 18. Thus, the operators J α + K and J α are similar in each Lp [0, l], p ∈ [1, ∞].

As an immediate consequence of Theorem 18 one obtaines unicellularity of convolution Volterra operators (74) satirfying conditions either of this theorem or Corollary 11. To improve this statement let us recall a localization principle due to G. Kisilevskii [44]. Theorem 19 ([44]). Let the kernels k1 , k2 ∈ L1 [0, l] define convolution Volterra operators K1 and K2 of the form (74). If k1 (x) = k2 (x) for a.e. x ∈ (0, δ) with some δ < 1, then operators K1 and K2 are unicellular in each Lp [0, l], p ∈ [1, ∞), or C[0, 1] only simultaneusly. Combining Theorem 18 with Theorem 19 we arrive at the following result. α Corollary 12. Let 0 < α = n − ε, ε ∈ [0, 1), and let k(⋅) ∈ W1α+1 [0, δ] ∩ W1,0 [0, δ] with some α p δ ∈ (0, 1). Then the operator J + K is unicellular in each space L [0, l], p ∈ [1, ∞), and C[0, 1].

Remark 13. (i) Theorem 18 was proved in [48, 49]. For integer α = n(⇔ ε = 0) it strengthens results of Kalisch [32] and Sakhnovich [68] who assumed that k ∈ C n+1 [0, l]. Besides, a weaker version of Corollary 11 was proved for L2 [0, l] earlier by Frankfurt and Rovnyak [20] (see also [21]). Namely, they have assumed that k1 ∈ C [α]+2 [0, l] and applied complex-analytic technique based on Fourier transform. Corollary 11 improves their result from [20] and is sharp in a certain class of convolution operators (see Section 5.3.2, especially Theorem 20 and Example 3). (ii) Note also that the similarity of the convolution Volterra operators (74) to the fractional integrals J α defined on the Sobolev spaces Wpk [0, 1] was investigated in [65]. 5.3.2 Necessary conditions for similarity Consider a perturbation of the operator J α given by α

α

x

α

x

(J + K1 + K2 )f = J f + ∫(x − t) k1 (x − t)f (t) dt + ∫ k2 (x, t)f (t) dt. 0

0

(80)

456 | M. M. Malamud Theorem 20 ([49]). Let α > 0 and c ∈ ℂ. Assume that k1 (x) is a nonnegative decreasing function on [0, 1] and k1 (+0) = ∞. Assume also that k2 (x, t) ≥ 0 be a bounded nonnegative measurable function in Ω = {0 ≤ t ≤ x ≤ 1}. Then the operator J α + K1 + K2 of the form (80) is not similar to the operator J α in Lp [0, 1] for any p ∈ [1, ∞]. Example 3. Let α > 0 and p ∈ [1, ∞]. Then the Volterra operator m

K = J α + ∑ cj J α+εj j=1

with positive cj > 0 is similar to the operator J α in Lp [0, 1] if and only if εj ≥ 1. Sufficiency is immediate from Theorem 18, while the necessity is implied by Theorem 20. Note that the absence of similarity between operators J + J 1+ε , ε ∈ (0, 1), and J was first established by G.K. Kalisch [33]. Making use of the operators J α for α < 1 we refine a result due to Foias and Williams [19]. Corollary 13 ([53]). Let α > 0 and p ∈ [1, ∞]. Then the operator J α (I − J α )−1 is similar to the operator J α in Lp [0, 1], if and only if α ≥ 1. Moreover, the operator J α (I − J α )−1 is not similar to cJ α in Lp [0, 1] whenever α ∈ (0, 1). Proof. It is easily seen that J α (I − J α ) 2α

2α

−1

∞

∞

k=1

k=3

= ∑ J αk = J α + J 2α + J 2α ( ∑ J α(k−2) ).

α(k−2) . ∑∞ k=3 J

Set K1 = J and K2 = J Clearly, K = K1 + K2 satisfy the conditions of Theorem 18 provided that α ≥ 1, hence J α + K is similar to J α in each Lp [0, 1]. Now let α ∈ (0, 1). The kernel k2 (x − t) of K2 is nonnegative while the kernel of K1 admits a representation x2α−1 = xα xα−1 , i. e. k1 (x) = xα−1 . Since α < 1, the kernels k1 (x−t) and k2 (x − t) meet the conditions of Theorem 20 and, hence the operator J α (I − J α )−1 is not similar to cJ α in any Lp [0, 1]. Remark 14. Foias and Williams [19] proved that there exists a quasi-nilpotent operator Q in the commutant of the integration operator J such that Q and Q(I − Q)−1 are not similar in L2 [0, 1]. According to Corollary 13, one can choose Q = J α with any α ∈ (0, 1). This refines the result from [19] and makes it explicit. Note also that the s-numbers α sj (J ) of Q = J α satisfy sj (Q) = O(j−α ), in particular, Q ∈ ⋂p>1/α Sp .

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Anatoly N. Kochubei

Fractional differentiation in p-adic analysis Abstract: The paper is a brief survey of results on the Vladimirov fractional differentiation operator acting on complex-valued functions on the field of p-adic numbers, its spectral properties, and some related equations. Keywords: p-adic numbers, Vladimirov operator, heat kernel, wavelets, radial functions MSC 2010: 11S80, 26A33

1 Introduction Pseudo-differential equations for complex-valued functions defined on a nonArchimedean local field are among the central objects of contemporary harmonic analysis and mathematical physics; see the monographs [1, 6, 14, 16]. The simplest example is the fractional differentiation operator Dα , α > 0, on the field ℚp of p-adic numbers (here p is a prime number). It can be defined as a pseudo-differential operator with the symbol |ξ |αp where | ⋅ |p is the p-adic absolute value or, equivalently, as an appropriate convolution operator. This operator is often called the Vladimirov operator. As first shown by Vladimirov (see [14]), already in this case properties of a p-adic pseudo-differential operator are much more complicated than those of the classical counterparts. It suffices to say that, as an operator on L2 (ℚp ), 𝔻(α) has a point spectrum of infinite multiplicity. It was found later [8] that 𝔻(α) behaves much simpler on the class of radial functions. In this survey, we describe the main properties of the Vladimirov operator. In Section 2, we recall the definition of p-adic numbers, describe the Fourier transform of complex-valued functions on ℚp , Bruhat–Schwartz distributions on ℚp . Section 3 is devoted to the fractional differentiation operator Dα and the related “heat kernel” and Markov process. In Section 4, we describe Kozyrev’s construction of eigenfunctions of 𝔻(α) in L2 (ℚp ) and their interpretation as p-adic wavelets. In Sections 5 and 6, which are based on the results from [8], we explain how the properties of 𝔻(α) are simplified, if this operator is considered on radial functions. In this case, some linear equations containing 𝔻(α) can be reduced to integral equations resembling classical Volterra equations of real analysis. In other words, we arrive at a p-adic analog of ordinary differential equations. Anatoly N. Kochubei, Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kyiv, 01004, Ukraine, e-mail: [email protected] https://doi.org/10.1515/9783110571622-019

462 | A. N. Kochubei Note that all the above results can be extended to general non-Archimedean local fields. We confined ourselves to the case of ℚp just to simplify our exposition.

2 p-Adic numbers and harmonic analysis In this section we follow [1, 14]. Let p be a prime number. The field of p-adic numbers is the completion ℚp of the field ℚ of rational numbers, with respect to the absolute value |x|p defined by setting |0|p = 0, |x|p = p−ν

if x = pν

m , n

where ν, m, n ∈ ℤ, and m, n are prime to p. ℚp is a locally compact topological field. Note that by Ostrowski’s theorem there are no absolute values on ℚ, which are not equivalent to the “Euclidean” one, or one of | ⋅ |p . The absolute value |x|p , x ∈ ℚp , has the following properties: |x|p = 0

if and only if x = 0;

|xy|p = |x|p ⋅ |y|p ;

|x + y|p ≤ max(|x|p , |y|p ). The last property, called the ultra-metric inequality (or the non-Archimedean property) implies the total disconnectedness of ℚp in the topology determined by the metric |x − y|p , as well as many unusual geometric properties. Note also the following consequence of the ultra-metric inequality: |x + y|p = max(|x|p , |y|p ) if |x|p ≠ |y|p . The absolute value |x|p takes the discrete set of non-zero values pN , N ∈ ℤ. If |x|p = pN , then x admits a (unique) canonical representation x = p−N (x0 + x1 p + x2 p2 + ⋅ ⋅ ⋅),

(1)

where x0 , x1 , x2 , . . . ∈ {0, 1, . . . , p − 1}, x0 ≠ 0. The series converges in the topology of ℚp . For example, −1 = (p − 1) + (p − 1)p + (p − 1)p2 + ⋅ ⋅ ⋅ ,

| − 1|p = 1.

We denote ℤp = {x ∈ ℚp : |x|p ≤ 1}. ℤp , as well as all balls in ℚp , is simultaneously open and closed. Proceeding from the canonical representation (1) of an element x ∈ ℚp , we define the fractional part of x as the rational number 0, {x}p = { −N p (x0 + x1 p + ⋅ ⋅ ⋅ + xN−1 pN−1 ),

if N ≤ 0 or x = 0;

if N > 0.

Fractional differentiation in p-adic analysis | 463

The function χ(x) = exp(2πi{x}p ) is an additive character of the field ℚp , that is, a character of its additive group. It is clear that χ(x) = 1 if |x|p ≤ 1. Denote by dx the Haar measure on the additive group of ℚp normalized by the equality ∫ℤ dx = 1. p

The above additive group is self-dual, so that the Fourier transform of a complexvalued function f ∈ L1 (ℚp ) is again a function on ℚp defined as (ℱ f )(ξ ) = ∫ χ(xξ )f (x) dx. ℚp

If ℱ f ∈ L1 (ℚp ), then we have the inversion formula f (x) = ∫ χ(−xξ )̃f (ξ ) dξ . ℚp

It is possible to extend ℱ from L1 (ℚp ) ∩ L2 (ℚp ) to a unitary operator on L2 (ℚp ), so that the Plancherel identity holds in this case. In order to define distributions on ℚp , we need a class of test functions. A function f : ℚp → ℂ is called locally constant if there exists an integer l ≥ 0 such that for any x ∈ ℚp 󵄩 󵄩 f (x + x󸀠 ) = f (x) if 󵄩󵄩󵄩x󸀠 󵄩󵄩󵄩 ≤ p−l . The smallest number l with this property is called the exponent of local constancy of the function f . Typical examples of locally constant functions are additive characters, and also cutoff functions like 1, if ‖x‖ ≤ 1; Ω(x) = { 0, if ‖x‖ > 1. In particular, Ω is continuous, which is an expression of the non-Archimedean properties of ℚp . Denote by 𝒟(ℚp ) the vector space of all locally constant functions with compact supports. Note that 𝒟(ℚp ) is dense in Lq (ℚp ) for each q ∈ [1, ∞). In order to furnish 𝒟(ℚp ) with a topology, consider first the subspace DlN ⊂ 𝒟(ℚp ) consisting of functions with supports in a ball BN = {x ∈ ℚp : |x|p ≤ pN },

N ∈ ℤ,

and the exponents of local constancy ≤ l. This space is finite-dimensional and possesses a natural direct product topology. Then the topology in 𝒟(ℚp ) is defined as the double inductive limit topology, so that l

𝒟(ℚp ) = lim lim DN .

󳨀󳨀→ 󳨀󳨀→

N→∞ l→∞

464 | A. N. Kochubei If V ⊂ ℚp is an open set, the space 𝒟(V) of test functions on V is defined as a subspace of 𝒟(ℚp ) consisting of functions with supports in V. The space 𝒟󸀠 (ℚp ) of Bruhat–Schwartz distributions on ℚp is defined as a strong conjugate space to 𝒟(ℚp ). In contrast to the classical situation, the Fourier transform is a linear automorphism of the space 𝒟(ℚp ). By duality, ℱ is extended to a linear automorphism of 𝒟󸀠 (ℚp ). There exists a detailed theory of convolutions and direct products of distributions on ℚp closely connected with the theory of their Fourier transforms; see [1, 6, 14].

3 p-Adic fractional differentiation The Vladimirov operator Dα , α > 0, of fractional differentiation is defined first as a pseudo-differential operator with the symbol |ξpα : (Dα u)(x) = ℱξ−1→x [|ξ |αp ℱy→ξ u],

u ∈ 𝒟(ℚp ),

where we show arguments of functions and their direct/inverse Fourier transforms. There is also a hypersingular integral representation giving the same result on 𝒟(ℚp ) but making sense on much wider classes of functions (for example, bounded locally constant functions): (Dα u)(x) =

1 − pα [u(x − y) − u(x)] dy. ∫ |y|−α−1 p 1 − p−α−1

(2)

ℚp

It is useful to introduce the subspaces of 𝒟(ℚp ): Ψ(ℚp ) = {ψ ∈ 𝒟(ℚp ) : ψ(0) = 0}, Φ(ℚp ) = {φ ∈ 𝒟(ℚp ) : ∫ φ(x) dx = 0}. ℚp

The Fourier transform ℱ is a linear isomorphism from Ψ(ℚp ) onto Φ(ℚp ), thus also from Φ󸀠 (ℚp ) onto Ψ󸀠 (ℚp ). The spaces Φ(ℚp ) and Φ󸀠 (ℚp ) are called the Lizorkin spaces (of the second kind) of test functions and distributions respectively; see [1]. Note that two distributions differing by a constant summand coincide as elements of Φ󸀠 (ℚp ). The operator Dα is defined on 𝒟(ℚp ); however, Dα does not act on the space 𝒟(ℚp ), since the function ξ 󳨃→ |ξ |α is not locally constant. On the other hand, Dα : Φ(ℚp ) → Φ(ℚp ) and Dα : Φ󸀠 (ℚp ) → Φ󸀠 (ℚp ); see [1], and that was a motivation to introduce these spaces.

Fractional differentiation in p-adic analysis | 465

In a certain sense, Dα is meromorphic with respect to α; for the details regarding the procedure of analytic continuation see [14]. In particular, this analytic continuation leads to the following operators D−β , β > 0: (D−β φ)(x) =

1 − p−β ∫ |x − y|β−1 p φ(y) dy, 1 − pβ−1

φ ∈ 𝒟(ℚp ), β ≠ 1,

(3)

ℚp

and (D−1 φ)(x) =

1−p ∫ log |x − y|p φ(y) dy. p log p

(4)

ℚp

Then Dα D−α = I on 𝒟(ℚp ), if α ≠ 1. This property remains valid on Φ(ℚp ) also for α = 1. The Cauchy problem for the heat-like equation 𝜕u + Dα u = 0, 𝜕t

u(0, x) = ψ(x),

x ∈ ℚp , t > 0,

possesses many properties resembling classical parabolic equations. If ψ is regular enough, for example, ψ ∈ 𝒟(ℚp ), then a classical solution is given by the formula u(t, x) = ∫ Z(t, x − ξ )ψ(ξ ) dξ ℚp

where Z is, for each t, a probability density and Z(t1 + t2 , x) = ∫ Z(t1 , x − y)Z(t2 , y) dy,

t1 , t2 > 0, x ∈ ℚp .

ℚp

Explicitly, ∞

Z(t, x) = ∑ pk ck (t)Δ−k (x) k=−∞

where Δl (x) is the indicator function of the ball Bl , ck (t) = exp(−pkα t) − exp(−p(k+1)α t). Another expression for Z(t, x), valid for x ≠ 0, is 1 − pαm m −αm−1 (−1)m ⋅ t |x|p . 1 − p−αm−1 m=1 m! ∞

Z(t, x) = ∑

The “heat kernel” Z satisfies the estimate 0 < Z(t, x) ≤ Ct(t 1/α + |x|p )

−α−1

,

t > 0, x ∈ ℚp .

(5)

466 | A. N. Kochubei The fundamental solution Z(t, x) is a transition density of a Markov process, a p-adic analog of the symmetric stable process on ℝ. The difference between the exponential estimate for the transition density of the Brownian motion on ℝ and the estimate (5) for Z reflects the topological contrast between ℝ and ℚp . A Markov process on ℚp cannot have continuous trajectories (since ℚp is totally disconnected). By the Kinney–Dynkin theorems [2], the exponential estimate implies the continuity of trajectories, while the power-like decay given by (5) is sufficient only for the cádlág property of the process on ℚp . Various generalizations and analogs of the operator Dα appear in p-adic counterparts of parabolic, Schrödinger, wave and Klein–Gordon equations; see [6, 7, 15, 16] and the references therein.

4 Spectral theory As we see from the next theorem, as a selfadjoint operator in L2 (ℚp ), Dα has a pure point spectrum of infinite multiplicity with a unique accumulation point. In this section we follow [1, 9, 10]. Below we identify the group ℚp /ℤp with the set of rational numbers −1

n = ∑ nl pl , l=ν

nl ∈ {0, 1, . . . , p − 1}

where ν is a negative integer. The next two theorems were first proved by Kozyrev [9] Theorem 1. The set of functions {ψγnj }: 󵄨 󵄨 ψγnj (x) = p−γ/2 χ(pγ−1 j(x − p−γn ))Ω(󵄨󵄨󵄨pγ x − n󵄨󵄨󵄨p ), γ ∈ Z,

n ∈ Qp /Zp ,

j = 1, . . . , p − 1,

(6)

is an orthonormal basis in L2 (Qp ) of eigenvectors of the operator Dα : Dα ψγnj = pα(1−γ) ψγnj . The next result shows the wavelet character of the eigenfunctions (6). Let us consider the unitary representation G in L2 (ℚp ) of the p-adic affine group acting by shifts and dilations: G(a, b)f (x) = f a,b (x) = |a|−1/2 p f(

x−b ), a

a, b ∈ ℚp , a ≠ 0.

Theorem 2. Each function from the sequence (6) can be obtained by a shift and dilation from the elementary wavelet ψ(x) = χ(p−1 x)Ω(|x|p )

Fractional differentiation in p-adic analysis | 467

as follows: ψγnj (x) = G(p−γ j−1 , p−γ n)ψ(x) = ψp

j ,p−γ n

−γ −1

(x).

Conversely, every function ψa,b , obtained by shift and dilation from ψ, is proportional to a function from the system {ψγnj }. The Monna map ℚp → ℝ+ [12], which is continuous, surjective but not one-toone, transforms the above basis into a system of functions from L2 (ℝ+ ) coinciding, if p = 2, with the classical Haar wavelet system. A systematic development of the p-adic wavelet theory was given in a series of papers by Albeverio, Khrennikov, Shelkovich, Skopina; see [1, 3, 4, 11, 13]. On the other hand, the first explicit description of eigenfunctions of Dα was found by Vladimirov (see [14]) and was not connected with the wavelet theory. See [6, 14] for the spectral theory of some perturbations of Dα , operators on bounded sets etc.

5 Operators on radial functions Following [8] we describe special features of p-adic fractional differentiation for the case of radial functions. Let u be a radial function, that is, u = u(|x|p ), x ∈ ℚp . In order to make the notation concise, we identify the functions ℚp ∋ x 󳨃→ u(|x|p ) and pℤ ∋ |x|p 󳨃→ u(|x|p ). This abuse of notation will not lead to confusion. 1−pα Let us find an explicit expression of Dα u, α > 0. Below we write dα = 1−p −α−1 . Proposition 1. If a function u = u(|x|p ) is such that m

󵄨 󵄨 ∑ pk 󵄨󵄨󵄨u(pk )󵄨󵄨󵄨 < ∞,

k=−∞

∞

󵄨 󵄨 ∑ p−αl 󵄨󵄨󵄨u(pl )󵄨󵄨󵄨 < ∞,

l=m

(7)

for some m ∈ ℤ, then for each n ∈ ℤ the expression in the right-hand side of (2) with φ(x) = u(|x|p ) exists for |x|p = pn , depends only on |x|p , and (Dα u)(pn ) = dα (1 −

1 −(α+1)n n−1 k pα + p − 2 u(pn ) )p ∑ p u(pk ) + p−αn−1 −α−1 p 1 − p k=−∞

+ dα (1 −

1 ∞ −αl ) ∑ p u(pl ). p l=n+1

(8)

Definition. We say that the action Dα u, α > 0, on a radial function u is defined in the strong sense, if the function u satisfies (7), so that Dα u(|x|K ), |x|K ≠ 0, is given by (8), and the following limit exists: def

Dα u(0) = lim Dα u(|x|p ). x→0

468 | A. N. Kochubei It is evident from (2) that Dα annihilates constant functions (recall that in Φ󸀠 (ℚp ) they are equivalent to zero). Therefore D−α is not the only possible choice of the right inverse to Dα . In particular, we will use (I α φ)(x) = (D−α φ)(x) − (D−α φ)(0).

(9)

This is defined initially for φ ∈ 𝒟(ℚp ). It is seen from (3), (4), and the ultra-metric property of the absolute value that (I α φ)(x) =

1 − p−α 1 − pα−1

α−1 ∫ (|x − y|α−1 p − |y|p )φ(y) dy,

α ≠ 1,

(10)

|y|p ≤|x|p

and (I 1 φ)(x) =

1−p p log p

∫ (log |x − y|p − log |y|p )φ(y) dy.

(11)

|y|p ≤|x|p

In contrast to (3) and (4), in (10) and (11) the integrals are taken, for each fixed x ∈ ℚp , over bounded sets. Let us calculate I α u for a radial function u = u(|x|p ). Obviously, (I α u)(0) = 0 whenever I α is defined. Proposition 2. Suppose that m

󵄨 󵄨 ∑ max(pk , pαk )󵄨󵄨󵄨u(pk )󵄨󵄨󵄨 < ∞,

k=−∞

if α ≠ 1,

and m

󵄨 󵄨 ∑ |k|pk 󵄨󵄨󵄨u(pk )󵄨󵄨󵄨 < ∞,

k=−∞

if α = 1,

for some m ∈ ℤ. Then I α u exists, it is a radial function, and for any x ≠ 0, (I α u)(|x|p ) = p−α |x|αp u(|x|p ) +

1 − p−α 1 − pα−1

α−1 ∫ (|x|α−1 p − |y|p )u(|y|p ) dy,

(12)

|y|p 1. Then u = I α v is a strong solution of the Cauchy problem (14)–(15). For some generalizations see [8].

Bibliography [1] [2] [3] [4] [5]

S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, Theory of p-Adic Distributions. Linear and Nonlinear Models, Cambridge University Press, 2010. E. B. Dynkin, Foundations of the Theory of Markov Processes, Prentice-Hall, Englewood Cliffs, NJ, 1961. A. Khrennikov, V. Shelkovich, and M. Skopina, p-adic refinable functions and MRA-based wavelets, J. Approx. Theory, 161 (2009), 226–238. A. Yu. Khrennikov and V. M. Shelkovich, Non-Haar p-adic wavelets and their application to pseudo-differential operators and equations, Appl. Comput. Harmon. Anal., 28 (2010), 1–23. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

472 | A. N. Kochubei

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

A. N. Kochubei, Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields, Marcel Dekker, New York, 2001. A. N. Kochubei, A non-Archimedean wave equation, Pac. J. Math., 235 (2008), 245–261. A. N. Kochubei, Radial solutions of non-Archimedean pseudodifferential equations, Pac. J. Math., 269 (2014), 355–369. S. V. Kozyrev, Wavelet analysis as a p-adic spectral analysis, Izv. Math., 66(2) (2002), 367–376. S. V. Kozyrev, Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics, Proc. Steklov Inst. Math., 274, Suppl. 1 (2011), 1–84. S. V. Kozyrev, A. Yu. Khrennikov and V. M. Shelkovich, p-adic wavelets and their applications., Proc. Steklov Inst. Math., 285 (2014), 157–196. A. F. Monna, Analyse Non-Archimédienne, Springer, Berlin, 1970. V. Shelkovich and M. Skopina, p-adic Haar multiresolution analysis and pseudo-differential operators, J. Fourier Anal. Appl., 15 (2009), 366–393. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994. W. A. Zúñiga-Galindo, Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields, p-Adic Numbers Ultrametric Anal. Appl., 3 (2011), 344–358. W. A. Zúñiga-Galindo, Pseudodifferential Equations Over Non-Archimedean Spaces, Lect. Notes Math., vol. 2174, 2016.

Index additive character 463 aging 73 anomalous diffusion 75 bibliographic information 4 bibliographic metrics 1 binomial coefficient 48, 66 binomial series 66 brachistochrone 328 Brownian motion 65, 169, 175, 399, 408, 466 – delayed 407, 415, 416 – fractional – delayed 416 – iterated 417 Brownian semigroup 65 calculus – extended Riesz–Dunford 62 – fractional – generalized 136 – fractional variational 347 – functional 59, 420 – Hille–Phillips 63 – Mikusinski 58 – operational 151 – rational functional 60 – Riesz–Dunford 61 – spectral 60 calculus of variations 347 canonical representation 462 Cauchy integral 61 classification of phase transitions 70 coefficients – binomial – generalized 29 composite CTRW 74 continuous time random maximum 391 continuous time random walk (CTRW) 73 contraction 342 convolution – integral 54 – kernel 51, 54 – Laplace 32 – Mellin 139, 198, 205 – operators 54 – product 54 https://doi.org/10.1515/9783110571622-020

convolution quotient 58 convolution ring 57 convolution theorem 198 CTRW integral equation 73 CTRW interpretation 73 cycloid 328 decomposition – Lévy–Itô 170 derivative 50 – fractional – at lower limit 56 – Caputo 33, 39, 90, 130, 148, 214, 257, 364, 408 – Caputo type 97, 150, 152 – Caputo type Erdélyi–Kober 39, 135, 261 – Caputo type multiple Erdélyi–Kober 146 – Caputo–Djrbashian 469 – Cauchy integral interpretation 61 – convolution quotient interpretation 57 – coupled space-time 398 – distributed order 40, 118, 422 – distributional interpretation 58 – Djrbashian–Nersessian 40 – Erdélyi–Kober 38, 91, 134, 206, 207 – Fourier transform interpretation 64 – functional calculus interpretation 59 – general 40, 111 – generalized 97, 143 – generalized Riemann–Liouville 56 – generalized Riemann–Liouville definition 56 – generalized Riemann–Liouville interpretation 56 – generalized Riesz interpretation 56 – generalized Weyl interpretation 56 – Grünwald–Letnikov 38, 55, 356 – Grünwald–Letnikov definition 55 – Grünwald–Letnikov interpretation 55 – Hadamard 41 – Hilfer 35, 90, 130, 155 – Hille–Phillips interpretation 63 – interpolation 56 – Laplace transform interpretation 62 – Liouville 228, 229, 234, 238 – Liouville(-Caputo) 56 – local 56

474 | Index

– localized Riemann–Liouville interpretation 56 – lower limit definition 54 – Marchaud 36 – Marchaud–Hadamard 55 – Marchaud–Hadamard definition 55 – Marchaud–Hadamard interpretation 55 – material 400 – Mellin transform interpretation 64 – multiple Erdélyi–Kober 96, 142, 143 – notation 55 – order of 54–56 – quantum Riesz–Feller 234, 237 – Riemann–Liouville 26, 31, 39, 54, 56, 89, 129, 282, 363, 413, 450 – Riemann–Liouville interpretation 54 – Riemann–Liouville type 97, 143, 148 – Riesz 42, 160, 234 – Riesz–Dunford interpretation 62 – Riesz–Feller 234 – sequential 149 – spectral projection interpretation 60 – standard interpretations 54 – Stieltjes transform interpretation 64 – symbols 55 – tempered 421 – type 56 – upper limit definition 54 – Weyl 37 – integer power 51 – iterated 51 – weak 420 differ-integral – fractional – generalized 102, 103 – multiple Erdélyi–Kober 145 difference quotient 50 differentiation 50 – fractional – p-adic 467 – iterated 51 diffusion – anomalous 75, 421 – fractional 75, 385 – tempered 422 – ultraslow 422 dimension – Hausdorff 172

Dirac distribution – derivatives 59 – fractional derivatives 59 Dirichlet problem 69 – fractional 69 distribution – α-stable 393 – fractional derivative 58 – fractional integral 58 – Lévy stable 280 – Mittag-Leffler 292 – tempered 176 – zeros 277 distribution of zeros 249 distributions – Bruhat–Schwartz 464 dynamic equations 339 Ehrenfest order 71 Ehrenfest’s classification 70 equality – Parseval 198 equation – Chapman–Kolmogorov 170 – differential – fractional 286, 291, 330 – nonlinear fractional 330 – time-fractional 407 – diffusion 261, 399, 408 – distributed order fractional 422 – fractional 261, 288 – space-fractional 265 – space-time fractional 214, 235, 388, 399 – tempered fractional 421 – diffusion-wave – fractional 262, 290 – space-fractional 258 – space-time fractional 218, 264 – time-fractional 257 – Euler–Lagrange 348, 365, 366, 371 – evolution – abstract fractional 214, 262 – fractional – Euler–Lagrange 351 – multi-order 153 – governing 397, 398, 420 – Hamilton 372 – heat-like 465 – integral 221, 470

Index | 475

– Abel 283–285, 327, 330 – fractional 328 – Volterra 330 – master 387 – oscillation – fractional 291 – pseudo-differential 395, 461 – relaxation – fractional 291 – Schrödinger – fractional 237, 238 – time-fractional – Schrödinger 68 – wave 261, 418 – time-fractional 417 equations – integral – dual 218 ergodic theory 72 ergodicity 73 ergodicity breaking 73 excessive functions 65 expansion – algebraic 299, 305, 306, 311, 312, 314, 318 – exponential 299, 306, 311, 312, 314, 317 extended eigenvalue 437 extended eigenvector 437 extension technique 179, 180, 188 first passage time 394, 395, 409, 417 form – Dirichlet 178 formula – asymptotic 278 – differentiation 245 – Dynkin 172 – expansion 374 – Hankel 244 – Ikeda–Watanabe 173 – integration by parts 26, 28, 30, 230 – fractional 364 – interpolation 48 – inversion 463 – Lagrange 48 – Leibniz type 196 – Lévy–Khintchine 393, 418 – Montroll–Weiss 387 – Post–Widder 196 – Post–Widder type 209

– Stirling 276, 375, 435 – subordination 214, 215, 262 Fourier multiplier 53 fractal dimension 70 fractance device 328 fractional boundary value problem 70 fractional calculus 1, 87, 388 – axioms 88, 101 – book 5, 87 – conference 1, 5, 87 – generalized 87 – history 1, 47, 87 – journal 8 – on time scales 340 – round table 4 fractional curl 66 fractional derivative on time scales 340 fractional differential form 66 fractional diffusion – distinct from CTRW 75 fractional diffusion coefficient 74 fractional divergence 66 fractional duality 68 fractional ergodicity 73 fractional gradient 66 fractional integral – Riemann–Liouville interpretation 52 – standard interpretations 52 fractional integral on time scales 340 fractional integration by parts 58 fractional master equation 73 fractional operator on time scales 339 fractional order phase transition 71 – Heisenberg model 71 – Ising model 71 – spherical model 71 fractional potential theory 69 fractional quantum mechanics 70 fractional spectral calculus 60 fractional stationarity 73 fractional time 72 fractional time evolution 72 fractional vector calculus 66 Fredholm resolvent 433 function – 1 Ψ2 (z) 314 – α-harmonic 164, 165, 173, 175, 182, 184 – regular 165, 173 – singular 173

476 | Index

– analytic 95, 107, 131, 140 – Bernstein 113, 180, 213, 254, 292 – Bessel 243, 249 – generalised 306, 309 – Beta 26 – characteristic 393 – complete Bernstein 113, 115 – completely monotone 113, 209, 210, 213, 253–255, 280, 292 – cumulative distribution 391 – distribution – probability 292 – entire 132, 247, 276 – error 274 – excessive 65 – FC special 131, 153, 200 – Fox H 93, 95, 99, 128, 131, 153, 200, 211, 219, 238, 243, 252, 272, 316, 319 – Gamma 25, 48, 197, 208, 244 – generalized – Wright 93 – Green 164, 166, 171 – Heaviside step 51 – holomorphic 61 – hyper-Bessel 151 – hypergeometric 298 – Gauss 29, 196 – generalized 168, 244, 279, 280 – hypergeometric type 298 – hyperharmonic 65 – indicator 248, 465 – Kilbas–Saigo 271 – Lebesgue integrable 95, 131 – locally integrable 49, 51 – Luchko–Kilbas–Kiryakova 272 – Mainardi 245 – Meijer G 93, 95, 97, 99, 128, 132, 201, 219 – Meijer G- 167 – Mittag-Leffler 74, 93, 107, 132, 148, 200, 202, 212, 213, 237, 247, 253, 254, 269, 274, 277, 288, 305, 412, 433 – binomial 74 – generalized 74, 246 – multi-index 93, 107, 132, 151–153, 155 – three-parameter 312 – two-parameter 281, 282, 303 – negative definite 65 – non-negative definite 77 – periodic 53

– power 48 – Prabhakar 271, 312 – probability density 212, 215, 253, 254, 264, 386, 408, 418, 465 – radial 467 – slowly varying definite 77 – special 87 – Stieltjes 114 – two-parametric – Mittag-Leffler 269 – vector index – Mittag-Leffler 132 – Volterra 281 – Wright 200, 202, 204, 212, 214, 217, 242, 248–250, 261, 264, 265, 272, 290, 311, 312 – four parameter 218, 250, 251, 254, 261, 264 – generalized 132, 153, 211, 252, 255, 258, 259 – Wright–Fox 279 functional 365 – complementary 373 – dual 373 – primal 371, 373 functional calculus 59 – Bochner–Phillips 121 – for operators 59 – for polynomials 59 Gelfand isomorphism 50 glassy dynamics 73 Gohberg–Krein conjecture 445 Hamiltonian 372, 373 heat kernel 465 Heisenberg model – fractional order transition 71 Hille–Phillips calculus 63 Hurst index 416 hyperharmonic functions 65 hyperinvariant subspace 432 induced automorphism 72 inequality – Gronwall 332 – Hardy–Littlewood–Sobolev 161 – Harnack – boundary 175, 185 – ultra-metric 462 infinitesimal generator 62

Index |

initial condition – Cauchy-type 286 integral 49 – Cauchy 61 – fractional – Cauchy integral interpretation 61 – common symbols 53 – convolution quotient interpretation 57 – distributed order 122 – distributional interpretation 58 – domain of definition 52 – Erdélyi–Kober 38, 90, 133, 137, 205, 206, 261 – Fourier transform interpretation 64 – functional calculus interpretation 59 – general 117 – generalized 96 – generalized Riemann–Liouville 137 – Hadamard 41 – Hille–Phillips interpretation 63 – hypergeometric 106, 137, 149 – Laplace transform interpretation 62 – Liouville 228, 229 – lower limit definition 52 – Mellin transform interpretation 64 – multiple Erdélyi–Kober 95, 96, 136, 141, 151 – notation 53 – Riemann–Liouville 25, 29, 30, 52, 89, 129, 205, 215, 282, 362, 427, 430 – Riemann–Liouville definition 52 – Riesz 54 – Riesz interpretation 54 – Riesz–Dunford interpretation 62 – spectral projection interpretation 60 – Stieltjes transform interpretation 64 – upper limit definition 52 – Weyl 37, 53 – Weyl definition 53 – Weyl interpretation 53 – generalized – Laplace–Abel 270 – hypersingular 42, 259 – improper 201 – iterated 51 – Mellin–Barnes 131, 205, 207, 210, 212, 216, 243, 254, 272, 299 – Mellin–Barnes type 200 – n-fold definite 25 integral equations of fractional order 342

integral representation 341 – hypersingular 464 integration 49 – fractional 52 interpolation spaces 60 interpretation 52 – geometric 66 – mathematical 52 – physical 67 – physical and geometrical 154, 155 – Riemann–Liouville 52 interpretation of α – anomalous diffusion 74 – as critical exponent 71 – continuous time random walk 73 – geometric 66 – physical 67 – statistical mechanics 71 – stochastic 65 – thermodynamic 70 interpretation of Dα 52 – anomaous diffusion 74 – Balakrishnan 64 – Cauchy integral 61 – convolution quotient 57 – CTRW 73 – distributional 58 – Fourier transform 64 – functional calculus 59 – fundamental constraints 68 – generalized Riemann–Liouville 56 – generalized Riesz 56 – generalized Weyl 56 – Grünwald–Letnikov 55 – Hille–Phillips 63 – Laplace transform 63 – Liouville–Caputo 56 – localized Riemann–Liouville 56 – Marchaud–Hadamard 55 – mathematical 52 – mathematical vs. physical 68 – Mellin transform 64 – Mikusinski 58 – physical 67 – Riemann–Liouville 54 – Riesz–Dunford 61 – spectral projection 60 – Stieltjes transform 64 – thermodynamic 70

477

478 | Index

– type changing 66 interpretation of Iα 52 – mathematical 52 – Riemann–Liouville 52 – Riesz 54 – Weyl 53 interval – maximum existing 336 irreversibility 73 Ising model – fractional order transition 71 iteration 50 Jacobson radical 50 kernel – Gauss–Weierstrass 175 – logarithmic 162 – Martin 165–167, 172, 174 – Poisson 164, 166, 167, 172, 174 – Riesz 161, 162 – Sonine 117 Lagrangian 366, 378 Laplacian – fractional 42, 65, 69, 215, 259, 263, 413, 416 – fractional power 69 Leibniz formula 48 Leibniz rule 48, 49, 66 lemma – fundamental 363 – Jordan 203, 259 Lévy process 65 Lévy walk 400 line – Stokes 309 linear visco-elasticity 292 local equilibrium 71, 73 local fractional derivative 56 long-time limit 71 map – Dirichlet-to-Neumann 178 – Monna 467 measure – α-harmonic 171, 174 – Dirac 63 – Lévy 170, 176, 394, 399, 418 measure space 49

method of successive approximations 331 Mikusinski calculus 58 momentum – generalized 372 multi-order 95, 96, 103, 143 – fractional 136 multiplier – Fourier 176 n-fold integration 88 non-locality 69 notations for Iα 53 nth derivative 27, 31 operational analogy 48 operator – accretive 60, 430 – bounded 76 – closed 76 – composition 76 – continuous 76 – convolution 54 – cyclic 431 – difference 50 – differential – hyper-Bessel 97, 106, 138, 144, 150 – dissipative 444 – domain 75 – Dynkin characteristic 173 – Dzrbashjan–Gelfond–Leontiev – multiple 152 – embedding 436 – Erdélyi–Kober 148 – multiple 128 – Fourier multiplication 53 – fractional differentiation – generalized 151 – fractional integration – generalized 93, 137 – hypergeometric 91 – Wright–Erdélyi–Kober 92 – fractional powers 63 – Gelfond–Leontiev 107, 152 – initial conditions 145 – injective 76 – integer power 50, 59 – integral – Marichev-Saigo-Maeda fractional 106 – Saigo fractional 106

Index | 479

– Volterra 428 – integration – generalized fractional 87, 136 – iteration of 50 – jump – forward 341 – kernel 75 – Laplace 159, 185 – Dirichlet fractional 186, 187, 189 – fractional 159, 164, 167, 169, 170, 176, 185 – Navier fractional 187 – Neumann fractional 187 – regional fractional 187 – spectral fractional 187 – left-inverse 27, 31, 34, 134, 144, 147 – linear 26, 34, 75 – Marichev–Saigo–Maeda 137 – non-negative 76 – nonlocal 68 – positive 76 – pseudo-differential 43, 215, 234, 394, 420, 464 – quasisimilar 434 – range 75 – resolvent 76 – Saigo 149, 150 – sectorial 61, 77, 430 – surjective 76 – transition 170 – translation 50 – transmutation 221 – unicellular 431 – Vladimirov 461, 464 – Volterra 430, 442, 453 – sectorial 448 – Wright–Erdélyi–Kober 103 – multiple 103–105 p-adic numbers 461, 462 paradox – irreversibility 73 – reversed irreversibility 73 phase transition 70, 71 – Ehrenfest order 70 – fractional order 70 Poisson integral 69 polar set 172 polynomial – harmonic 167, 168

– Jacobi 168 potential – Bessel 162, 182 – Feller 43 – fractional 69 – Riesz 42 – Riesz 161, 162, 229 precompact 332 principle – Banach 342 – Hamilton 366, 370 – generalized 367 – maximum 179, 182 – strong 180 – subordination 214 – variational 365 – complementary 371, 373 – dual 373 – primal 373 problem – Cauchy 122, 148, 150, 214, 215, 236, 258, 287, 290, 411, 414, 465, 469, 470 – fractional 408, 411, 413, 414 – tempered fractional 422 – Cauchy-type – fractional order 331 – initial value 148 – initial-value – abstract 214 – isoperimetric 353 – Poisson 165, 180 – signaling 258 – variational – fractional 365 process – α-stable – censored 188 – reflected 188 – α-stable Lévy 416 – killed 186, 187 – Lévy 280, 388, 394, 409 – α-stable 169, 175, 186 – stable 389 – Markov 387, 412, 417, 466 – non-Markovian 412 – Poisson 293, 414 – fractional 293, 414 – tempered fractional 422 – renewal 389, 415

480 | Index

– stochastic 293, 409 – self-similar 396 – symmetric stable 466 product rule 49 projection 436 projector 145 property – composition 98, 99, 103 – mapping 140 – non-Archimedean 463 – operational 104 – semigroup 26, 28, 30, 101, 104, 129, 142, 206 quantum mechanics – fractional 70 random fractal 416 random variable 386 random walk 414 – continuous time 385, 392, 414 – coupled 392 – overshooting 392 – uncoupled 386 rational functional calculus 60 regular variation 77 regularity – boundary 184 relation – asymptotic 277 – differential 275 – integral 276 – operational 233, 235 – recurrence 245, 274 representation – integral – Mellin–Barnes 281 resolvent 76, 122 resolvent set 76 Riesz kernel 54 Riesz potential 54 – conjugate 54 Riesz–Dunford calculus 61 – bounded operators 61 – extended 62 – sectorial operators 62 rule – Leibniz 29 – Leibniz type 28, 35 – operational 141

scaling limit 71 Schrödinger equation – fractional 68 sectorial operator 77 semigroup 62, 121, 186 – bounded 62 – infinitesimal generator 62 – strongly continuous 62 series – Neumann 284 similarity of operators 448 slowly varying function 77 solution – fundamental 151, 162, 215, 236, 258, 259, 263, 291, 466 – global 337, 338 – local 338 – noncontinuable 336 – scale-invariant 260–262 space of functions – Lizorkin 32, 177, 227, 464 – Sobolev 182, 441 – Zygmund–Hölder 183 spectral calculus 60 spectral multiplicity 431 spherical model – fractional order transition 71 stable subordinator 72 stationarity 73 stationarity breaking 73 Stokes line 301, 303, 305 Stokes phenomenon 302 strong solution 471 subordination – Bochner 175, 187 subordinator 396, 409 – β-stable 398, 415, 418 – Brownian 417 – inverse – distributed order 423 – stable 409, 416 – inverse 409, 410, 413, 414, 418 – inverse tempered 421 – tempered 420, 421 symbol – Fourier 419 – Pochhammer 279 tautochrone 328

Index |

tautochrone problem 327, 330 theorem – Bernstein 209, 253, 280 – central limit 414 – composition/decomposition 139 – continuation 336 – fixed-point – Schauder 332, 335, 342 – residue 259 – Cauchy 203 – Titchmarsh convolution 431 thermistor problem – nonlocal 334 – fractional-order 339 – nonlocal Riemann–Liouville 329 thermistor-type problem 328 thermodynamic potential 71 time scales 329, 339 topological pressure functional 71 transform – integral – Borel–Dzrbashjan type 152, 153 – Fourier 31, 64, 176, 215, 227, 230, 281, 289, 386, 419, 463 – fractional Fourier 231, 235, 236 – fractional inverse Fourier 232

481

– Hilbert 235 – inverse Fourier 177, 227, 289 – inverse Laplace 208, 289 – inverse Mellin 198 – Laplace 32, 34, 62, 115, 130, 203, 208, 246, 280, 386, 410, 418 – Laplace type 145, 153 – Mellin 33, 64, 94, 100, 133, 139, 140, 198, 200, 253, 281 – Obrechkoff 145 – Stieltjes 64 – Kelvin 163, 166, 167 – quasiaffine 434 transformation – scaling 260 transversality conditions 351, 352 types of fractional derivative 55 uniqueness 342 variable – similarity 260 variation 350 wavelets – p-adic 466