Handbook of Fractional Calculus with Applications: Volume 8 Applications in Engineering, Life and Social Sciences, Part B 9783110571929, 9783110570922

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Table of contents :
Preface
Contents
Economic models with power-law memory
Four-quadrant fractors and their applications in fractional order circuits
Energy harvesting in dynamical systems with fractional-order physical properties
Fractional kinetics of charge carriers in supercapacitors
Recursive-operational method for fractional systems
Discrete-time fractional signals and systems
Signal prediction using fractional derivative models
Spectral methods within fractional calculus
Design and generation of fractional-order multi-scroll chaotic attractors
Discrete fractional masks and their applications to image enhancement
Existence theory for fractional differential equations with nonlocal integro-multipoint boundary conditions with applications
Index
Recommend Papers

Handbook of Fractional Calculus with Applications: Volume 8 Applications in Engineering, Life and Social Sciences, Part B
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Dumitru Baleanu, António Mendes Lopes (Eds.) Handbook of Fractional Calculus with Applications

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

Volume 1: Theory Anatoly Kochubei, Yuri Luchko (Eds.), 2019 ISBN 978-3-11-057081-6, e-ISBN (PDF) 978-3-11-057162-2, e-ISBN (EPUB) 978-3-11-057063-2 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 Baleanu, 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

Dumitru Baleanu, António Mendes Lopes (Eds.)

Handbook of Fractional Calculus with Applications |

Volume 8: Applications in Engineering, Life and Social Sciences, Part B Series edited by Jose Antonio Tenreiro Machado

Editors Prof. Dr. Dumitru Baleanu Çankaya University Faculty of Arts and Sciences Department of Mathematics Öğretmenler Caddesi 14 06530 Ankara Turkey [email protected]

Prof. Dr. António Mendes Lopes University of Porto Faculty of Engineering Dept. of Mechanical Engineering Rua Dr. Roberto Frias 4003-465 Porto Portugal [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 978-3-11-057092-2 e-ISBN (PDF) 978-3-11-057192-9 e-ISBN (EPUB) 978-3-11-057107-3 Library of Congress Control Number: 2019934659 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) has 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 original contributions of important mathematicians, physicists and engineers. FC had a rapid development during the last few decades, both in mathematics and applied sciences, 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 on theoretical aspects and applications of FC in areas such as physics, engineering, biology, medicine, economy, and 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. The Handbook of Fractional Calculus with Applications (HFCA) intends to fill that gap and provides the reader 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 encyclopedia of FC and its applications, reflecting the core of the present day scientific knowledge, is now available with the HFCA. This work is distributed in eight distinct volumes, each one developed under the supervision of its editors. The eighth volume of HFCA includes 11 chapters on FC applications in engineering, life and social sciences. The volume starts with an interesting chapter on economic models with power-law memory. The next three chapters are top level contributions addressing fractors and their applications in fractional-order circuits, energy harvesting in dynamical systems, and fractional models of charge kinetics in supercapacitors. The following is a group of four chapters on signal processing. The reader will find state of the art techniques for fractional signals and systems, signal prediction using fractional models, and spectral methods within FC. The next two chapters address the design and generation of the fractional-order multi-scroll chaotic attractors, and discrete fractional masks with applications to image enhancement. One chapter discussing the existence theory for fractional differential equations with non-local integro-multipoint boundary conditions with applications ends the volume. Our special thanks go to the authors of individual chapters, which are excellent surveys of selected classical and new results in several important fields of FC. The https://doi.org/10.1515/9783110571929-201

VI | Preface editors believe that the HFCA will represent a valuable and reliable reference work for all scholars and professionals willing to engage in research in this challenging and timely scientific area. Dumitru Baleanu and António Mendes Lopes

Contents Preface | V Vasily E. Tarasov Economic models with power-law memory | 1 Avishek Adhikary and Karabi Biswas Four-quadrant fractors and their applications in fractional order circuits | 33 Grzegorz Litak, Cedrick A. K. Kwuimy, and Benjamin Ducharne Energy harvesting in dynamical systems with fractional-order physical properties | 63 Renat T. Sibatov and Vladimir V. Uchaikin Fractional kinetics of charge carriers in supercapacitors | 87 Gabriel Bengochea, Manuel Ortigueira, Luis Verde-Star, and António M. Lopes Recursive-operational method for fractional systems | 119 Manuel D. Ortigueira, J. Tenreiro Machado, Fernando J. V. Coito, and Gabriel Bengochea Discrete-time fractional signals and systems | 149 Tomas Skovranek and Vladimir Despotovic Signal prediction using fractional derivative models | 179 E. H. Doha, M. A. Zaky, and M. A. Abdelkawy Spectral methods within fractional calculus | 207 Liping Chen and Ranchao Wu Design and generation of fractional-order multi-scroll chaotic attractors | 233 Guo-Cheng Wu, Dumitru Baleanu, and Yun-Ru Bai Discrete fractional masks and their applications to image enhancement | 261 Bashir Ahmad, Sotiris K. Ntouyas, and Ahmed Alsaedi Existence theory for fractional differential equations with nonlocal integro-multipoint boundary conditions with applications | 271 Index | 283

Vasily E. Tarasov

Economic models with power-law memory Abstract: Macroeconomic models, which take into account the effects of power-law fading memory, are considered. The power-law long memory is described by using the mathematical tool of fractional calculus. The model equations are fractional differential equations with derivatives of non-integer order. Solutions of the fractional differential equations of these macroeconomic models are suggested. Examples of the dependence of macroeconomic dynamics on long memory are considered. The asymptotic behaviors of the solutions, which characterize the rate of technological growth with memory, are described. The principles of economic dynamics with long memory are proposed. It is shown that the effects of a long memory can change the economic growth rate and dominant parameters, which determine the growth rates. Accounting of memory effect in the model can lead to qualitative changes, including economic growth appearing instead of decrease. Keywords: Macroeconomics, economic growth model, long memory, dynamic memory, fading memory, fractional calculus, fractional dynamics, fractional derivative, derivative of non-integer order MSC 2010: 26A33, 34A08, 91B55

1 Introduction Mathematical models of national, regional and global economy are powerful tools for theoretical studies of real macroeconomic processes [1, 2, 20, 33, 5, 53]. Macroeconomic models are constructed to explain the relationship between national income, production, investment, consumption, employment, saving, and other factors and indicators. Important advantages of macroeconomic models are their accessibility for detailed mathematical analysis and the possibility of studying macroeconomic processes by using a small number of input data. These models also used to consider possible alternatives of economic policy and their long-term consequences. The “marginal revolution” and the “Keynesian revolution” introduced fundamental economic concepts into economic analysis, which led to the use of mathematical tools based on derivatives and integrals of integer order, differential and finite difference equations. Modern macroeconomic growth models with continuous time are Acknowledgement: The author thanks Valentina V. Tarasova (Lomonosov Moscow State University Business School, Lomonosov Moscow State University). Vasily E. Tarasov, Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia, e-mail: [email protected] https://doi.org/10.1515/9783110571929-001

2 | V. E. Tarasov mathematically described by differential equations with derivatives of integer order. The models use various simplifying assumptions, which lead to disadvantages, since models do not take into account some important economic effects. Some of these disadvantages are partly caused by restrictions of the mathematical tools. It is well known that the derivatives of integer order are determined by the properties of the differentiable functions only in infinitesimal neighborhoods of the considered points. As a result, the differential equations with derivatives of integer order with respect to time cannot describe processes with long memory. In fact, these equations describe only such economic processes, in which agents actually have a total amnesia. In other words, economic models which use derivatives of integer order can be applied, when economic agents forget the history of changes of economic indicators and factors during an infinitesimally small period of time. Obviously, the assumption as regards the absence of memory of economic agents is a strong restriction. Methods which describe processes with long memory can be conditionally divided into the following three approaches: (A) the approach based on integral equations and integro-differential equations; (B) the approach based on statistics and time series analysis; (C) the approach based on fractional calculus. The first one who mathematically described processes with long memory in physics was Boltzmann, who published work on this topic in 1874 and 1876 [9, 10]. Boltzmann proposed an integral equation to describe the dynamics of isotropic viscoelastic media [9, 10], whose behavior demonstrates memory effects. He describes the stress at time t as a function that depends on the history of the process at τ < t. Boltzmann formulated the linear superposition principle and the memory fading principle. The Boltzmann theory has been developed by Volterra in 1928 and 1930 [110, 111] in the form of the heredity concept and its application to physics [111]. Volterra formulated the principles, called by him the general laws of heredity [111]. Volterra also developed the theory of integral equations. The integral and integrodifferential equations are actively used to describe processes with memory in physics. The general theory of the integral and integro-differential equations is rarely used in the economic and financial models with memory. Note that a wide class of integral and integro-differential equations refer to fractional calculus and the theory of equations with derivatives and integrals of non-integer order. The processes with long memory can be described by statistical methods that allow one to analyze the behavior of a time series [6, 49, 7]. The first group of statistical methods is connected with analysis in time domain such as correlation analysis. The second group is related with analysis in the frequency domain such as spectral analysis. The processes with memory have empirically been observed as long-range time dependencies, for which the correlations decay to zero more slowly than it can be expected from independent data or data resulting from classical Markov and ARMA models [6, 49, 7]. The processes, which are characterized by slowly decaying autocorrelations or spectral density, significantly change the statistical estimates and predictions. Therefore standard methods, which are used for analyzing classical time se-

Economic models with power-law memory | 3

ries of the Markov and ARMA processes, cannot be applied to processes with memory [6, 49, 7, 54, 108]. The effect of a long memory in economics was discovered by Granger in 1964/1966, who received the Nobel memorial prize in economic sciences in 2003 “for methods of analyzing economic time series with common trends (cointegration)”. The importance of long-term time dependence in economics was indicated by Granger in his technical report [21] in 1964, and then in his article [22] in 1966 (see also [23, 24]). Granger showed that a number of spectral densities, which are estimated from economic time series, have a similar form. Then, to describe economic processes with long memory, Granger and Joyeux in 1980 and Hosking in 1981 proposed the fractional ARIMA models [25, 31], which are also called ARFIMA(p, d, q). The fractional ARIMA models are generalizations of the classical ARMA and ARIMA models, which are the most popular types of linear models in time series analysis. The fractional ARIMA(p, d, q) models mathematically can be considered as a generalization of the ARIMA(p, d, q) model from a positive integer order d to non-integer (positive and negative) orders d. This generalization is analogous to the generalization of derivatives and integrals from integer to non-integer order. To generalize ARMA and ARIMA models Granger, Joyeux, and Hosking [25, 31] proposed the so-called fractional differencing and integrating in the framework of the discrete time models [3, 50, 4, 18]. In economics these fractional differencing and integrating methods were proposed and then are used without any connection with the fractional calculus and the well-known fractional differences of non-integer order. In [66, 86], it was shown that these fractional differencing and integrating methods are concerned with the well-known Grunwald–Letnikov fractional differences, which have been suggested in 1867 and 1868 by Grunwald [26] and Letnikov [45], i. e., 150 years ago. These fractional differences of non-integer order are actively used in the fractional calculus [55, 52, 38]. In the continuous time limit these fractional differences of positive orders give the Grunwald–Letnikov fractional derivatives that coincide with the Marchaud fractional derivatives of non-integral order (see Theorems 4.2 and 4.4 of [55], p. 380 and 382) and they have the same domain of definition [55]. To describe processes with long memory, it is possible to use the fractional calculus [55, 52, 38, 14, 87]. The fractional derivatives and integrals of non-integer order have a set of nonstandard properties [58, 61, 62, 48, 64]. For example, such properties include a violation of the standard Leibniz rule [58, 61]; it describes the derivative of the product of functions and shows a violation of the standard chain rules [62], which described the derivative of the composition of functions. It should be emphasized that the violation of the standard form of the Leibniz rule is a characteristic property of derivatives of non-integer order [58]. These nonstandard properties of the fractional derivatives are important for describing complex properties of processes with long memory and spatial nonlocality. At present, the fractional calculus is actively being used to describe wide class of processes with memory in physics (see [57, 65] and the references therein).

4 | V. E. Tarasov Recently, the fractional calculus approach has been applied to finance in [56, 46, 42, 51, 11, 47, 8, 106, 107, 41, 32, 27, 17] in the framework of econophysics. Unfortunately, the papers referred to consider only the financial processes. The basic economic concepts and principles for economic processes with memory are not considered in this work. Fractional calculus approach has been used to define basic concepts of economic processes with memory in References [69–73, 89, 90, 74–78, 91, 99, 79], and to describe the dynamics of these processes [80–82, 92, 100, 98, 83–85, 101, 93, 67, 94–96] in the framework of continuous time models. The concept of memory itself for economic processes is discussed in [88, 98, 68]. Using the fractional calculus approach to describe the long memory, generalizations of some basic economic notions have been proposed, such as an elasticity of fractional order [69–72], an accelerator and a multiplier with memory [73, 89, 90, 74], a marginal value of non-integer order [75–77, 87], measures of risk aversion [78, 91], and methods of deterministic factor analysis [99, 79]. The use of the concepts of accelerator and multiplier with memory, which is proposed in [73, 89, 90, 74], allows us to generalize the classical macroeconomic growth models that have been suggested by Harrod and Domar [28–30, 15, 16], Keynes [34, 35], Leontief [43, 44] and others. Tarasova and Tarasov proposed macroeconomic models with long memory, which are generalizations classical growth models, including the natural growth model [80], the Harrod–Domar model [81, 82, 92, 100, 98], the Keynes model [83–85], the dynamic Leontief (intersectoral) model [101, 93, 67, 94], the growth model with constant pace [95], the logistic model [96] and other. Note also an application of fractional calculus for modeling the economy in recent papers [13, 12] and [105, 104, 103, 102]. In this chapter, we consider an example of simple macroeconomic model that allows us to demonstrate an application of fractional calculus to describe economic processes with long memory. For simplification, we assume that the fading of the long memory has a power-law form. We find solutions of the fractional differential equations that describe macroeconomic models with power-law memory. Using these solutions and its asymptotic behavior, some principles of economic dynamics with long memory are proposed. We prove that memory effects can lead to qualitative changes of economic dynamics.

2 Macroeconomic growth model with power-law memory To demonstrate the effects of a long memory in economics, we will consider a generalization of the Harrod–Domar model with continuous time [28–30, 15, 16, 109], which is one of the simplest classical models of economic growth ([1], p. 64–69, [2], p. 197– 203, [20], p. 46–59). This classical model does not take into account memory effects. The generalization of this model, which has been suggested in [81, 82, 92, 98, 100],

Economic models with power-law memory | 5

allows us to demonstrate the impact of memory effects on the macroeconomic dynamics. Note that the Harrod–Domar model can be considered as a one-sectoral form of the dynamic Leontief model [43, 44], which is also called the input–output model. A generalization of the dynamic Leontief model, in which long memory is taken into account, has been proposed in References [101, 93, 67, 94]. Let us give the main assumptions that will be used in the macroeconomic model considered. The first assumption of the model is that exports and imports are equal; losses will be included in the used national income (input). Under these assumptions, the values of the produced national income (output) and the used national income (input) will be equal. As a result, the balance of production and distribution of gross product at any given time moment t ≥ 0 has the form X(t) = AX(t) + Y(t),

(1)

where the function X(t) describes the gross product, Y(t) is the national income (output) and the coefficient A is the material consumption of the gross product (0 < A < 1). The coefficient A includes not only the current production costs, but also the replacement investment and the costs of capital repairs of basic production assets. Equation (1) can be written as the equation of a linear multiplier, X(t) = mY(t),

(2)

where the coefficient m = 1/(1 − A) is the multiplier of the gross product, which characterizes the ratio of gross product and national income. In general, we can take into account a long memory by using the concept of multiplier with memory, which is proposed in [73, 89, 90, 74], instead of the standard multiplier (2) and (1). For simplicity, we will not consider this generalization in this chapter. The second assumption of the macroeconomic model is that the used national income Y(t) is divided into two parts: the investment I(t) (accumulation of basic production assets) and the non-productive consumption C(t) (non-productive accumulation, growth of material current assets, state material reserves, losses). For brevity, the function I(t) will be called “accumulation”, and C(t) will be called “consumption”. As a result, we have the balance equation Y(t) = I(t) + C(t),

(3)

where the independence of consumption behavior is assumed. The third assumption of the standard model without memory is the direct proportionality of production accumulation and the growth of the national income. This assumption gives a relationship between investment (accumulation) and the growth of the gross product. This relationship is realized by the accelerator equations with the coefficients of capital intensity. In the standard macroeconomic models, an instantaneous transformation of investment (capital investment) in the growth of gross

6 | V. E. Tarasov product is also assumed, that is, these models assume we may neglect the delay and memory. The direct proportionality of the production accumulation (the gross product X(t)) and the growth of the national income (the capital investment I(t)) is described by the linear accelerator equation I(t) = b

dX(t) , dt

(4)

where b is the capital intensity of the gross product. Using equations (2), we can write equality (4) in the form of a direct proportionality between investment (productive accumulation) and the growth of the national income I(t) = B

dY(t) , dt

(5)

where B = bm = b(1 − A)−1 is the accelerator coefficient, which describes the capital intensity of the national income (incremental capital intensity, differential capital intensity). The fourth assumption, which is usually used in the standard macroeconomic models, is the absence of a long memory, i. e. full amnesia of all economic agents. Equation (4) means that the capital investment I(t) at time t is determined by the change of the gross product X(t) in an infinitesimal neighborhood of the instant of time (that is, it is determined by an infinitesimally close past). This assumes neglecting the memory effects. In other words, instant amnesia of economic agents is assumed in equations (4) and (5). This means that these equations assume the instant forgetting of the history of changes in the gross product and the investments made. Let us give the main equation of the standard macroeconomic model of national income dynamics. Substituting equation (5) into equation (3), we obtain Y(t) = B

dY(t) + C(t). dt

(6)

Equation (6) describes the dynamics of the national income within the framework of the proposed macroeconomic model. Equation (6) shows that, for a given parameter B, the dynamics of the national income Y(t) is determined by the behavior of the function C(t). Equation (2) allows us to describe the change of the gross product by using the dynamics of national income, which is described by solutions of equation (6). Mathematically, the standard model of the national income dynamics is described by linear nonhomogeneous differential equations of the first order. We can state that one of the strong limitations imposed on the standard model, which is described by equation (6), is the absence of memory. This assumption is a significant drawback of standard macroeconomic models. This drawback is due to the shortcomings of the mathematical apparatus used in standard macroeconomic models. Derivatives of the first order, which are used in the linear accelerator equations (4)

Economic models with power-law memory | 7

and (5), imply an instantaneous change of the endogenous variable I(t), when changing the growth rate of the gross product X(t) and national income Y(t). In other words, the investments at time t are determined by the properties of the national income in an infinitesimal neighborhood of this time point. Because of this, the accelerator equation (5) does not take into account the effects of a long memory. As a result, the differential equation (6) can be used only to describe an economy in which all economic agents have an instant amnesia. This restriction substantially narrows the field of application of macroeconomic models to describing real economic processes. In many cases, economic agents can remember the history of changes of the gross product, national income and investment. As a result, this memory of the history of changes can influence decision-making by economic agents. To take into account the effects of a power-law memory [98, 89] in macroeconomic models, we can apply the fractional calculus [55, 52, 38, 14]. The use of derivatives and integrals of non-integer order allows us to generalize the standard equation (4), which describes the relationship between the investment I(t) and the gross product X(t). There are different types of derivatives of non-integer order [55, 52, 38]. We will use the left-sided Caputo derivative [52, 38, 14] with respect to time, since the action of this derivative on a constant function gives zero. To take into account the history of changes of dynamic variables in the past, we use the left-sided fractional derivative. Using the notion of marginal value of non-integer order [75–77, 87], it is possible to obtain the accelerator equation with memory. Note that the concept of economic accelerator with memory has been suggested in [73, 89, 90, 74]. In the case of a long memory with one-parameter power-law fading, the linear equation of the accelerator with memory is written in the form I(t) = bDα0+ X(t),

(7)

where (Dα0+ X)(t) is the Caputo derivative of the order α ≥ 0 [55, 52, 38, 14]. Equation (7) assumes that the function X(τ) has integer-order derivatives up to (n − 1)th order, which are absolutely continuous functions on the interval [0, t]. Equation (2) allows us to write equation (7) for the national income in the form I(t) = BDα0+ Y(t),

(8)

where B = mb is the capital intensity of the national income (incremental capital intensity, differential capital intensity), when the memory effects are taken into account. In general, the capital intensity depends on the parameter of memory fading, i. e. B = B(α). In order to have an easily interpretable dimension of variables, we can use time t as a dimensionless variable. For α = 1 equation (8) gives equation (5), since the Caputo fractional derivative of the order α = 1 coincides with the first derivative (D10+ Y)(t) = dY(t)/dt (see [38], p. 92).

8 | V. E. Tarasov Substituting the expression for the investment I(t), which is given by equation (8), into the balance equation (3), we obtain the fractional differential equation Y(t) = BDα0+ Y(t) + C(t).

(9)

For α = 1 equation (9) gives equation (6). Equation (9) describes the economic dynamics with one-parameter power-law memory. If the parameter B is given, then the dynamics of the national income Y(t) is determined by the behavior of the function C(t). As a result, this macroeconomic dynamics with memory is described by fractional differential equations of the noninteger order α > 0.

3 Macroeconomic dynamics with long memory 3.1 General solution of model equation To solve equation (9), we can use Theorem 5.15 of [38], p. 323. The fractional differential equation (9) can be written in the form (Dα0+ Y)(t) − B−1 Y(t) = −B−1 C(t),

(10)

where n − 1 < α ≤ n. If C(t) is a continuous function defined on the positive semiaxis (t > 0), then equation (10) has the solution n−1

Y(t) = YC (t) + ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ], k=0

(11)

where Y (k) (0) are for the derivatives of integer order k > 0 of the function Y(t) at t = 0, and t

YC (t) = −B−1 ∫(t − τ)α−1 Eα,α [B−1 (t − τ)α ]C(τ)dτ.

(12)

0

Here Eα,β [z] is the two-parameter Mittag-Leffler function [38, 19], which is defined by the expression zk , Γ(αk + β) k=0 ∞

Eα,β [z] := ∑

(13)

where Γ(α) is the gamma function, α > 0, and β is an arbitrary real or complex number.

Economic models with power-law memory | 9

For 0 < α ≤ 1, (n = 1), the solution (11) of equation (10) has the form t

Y(t) = −B−1 ∫(t − τ)α−1 Eα,α [B−1 (t − τ)α ]C(τ)dτ 0

+ Y(0)Eα,1 [B−1 t α ].

(14)

In the macroeconomic model, which is described by equation (10), the nonproductive consumption C(t) is considered as a function which is independent of national income and investment. This function can be a constant in time, a powerlaw function, or it can change in a more complicated way. If the non-productive consumption is represented as a fixed part of income C(t) = cY(t), where c is the marginal propensity to consume, then equation (10) can be written in the form (Dα0+ Y)(t) = B−1 (1 − c)Y(t).

(15)

In this case, the macroeconomic model coincides with the natural growth model with memory [80]. Let us analyze the behavior of the solution for various functions C(t) that describe the consumption dynamics. Two special cases can be distinguished. These cases can be identified by the terms of closed and open economic models, which have been suggested for models without memory by Leontief, who won the Nobel Laureate in Economics in 1973. The closed model of Leontief assumed that there is no external demand, when all inputs and outputs are spent on the production. In our case, the macroeconomic model will be called closed if it assumes the absence of non-productive consumption, that is, C(t) = 0. The consideration of this case allows us to estimate the greatest possible growth rate in national income, which is limited only by the value of the incremental capital intensity, i. e. by the material intensity and capital intensity of production.

3.2 Closed model with memory: Technological growth rate with memory A macroeconomic model with memory is called closed if it assumes the absence of non-productive consumption, C(t) = 0. The assumption C(t) = 0 is unrealistic, but the consideration of this case allows us to estimate the greatest possible growth rate of the national income, which is limited only by the incremental capital intensity. For the case C(t) = 0, all resources are directed to investments, as a result of which the maximum technically possible rates of growth can be determined from the solution of the model equation. The solution (11) of equation (10) with C(t) = 0 has the form n−1

Y(t) = ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ], k=0

(16)

10 | V. E. Tarasov where Eα,k+1 [z] is defined by equation (13). For 0 < α < 1, solution (16) takes the form Y(t) = Y(0)Eα,1 [B−1 t α ]. For α = 1, solution (16) gives the equation Y(t) = Y(0) exp{−t/B},

(17)

where the value λ = B−1 is often called the technological growth rate [20], p. 49 (or Harrod’s “warranted” rate of growth [1], p. 67), which is fixed by the structural constants B and m. Let us now analyze the behavior of the solution (16) at t → ∞ to obtain the asymptotic behavior that determines the technological growth rate. To describe the behavior of the two-parameter Mittag-Leffler function Eα,k+1 [λt α ] at t → ∞, we can use equation (1.8.27) of [38], p. 43, for 0 < α < 2 in the form Eα,β+1 [λt α ] =

λ−β/α −β t exp(λ1/α t) α m λ−j 1 1 −∑ + O( α(m+1) ) αj Γ(β + 1 − αj) t t j=1

(18) (19)

for t → ∞, where λ is a real number and we use the big O notation (asymptotic notation), which provides an upper bound on the growth rate of the following terms of the asymptotic series. The asymptotic equation (19) allows us to describe the behavior at t → ∞ in a macroeconomic model with power-law memory fading parameter 0 < α < 2. Substitution of equation (19) with λ = B−1 and β = k into (16) gives n−1

Y(t) = ∑ Y (k) (0) k=0

n−1

m

+ ∑ (∑ k=0

j=1

Bk/α exp(B−1/α t) α

1 Y (k) (0)Bj k−αj t + O( α(m+1)−k )), Γ(k + 1 − αj) t

(20)

where 0 < n − 1 < α < n. Equation (20) describes the behavior of the solution (16) at t → ∞. As a result, the technological growth rate with memory is equal to the value λeff (α) := λ1/α = B−1/α ,

(21)

which will be called the effective technological growth rate (or the rate of technological growth with memory) of the macroeconomic model with one-parameter power-law memory. We see that the technological growth rates of macroeconomic models with one-parameter memory do not coincide with the growth rates λ = B−1 of standard models without memory. Note that the effective technological growth rate for the parameter α = 1 is equal to the standard rate of growth without memory, λeff (1) = λ.

Economic models with power-law memory | 11

The effective technological growth rate (21) is the rate of technological growth with memory of which the fading parameter is equal to α > 0. Using equation (1.8.29) of [38], p. 43, for α ≥ 2 and t → ∞, we get the asymptotic expression Eα,β+1 [λt α ] =

2πβN 2πN λ−β/α −β t ∑ exp(λ1/α te(i α ) )e(−iβ α ) α N

m

1 1 λ−k + O( α(m+1) ), αk Γ(β + 1 − αk) t t k=1

−∑

(22)

where λ = B−1 is a real number (arg(λ) = 0) and the first sum is taken over all integer values of N that satisfy the condition |N| ≤ α/4. Using equation (22) and the inequality |N| ≤ α/4, we see that equation (19) can be used for α < 4. For α ≥ 4, the use of the Euler equation and equation (22) leads us to the rate of technological growth with memory in the form λeff (α; N) := B−1/α cos(

2πN ), α

(23)

where N runs through all integer values that satisfy the condition |N| ≤ α/4. Note that for α ≥ 4 there are several effective growth rates. However, the inequality cos(x) ≤ 1 leads to the largest value, λeff (α; 0) = λeff (α) = B−1/α ,

(24)

which is an effective technological growth rate (the rate of technological growth with memory) for memory with α ≥ 4. Let us compare the technological growth rates in the macroeconomic model with memory and the rates of growth for the standard model without memory in Table 1, where λeff (α) is the rate of technological growth with memory, and λeff (1) = λ is the rate of technological growth without memory. Table 1: Comparison of technological growth rates for models with memory and without memory. λ

B

0 0 lead to the dependence of the technological growth rates on the parameter α, which are determined by the expression λeff (α) = B−1/α ,

(25)

where B is the incremental capital intensity that characterizes the amount of investment corresponding to the change of the growth rate of the national income. Principle of changing of technological growth rates by memory: For small technological growth rates, which are described by the standard model, the effects of oneparameter memory with fading parameter α lead to decrease of the growth rates of the economy for 0 < α < 1, and it leads to an increase of the growth rates for α > 1. For the large rates of technological growth, which are described by the standard model, the effects of power-law memory with fading α lead to an increase of the growth rates for 0 < α < 1, and it leads to a decrease of the growth rates for α > 1. We can state that neglecting the memory effects in macroeconomic models can greatly change the result. Accounting of long memory can lead to new results for the same parameters of macroeconomic models.

3.3 Open model with memory: The power-law change of consumption The case of the absence of non-productive consumption C(t) = 0, which describes the closed macroeconomic model, is an unrealistic case. We have considered this case to calculate the maximum possible (technological) rate of growth with memory. An important realistic particular case, which can be used in macroeconomic growth models, is the power law and constancy of the non-productive consumption. Let us describe important particular cases of the power-law behavior of the nonproductive consumption C(t). The constancy of the consumption can be considered as a special case of the power-law function.

Economic models with power-law memory | 13

In the power-law case one can obtain an explicit form of the analytical solution. In addition, this analytical solution allows us to describe the wide class of the consumption functions C(t), which are described by power series and finite polynomials. Let us consider the consumption function in the form C(t) = Ct μ−1 ,

(26)

where μ > 0. The constant consumption function (C(t) = C) is a special case of (26), when μ = 1. Let us obtain the general solution of equation (10) with the function (26). We note that equation (12) can be written in the form t

YC (t) = −B−1 ∫ τα−1 Eα,α [B−1 τα ]C(t − τ)dτ.

(27)

0

To calculate the integral (27) with the power-law function (26), we can use equation (4.10.8) from [19], p. 86, or equation (4.4.5) of [19], p. 61, where Γ(μ) should be used instead of Γ(α), in the form t

∫ τβ−1 Eα,β [λτα ](t − τ)μ−1 dτ = Γ(μ)t β+μ−1 Eα,β+μ [λt α ],

(28)

0

where β > 0 and μ > 0. Note that in equation (28) the exponent μ can be considered as a positive integer, or as a real positive number. For β = α, equation (28) has the form t

∫ τα−1 Eα,α [λτα ](t − τ)μ−1 dτ = Γ(μ)t α+μ−1 Eα,α+μ [λt α ].

(29)

0

Using (29), equation (27) with the function (26) takes the form YC (t) = −CB−1 Γ(μ)t α+μ−1 Eα,α+μ [B−1 t α ],

(30)

where μ > 0. As a result, the solution (11), (12) of equation (10) with the consumption function (26) has the form n−1

Y(t) = ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ] k=0

− CB−1 Γ(μ)t α+μ−1 Eα,α+μ [B−1 t α ],

(31)

where 0 < n − 1 < α ≤ n. To describe the behavior of equation (30) at t → ∞, we can use equation (19) (see also [38], p. 43, and [19]). Substitution of (19) with λ = B−1 and β = α + μ − 1 into (30) gives

14 | V. E. Tarasov CΓ(μ)B(1−2α−μ)/α exp(B−1/α t) α m CΓ(μ)Bj−1 α(1−j)+μ−1 1 +∑ + O( αm+1−μ ), t Γ(α(1 − j) + μ) t j=1

YC (t) = −

(32)

where μ > 0 and we use the big O notation (asymptotic notation), which provides an upper bound on the growth rate of the following terms of the asymptotic series. Using equations (32) and (20), we can see that the rate of growth with memory, which is described by equation (31) is defined by equation (21), i. e. λeff (α) := λ1/α = B−1/α . This growth rate coincides with the rate of technological growth with memory. For equation (30), we can use equation (4.2.3) of [19], p. 57, which has the form Eα,β [z] =

1 + zEα,α+β [z]. Γ(β)

(33)

Using z = λt α and β = μ, equation (33) takes the form t α Eα,α+μ [λt α ] =

1 1 (Eα,μ [λt α ] − ). λ Γ(μ)

(34)

Then equation (28) can be written as t

∫ τα−1 Eα,α [λτα ](t − τ)μ−1 dτ = 0

1 μ−1 t (Γ(μ)Eα,μ [λt α ] − 1). λ

(35)

Using equation (35) with λ = B−1 , equation (30) can be written in the form YC (t) = Ct μ−1 (1 − Γ(μ)Eα,μ [B−1 t α ]).

(36)

As a result, the equation of the macroeconomic model with power-law memory, whose fading parameter is α (0 < n − 1 < α ≤ n), and with the consumption function (26), has the solution Y(t) = Ct μ−1 (1 − Γ(μ)Eα,μ [B−1 t α ]) n−1

+ ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ]. k=0

(37)

Solution (37) of equation (10) describes the dynamics of the national income in the framework of the model of economic growth with a power-law increase of consumption with one-parameter fading memory. The obtained results can be generalized to the consumption functions C(t), which are described by power series and finite polynomials. For example, using N

C(t) = ∑ Cm t m , m=1

(38)

Economic models with power-law memory | 15

the dynamics of national income is described by the equation N

Y(t) = ∑ Cm t m (1 − Γ(m + 1)Eα,m+1 [B−1 t α ]) m=1

n−1

+ ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ]. k=0

(39)

The interpolation by polynomials of finite degree allows one to apply the suggested results for a wide class of consumption functions that describe real economic processes. Let us consider the constant consumption function (C(t) = C) for solution of equation (10). This case will allow us to clearly illustrate the dependence of economic growth on memory effects by the numerical simulations in the form of graph (see Figures 1–5). In this case, solution of equation (10) has the form n−1

Y(t) = C(1 − Eα,1 [B−1 t α ]) + ∑ Y (k) (0)t k Eα,k+1 [B−1 t α ]. k=0

(40)

Solution (40) describes economic growth with one-parameter power-law memory and a constant non-productive consumption. For 0 < α ≤ 1 (n = 1) solution (40) takes the form Y(t) = C(1 − Eα,1 [B−1 t α ]) + Y(0)Eα,1 [B−1 t α ].

(41)

For 1 < α ≤ 2, (n = 2), solution (40) has the form Y(t) = C(1 − Eα,1 [B−1 t α ])

+ Y(0)Eα,1 [B−1 t α ] + Y (1) (0)tEα,2 [B−1 t α ].

(42)

For α = 1, the equality Eα,1 [z] = exp{z} renders solution (41) into the form Y(t) = C(1 − exp(t/B)) + Y(0) exp(t/B),

(43)

which coincides with the solution of the standard model (6). Equation (43) describes economic growth in the framework of the standard macroeconomic model with constant consumption (C(t) = C), which does not take into account memory effects (α = 1). Equations (40), (41), and (42) represent the model of economic growth with constant consumption (C(t) = C), where we take into account the long memory with the fading parameter α > 0. Let us consider the behavior of the solution for the case of constant function C(t) = C > 0, where we will use λ = B−1 > 0. The behavior of the solution (40) at t → ∞ can be described by using the same asymptotic equations that is used for closed model (C(t) = 0).

16 | V. E. Tarasov For 0 < α < 1 the behavior of the solution is described by the equation 1 Y(t) = C + (Y(0) − C) exp(λ1/α t) α m

− (Y(0) − C)(∑ j=1

λ−j t −αj 1 + O( α(m+1) )). Γ(1 − αj) t

(44)

For 1 < α < 2 the behavior of the solution is described by the expression 1 Y(t) = C + (Y(0) − C + λ−1/α Y (1) (0)) exp(λ1/α t) α m 1 Y (1) (0)t λ−j t −αj ) + O( α(m+1)−1 ). − ∑(Y(0) − C + 1 − αj Γ(1 − αj) t j=1

(45)

The conditions of growth and decline are important for macroeconomic models. Let us give these conditions for the considered model with power-law memory, where we will use λ = B−1 > 0. The condition of growth with memory with order 1 < α ≤ 2 is represented by the inequality Y(0) − C + λ−1/α Y (1) (0) > 0.

(46)

The condition of decline with memory is represented by the inequality Y(0) − C + λ−1/α Y (1) (0) < 0.

(47)

For the case 0 < α ≤ 1, we can consider these inequalities, when we use Y (1) (0) = 0. In the case of absence of memory (α = 1) these conditions of growth and decline take the following form. The condition of growth without memory (α = 1) and with memory 0 < α ≤ 1 is represented by the inequality Y(0) − C > 0. The condition of decline without memory (α = 1) and with memory 0 < α ≤ 1 is represented by the inequality Y(0) − C < 0. Let us note the case Y (1) (0) > 0, where we take into account C > 0 and λ = B−1 > 0. If the inequalities hold, Y(0) < C < Y(0) + λ−1/α Y (1) (0)

(48)

holds, and the process without memory shows a decline, while the process with memory and the other parameters the same demonstrates growth. This condition means that the decline is replaced by growth, when the memory effect is taken into account.

Economic models with power-law memory | 17

Let us formulate some properties of the growth processes with memory, when C > 0 and λ = B−1 > 0. 1. The effects of memory can lead to a decrease of the growth rates. The memory with α < 1 can slow down the rate of economic growth. 2. The effects of memory can lead to a decrease of the rates of decline. We see that for α < 1 the memory effects can slow down the pace of decline. The effects of memory can slow the rate of decline. 3. The effects of memory with α > 1 can lead to an increase of the growth rates. 4. The effects of memory with α > 1 can leads to growth instead of decrease and decline. The decline of processes can be replaced by growth, when the memory effect is taken into account. 5. The effects of memory with α > 1 can lead to a slowdown in the rate of economic decline. Memory effects may reduce the rate of decline. Memory with α > 1 can decrease the rate of slowdown of a process. As a result, we can formulate the following principles of economic growth. Principle of long memory with fading less than unity: Long memory with α < 1 leads to a slowdown in the growth and decline of the economy. The effect of a long memory with α < 1 leads to inhibition of growth and economic decline. The memory with small value of the parameter α < 1 leads to stagnation of the economy. Principle of long memory with fading greater than unity: Long memory with α > 1 leads to an improvement in economic dynamics. Long memory with α > 1 leads to positive results, such as a slowdown in the rate of decline, a replacement of the economic decline by its growth, and an increase in the rate of economic growth. As a result, we conclude that accounting of the memory effects can give a new type of behavior for the same parameters of the macroeconomic models. The neglecting of the long memory can lead to qualitatively different results, conclusions and predictions. In studies of economy and the construction of macroeconomic models, we should take into account the effects of a long memory to obtain correct results and predictions. The effect of memory with the fading parameter α > 1 leads to faster economic growth than the standard model, which does not take into account memory effects. Using the well-known stock symbols, we can say that “long memory, whose fading parameter is greater than one, is a bull, who raises bear and bull with amnesia”. The effects of memory with the fading parameter α < 1 lead to suppression of economic growth and decline. Using the same stock symbols one can say that “long memory, whose fading parameter is less than one, is a bull for bear with amnesia and it is bear for bull with amnesia”.

18 | V. E. Tarasov

4 Example of memory effects for growth model Let us give examples of behavior of the economy, when we take into account the long memory. For this purpose, we give the plots of the solution (40) for the cases 0 < α < 1 and 1 < α < 2, which are given by (41) and (42), in comparison with the plots of the solution (43) of the standard model without memory (α = 1). Let us consider solution (41) with C = 19, Y(0) = 20, and B−1 = 0.1. Then equation (41) has the form Y(t) = 19 + Eα,1 [0.1t α ].

(49)

For α = 0.1 this expression describes the growth with memory. For α = 1 equation (49) describes growth process without memory. The comparison of these processes is presented by Figure 1. It shows that the effects of memory can lead to a decrease of the growth rates (growth recession). We see that the memory with α < 1 can slow the growth rate. Memory effects can slow down the rate of economic growth.

Figure 1: The functions Y (t) for the model without memory (Plot 1) and the model with memory for α = 0.8 (Plot 2) with C = 19, Y (0) = 20, B−1 = 0.1.

Now we address the solution (41) with C = 21, Y(0) = 20, and B−1 = 0.1. Then equation (41) has the form Y(t) = 21 − Eα,1 [0.1t α ].

(50)

For α = 0.1 this expression describes the decline with memory. For α = 1 equation (50) describes a growth process without memory. The comparison of these processes is presented by Figure 2. It shows that the effects of memory can lead to a decrease of the rates of decline. We see that for α < 1 the memory effects can slow down the pace of economic decline. The effects of memory can slow the rate of economic decline.

Economic models with power-law memory | 19

Figure 2: The functions Y (t) for the model without memory (Plot 1) and the model with memory for α = 0.8 (Plot 2) with C = 21, Y (0) = 20, B−1 = 0.1.

Figure 3: The functions Y (t) for the model without memory (Plot 1) and the model with memory for α = 1.1 (Plot 2) with C = 19, Y (0) = 20, Y (1) (0) = 0.1, B−1 = 0.1.

Let us consider solution (42) with C = 19, Y(0) = 20, Y (1) (0) = 0.1, and B−1 = 0.1. Then equation (42) has the form Y(t) = 19(1 − Eα,1 [0.1t α ]) + 20Eα,1 [0.1t α ] + 0.1tEα,2 [0.1t α ].

(51)

For α = 1.1 this expression describes the growth with memory. In this case, condition (46) holds. The comparison of this processes and the growth without memory (α = 1) is given in Figure 3. In Figure 3, we see that the effects of memory with α > 1 can lead to an increase of the growth rates. Memory effects may increase the rate of economic growth. Solution (42) with C = 19, Y(0) = 20, Y (1) (0) = 0.1, and B−1 = 0.1. In this case equation (42) has the form Y(t) = 19(1 − Eα,1 [0.1t α ]) + 20Eα,1 [0.1t α ] + 0.3tEα,2 [0.1t α ].

(52)

20 | V. E. Tarasov

Figure 4: The functions Y (t) for the model without memory (Plot 1) and the model with memory for α = 1.1 (Plot 2) with C = 21, Y (0) = 20, Y (1) (0) = 0.3, B−1 = 0.1.

For α = 1.1 this expression describes the growth with memory. In this case, condition (48) holds. The comparison of this processes and the growth without memory (α = 1) is given in Figure 4. We see that the memory effect can give growth instead of decrease and decline. The effects of power-law memory with α > 1 can lead to economic growth instead of a recession economy as follows from the condition (48). The decline of the economy is replaced by the growth of the economy, when the memory effect is taken into account. Let us consider the solution (42) with C = 19, Y(0) = 20, Y (1) (0) = 0.1, and B−1 = 0.1. In this case equation (42) has the form Y(t) = 21(1 − Eα,1 [0.1t α ]) + 20Eα,1 [0.1t α ] + 0.1tEα,2 [0.1t α ].

(53)

For α = 1.1 this expression describes the growth with memory. In this case, condition (47) holds. The comparison of this process and growth without memory (α = 1) is given in Figure 5. It shows that the effects of power-law memory with α > 1 can lead to a slowdown in the rate of economic decline. Memory effects may reduce the rate of decline. Long memory with α > 1 can decrease of the rate of recession and slowdown in economic activity. From Figures 1–5, it can be seen that the behavior of the income function is essentially dependent on the presence or absence of a long memory. Figures 1–5 show that neglecting of the memory effects in macroeconomic models can change the result. We see (Figures 1 and 2) that memory with α < 1 leads to a slowdown in the growth and decline of the economy. The effect of a long memory with α < 1 leads to inhibition of growth and economic decline. We can say that memory with a small value of the parameter α < 1 leads to stagnation of the economy. It is also clear (Figures 3–5) that taking into account the memory effect with α > 1 leads to an improvement in economic dynamics. Long memory with α > 1 leads to positive results, such as a slowdown in the rate of decline, a replacement of the economic decline by its growth, and an increase in the rate of economic growth.

Economic models with power-law memory | 21

Figure 5: The functions Y (t) for the model without memory (Plot 1) and the model with memory for α = 1.1 (Plot 2) with C = 21, Y (0) = 20, Y (1) (0) = 0.1, B−1 = 0.1.

5 Model with multi-parameter long memory In previous sections, we considered a macroeconomic model in which the power-law fading of memory was characterized only by one parameter, α. In real economy the memory fading might be different for different types of economic agents and for different situations [98, 88, 89, 68]. In addition, for complex types of fading memory, one can consider interpolations in the form of linear combinations of power-law memory functions. In these cases, to describe memory fading, we can use the kernel of a fractional integral operator in the form of a linear combination of power-law functions with different exponents of memory fading. For a two-parameter description of memory fading, we can use the parameters α > 0 and β > 0, and the accelerator equation (8) of the form β

I(t) = B((Dα0+ Y)(t) − θ(D0+ Y)(t)),

(54)

where α > β, B > 0 and θ ≠ 1. The parameter θ characterizes the influence of the memory, of which the fading parameter is β > 0, on the investment. The value of θ can be either positive or negative. Substituting the expression of the investment I(t), which is given by equation (54), into the balance equation (3), we obtain the fractional differential equation β

Y(t) = B(Dα0+ Y)(t) − θB(D0+ Y)(t) + C(t).

(55)

Equation (55) can be rewritten as β

(Dα0+ Y)(t) − θ(D0+ Y)(t) − B−1 Y(t) = −B−1 C(t),

(56)

where α > β, B−1 > 0, and the parameter θ is a real number, which can be either positive or negative, in the general case. The fractional differential equation (56) determines the economic dynamics with two-parameter power-law memory.

22 | V. E. Tarasov To solve equation (56) we can use Theorem 5.16 of [38], p. 323–324. Equation (56) coincides with equation (5.3.73) of [38], p. 323, if we will use the notation λ = θ, μ = B−1 and f (t) = −B−1 C(t). As a result, for a continuous function C(t), which is defined on the positive semiaxis (t > 0), equation (54) with the parameters 0 < n − 1 < α ≤ n, m − 1 < β ≤ m (where 0 < β < α, m ≤ n, α − n + 1 ≤ β), has the general solution n−1

Y(t) = ∑ Cj Yj (t) + YC (t), j=0

(57)

where Cj (j = 0, . . . , n − 1) are real constants that are determined by the initial conditions, the function YC (t) is defined as t

YC (t) = −B−1 ∫(t − τ)α−1 Gα,β,θ,B [t − τ]C(τ)dτ,

(58)

0

and the function Gα,β,θ,B [τ] is given by the equation τkα |θτα−β ]. B−k Ψ1,1 [(n+1,1) (αk+α,α−β) Γ(k + 1) k=0 ∞

Gα,β,θ,B [τ] = ∑

(59)

The functions Yj (t) with j = 0, . . . , m − 1 are represented by the expressions t kα+ |θt α−β ] B−k Ψ1,1 [(n+1,1) (αk+j+1,α−β) Γ(k + 1) k=0 ∞

Yj (t) = ∑

t kα+j+α−β −k |θt α−β ]. B Ψ1,1 [(n+1,1) (αk+j+1+α−β,α−β) Γ(k + 1) k=0 ∞

−θ ∑

(60)

For j = m, . . . , n − 1 the functions Yj (t) are defined by the equations t kα+j −k B Ψ1,1 [(n+1,1) |θt α−β ]. (αk+j+1,α−β) Γ(k + 1) k=0 ∞

Yj (t) = ∑

(61)

Here Ψ1,1 is the generalized Wright function (the Fox–Wright function) [37, 36], which is defined by the equation Γ(αk + a) z k . Γ(βk + b) k! k=0 ∞

|z] := ∑ Ψ1,1 [(a,α) (b,β)

(62)

Note that the two-parameter Mittag-Leffler function is a special case of the Fox–Wright function [38], p. 59, for a = α = 1, that is, ∞ Γ(k + 1) z k zk = ∑ = Eα,β [z]. Γ(αk + β) k! k=0 Γ(αk + β) k=0 ∞

|z] = ∑ Ψ1,1 [(1,1) (β,α)

(63)

Economic models with power-law memory | 23

Equation (56) and its solution (57)–(61) describe the macroeconomic dynamics of the national income, where the long memory is characterized by two-parameter powerlaw fading. Let us consider the case when all national income is used to expand the production, that is, the consumption is absent (C(t) = 0). Equation (56) with C(t) = 0 has the form β

(Dα0+ Y)(t) − θ(D0+ Y)(t) − B−1 Y(t) = 0,

(64)

where 0 < β < α ≤ 2. The solution of equation (64) can be used to estimate the greatest possible growth rate of the national income, when we take into account the twoparameter power-law fading memory. Let us first consider equation (64) for the case 0 < α ≤ 1 and 0 < β < α, by using Corollary 5.8 of [38], p. 317. In this case, the solution of equation (64) with 0 < β < α ≤ 1 has the form Y(t) = C1 Y1 (t),

(65)

where the constant C1 is determined by the initial condition, and Y1 (t) is given by the equation t kα |θt α−β ] B−k Ψ1,1 [(n+1,1) (αk+1,α−β) Γ(k + 1) k=0 ∞

Y1 (t) = ∑

t kα+α−β −k |θt α−β ]. B Ψ1,1 [(n+1,1) (αk+1+α−β,α−β) Γ(k + 1) k=0 ∞

−θ ∑

(66)

(a,α) |0] = 1/Γ(β) and equations (65), we get Using Ψ1,1 [(b,β)

C1 =

Γ(β)Y(0) , 1−θ

(67)

where θ ≠ 1. Let us now consider equation (56) for the case 1 < α ≤ 2 and 0 < β < α. Using Corollary 5.9 of [38], p. 317, the solution of equation (56) can be represented as a linear combination of Y1 (t), which is given by (58), and Y2 (t). If 0 < β ≤ 1 < α ≤ 2, Y2 (t) is defined by the expression t kα+1 B−k Ψ1,1 [(n+1,1) |θt α−β ]. (αk+2,α−β) Γ(k + 1) k=0 ∞

Y2 (t) = ∑

(68)

If 1 < β < α ≤ 2, Y2 (t) is defined by the equation t kα+1 −k B Ψ1,1 [(n+1,1) |θt α−β ] (αk+2,α−β) Γ(k + 1) k=0 ∞

Y2 (t) = ∑

t kα+1+α−β −k |θt α−β ]. B Ψ1,1 [(n+1,1) (αk+2+α−β,α−β) Γ(k + 1) k=0 ∞

−θ ∑

(69)

24 | V. E. Tarasov Below, we will consider some properties of the two-parameter model with 0 < β < α ≤ 2. To determine the effective technological growth rates (the rate of technological growth with memory) for the two-parameter macroeconomic model (56), it is necessary to consider the asymptotic behavior of solutions (66), (68) and (69) at t → ∞, which are represented as infinite sums of the Fox–Wright functions. Unfortunately, such asymptotic expressions are unknown. Equations (66), (68), and (69) in the form of sums of the Fox–Wright functions also contain the argument θt α−β , which depends on the difference α − β. This allows us to assume that the asymptotic behavior of solutions (66), (68) and (69) at t → ∞ should be described by α − β. As a result, it can be assumed that the rate of technological growth with memory for models with two-parameter power-law memory are determined by the difference α − β in the general case. To take into account memory with m ≥ 2 different power-law fading parameters, we can use the accelerator equation in the form m−2

β

α

I(t) = B((Dα0+ Y)(t) − θ(D0+ Y)(t) − ∑ θk (D0+k Y)(t)), k=1

(70)

where α > β > αm−2 > ⋅ ⋅ ⋅ > α1 > 0. Here the parameters θk characterize how memory with exponents αk > 0 influences the investment. We can consider θk and αk with k = 1, 2, . . . , m, by setting θm = 1, θm−1 = θ, αm = α, αm−1 = β. Macroeconomic models with multi-parameter memory can more accurately describe the real economic processes with long memory. In particular, the memory function can be represented by an interpolation method as a linear combination of power-law memory functions. Equation (70) can be substituted into the balance equation (3). As a result, we obtain the fractional differential equation β

m−2

α

(Dα0+ Y)(t) − θ(D0+ Y)(t) − ∑ θk (D0+k Y)(t) = −B−1 C(t), k=1

(71)

where we use the notation θ0 = B−1 , α0 = 0 and the property (D00+ Y)(t) = Y(t). The solutions of the equations of the macroeconomic model with m-parameter fading memory can be described by using Theorem 5.17 of [38], p. 324. The solution of equation (71) with C(t) = 0 is described by equations (5.3.58–5.3.60) of Theorem 5.14 of [38], p. 319–320. Note that despite the presence of m ≥ 2 different memory fading parameters in equation (71), the solutions are determined only by the sums of the Fox– Wright functions with the argument θt α−β , which depends only on the difference α − β of the two leading fading parameters. In other words, the argument of the Fox–Wright functions does not depend on αm−2 , . . . , α1 . It depends on the same argument as the solution of the two-parameter model. As a result, it can be assumed that the rate of technological growth with m-parameter long memory is determined by the difference

Economic models with power-law memory | 25

α − β. This allows us to propose the following principles for macroeconomic processes with multi-parameter power-law memory. Principle of seniority of memory fading parameters: In macroeconomic models with multi-parameter power-law memory with fading parameters α > β > αm−2 > ⋅ ⋅ ⋅ > α1 > 0, the rate of technological growth with memory depends only on the difference of the two leading parameters α − β, that is, λeff (α, β, αm−2 , . . . , α1 ) = λeff (α − β).

(72)

Principle of changing of dependence in growth by memory: The second largest parameter β of the power-law memory fading leads to a change of the dependence of the rate of technological growth with memory, which expresses, through the inverse value of incremental capital B−1 , the influence factor θ, which corresponds to the second fading parameter β that characterizes the influence of memory on the value of investment. In symbolic form, the change of the dependence can be represented as λeff (α) = B−1/α 󳨐⇒ λeff (α − β) = θ1/(α−β) .

(73)

The proposed principles allow us to describe the qualitative changes caused by taking into account the multi-parameter power-law memory in macroeconomic processes. These principles of multi-parameter memory can be corrected when the exact expressions for the asymptotic behavior of the solution, which is described by Theorem 5.17 of [38], p. 324, of equation (71) are obtained.

6 Conclusion The fractional calculus approach allows us to take into account the long memory of power-law type in macroeconomic models. It can serve as a theoretical basis to develop the concept of economic growth, to study possible alternatives of economic policy and their long-term consequences, and to predict the behavior of economic processes. This approach allows us to build more adequate macroeconomic models of real economy, since these models can take into account the circumstance that economic agents remember the history of changes in economic processes and take these changes into account, when they make decisions. The suggested principles of economic dynamics with long memory allow us to give qualitative descriptions of macroeconomic processes with power-law memory and to realize qualitative analysis of different ways of economic development. These principles allow us to study macroeconomic processes with memory by using this small number of initial data. Using these principles, we can draw conclusions only on the basis of the parameters of memory fading and the rate of technological growth without memory. These principles can be used for a qualitative analysis of different ways

26 | V. E. Tarasov of economic development of global and national economy. The proposed principles allow us to draw conclusions about the rates of economic growth if the fading parameters of a long memory were determined from economic data. The memory fading parameters can be considered as controlling parameters for increasing economic growth in the construction of economic policies and measures of influence of a long memory on the real economy. The fractional calculus approach can be used not only for constructing economic models with continuous time, but also for models with discrete time. The discrete (lattice) fractional calculus, which is based on the exact fractional differences [59, 60, 63, 97, 90, 66, 86], can be used to consider the exact correspondence between the discrete time and continuous time macroeconomic models. The fractional calculus approach to describing economic processes with powerlaw memory can also be generalized to a more general type of memory [98, 73, 89, 68]. For this purpose, we can use the methods of generalized fractional calculus [39, 40]. For example, the fractional integration and differentiation of variable order allow us to describe economic processes with memory in which the fading parameter changes with time. Note that financial processes were modeled by the fractional differential equations of varying order in the work of Korbel and Luchko in [41]. In the general case, the parameter of memory fading can be distributed over a certain interval, characterizing the distribution of the memory fading on a set of economic agents. This case is important, since various types of economic agents may have different parameters of memory fading [98, 98, 73, 89, 68]. In this case, we should use fractional derivatives and integrals of distributed order.

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[103] I. Tejado, D. Valerio, E. Perez, and N. Valerio, Fractional calculus in economic growth modelling: the Spanish and Portuguese cases, International Journal of Dynamics and Control, 5(1) (2017), 208–222, 10.1007/s40435-015-0219-5. [104] I. Tejado, D. Valerio, and N. Valerio, Fractional calculus in economic growth modeling. The Portuguese case, in Conference: 2014 International Conference on Fractional Differentiation and Its Applications (FDA’14), IEEE, 2014, ISBN 978-1-4799-2591-9, 10.1109/ICFDA.2014.6967427. [105] I. Tejado, D. Valerio, and N. Valerio, Fractional calculus in economic growth modelling. The Spanish case, in A. P. Moreira, A. Matos, and G. Veiga (eds.), Proceedings of the 11th Portuguese Conference on Automatic Control (CONTROLO’2014), Lecture Notes in Electrical Engineering, vol. 321, pp. 449–458, Springer International Publishing, Berlin, 2015, 10.1007/978-3-319-10380-8_43. [106] J. A. Tenreiro Machado, F. B. Duarte, and G. M. Duarte, Fractional dynamics in financial indices, International Journal of Bifurcation and Chaos, 22(10) (2012), 1250249, 12 p., 10.1142/S0218127412502495. [107] J. A. Tenreiro Machado, M. E. Mata, and A. M. Lopes, Relative fractional dynamics of stock markets, Nonlinear Dynamics, 86(3) (2016), 1613–1619, 10.1007/s11071-016-2980-1. [108] G. Teyssiere and A. P. Kirman (eds.), Long Memory in Economics, Springer-Verlag, Berlin, Heidelberg, 2007, 390 p., 10.1007/978-3-540-34625-8. [109] A. P. Thirlwall, G. Rampa, and L. Stella, Economic Dynamics, Trade and Growth: Essays on Harrodian Themes, Springer, 2015, 310 p., ISBN 9781349269310, Palgrave Macmillan UK, London, 1998, 10.1007/978-1-349-26931-0. [110] V. Volterra, On the mathematical theory of hereditary phenomena [Sur la theorie mathematique des phenomenes hereditaires], Journal de Mathematiques Pures et Appliquees, 9(7), 1928, 249–298 (French), http://gallica.bnf.fr/ark:/12148/bpt6k107620n/f257n50.capture. [111] V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Blackie and Son Ltd., London and Glasgow, 1930, 226 p.

Avishek Adhikary and Karabi Biswas

Four-quadrant fractors and their applications in fractional order circuits Abstract: Fractors show a constant phase angle behavior, but they are generally capacitive in nature, i. e., the phase of their impedance lie in the fourth quadrant. However, using typical generalized impedance converter (GIC) circuits, one can obtain fractors in any of the four quadrants, thus, broadening the scope of the applications. The design and realization of these four-quadrant fractors from the given specifications and their application in electrical circuits are elaborated in this chapter. Keywords: Fractor, fractional order element, four quadrants, ladder circuit, generalized impedance converter, resonator, filter, tuning, specifications, design PACS: 84.30.Bv, 84.30.Vn

In this chapter we present a realization of fractional order (FO) elements or fractors in all the four quadrants of the impedance plane and discuss the applications of such four-quadrant fractors in FO circuits. The fractor is an elementary FO immittance. A fractor or an FO element is also known as a fractional capacitor [9] and fractance device [13]. Its impedance function relates to non-integer order integration or differentiation. In the Laplace domain, the fractor impedance ZF (s) can be expressed in different ways [2], like ZF (s) =

1 CF sα

R 1 Q 1 ≡ ≡ ≡ . (τs)α CF sα sα Fsα

(1)

Here, s is the Laplace operator. Many references prefer the representation ZF (s) = as they interpret fractors as ‘fractional capacitors’ [20]. In these cases, CF is termed

a fractional capacitance and its unit is F/s1−α (s is for the unit of second here). The main advantage of such a representation is its resemblance to the conventional capacitor. However, there can be fractional inductors [15]. So, it is better to adopt a generic representation for a fractor impedance. From that point of view, we represent the fractor as ZF = Fs1 α . Here, the coefficient F is termed a fractance and its unit is ℧sα [3]. Next, we discuss the impedance characteristics of a typical fractor, its parameters and how to determine its parameter values in practice. Avishek Adhikary, Indian Institute of Technology Bhilai, Bhilai, India-492015, e-mail: [email protected] Karabi Biswas, Indian Institute of Technology Kharagpur, Kharagpur, India-721302, e-mail: [email protected] https://doi.org/10.1515/9783110571929-002

34 | A. Adhikary and K. Biswas

1 Characteristics, parameters and unit Phase and magnitude characteristics of an ideal fractor From equation (1) we see that, in the frequency domain, the impedance of the fractor becomes ZF (jω) =

1 1 απ απ 1 απ = (cos − sin )= ∠− . F(jω)α Fωα 2 2 Fωα 2

(2)

This indicates that the phase of a fractor, θ, is independent of frequency and ideally remains constant at − απ radian or −90α degree at any frequency. For example, if 2 α = 0.5, the phase is always −45°; viz. Figure 1(a). For these, a fractor is often called a constant phase element (CPE) [6] and its phase is called a constant phase (CP). It can be mentioned here that the resistor, capacitor and inductors all are, basically, CPEs where α = 0, 1 or − 1, respectively. However, only in the case of fractors, one can have for CP something other than 0° or ±90°. On the other hand, the magnitude of the fractor impedance is |ZF | = Fω1 α . Therefore, ⇒ log |ZF | = − log F − α log ω.

Figure 1: (a) Phase plot and (b) magnitude plot of an ideal fractor, for α = 0.5, (c) phase plot and (d) magnitude plot of a typical practical fractor, for α = 0.5.

(3)

Four-quadrant fractors and their applications in fractional order circuits | 35

Thus, in the log–log plane, |ZF | versus ω is a straight line whose gradient is equal to −α (Figure 1(b)). The y-axis intercept is a measure of the fractance, F.

Phase and magnitude characteristics of a practical fractor The characteristics of a practically realized fractor deviate substantially from that of an ideal fractor. First, a practical fractor shows its CP nature only for a limited zone of frequency. This zone is called the constant phase zone (CPZ); it starts from a lower frequency limit fl and ends at an upper frequency limit fh . Second, even within the CPZ, the phase of the practical fractor is not exactly constant. Rather, it varies between two certain phase boundaries, θl (lower boundary) and θu (upper boundary); viz. Figure 1(c). The amplitude of this variation is named the phase band (PB) and it is defined as PB = ±

θh − θl . 2

(4)

Not only the phase characteristics, the magnitude characteristics of the practical fractor also differ from the ideal ones. Only a limited zone of the magnitude plot is linear in the case of practical fractors (Figure 1(d)). This means that the gradient can be considered to be constant only for this zone. So, this is the CPZ which is determined from the magnitude plot. Ideally, the CPZ determined from the phase plot and that from the magnitude plot are the same, but in practice, they may differ slightly, depending upon how the linear zone is selected. To avoid this ambiguity, CPZ is determined here only from the phase plot.

Parameters of fractor Thus, ideally, a fractor has two parameters: its order and its coefficient, i. e., α and F. But, in practice, there are at least four significant parameters: α, F, CPZ and PB. It is not only so that the parameter evaluation techniques differ in the cases of ideal and practical fractors. In the case of ideal fractors, as phase and magnitude responses are similar at every frequency point, α and F can be determined from any of the frequency points. In this case, α is determined from the phase plot and F is determined from the magnitude plot by using the following formula: θi (in degree) θi (in radian) ≡ , 90 π/2 1 F= . |ZFi |ωαi α=

(5) (6)

Here, θi and |ZFi | are the impedance phase and impedance magnitude of the fractor at any frequency point ωi (rad/s). However, in the case of practical fractors, the

36 | A. Adhikary and K. Biswas phase value and magnitude gradient vary from frequency to frequency and they are approximately constant only within the CPZ. Therefore, in the case of practical fractors, first, a CPZ is determined. Then the α and F values are determined by the following formulas, which include the data of the entire CPZ: α= F=

1 N θi (in degree) , ∑ N 1 90 1 N 1 . ∑ N 1 |ZFi |ωαi

(7) (8)

Here N is the total number of data points (the frequency points in which impedance data are collected/measured) within the chosen CPZ. Finally, the PB is determined on the basis of these data points as per equation (4). Two more points can be mentioned in this context. First, we see that α and CP are basically equivalent, hence, often in this chapter only one of them is mentioned. Second, α can also be determined from the gradient of the magnitude plot. The exponent α determined from the phase plot and from the magnitude plot are almost the same, even in practice. In this work, we present the α values which are determined from the phase plot, using equation (7).

Unit of fractance From equation (6), we see that the unit of F is ℧sα . Ambiguity may arise from the fact that the unit of F is dependent on the order of the fractor α. As the order changes, the unit changes, e. g., the fractor whose α = 0.2, has fractance unit ℧s0.2 and the fractor whose α = 0.7 has fractance unit ℧s0.7 . Both elements are fractors but they have different units for their fractance values. This can be explained with the help of the generic representation of impedance elements, given in Figure 2. We see that the expression Fs1 α is a generic expression for any impedance element. For example, if α = 0, it represents a resistor. Similarly, for α = 1, −1, ±2, we get capacitor, inductor and frequency-dependent-negative resistor (FDNR). These four basic elements are represented by four continuous lines in the

Figure 2: Generic impedance plane and four-quadrant fractors.

Four-quadrant fractors and their applications in fractional order circuits | 37

generic impedance plane of Figure 2. Now, when α is a fractional number we get a fractor, but we see that for different fractional values of α there exist different lines in the said impedance plane (black dotted lines). As each line represents a different impedance element, therefore, we can say that the word ‘fractor’ does not indicate a single type of element. Fractors having different α are actually different types of element with a common name ‘fractor’. So, it is logical that they have different units.

2 Four-quadrant fractors From the same picture of the generic impedance plane of Figure 2, we get the idea of four-quadrant fractors. We see that, depending upon the value of α, one can have a fractor phase (CP) in any of the four quadrants of the geometric angle. For example, if α lies between 0 and 1, the corresponding CP lies between 0° and −90°, the geometrical fourth quadrant of the angles. Thus, we can call such fractors the fourth quadrant fractors. They are capacitive fractors or more specifically acute angle capacitive fractors (ACFs). Similarly, if α lies between 0 and −1, the corresponding CP lies between 0° and +90°, the first quadrant. Therefore, such fractors are the first-quadrant fractors or the acute angle inductive fractors (AIFs). In this line, we can also have second- and thirdquadrant fractors, which means obtuse angle inductive fractors (OIFs) (−2 < α < −1) and obtuse angle capacitive fractors (OCFs) (1 < α < 2). In this context, a set of new nomenclature is introduced in the literature [3]. Fractors of different quadrants are named Type I, Type II, Type III and Type IV fractors, according to their quadrants; viz. Figure 2. For example, the fourth-quadrant fractors are termed Type IV fractors. However, a practical realization of such a four-quadrant fractor is a challenge. At present, there is no commercially available fractor or FO element, though researchers have developed several multi-component- and single component-based techniques in the past 50 years to realize fractors. In multi-component-based techniques, fractors are realized by electronic networks, made by some resistors, and capacitors [9, 14], which are occasionally accompanied by op-amps [3, 19] or OTAs [10]. In single component-based techniques, a typical physical or chemical system itself behaves as a fractor [8, 11, 12]. A brief review of different multi-component and single componentbased realizations is available in [2]. Now, most of these techniques realize fourthquadrant fractors, i. e., Type IV fractors or ACF. However, some FO circuits also require inductive type fractors [4, 15–17], i. e., the first- or second-quadrant fractors (Type I and Type II). Thus, researchers have designed different types of FO gyrator circuits to realize fractors in other quadrants using a Type IV fractor [3, 10]. In this chapter, we discuss one such technique to realize four-quadrant fractors using a typical FO generalized impedance converter (GIC). In such an FO GIC, a Type

38 | A. Adhikary and K. Biswas IV fractor (ACF) is placed in a suitable position so that the overall GIC behaves as a fractor in the desired quadrant. We shall call this Type IV fractor the ‘constituent fractor’. We shall see later that the FO parameters of GIC-emulated four-quadrant fractors are closely related to the parameters of this constituent Type IV fractor. Therefore, to design four-quadrant fractors of given specifications, the first step is to design a Type IV fractor with given specifications. This will be discussed next.

3 Design of a RC ladder-based Type IV fractor from the given specifications The most common technique to realize Type IV fractor is via an RC ladder network. Theoretically, an infinitely long RC ladder realizes a fractor. In practice, a truncated ladder is used. There are many techniques to design such truncated RC ladder. The technique which is presented is based on the work of Oldham and Zoski of 1983 [14] and the work of Adhikary et al. of 2015 [3]. The peculiarity of the presented technique is that it can realize a Type IV fractor with the specifications of all four significant FO parameters, i. e., α, F, CPZ (both upper limit, fh and lower limit, fl ) and PB. In this technique, the truncated RC ladder has a structure as shown in Figure 3. It has in total N + 3 number of blocks, connected together in series and indexed as 0, 1, 2, . . . , N +2. The first N +2 blocks are made of ‘a resistor in parallel with a capacitor’, but the last one is only a resistor. Now, except for the first (block 0) and the last two blocks (block N +1 and block N +2), the resistor and capacitor values are in geometrical progression, as follows: Ri = a i R i

Ci = b C

for i ∈ [1, N],

for i ∈ [1, N].

Here Ri = the resistance of the ith block,

Ci = the capacitance of the ith block,

a = the geometric progression ratio for resistances,

Figure 3: The truncated RC ladder structure for realizing a Type IV fractor.

(9) (10)

Four-quadrant fractors and their applications in fractional order circuits | 39

b = the geometric progression ratio for capacitances,

R, C = two constant terms.

However, truncation of blocks results in a large deviation in frequency characteristics of this RC ladder from the desired CP nature. To overcome this issue, Oldham and Zoski have modified the R and C values of the terminal blocks: 1 1 R0 = [ − ]R, 2 ln(b) C C0 = [ 1 ], 1 − 2 ln(b)

(11) (12)

1 RN+1 = aN+1 R, 2 CN+1 = 2bN+1 C,

(14)

RN+2 = −

(15)

(13)

N+1

a R . ln(a)

So, we see from equations (9)–(15) that we have five design variables, i. e., a, b, N, R and C on which the values of resistors and capacitors of the ladder network depend. On the other hand, we have five specifications, i. e., α, F, fl , fh and PB, to meet. Therefore, we need some guidelines which can evaluate the said design variables while meeting the given specifications.

3.1 Guidelines to design ladder-based Type IV fractor Step 1: choice of geometric ratios: a and b The choice of a and b is governed by the specifications of α and PB (ϵ), 2 1 (ϵ + 2(1 − α)α 3 )] α(1 − α) α ln(a) ≈ ln(b). 1−α

−(1−α)

b=[

,

(16) (17)

Step 2: choice of time constant, τ, where τ = RC The time constant τ = RC is actually the time constant of the first block and can be regarded as the fundamental time constants of the circuit. In this step, τ value is set as per the specification of the CPZ lower limit fl , τ = RC = (

111 b2 ) . 2π fl

(18)

40 | A. Adhikary and K. Biswas Step 3: choice of number of blocks: Choice of N Next, N is determined from the following expression. Through this step, we meet the specification regarding upper limit of CPZ, fh . We have N +1≥

5.5 + ln(fh /fl ) − 3(1 − α)2/3 . − ln(ab)

(19)

Step 4: evaluation of resistance constant R, capacitance constant C In this step, the constant term R is determined from the specification of F: R≈

and

1 . F|K(jωC )|(ωC )α

(20)

Here, ωC is the geometric mean frequency in the specified CPZ, i. e., ωC = 2π√fl fh

K(jω) =

( 21 −

1 ) ln(b)

1 + jωτ

+

1 N+1 a a aN aN+1 2 + ⋅ ⋅ ⋅+ + + N N+1 1 + jωpτ 1 + jωp τ 1 + jωp τ − ln(a)

(p = ab). (21)

Once R is evaluated, the other constant C is found from equation (17) by C = τ/R. Step 5: evaluation of all resistors and capacitors value for each block Now, a, b, N, R, C are all known, we can find all elements of all the (N + 3) number of blocks of the ladder (Figure 3) following equations (9)–(15) and can realize a particular fractor for given F, α (0 < α < 1), CPZ (both fl and fh ) and PB.

3.2 Practical realization of RC ladder-based fractors Practical examples of some Type IV fractors, realized as per the presented guidelines, are provided here. These are named ACF1 , ACF2 , . . . , ACF7 ; viz. Table 1. Their detailed design specifications are given in the table. The R and C values with which the corresponding ladders are designed are given in Table 2. In practice, resistors and capacitors exactly equal to the values deduced by the guidelines are not available on the market. So, in practical circuits, closely available valued resistors and capacitors are used. In Table 1, the theoretically calculated values (Rt , Ct ) and the practically used values (Rp , Cp ), are given. Heuristically, it is found that the practical choice of R and C should follow the two rules, given by: 1. The practically chosen values for each resistors and capacitors should be within ±10 % margin from the theoretical values.

Four-quadrant fractors and their applications in fractional order circuits | 41 Table 1: The FO characteristics of RC ladder-based Type IV fractors, all designed for CPZ: 1.6 Hz to 1.6 MHz (Theo: Theoretical results, Expt: Experimental results). Fractor name

Design specifications

CP ± PB (degree)

CPZ (Hz)

F (℧sα )

ACF1

α = 0.2, F = 25.0 μ℧sα , PB = 0.25°,

Theo. Expt.

−17.9 ± 0.3 −17.8 ± 1.6

1.0 to 708 k 0.4 to 3.55 M

24.95 μ 25.11 μ

ACF2

α = 0.3, F = 13.3 μ℧sα , PB = 0.50°,

Theo. Expt.

−26.9 ± 0.5 −26.6 ± 1.5

1.0 to 1.26 M 0.5 to 1.00 M

13.22 μ 13.45 μ

ACF3

α = 0.4, F = 7.50 μ℧sα , PB = 0.75°,

Theo. Expt.

−35.9 ± 0.7 −36.6 ± 2.4

0.8 to 1.41 M 0.4 to 2.24 M

7.44 μ 7.09 μ

ACF4

α = 0.5, F = 3.75 μ℧sα , PB = 1.00°,

Theo. Expt.

−44.9 ± 1.0 −44.3 ± 1.9

0.7 to 1.58 M 0.5 to 708 k

3.77 μ 3.64 μ

ACF5

α = 0.6, F = 2.00 μ℧sα , PB = 1.25°,

Theo. Expt.

−53.9 ± 1.2 −54.1 ± 3.4

0.6 to 2.00 M 0.3 to 891 k

2.02 μ 1.79 μ

ACF6

α = 0.7, F = 130 n℧sα , PB = 1.50°,

Theo. Expt.

−62.9 ± 1.5 −62.2 ± 3.9

0.6 to 1.78 M 0.4 to 1.58 M

134 n 122 n

ACF7

α = 0.8, F = 45.0 n℧sα , PB = 1.75°,

Theo. Expt.

−71.7 ± 1.9 −73.3 ± 2.8

0.6 to 1.78 M 0.6 to 1.41 M

46.4 n 43.2 n

2.

The time constant value of each blocks (i. e., Ri Ci for ith block) in practical circuits should be within ±10 % margin from the theoretical values.

Figure 4(a) presents the experimentally obtained phase plots and Figures 4(b) and 4(c) present the experimental magnitude plots of realized Type IV fractors. Experimental plots are obtained from the impedance data, which are collected by a ‘Novocontrol Impedance Analyzer’. Theoretical and experimentally obtained values of the FO parameters (CP, PB, CPZ and F) of these ladder-based Type IV fractors are given in Table 1. Here, the CP values have been presented instead of α to have a better visual correlation with fractor phase plots and PB. We see that the experimentally obtained FO parameter values closely match the given specifications. Next, we shall see how fractors in other quadrants can be realized using these ladder-based fractors.

4 Realization of four-quadrant fractors To realize the fractors in other quadrants, we here adopt a typical grounded GIC circuit, as shown in Figure 5. Ideally, such a GIC circuit offers an impedance (ZGIC ) which depends on the impedance of its five constituent elements, i. e., Z1 , Z2 , . . . , Z5 , as per

ACF5

ACF4

ACF3

ACF2

ACF1

Fractor

14.9 k 56.2 μ

15.0 k 56.0 μ

44.8 k 23.3 μ

47.0 k 22.0 μ

116 k 11.3 μ

120 k 10.0 μ

322 k 5.21 μ

330 k 4.70 μ

842 k 2.59 μ

1.00 M 2.2 μ

Rp (Ω) Cp (F)

Rt (Ω) Ct (F)

Rp (Ω) Cp (F)

Rt (Ω) Ct (F)

Rp (Ω) Cp (F)

Rt (Ω) Ct (F)

Rp (Ω) Cp (F)

Rt (Ω) Ct (F)

Rp (Ω) Cp (F)

0

Rt (Ω) Ct (F)

Block →

150 k 2.00 μ

144 k 1.98 μ

82.0 k 3.30 μ

80.3 k 3.16 μ

39.0 k 5.17 μ

39.6 k 5.61 μ

20.0 k 10.0 μ

20.1 k 9.60 μ

8.20 k 20.0 μ

8.46 k 19.7 μ

1

47.0 k 680 n

42.4 k 877 n

33.0 k 1.00 μ

31.2 k 1.23 μ

20.0 k 2.00 μ

19.4 k 1.93 μ

12.0 k 2.67 μ

12.1 k 2.94 μ

5.60 k 5.60 μ

6.12 k 5.40 μ

2

12.0 k 330 n

12.5 k 388 n

12.0 k 470 n

12.1 k 477 n

10.0 k 620 n

9.54 k 664 n

6.80 k 1.00 μ

7.28 k 902 n

4.70 k 1.47 μ

4.43 k 1.48 μ

3

3.90 k 168 n

3.68 k 172 n

4.70 k 150 n

4.71 k 185 n

4.70 k 220 n

4.68 k 228 n

4.70 k 220 n

4.38 k 276 n

3.30 k 470 n

3.21 k 407 n

4

1.00 k 68.0 n

1.09 k 76.2 n

1.80 k 68.0 n

1.83 k 72.1 n

2.20 k 94.0 n

2.30 k 78.5 n

2.40 k 94.0 n

2.64 k 84.7 n

2.40 k 100 n

2.32 k 112 n

5

330 33.0 n

320 33.7 n

680 23.5 n

712 28.0 n

1.00 k 33.0 n

1.13 k 27.0 n

1.50 k 22.0 n

1.59 k 26.0 n

1.80 k 33.0 n

1.68 k 30.7 n

6

100 14.7 n

94.0 14.9 n

270 10.0 n

277 10.9 n

560 12.2 n

554 9.30 n

1.00 k 8.20 n

959 7.96 n

1.20 k 8.20 n

1.22 k 8.42 n

7

22.0 5.60 n

27.8 6.62 n

100 4.40 n

108 4.23 n

270 4.70 n

272 3.20 n

560 2.20 n

577 2.44 n

890 2.20 n

881 2.31 n

8

4.70 5.60 n

4.10 5.86 n

47.0 1.68 n

41.8 1.64 n

150 1.50 n

134 1.10 n

330 680 p

348 747 p

660 680 p

638 634 p

9

6.80

6.71

8.20 1.50 n

8.13 1.28 n

47.0 680 p

33.0 757 p

180 220 p

210 229 p

470 150 p

462 174 p

10

Table 2: The resistors and capacitors of RC ladders for different Type IV fractors [both theoretical (Rt , Ct ) and practically used (Rp , Cp ) values].

18.0

17.2

10.0

92.0

68.0 150 p

53.0 140 p

180 100 p

167 95.6 p

11

270

249

1.00 k

1.03 k

12

42 | A. Adhikary and K. Biswas

ACF7

ACF6

Fractor

19.7 M 147 n

20.0 M 150 n

77.1 M 51.6 n

40.0 M 47.0 n

Rp (Ω) Cp (F)

Rt (Ω) Ct (F)

Rp (Ω) Cp (F)

0

Rt (Ω) Ct (F)

Block →

Table 2: (continued)

4.70 M 82.0 n

4.04 M 75.7 n

2.20 M 150 n

2.07 M 149 n

1

560 k 47.0 n

519 k 45.3 n

470 k 82.0 n

431 k 76.1 n

2

68.0 k 30.0 n

66.7 k 27.1 n

100 k 39.0 n

89.7 k 38.8 n

3

8.20 k 20.0 n

8.58 k 16.2 n

20.0 k 20.0 n

18.7 k 19.8 n

4

1.00 k 12.2 n

1.10 k 9.73 n

3.90 k 10.0 n

3.90 k 10.1 n

5

180 6.60 n

142 5.82 n

1.00 k 4.70 n

812 5.17 n

6

18.0 4.70 n

18.2 3.49 n

180 2.20 n

169 2.64 n

7

2.20 3.30 n

1.17 4.17 n

39.0 1.22 n

35.0 1.35 n

8

1.00

1.14

4.70 1.22 n

4.00 1.38 n

9

4.70

5.00

10

11

12

Four-quadrant fractors and their applications in fractional order circuits | 43

44 | A. Adhikary and K. Biswas

Figure 4: (a) Phase plots and (b), (c) magnitude plots of different RC ladder-based Type IV fractors of Table 2 (experimental data).

Figure 5: A typical general impedance converter (GIC) circuit.

the following relation: ZGIC (s) =

Z1 (s)Z3 (s)Z5 (s) . Z2 (s)Z4 (s)

(22)

This GIC circuit is conventionally used to realize an inductor from a capacitor in the integer order domain [7]. Here, we use a Type IV fractor i. e., an acute angle capacitive fractor (ACF) instead of the capacitor. We shall also see that one can tune the fractance value of such a GIC-based four-quadrant fractor using GIC resistors. Or, in other words, using a GIC one can realize tunable fractors in practice.

Four-quadrant fractors and their applications in fractional order circuits | 45

Figure 6: FO GIC configurations to realize (a) Type I (acute angle inductive), (b) Type II (obtuse angle inductive), (c) Type III (obtuse angle capacitive), and (d) Type IV (acute angle capacitive) fractors.

Realization of tunable Type I fractor To realize a Type I fractor, the constituent Type IV fractor (say, its fractance is F and the order is α) is put as the Z2 element of the GIC. Resistors are connected in place of other elements; they are R1 , R2 , R3 and R5 , accordingly; viz. Figure 6(a). In that case, the overall GIC impedance becomes ZGIC (s) =

Z1 Z3 Z5 R1 R3 R5 F α 1 . = s ≡ Z2 Z4 R4 FI sαI

(23)

We see that the overall GIC impedance has a form similar to a fractor whose fractance (FI ) and order (αI ) can be written as R4 , R1 R3 R5 F αI = −α.

FI =

(24) (25)

As α (the order of the constituent Type IV fractor) lies between 0 and 1, the αI (the order of the GIC-emulated fractor) lies between 0 and −1, which satisfies the condition for a Type I fractor. Therefore, this configuration realize a Type I fractor (first-quadrant fractor or acute angle inductive fractor). Also, we see that one can adjust the FI value as per the requirement by varying one of the GIC resistors. This means that this technique not only realizes a Type I fractor but also provides the means to tune it. This means that the GIC-based Type I fractor is a tunable fractor.

46 | A. Adhikary and K. Biswas There also exists an alternate configuration, where the constituent fractor is put in place of Z4 and resistors are placed in the rest of the places. The fractance value and order of this alternate configuration are similar to those of the presented configuration. Therefore, we discuss here only one configuration. Similarly, for Type II or Type III, more than one configuration exists, but for brevity we discuss only one in each case.

Realization of tunable Type II fractor To realize a Type II fractor, the constituent Type IV fractor is put as the Z2 element, a capacitor (C) is put as Z4 element and resistors are connected in rest of the three places; viz. Figure 6(b). In this case, the overall GIC impedance becomes ZGIC (s) =

Z1 Z3 Z5 1 . = R1 R3 R5 FCs1+α ≡ Z2 Z4 FII sαII

(26)

So, here also the overall GIC impedance acts as fractor or FO element. Its fractance value (FII ) and order (αII ) are, therefore, 1 , R1 R3 R5 FC αII = −(1 + α).

FII =

(27) (28)

We have 0 < α < 1, thus −2 < αII < −1, which is the condition of any Type II fractor. So, this configuration results in a Type II fractor. Also, adjusting one of the GIC resistors (R1 , R3 , R5 ), one can tune its fractance value, just like the Type I fractor presented above.

Realization of tunable Type III fractor We can realize the Type III fractor following the configuration given in Figure 6(c). Here, the constituent fractor and the capacitor (C) are put in place of Z3 and Z5 , respectively, keeping resistors in other places. This configuration gives Z1 Z3 Z5 R1 1 = , ≡ Z2 Z4 FCR2 R4 s1+α FIII sαIII R R FC = 2 4 , R1 = (1 + α).

ZGIC (s) =

(29)

⇒ FIIIA

(30)

αIIIA

(31)

We have 0 < α < 1; therefore 1 < αIII < 2, which is the condition of any Type III fractor. Besides, varying the GIC resistors, we can tune FIII as per the requirement.

Four-quadrant fractors and their applications in fractional order circuits | 47

Realization of tunable Type IV fractor (GIC-based) Similarly, the configuration of Figure 6(d) gives a GIC-based, tunable Type IV fractor, whose fractance FIV and order (αIV ) are R2 R4 F , R1 R5 = α.

FIV =

(32)

αIV

(33)

It is important to mention here that we can directly design a Type IV fractor by an RC ladder network; viz. Section 3.1. However, with the GIC-based realization we can tune its fractance value. There is always a limitation in realizing the fractance value by an RC ladder-based network due to limited market availability of resistors and capacitors [3]. With the GIC-based Type IV fractors, we can realize much wider range of fractance value.

5 Guidelines to choose GIC resistors In Section 4, it is explained how one can realize a fractor in any of the four quadrants with proper placement of a constituent fractor and a capacitor (in the case of an obtuse angle fractor). However, it is not discussed how the GIC resistors should be chosen. It is found that the choice of GIC resistors severely affects the FO parameters of the GIC-based fractors [1]. Ideally, parameters like the constant phase zone (CPZ) or phase band (PB) of the GIC-based fractors should be equal to their constituent fractors, but, in practice, they are different. Besides, it is also found that the order and fractance value of the practically realized GIC-emulated fractor deviates from the ideal values as obtained from equations (23)–(33). One key factor behind these deviations is in the non-ideal characteristics of the op-amp, which are not considered in the derivation of the ideal GIC impedance expression of equation (22). Basically, practical op-amps have many non-ideal characteristics, e. g., a finite open loop gain and finite unity gain frequencies. To minimize the said deviations, one needs to derive the impedance expression for an FO GIC with non-ideal op-amps. A typical non-ideality analysis for FO GIC has been carried out by Adhikary et al. in [1]. The analysis results in the conditions with which we can minimize the effects of non-ideal op-amps on phase and magnitude response of the GIC-emulated fractors. Based on these conditions, guidelines can be developed to choose GIC resistors for different design criteria. Details of a non-ideality analysis are not included here, but the guidelines to choose GIC resistors for two different cases are discussed below with some practical examples.

48 | A. Adhikary and K. Biswas

Design Case 1: guidelines to design a tunable Type I fractor with minimum phase and magnitude error with given αI , FI and CPZ upper limit, fh These guidelines are for the configuration presented in Figure 6(a). The first step is to choose a constituent fractor whose order is α = −αI (as per equation (25)) and the fractance F is as per availability for that α. The next step is to choose the resistors as per the following guidelines: 1 3+2α 41 1. The resistor R3 should be equal to F(2πf α ( 3−2α ) . h) 2. The resistors R4 and R5 should be of equal value. 3. The resistor R1 should be adjusted to tune it to the desired FI or to re-tune it in the future to any other values. The above guidelines are applicable for the cases where the op-amps of GIC are from the same batch and their unity gain frequencies are much higher than the desired CPZ upper limit. For example, we here present a realization of seven different Type I fractors with different αI specifications (Table 3) and a desired CPZ upper limit, fh = 100 kHz. In this practical realization, the GIC resistors are chosen as per the given guidelines; their values are given in the table. Two LF411N op-amps of the same batch have been used to make the desired GIC whose unity gain frequency is 3 MHz (much higher than the desired fh ). The RC ladder-based Type IV fractors, ACF1 , . . . , ACF7 , whose characteristics are given in Table 2, are used here as the constituent fractors. The experimentally obtained phase plots of these Type I fractors are shown in Figure 7. There is another point to note. These guidelines are for a minimum phase error and minimum magnitude error. That means that the deviations in the practical values of αI and FI (with respect to their ideal values, as per equations (24)–(25)) are less if these guidelines are followed [1].

Table 3: GIC configuration parameters for different Type I fractor realizations. Type I fractors

Design specifications

Constituent fractors

AIF1 AIF2 AIF3 AIF4 AIF5 AIF6 AIF7

αI αI αI αI αI αI αI

ACF1 ACF2 ACF3 ACF4 ACF5 ACF6 ACF7

= −0.2, fh = −0.3, fh = −0.4, fh = −0.5, fh = −0.6, fh = −0.7, fh = −0.8, fh

= 100 kHz = 100 kHz = 100 kHz = 100 kHz = 100 kHz = 100 kHz = 100 kHz

GIC resistors R1 (Ω) R3 (Ω) 1k 1k 1k 1k 1k 1k 1k

3.15 k 1.50 k 0.76 k 0.47 k 0.21 k 0.84 k 0.72 k

R4 , R5 (Ω) 1k 1k 1k 1k 1k 1k 1k

Four-quadrant fractors and their applications in fractional order circuits | 49

Figure 7: Experimental phase plots of Type I fractors corresponding to Table 3.

Design Case 2: guidelines to designing a tunable Type II fractor with wide CPZ for given αII , FII , CPZ lower limit, fl and PB These guidelines are for the configuration presented in Figure 6(b). The first step is to choose a constituent fractor whose order α = −αII − 1 (as per equation (28)) and fractance F is as per availability for that α. We also choose a suitable capacitor to be used as Z4 element (a ceramic disc type capacitor is preferable). Next, we define two characteristic frequencies, i. e., ωF and ωC , where ωαF = FR1 and ωC = CR1 . Then we 3 5 find the suitable value of these characteristic frequencies using the following two relationships: 1+α

1

2+α (Δϵ ωu )2 2 + α ωl sin θ 2+α ωC = [ ][ ] [ ] , cos θ 1+α kl ω0 cos θ + ωl sin θ

(34)

2+α Δϵωu ωl 1 ωF = α [ ] . ωC kl ω0 cos θ + ωl sin θ

(35)

1+α

1

ω

Here, ω0 is the cut-off frequency of the op-amps (ω0 = A u ), ωl is the lower limit 0 of the CPZ (rad/s) and Δϵ = (specified PB for the GIC-based Type II fractor – PB of the constituent Type IV fractor). Here, kl is a constant whose value is to be determined iteratively. In the initial iteration it should be put to 1. Once ωF and ωC are determined, we can choose the GIC resistors as per the following guidelines: 1. Choose a resistor as R3 whose value is equal to = Fω1 α .

2. 3.

F

Choose a resistor as R5 whose value is equal to Cω1 C Adjust R1 to tune or re-tune the fractor for the desired FII value.

With these guidelines we have designed several Type II fractors as shown in Table 4. The corresponding design specifications and the GIC resistor values are also mentioned in the table. The experimentally obtained phase plots are given in Figure 8. These design guidelines are to realize Type II fractors, i. e., obtuse angle inductive fractors, with a wide constant phase zone for a given phase band. Here, in this chapter,

50 | A. Adhikary and K. Biswas Table 4: The GIC configuration parameters for Type II fractor realization.

Type II fractors

Design specifications αII fl (Hz)

PB (°)

OIF1󸀠 OIF2󸀠 OIF3󸀠 OIF4󸀠 OIF5󸀠 OIF6󸀠 OIF7󸀠

−1.2 −1.3 −1.4 −1.5 −1.6 −1.7 −1.8

10 10 10 10 10 10 10

3.1 3.0 3.9 3.4 4.9 5.4 4.3

GIC configuration parameters R1 Constituent R3 (Ω) fractors (Ω) 1k 1k 1k 1k 1k 1k 1k

ACF1 ACF2 ACF3 ACF4 ACF5 ACF6 ACF7

15.6 k 16.6 k 18.9 k 22.6 k 29.7 k 276 k 700 k

C4 F

R5 (Ω)

100 n 100 n 100 n 100 n 10 n 10 n 10 n

65 70 75 78 780 720 604

Figure 8: Experimental phase plots of Type II fractors corresponding to Table 4.

guidelines for two possible design problems are discussed. Similarly, guidelines can be developed for Type III and Type IV (GIC-based) fractor realizations as well as for different types of design criteria. For the reader’s interest, some typical phase plots (experimental) of Type III fractors are presented in Figure 9.

Tuning of fractor It has already been mentioned that the GIC-based fractors are tunable, because by varying one of the GIC resistors we can adjust the fractance value as per our requirement. It is also seen that the resistor R1 is the most suitable resistor to tune the fractance value. This is so because a non-ideality analysis shows that the errors in phase and magnitude responses due to the non-ideal op-amps do not depend on the R1 value. Hence, in the above-mentioned guidelines, R1 is selected to tune the fractance value. Here, we present a practical example of fractance tuning. In this case, we tune a typical Type II fractor, OIF3 , by varying the resistance R1 . We see that, with the change

Four-quadrant fractors and their applications in fractional order circuits | 51

Figure 9: Experimental phase plots of GIC-based Type III fractors.

Figure 10: Tuning of Type II fractor OIF3 by R1 : phase, magnitude, and fractance plots.

in R1 value, the CP nature of OIF3 does not vary much, but its magnitude plot shifts

parallelly (Figure 10). As a result, its fractance value changes. We see in Figure 10 that we can tune the fractance of OIF3 from 0.7 m℧s−1.4 to 154 ℧s−1.4 by changing the R1

resistor from 1 MΩ to 10 Ω.

Next we shall see how such tunable four-quadrant fractors can be used in design-

ing tunable FO circuits with a high quality factor.

52 | A. Adhikary and K. Biswas

Figure 11: Proposed FOPR: (a) the circuit and (b) its admittance phasor diagram.

6 Application of tunable four-quadrant fractors: fractional order parallel resonators This section presents a typical FO parallel resonator (FOPR) using the above-discussed GIC-based tunable fractor. The presented FOPR is basically a parallel combination of a resistor (RP ) and an acute angle capacitive fractor (Type IV) (ZFC ) and an obtuse angle inductive fractor (Type I fractor) (ZFL ), as shown in Figure 11(a). In general, it is basically an FO version of a conventional parallel RLC circuit but with a special feature. The inductive fractor is an obtuse angle fractor (Type II fractor). In the Laplace domain, the impedances and admittances of these fractors can be represented by ZFC (s) = ( ZFL (s) = (

1 −α )s FC

1 1+β )s FL



YFC (s) = FC sα



YFL (s) = FL s−(1+β)

(0 < α < 1), (0 < β < 1).

(36) (37)

Unlike in the previous sections, notations like FL and FC are used instead of FII and FIV for the reader’s convenience to compare them with conventional integer order resonators. Now, as per equations (36) and (37) the admittance of an FOPR block

Four-quadrant fractors and their applications in fractional order circuits | 53

(admittance excluding source and line resistance r) of Figure 11(a) becomes Y(jω) =

1 απ απ + FC ωα [cos( ) + j sin( )] RP 2 2 +

FL (1 + β)π (1 + β)π ) − j sin( )] ≡ M + jN. [cos( 2 2 ω(1+β)

(38)

So, for N = 0 (i. e., for no imaginary admittance), ω1+α+β = (

βπ

FL cos 2 ). × FC sin απ 2

(39)

That means the resonating frequency, ωR , is given by 1

βπ

F cos 2 1+α+β ] ωR = [ L . FC sin απ 2

(40)

This ωR is the resonating frequency corresponding to “the imaginary part of the admittance is zero” or “the admittance phase is zero”. But, in the case of an FO resonator, the zero phase does not guarantee a minimum admittance. This is so because 󸀠 󸀠 the fractors’ admittances also contribute to the real part (viz. YFC , YFL in Figure 11(b)). However, the obtuse angle fractor provides here an unique advantage. It contributes to the −ve real impedance when the acute angle capacitive fractor gives a +ve real impedance. So, one can adjust the parallel resistance RP to such a value that the total real part of Y(jω) also becomes zero, i. e. M = 0. And this can be done at the same resonating frequency ωR if RP = −

βπ

cos 2 1 [ ] = RC . α FC ωR cos (α+β)π

(41)

2

Thus, by adjusting the values of FL and RP separately, both the real and the imaginary part of Y(jω) (i. e., N and M) can be made zero together. So we see that in this typical configuration of FO resonator (using an obtuse angle fractor), we can achieve a zero impedance phase and minimum admittance simultaneously. Besides, we see in the proposed FO circuits that, despite the presence of a finite parallel resistance (RP ), the circuit impedance becomes infinite at a particular frequency. As a result, the resonator’s bandwidth (BW) becomes zero. So if the quality fR factor (Q) is defined as Q = BW , we see that Q becomes theoretically infinite for the presented FOPR.

Tunability of FOPR and the tuning rules According to the above discussion, the presented FOPR can be tuned to a desired resonating frequency by the following two steps:

54 | A. Adhikary and K. Biswas sin

απ 2 βπ 2

1+α+β

1.

First, FL is set to FL = [FC

2.

Next, the resonator resistance RP is set equal to RC as per equation (41) to make the admittance zero at ωR (i. e. to achieve an infinitely large Q factor).

cos

ωR

], to get the desired resonating frequency, ωR .

So, FL and RP are the tuning parameters: FL is used to adjust ωR and RP is used to adjust Q. Here, FC is a fixed parameter and can be chosen such a way that the variation of RP can be kept within a desired limit for the specified frequency range. In practice, the said obtuse angle inductive fractor (Type II fractor) is realized by the GIC circuits. So, its fractance, FL , can be tuned by merely adjusting the GIC resistor R1 , as discussed in Section 5. Therefore, we can say that the presented FOPR can be tuned by using two potentiometers: one as parallel resistor RP and the other as GIC resistor R1 . There is another point to be mentioned here. Although theoretically it is a twostep tuning, in practice, such two-step tuning requires multiple iterations. A detailed discussion of the practical aspects of FOPR tuning is given in [4].

Choice of α, β The presented FOPR has two extra design variables compared to its integer order counterpart. They are α and β, which define the order of the fractors. Ideally, α, β can have any value as long as 0 < α < 1, 0 < β < 1 and α + β > 1 (as per equations (36), (37) and (41)). However, the sensitivity of the FOPR is also dependent on α, β. We have seen that the CP nature of a practically realized fractor is oscillatory. This means that their α or β values are also fluctuating around a mean value. From a practical point of view, it is desirable that the resonating frequency be least sensitive to the fluctuation in α, β. Similarly, the critical value for zero admittance (RC ) should be least affected by the change in α, β. Therefore, to have an optimal zone for the choice of α, β, first we need to study the related sensitivity functions. Some of these sensitivity functions are 󵄨 󵄨 1 󵄨󵄨 ωR 󵄨󵄨 󵄨󵄨󵄨 𝜕ωR /ωR 󵄨󵄨󵄨 , (42) 󵄨󵄨SFL 󵄨󵄨 = 󵄨󵄨 󵄨= 󵄨󵄨 𝜕FL /FL 󵄨󵄨󵄨 1 + α + β 󵄨 󵄨 α απ 󵄨󵄨 ωR 󵄨󵄨 󵄨󵄨󵄨 𝜕ωR /ωR 󵄨󵄨󵄨 cot( ), (43) 󵄨= 󵄨󵄨Sα 󵄨󵄨 = 󵄨󵄨 󵄨󵄨 𝜕α/α 󵄨󵄨󵄨 1 + α + β 2 󵄨 󵄨 β βπ 󵄨󵄨 ωR 󵄨󵄨 󵄨󵄨󵄨 𝜕ωR /ωR 󵄨󵄨󵄨 tan( ), (44) 󵄨󵄨Sβ 󵄨󵄨 = 󵄨󵄨 󵄨󵄨 = 2 󵄨󵄨 𝜕β/β 󵄨󵄨 1 + α + β 󵄨 󵄨 (α + β)π 󵄨󵄨 RC 󵄨󵄨 󵄨󵄨󵄨 𝜕RC /RC 󵄨󵄨󵄨 πα tan , (45) 󵄨󵄨 ≈ 󵄨󵄨Sα 󵄨󵄨 = 󵄨󵄨 2 2 󵄨󵄨 𝜕α/α 󵄨󵄨 󵄨 󵄨 (α + β)π βπ 󵄨󵄨 RC 󵄨󵄨 󵄨󵄨󵄨 𝜕RC /RC 󵄨󵄨󵄨 πβ (tan − tan ). (46) 󵄨󵄨Sβ 󵄨󵄨 = 󵄨󵄨 󵄨= 󵄨󵄨 𝜕β/β 󵄨󵄨󵄨 2 2 2 A detailed study of such a system sensitivity is discussed in [4]. From this sensitivity study, the following guidelines are proposed for the choice of α and β.

Four-quadrant fractors and their applications in fractional order circuits | 55

Figure 12: Variation of (a) ZR and (b) Q factor of an FOPR, with respect to different RP , for different α and β combinations.

1. 2. 3.

The value of α should lie between 0.65 and 0.95. The value of β should lie between 0.25 and 0.45. The value of α + β should lie between 1.1 and 1.2.

Stability analysis and Q factor Stability analysis of the presented FOPR can be carried out by the technique described by Radwan et al. in [18]. It is found that the proposed FOPR becomes marginally stable if RP is set at RC (the critical value to get infinite Q; viz. equation (41)) and it is stable if RP < RC . However, if RP is less than the RC , the FOPR impedance at resonance (ZR ) is no more infinite, neither is its Q factor. Yet, both of them are still quite high, of the order of MΩ and hundreds, respectively (Figure 12). We also see that the closer is the RP to RC , the higher is the ZR and Q factor. For example, if RP = 0.98RC , the Q factor is 347, if RP = 0.995RC ; the Q factor is 1448.

Design and realization of FOPR: a practical example Next is presented a typical design of proposed FOPR with experimental results. Example. Design a FOPR for a tuning range 1 kHz to 2 kHz with Q factor more than 300 which can be tuned using a 200 kΩ potentiometer. Also we meet the following ω ω sensitivity specifications: |Sα R | ≤ 0.2, |Sβ R | ≤ 0.2. Choice of α, β. As per the guidelines to choose α, β, we can initially select α + β = 1.1. ω ω Now, for α + β = 1.1, |Sα R | < 0.2 if α > 0.622 (from equation (43)) and |Sβ R | < 0.2 if β < 0.467 (from equation (44)). Thus, we can choose β = 0.39 and α = 0.69. These

56 | A. Adhikary and K. Biswas values are chosen because already an acute angle capacitive fractor (ACF6 ) is realized with αIV = 0.69 (Table 1) and a tunable obtuse angle inductive fractor (OIF3 ) is realized ω ω with αII = −1.39 (Figure 8). For such choices, |Sα R | = 0.18 and |Sβ R | = 0.13, which closely match the given specifications. Choice of FC . It is given that the system should be tunable using a 200 kΩ potentiometer. This means that RC < 100 kΩ. The RC value is maximum at the lower limit of tuning frequency i. e. at 1 kHz. For fR = 1 kHz, RC < 200 kΩ if FC > 79 n℧/sα (from equation (41)). We see that, for ACF6 , FC is 134 n℧/sα ; therefore, it satisfies the requirement. For such a choice, the RC varies within 72 kΩ to 117 kΩ for the desired tuning range of 1 kHz to 2 kHz. Tuning variable FL and RC . All the fixed parameters, α, β, and FC have been selected. Now, to get a high Q factor at a desired resonating frequency we shall tune FL and RP as per the FOPR-tuning rules, mentioned above. Now, a tunable obtuse angle fractor is realized via GIC and its fractance value is adjusted by changing the GIC resistor, R1 . Therefore, we keep two potentiometers in place of R1 and RP and adjust them to get the desired fR and Q, respectively. Experimental results. Some practical examples of FOPR tuning at different frequencies are given in Table 5. Corresponding R1 and RP data and achieved Q factor are mentioned in the table. We see that practically a Q factor of 360 has also been achieved, which is higher than its integer order equivalent in this frequency range [4]. The corresponding phase and magnitude plots (experimentally obtained) are presented in Figures 13(a)–13(b).

Unique features of presented FOPR 1.

The presented FOPR uses an obtuse angle inductive fractor (Type II fractor) which results in a theoretically infinite Q factor and an infinite impedance at resonance.

Table 5: Experimental data of practically realized FOPR circuits with α = 0.69, β = 0.39, FC = 134 ℧s0.69 : see Figures 13(a)–13(b). Expt. Case No. 1 2 3 4

Desired fR (Hz) 1000 1300 1550 1800

Theoretical values FL R1 ℧s1+β (kΩ) 11.50 19.85 28.61 39.05

86.5 50.1 34.8 25.5

RP (kΩ) 115.9 96.7 85.7 77.3

Practical realization data R1 RP fR (kΩ) (kΩ) (Hz) 83.2 45.9 30.1 22.4

95.2 77.0 69.7 64.7

994 1307 1537 1780

Q value

ZR (MΩ)

247 291 223 360

4.73 3.78 3.47 3.34

Four-quadrant fractors and their applications in fractional order circuits | 57

Figure 13: Experimentally obtained impedance response of practically realized FOPR: (a) magnitude plots, (b) phase plots.

2.

3.

In practice, they are finite, yet quite high, of the order of hundreds and tens of MΩ, respectively. In the presented FOPR, zero phase and maximum impedance occur at almost the same frequency. This is not possible in the FOPR circuits where the obtuse angle fractor is not used [19]. The FOPR is tunable in practice. Using only two potentiometers it can be tuned to the desired frequency with the desired Q factors.

Using this tunable high Q factor FOPR, we can realize tunable FO bandpass and notch filters with high Q factor. This will be discussed next. Also, in a similar manner we can realize a tunable high Q factor FO series resonator. Such an FO series resonator is described in [5].

7 Tunable fractional order bandpass and notch filters using Type II fractor Another application of such a tunable four-quadrant fractor is the FO bandpass and notch filter. Such FO filters can be designed by simply adding a high value series resistor (RO ) with the previously discussed FOPR block; viz. Figure 14. If VB is the voltage across the parallel resistor RP and GB (s) = VB (s)/V(s), then GB (s) acts as an FO bandpass transfer function, with a grounded output: RP GB (s) = r + RO

1 s1+β RP FC

s1+α+β

R

+

1+ r+RP

O

RP FC

s1+β

+

FL FC

.

(47)

Besides we also get a band-reject filter output as a complement to this FOBP output. If VN is the voltage across the series resistor RO and GN (s) = VN (s)/V(s) then GN (s)

58 | A. Adhikary and K. Biswas

Figure 14: Proposed FO filters using FOPR, VB : bandpass output, VN : notch output.

acts as an FO band-reject transfer function: s1+α+β + R 1F s1+β + FFL RO P C C GN (s) = . . RP r + RO 1+α+β 1+ r+R F O 1+β L s + RF s +F P C

(48)

C

However, the output of this band-reject filter is floating, not grounded. Now, as the FOPR blocks within these filters have a very high Q factor, both of these filters are narrow-band filters and the band-reject filter actually becomes a FO notch filter (FON). It is also evident that fB and fN are close to fR (where fB is the center frequency of FOBP and fN is the notch frequency of FON). We have βπ

1

F cos( 2 ) 1+α+β . ] ωB ≈ ωN ≈ ωR = [ L FC sin( απ ) 2

(49)

Therefore, we can tune such filters to the desired frequency following the same tuning rule as of FOPR.

Variation in Q factor and notch attenuation with respect to RO It is found that the Q factors of both filters depend on RO . Also, the attenuation at the notch frequency (AN ) depends on it. To study these, AN versus RO /RP ratio is plotted in Figure 15(a) and QN versus the RO /RP ratio is plotted in Figure 15(b). Here, QN is the fN Q factor of the notch filter and it is defined as QN = BW where BW is the bandwidth of 3 dB attenuation from the nominal passband (0 dB ideally) of the notch filter. From Figure 15(a) it is clear that if RP is tuned at RC , AN is higher than 200 dB for any RO /RP ratio. However, such an exact tuning may not be possible in practice. So, we study the cases where RP is tuned within ±1 % margin (i. e., between 99 % to 101 % of RC ) and ±5 % margin (i. e., between 95 % to 105 % of RC ). In all these cases, we find that the lower is the RO /RP ratio the higher is AN . On the contrary, from Figure 15(b) we see that the lower is the RO /RP ratio, the lower is QN . Therefore, the designer needs to choose RO according to the requirement. If high AN is required, one can set RO /RP = 0.1, then the notch attenuation will be 62 dB (but QN = 0.2); if high QN is required, RO /R can be set at 125, then QN = 500 (AN = 5.7 dB). Again, if RO /R = 3, then AN = 32 dB and QN = 14.5, an adjustment for both values.

Four-quadrant fractors and their applications in fractional order circuits | 59

Figure 15: (a) Variation of notch frequency attenuation w.r.t. RO /RP ratio, (b) variation of the notch Q factor w.r.t. RO /RP ratio.

Table 6: Experimentally obtained specifications of tunable FO filters (Ref. Figures 16(c)–16(b)) designed with α = 0.68, β = 0.42, FC = 135 × 10−9 ℧/sα , RO = 200 kΩ. Sl. No. 1 2 3 4

R1 (kΩ)

R (kΩ)

Desired fN (Hz)

83 46 30 22

95 77 70 65

1000 1300 1550 1800

Filter characteristics fN (Hz) AN (dB) 992 1292 1524 1750

30.6 34.9 29.4 28.2

QN

fB (Hz)

QBP

10.7 13.5 18.2 15.0

973 1274 1524 1750

11.5 10.5 14.1 11.7

It is important to mention here that the FO filters are in the stable zone. The system moves to marginal stability only if RO /RP → ∞. This makes the tuning easier in practice. More discussion of the design of the presented FO filters is available in [4]. Some practical examples of such FO filters (magnitude and phase plots) are presented in Figures 16(a)–16(d). The corresponding data are given in Table 6.

Advantages of proposed FO filters The proposed FO notch filter is simple in structure. Yet, it is found that it has higher Q factor for same attenuation than different integer order and fractional order notch filters [4]. On the other hand, the realized FOBP is a narrow-band filter. It has Q factor QBP between 10 and 15. In earlier work, the Q factor of a practically realized FOBP was 0.4 to 0.6 [19]. So we can say that with the proposed FOBP, the Q factor is improved by 20 to 35 times. The high quality FOBP and FON can be useful in the domain of biomedical and acoustic instruments. Future research needs to be done along this line.

60 | A. Adhikary and K. Biswas

Figure 16: (a) Gain magnitude plots, FON filter, (b) phase plots, FON filter, (c) gain magnitude plots, FOBP filter, and (d) phase plots, FOBP filter.

8 Conclusion In this chapter, we have presented the design and practical realization of fractors with given specifications in any of the four quadrants. Different design guidelines for RC ladder-based fractors and GIC-based fractors are presented in this context. We also have shown how the fractance of such four-quadrant fractors can be tuned to the desired value by changing a resistor, thus resulting in tunable fractors in practice. Next, using such four-quadrant fractors, different FO resonators and filters have been designed. Such FO resonators are practically tunable and have a high Q value.

Four-quadrant fractors and their applications in fractional order circuits | 61

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[20] S. Westerlund and L. Ekstam, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1(5) (1994), 826–839.

Grzegorz Litak, Cedrick A. K. Kwuimy, and Benjamin Ducharne

Energy harvesting in dynamical systems with fractional-order physical properties

Abstract: Energy harvesting from ambient sources shows much promise in modern solutions for powering small devices. In this chapter we report energy harvesting as obtained from mechanical vibration ambient sources after transformation into electrical power output. An energy harvesting system is composed of a mechanical resonator and a multiple field transducer. The fractional properties come from the material laws. Changes in ferroelectric, ferromagnetic domains as well as other material structure changes in visco-elastic materials cause hysteretic damping. This damping is characterized by a multiple relaxation rate and by the definition has a multiscale character. Such an effect could have positive and negative implications for the energy harvesting from the ambient sources. The evident drawback is coming from the fact that any additional damping will decrease electrical energy output. But if we start from the impedance matching principle then it is clear that damping will extend the efficiency. Multiscale smearing of the optimum frequency will lead to broadening of the vibration frequency interval useful for energy harvesting. In the present note we study a few examples of such systems. Keywords: Energy harvesting, fractional derivatives, nonlinear systems PACS: 43.40.+s, 43.38.+n, 05.45,-a, 02.30.-f

1 Introduction Energy harvesting from ambient mechanical vibration may be used to power small devices [12, 20, 43, 48, 63]. For example, sensors or modern electronic applications can Acknowledgement: Part of this work was done during the Postdoctoral fellow of C. A. K. Kwuimy at the VCADS. He would like to thank Dr. C. Nataraj and the US Office of Naval Research for the financial support (grant N00014-13-1-0485). G. Litak and B. Ducharne would like to thank for the support from the Polish–French collaboration project (Polonium). Grzegorz Litak, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, PL-20-618 Lublin, Poland; and Department of Process Control, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland, e-mail: [email protected] Cedrick A. K. Kwuimy, Department of Engineering Education College of Engineering and Applied Science, University of Cincinnati, 2901 Woodside Drive, Cincinnati, OH 45221, USA, e-mail: [email protected] Benjamin Ducharne, Laboratoire de Genie Electrique et Ferroelectricite, Institut National des Sciences Appliquees de Lyon, 8 rue de la Physique, 69621 Villeurbanne cedex, France, e-mail: [email protected] https://doi.org/10.1515/9783110571929-003

64 | G. Litak et al. work directly on harvested energy, or be powered by batteries with an energy recharging option using energy harvesting [43]. Earlier results were obtained by using simple cantilever beams with piezo-electric, electrostatic, or electromagnetic transducers [3]. In the case of linear dynamical systems, single frequency resonance-based energy harvesting was proposed [3, 23]. In nonlinear cases, the transduction band is much broader due to the nonlinear effects [8]. Among the leading nonlinear effects which influence the frequency band broadening, one may mention the inclination of the resonance band, appearance of sub- and super-harmonics resonances, and the existence of multiple solutions. This is important, as ambient vibrational sources like wind or road traffic noise are variable in amplitude and frequency. Therefore the multiple frequency, modulated vibrations [6] or even white noise excitations have to be considered [17, 21, 26, 37, 38]. One of the directions of development was the application of bistable or multistable mechanical resonators which were coupled to the piezo-electric transducer [4, 7, 13, 14, 16, 20, 25, 26, 29, 37, 38, 46–48, 56–58, 60, 61]. Similarly, in order to maximize the power output the switching technique was suggested. The switch is controlled during short moments in a synchronous way to the vibration creating the desired maximal damping or energy harvesting in a Synchronized Switch Harvesting on Inductor (SSHI) [1, 2]. On the other hand the miniaturization of energy harvesters favors the class of materials with higher density of potential power output, including piezo-electric and ferromagnetic materials. In the beginning of the development of energy harvesting [3], the material couplings were treated linearly; however, in recent years, in parallel with the needs of miniaturization and frequency broadband concepts, the nonlinear and hysteretic material properties were also considered [11]. In this spirit, fractional properties of the material were considered [4, 5, 14, 25, 30, 39, 60, 61, 67] In the present chapter we analyze the consequences of the material hysteresis, including the effects originating in dielectric, mechanical and magnetic properties, on the piezo-electric energy harvesting. The chosen examples show that fractional-order properties may play an important role in energy harvesting changing (increasing) the damping properties and shifting the working point. Such a shift, realized in the nonlinear system, may have consequences in choosing one of multiple solutions of a nonlinear system.

2 Material with fractional-order properties Ferroelectric materials are widely used in many areas of technology and science [22, 50, 51]. Sensors based on the piezo-electric effect transform mechanical signals into electrical signals and are used as accelerometers, or for mechanical measurements

Energy harvesting in dynamical systems with fractional-order physical properties | 65

(pressure and vibration). Except in memory applications, which are based on polarization switching and hysteresis polarization-electric field relationships, hysteresis is undesired in high-precision sensor, actuator and capacitor applications [18, 19]. Dynamic phenomena in ferroelectric ceramics and magnetic materials present similarities. Both materials have a hysteretic behavior, which induces increasing losses with frequencies. Another similarity is the domain structure of both materials, inducing losses caused by wall motion domains when the material is magnetized (for magnetic materials) or polarized (for ferroelectric ceramics). For the latter type of materials, dynamical effects are only due to wall motions. Few representations of this phenomenon exist; a novel approach based on fractional derivatives has been implemented and tested in the case of ferroelectric ceramics [40]. This approach leads to accurate results on a large frequency bandwidth. This approach is based on the following observation: beyond the quasi-static limit, ferroelectric materials exhibit hysteresis loops that are strongly dependent on the frequency. These states of dependence were clearly considered in the literature, indeed numerous theoretical studies have described the scaling law of the hysteresis area. As reported in various papers, the dynamical effect in a dielectric hysteresis model (evolution of the electric field versus the dielectric polarization) is usually implemented by considering an equivalent dissipative field derived from Ohm’s resistivity. The dynamical effect is then introduced by adding the product of a resistive term and the time derivation of the polarization to the quasi-static contribution. Recent research contributions have addressed mechanical elements and electrical components whose dynamics is well described by fractional-order differential equations or a fractional-order power law. In fact, a considerable amount of research in fractional calculus has been published in the engineering and applied science literature. As an illustration, from the contribution in References [9, 41, 46], the current– voltage relation of a condenser whose current–voltage characteristic has fractance is dκ U (1) dτκ where Cf is the capacitance of the fractance capacitor, i is the harvest current, τ is the time, and κ is the order of the derivative (κ = 1 for an ideal capacitor). Similarly, a fractional-order inductance has a current–voltage relation in the form [62] i = Cf

dκ i (2) dτκ where L is the value of the equivalent “inductance”. For the mechanical element, Debnath [9] reported that the stress–strain law of elasticity theory can be generalized using fractional-order differentiation. This contributes to the induction of viscous fractional-order damping into the system mathematical model. The fractional-order damping force is given by U=L

μ

Fd = ηDt y

(3)

66 | G. Litak et al. where η is the damping coefficient, μ is the order of derivative, and y is the mechanical displacement. The material also shows fractional-order elastic properties and the corresponding elastic force has a fractional-order power law generalized as [27, 28, 31] F(X) = K1 X + K3 X|X|α−1

(4)

where K1 , K3 and α > 1 are, respectively, the linear and nonlinear stiffness coefficient, and the fractional order. In practice, K1 > 0 for a beam, and under a sufficient axial load K1 < 0.

3 Energy harvester system with fractional-order electric elements Figure 1 shows a representation of an example of such energy harvester system. The transduction mechanism is due to the piezo-electric properties of the ceramic patches at the ends of the flexible beam. The dimensionless mathematical model is given as [25] d2 x dx + λ (1 + μx 2 ) + ω0 x + βx|x|α−1 − θv = f (t), 2 dt dt dx dκ v + γv + θ = 0, dt κ dt

(5) (6)

where the fractional derivative is defined according to the Caputo definition as [9] τ

dκ v 1 = Dκ v(τ) = ∫(τ − s)−κ v(s)ds. dτκ Γ(1 − κ)

(7)

0

Γ() is the Gamma function. Unless otherwise specified, the following values are used in this paper: β = 1, λ = 0.13, μ = 0.25, θ = 0.283, γ = 0.0078, and the initial conditions

Figure 1: Generic model of energy harvester.

Energy harvesting in dynamical systems with fractional-order physical properties | 67

̇ = 0) = 0 and v(t = 0) = 0. These physical values correspond to are x(t = 0) = −1, x(t an experimental model analyzed by Stanton et al. [57, 58]. For the analysis, we assume a harmonic perturbation of amplitude E0 and frequency Ω: f (t) = E0 cos Ωt

(8)

and the harmonic response of the system can be found in the form x = Am (t) cos(Ωt − ϕ1 ) v = Ae (t) cos(Ωt − ϕ2 )

(9)

where Am and Ae are amplitudes and the subscript m and e stand for mechanical and electric, ϕ1 and ϕ2 are the corresponding phases. The amplitudes equations are obtained using standard methods such as the multiple scale method [25, 45]: Ωθ

Ae =

√(Ωκ

)2 + (γ + Ωκ sin κπ 2 E02 = A2m [r12 + r22 ]

cos κπ )2 2

Am

(10) (11)

with r1 = ω0 − Ω2 [1 − r2 = Ωλ(1 +

θ2 Ωκ−1 sin κπ 2

)2 + (γ + Ωκ cos κπ )2 (Ωκ sin κπ 2 2

] + Aα−1 m J1 ,

Ω(γ + Ωκ cos κπ )θ2 μ 2 π Am ) + κ . 4 (Ω sin κπ )2 + (γ + Ωκ cos κπ )2 2 2

(12) (13)

Also we have 2π

1 J1 = ∫ cos2 Ψ| cos Ψ|α−1 dΨ 2π 0

α+2 α+2 2α+2 B( , ) = π 2 2

(14)

where B(a, b) denotes the Beta function. In order to verify the precision of the analytical solution, we also present the numerical results. The numerical formula for the fractional derivative used here is, following [10, 32, 33, 36, 54, 55, 65], j

Dκ v(tj ) ≈ h−κ ∑ Ciκ v(tj−i ) i=0

(15)

where h is the time step and C0κ = 0

and

Ciκ = (1 −

1+κ κ )Cj−i . i

(16)

68 | G. Litak et al.

Figure 2: Amplitude response of the system as a function of the driven frequency and E0 = 0.1, α = 3 and κ = 0.95 in (i), κ = 0.75 in (ii), κ = 0.5 in (iii). (a) Comparative analysis between analytical and numerical results for κ = 0.95. (b) Maximum mechanical displacement. (c) Maximum output voltage.

Figure 2 shows the effects of the order of the derivative on the amplitude of vibration of the mechanical arm (Figure 2(b)) and the output electrical voltage (Figure 2(c)). The first graph (Figure 2(a)) shows a comparison between the results from the analysis (curve with continuous line) and the results obtained from numerical simulation of equations (5)–(6) (curve with stars) using the definition of equation (15). The match between the results shows a good level of precision of the approximation made in obtaining equation (15). The graph of Figure 2(b) shows for the mechanical arm that, as the order of the derivative increases, the resonant amplitude of mechanical vibration decreases and the bending degree (hardening) remains constant. Similar results were obtained by Shen et al. [54] who analyzed the periodic vibration of a Duffing os-

Energy harvesting in dynamical systems with fractional-order physical properties | 69

Figure 3: Amplitude response of the system as a function of the driven frequency and E0 = 0.3, α = 3 and κ = 0.95 in (i), κ = 0.75 in (ii), κ = 0.5 in (iii). (a) Maximum mechanical displacement. (b) Maximum output voltage.

cillator with additional fractional damping and showed that increasing the order of the derivative (damping term) results in a decrease of the amplitude of vibration. For the device considered in this paper, the equivalent mathematical model (obtained by eliminating the electric variable) is a nonlinear oscillator with non-classic damping whose dynamics, in addition to the memory effects, is similar to the Duffing oscillator [53]. Interestingly, in Figure 2(c), similar observations are made. However, beyond resonance, the opposite happens for the electrical response (Figure 2(c)); i. e., a smallorder derivative leads to a relatively higher output voltage. Figure 3 shows the results in analogy to Figure 2 but for E0 = 0.3. The main difference here is in the severity of the hardening, which leads to multiple amplitude solutions, the intermediate one being unstable [54]. The effects of the fractional-order derivative are unchanged for the mechanical arm. However, for the electrical variable, the opposite of Figure 2(c) happens: a small-order derivative leads to relatively high output voltage at resonance and beyond. Figure 4 gives illustrations of the effects of the order of the deflection on the amplitude-frequency of the system. Increasing α and thus the nonlinear coefficient will increase the amplitude of vibration and the output electric voltage. The results obtained are quite interesting. In fact, they reveal that the impact of the fractance in the electric circuit is significant for a large amplitude of the perturba-

70 | G. Litak et al.

Figure 4: Amplitude response of the system as a function of the driven frequency and E0 = 0.1, κ = 1 and α = 6, α = 4, α = 3, α = 3/2. (a) Maximum mechanical displacement. (b) Maximum output voltage.

tion, while a huge impact of the fractional-order deflection is obtained even for a small amplitude of the perturbation. Moreover, the order of the deflection can be viewed as a bifurcation parameter to tune the system in its efficiency regime. Such effects are investigated in detail in the next section by analyzing the bifurcation diagram of the system. We emphasize the following. On the one hand it might be difficult to design a flexible beam with a tunable order of the deflection in order to adapt to the frequency change in the excitation. On the other hand the scope of these results is compatible with the multiple degree of freedom approach of energy harvesters where the technology design consists of having multiple mechanical arms to enlarge the pass band of the system. This is equivalent to having multiple mechanical arms with identical linear properties and different orders of deflection (nonlinear stiffness).

4 Energy harvester system with fractional-order mechanical element The considered energy harvesting system is shown in Figure 5. It is made of a cantilever beam with an unimorph piezo-ceramic (PZT), excited by an harmonic force F(t), which corresponds to the mechanical part. The fractional viscoelasticity accounting for the

Energy harvesting in dynamical systems with fractional-order physical properties | 71

Figure 5: Schematics of a fractional properties system.

structural damping due to the friction internal to the beam [9, 60] is taken into consideration. Three magnets are placed on the base plate of the device to create the tristable potential [35, 60] and the free end of the beam also wears a magnet. The set of dimensionless equations governing the harvester have a following form: {

μ ÿ + λẏ + ηDt y + k(y) = ϑv + f0 cos ωt, v̇ + γv = −βy,̇

(17)

with k(y) = ay + by3 + cy5 being the nonlinear elastic force. The periodic response of the system can be modeled by the harmonic functions y = A cos(ωt + ϕ),

v = B cos(ωt + ϕ),

(18)

where A and B are solutions of the following algebra equation (obtained using the method of balance of harmonics): f02 = (−E 1 + ϑB sin ϕ)2 + (E 2 − ϑB cos ϕ)2 ,

(19)

βA cos ϕ, B=− γ

(20)

with E 1 = λAω + ηAωμ sin

μπ , 2

μπ 3bA3 5cA5 + + ηAωμ cos 4 8 2 ω ϕ = arctan(− ). γ

E 2 = −(ω2 − a)A +

and

Figure 6 is the amplitude of the output voltage plotted as a function of the frequency ω, for different values of the viscoelasticity coefficient η. We observe that the

72 | G. Litak et al.

Figure 6: Effect of the viscoelasticity coefficient η on the output voltage when driving frequency ω: Analytical curves (-), numerical curve (o).

Figure 7: Effect of the fractional order μ on the output voltage when driving frequency ω for η = 0.015: Analytical curves (-), numerical curve (o).

output voltage decreases when increasing η, with a forward shift of the resonant frequency. Considering η = 0.015 and looking at the effect of the fractional order μ on the output voltage, we also realize a decreasing and a forward shift of the resonance frequency when increasing μ. This can be observed on Figure 7. Numerical results are also plotted on both figures, in Figure 6 for η = 0 and in Figure 7 for μ = 0.9. We have a perfect match between numerical and analytical results, but close to the resonance the instability of the system decreases the allowed energy. Table 1 and Table 2 clearly show the impact of the instability on the energy. The expected values of the energy are evidently reduced and always obey the law of increasing η and μ leading naturally to

Energy harvesting in dynamical systems with fractional-order physical properties | 73 Table 1: Effect of η in numerical results for μ = 0.5. η

vmax

ymax

ω

0 0.015 0.03

0.006806 0.006466 0.005994

0.6839 0.6498 0.6023

0.93 0.94 0.95

Table 2: Effect of μ in numerical results for η = 0.015. μ

vmax

ymax

ω

0.1 0.3 0.9

0.006697 0.006592 0.006238

0.673 0.6624 0.6269

0.94 0.94 0.94

the decreasing of the amplitude of the mechanical part and thus of the decreasing of the output voltage.

5 Fractional-order properties and fractal behavior in energy harvesting The potential energy V(y) given by dV = k(y) is plotted in Figure 8. Under the tristable dy potential, two types of dynamics can appear: dynamics between stable equilibrium p1 and p0 (or p4 and p0 ) and dynamics between p1 and p4 . These are known as homoclinic and heteroclinic bifurcations. The latter configuration yields a larger mechanical displacement. This is known to be advantageous for energy harvesting systems as the transduction mechanism will lead to higher electric energy. The mathematical conditions leading to such a bifurcation can be derived using the Melnikov theorem. The first published paper using the Melnikov theorem to investigate the complex dynamics of a bistable energy harvesting system was authored by Stanton et al. [57]. They established the condition of homoclinic bifurcation (periodic and complex inter-well dynamics) which corresponds in the context of energy harvesting to large mechanical displacement and thus to more electrical energy. Here, we present similar results in the context of a tristable energy harvesting systems with fractional-order properties. We highlight the effects of the fractional-order properties. For simplicity, we assume that the rate of change of the electrical voltage is small and thus, the modeling equation can be simply written in a vector notation as a perturbed Hamiltonian as U̇ = F[U] + ϵG[U]

(21)

74 | G. Litak et al.

Figure 8: Potential energy and separatrices of the system.

where f0 and ω are, respectively, the amplitude and frequency of the perturbation, and dV(y) is given by equation (21) or equation (17), and dy y U=[ ] ẏ

(22)

0 G=[ ] −γ0 ẋ + f0 cos(ω(t))

(23)

is the state vector and F=[



], − dV(y) dy

where γ0 = λ + γ. The framework to analyze homoclinic and heteroclinic bifurcations was established by Melnikov [35, 44, 45] and consists of analyzing the transverse intersection between the unperturbed and perturbed manifolds (that is, for ϵ = 0 and ϵ ≠ 0.) Structurally, if the unperturbed system ϵ = 0 has an initial impulsion around one specific equilibrium, the system will display an intra-well oscillation around that equilibrium. Clearly, this results in a small amplitude of the motion. When the initial impulsion is greater than a certain threshold two types of inter-well motion are observed: Inter-well motion amongst the two extreme potential wells as a consequence of the so

Energy harvesting in dynamical systems with fractional-order physical properties | 75

called heteroclinic bifurcation. Inter-well response amongst two consecutive wells as consequence of the so-called homoclinic bifurcation. These motions lead to a large mechanical excursion in comparison with the intra-well response. It can be shown that the unperturbed Hamiltonian system possesses homoclinic and heteroclinic orbits connecting the two unstable points of the potential. The homoclinic orbit is given by yho = ±

√2x1 cosh( χt ) 2

, (κ + cosh(χt))1/2

ẏho = ±

√2x1 χ(1 − κ) sinh( χt ) 2

2(κ + cosh(χt))3/2

(24)

,

while the coordinates of the heteroclinic orbit are given as yhe = ±

√2x1 sinh( χt ) 2

, (−κ + cosh(χt))1/2

ẏhe = ±

√2x1 χ(1 − κ) cosh( χt ) 2 2(−κ + cosh(χt))3/2

,

(25)

where x1 = √ ρ=

−b − √b2 − 4ac , 2c

x2 , x1

−b + √b2 − 4ac , 2c 5 − 3ρ2 . and κ = 2 3ρ − 1

x2 = √

χ = x12 √2c(ρ2 − 1)

Unfortunately, for real systems ϵ ≠ 0, and the ambient perturbations are, in general, of weak amplitude. The system response presents, in general, a complex response involving period-nT response, chaos, and multiple resonances. We are interested in this section in analyzing the effects of the perturbation on the appearance of inter-well motion and focus, without losing any generality, to the motion initially perturbed around the linear static equilibrium position y = 0. The Melnikov theorem provides a good framework to determine the condition of the appearance of homoclinic and heteroclinic bifurcations by examining the transverse intersection between perturbed and unperturbed manifolds. This transverse intersection manifests itself by the erosion of the basin of attraction of the system. The homoclinic and heteroclinic bifurcations (transversality) are obtained if one can find an initial angle t0 such that M(t0 ) = 0,

̇ 0 ) ≠ 0 M(t

(26)

where M(t0 ) is the Melnikov function, defined as +∞

M(t0 ) = ∫ F[Uh ] × G[Uh , t − t0 ]dt

(27)

−∞

where the subscript h indicates that the integral is evaluated along the separatrix equations (24)–(25). The condition for the appearance of the homoclinic bifurcation is therefore seen to be f0 ≥ f0ho = x1 χ(1 − κ)(

λI11 + ϑβI12 + √2I14

η I Γ(1−μ) 13

)

(28)

76 | G. Litak et al.

Figure 9: Threshold condition for appearance of Melnikov chaos.

where the coefficients I11 , I12 , I13 and I14 are defined in the appendix. The condition for the appearance of a heteroclinic bifurcation is similarly seen to be f0 ≥ f0he = x1 χ(1 − κ)(

λI21 + ϑβI22 + √2I24

η I Γ(1−μ) 23

)

(29)

where the coefficients I21 , I22 , I23 and I24 are defined in the appendix. The Melnikov threshold for homoclinic and heteroclinic bifurcations is plotted, respectively, in Figures 9(a) and (b) in the (ω, f0 ) parameters space. The upper domain corresponds to the appearance of the bifurcations, while in the lower domain the system displays intra-well motion. These graphs are typical for such a potential energy. The interest here is on the effects of the “gain” and order of the fractional derivative. The figures show that increasing the order of derivative contributes to lowering the threshold boundary for the appearance of homoclinic and heteroclinic bifurcations.

Energy harvesting in dynamical systems with fractional-order physical properties | 77

Figure 10: Erosion of the basin of attraction as a function of the order of derivative.

78 | G. Litak et al. For illustration, the basins of attraction are plotted in Figures 10(a)–(d) for motions inside the well x = 0. Simulations are made for ω = 1, f0 = 01 and η = 0.03, which correspond to a point in the regular region delimited by the threshold for heteroclinic chaos. The blue color corresponds to motion attracted inside the well x = 0. We observe that, for a fractional order μ = 1 (Figure 6(a)), some sets of initial conditions lead to motions attracted out of the central potential. The white domain within the central well and the blue domain out of the central well are illustrations of the existence of intra-well motion. That is, motions initiated in a specific well are attracted by a different well. This is done through a heteroclinic bifurcation (intra-well dynamics). The dynamics of the system is thus chaotic as shown by the fractal structure of the basin of attraction. When decreasing the order of derivative, one observes erosion of the basin of attraction in Figure 10(a): Persistence of the white color in the central well and disappearance of the blue color in the left and right wells. That is, trajectories initiated in the central well are not confined within the central well. This tendency increases as the order of derivative increases. Thus, lowering the order of the derivative beyond a certain threshold might have negative effects on the system performance, as it plays against the appearance of multiple wells response.

6 Ferroelectric domain walls reconfiguration in synchronized switch conditions Most of the energy harvesting systems based on piezo-ceramic conversion operates with levels of mechanical excitation sufficiently weak to consider a linear relation between the dynamical excitation quantities (stress T(t)) and the induced ones [52] (electrical polarization P(t)). In the case of the Synchronized Switch Harvesting on Inductor (SSHI) [1, 2] energy extractor described in this article, we intentionally enhance the excitation amplitude in order to increase the amount of energy harvested. Figure 11 shows the schematics of such an energy harvesting system with synchronized switchers. As the linear material relations are no longer suitable, a more sophisticated model, including saturation, hysteresis and dynamic changes consideration is necessary to derive results and for analysis. Based on experimental notations, a highprecision piezo-ceramic (ferroelectric) nonlinear model, suitable for large frequency bandwidth and providing good accuracy whatever the excitation nature (electrical or mechanical) has been developed [64, 66–68]. Following this direction of research, the present hysteresis model is constituted of two contributions: quasi-static and dynamics ones. Under external mechanical compressive stress the piezo-electric polarization is affected by the coupling term. In our case we account for the piezo-electric layer which is subjected to elongation and contraction. The simplified equations have a following

Energy harvesting in dynamical systems with fractional-order physical properties | 79

Figure 11: (a) Schematics of the piezo-electric beam and electrical circuit with switchers. (b) The active part.

form: ρ

dμ Pi (t) −1 (Pi ), = E(t) + sT(t)Pi (t) − fstatic dt μ

(30)

where T(t) is the imposed harmonically changing mechanical stress T(t) = T0 sin(ωt),

(31)

and s is a coupling constant. Pi is the ferroelectric domain polarization, P = fstatic (Pi , E) is the Preisach evolution function [42, 49, 59] for a quasi-static external electrical field. μ is the fractional order of the derivative related to damping through domain walls reconfigurations. Note that the above model has a general form, but it does not include the effects of instantaneous switching (Figure 12) at the maximum stress instants. This will be discussed in the following part [11]. Table 3: Simulation parameters used in equation (30). parameter value units parameter value units

α

μ

ρ

A

0.03 V N−1 C−1 m3

0.56 –

20000 V Sn C−1 m2n−1

4 × 10 m2

−4

l

T0

ω

ω0

τ

L

0.001 mm

2 × 10 N m2

2πf rad s−1

20πf rad s−1

0.1 s

160 H

−7

Using the above equations (equation (30)) we performed simulations of the voltage dynamical response. The results are presented in Figure 12, while the system parameters are collected in Table 3. The presented results compare the simulation of the system with nonlinear (hysteretic) material characteristics to the linear model results (equation (30)). Note that we start the simulation from the first polarization curve (see Figure 12, polarization start from P = 0).

80 | G. Litak et al.

Figure 12: (a) First polarization curve, P, versus applied electrical field, E. Time variations of the current and the voltage along the piezo-element. One can see that after an initial increase a decreasing oscillatory trend typical for a hysteretic system occurs. (b) Voltage output time series. These results are based on the time series presented in the previous figure for an external stress excitation of f = 10 Hz.

Figures 12(a) shows the electrical field E and polarization P for the frequency of stress T oscillations fixed at f = 10. Clearly, the system reaches the steady state vibrational solution. This is better visible on the voltage time series. Note that in the linear case the electrode capacity is constant, while in the hysteretic mode it is dependent on the dynamic susceptibility χ: χ=

P(t) . ε0 E(t)

(32)

Finally, the first polarization together with the working stationary point is plotted in Figure 2(a). Namely, the resulting steady state small hysteresis is appearing here. Comparing it with the usual large loop quasi-static hysteresis it is much smaller because of a variable electrical field limited to the switching mechanism with small external electrical field E. However, this hysteresis is still visible and leads to corrections in the harvested energy estimations. The main effects are related in additional hysteretic damping and modification with a specific frequency response of the harvester.

Energy harvesting in dynamical systems with fractional-order physical properties | 81

7 Other results in EHS A recent contribution [24] has considered the mathematical model of a piezo-electric wind flow energy harvester system for which the capacitance of the piezo-electric material has fractional-order current–voltage characteristics. Additionally the mechanical element is assumed to have fractional-order damping. The analysis showed that the derivatives affect the appearance and characteristics of the manifestation of limit circle oscillations (LCOs) of galloping motion. It was found that the order of derivatives enhances the amplitude of LCO and lowers the threshold condition leading to LCO. In addition of considering a piezo-electric material with fractional-order properties, an additional feedback loop was added to the system. For an appropriate choice of the system’s derivative orders, the performance of the system was obtained to be significantly enhanced. Specifically, the lower bound for an LCO with integer-order derivative can be reduced by fractional material properties and/or electric components. Nonlinear analysis and analog simulation of a piezo-electric buckled beam with fractional derivative was studied by Fokou et al. [14, 15]. It emerges from these results that the order of the fractional derivative play an important role in the response of the system. In a similar analysis, Li et al. [34] and Cao et al. [4, 5] considered a broadband excitation and showed that a fractional-order model of piezo-electric energy harvesting could capture the hysteresis characteristics of the visco-elastic beam with a piezoelectric layer.

8 Conclusions A fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Fractional-order systems are particularly useful in studying the anomalous behavior of dynamical systems in physics as they provide new degrees of freedom in the modeling equations. Energy harvesters from ambient mechanical vibrations as the ones described in this chapter are complex systems in which high nonlinear behaviors, frequency dependence can be found in every stages and elements of the system. As a consequence and as described in this chapter, fractional derivations are elegant mathematical tools that can be used to solve more realistic model issues or to increase the simulation domains of validity in almost every stage of the energy conversion. Although a fractional derivative, as damping contribution, is associated with the additional hysteresis suppression, its property of multiple relaxation times introduces the smearing of the optimal working point and thus the widening of the energy transfer bandwidth important for ambient vibration sources which are variable in frequencies and amplitudes.

82 | G. Litak et al. Fractional derivatives can be used to consider the frequency dependence of the dielectric losses through the piezo-electric active materials required to convert the mechanical energy into electricity. Similarly, fractional derivatives can be used to model both the electric and the mechanical parts of the harvester. Fractional derivatives can be solved in the time domain or in the frequency domain and their introduction into lump models avoids the use of time consuming and hazardous space discretization techniques such as finite elements. Fractional orders can finally also be used to define the potential energy of a system and from the simulation related one can obtain the best configuration in terms of efficiency and maximum amount of energy harvested. Different approaches of fractional calculus exist. The Grunwald–Letnikov method, the Riemman–Liouville or the Caputo [9] seem to be the most popular. The type of functions described by the Riemman–Liouville definition seems to be broader (functions must be integrable) than the one defined by Grunwald–Letnikov.

Appendix Coefficients for the homoclinic orbits +∞

I11 = ∫ −∞

=

sinh2 ( χt2 )

(κ + cosh(χt))3

dt

κ 2(1 + 2κ) 1 1−κ )), (arctan − arctan √ ((2 + κ) + √1 − κ2 √1 − κ 2 1+κ 2χ(1 − κ)(1 − κ 2 )

+∞

I12 = ∫ −∞ +∞

I13 = ∫ −∞ +∞

I14 = ∫ −∞

t

sinh( χt2 )e−γt

sinh( χs2 )eγs ( ds)dt, ∫ (κ + cosh(χt))3/2 (κ + cosh(χs))3/2 0

t

sinh( χs2 ) ( ds)dt, ∫ (κ + cosh(χt))3/2 (t − s)μ (κ + cosh(χs))3/2 sinh( χt2 )

0

sinh( χt2 ) sin(ωt) dt (κ + cosh(χt))3/2

=

2√2 2ω sin( ). (1 − κ)χ χ

Coefficients for the heteroclinic orbits +∞

I21 = ∫ −∞

=

cosh2 ( χt2 )

(−κ + cosh(χt))3

dt

1−κ 2(1 + 2κ) κ 1 ((2 + κ) + (arctan + arctan √ )), √1 − κ2 √1 − κ 2 1+κ 2χ(1 − κ)(1 − κ 2 )

Energy harvesting in dynamical systems with fractional-order physical properties | 83

+∞

I22 = ∫ −∞ +∞

I23 = ∫ −∞ +∞

I24 = ∫ −∞

cosh( χt2 )e−γt

(−κ + cosh(χt))3/2

t

(∫ 0

cosh( χs2 )eγs

(−κ + cosh(χs))3/2

ds)dt,

t

cosh( χt2 )

cosh( χs2 ) ( ds)dt, ∫ (−κ + cosh(χt))3/2 (t − s)μ (−κ + cosh(χs))3/2 0

cosh( χt2 ) cos(ωt) dt (−κ + cosh(χt))3/2

=

2√2πω . ) (1 − κ)χ 2 sinh( πω χ

The integrals I12 , I13 , I22 and I23 are evaluated numerically.

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Renat T. Sibatov and Vladimir V. Uchaikin

Fractional kinetics of charge carriers in supercapacitors Abstract: Fractional-order kinetic equations constitute a unifying tool for description of charge carriers behavior in disordered media. In this chapter, we review applications of these equations to mathematical description of specific processes in supercapacitors (ion transport in electrolytes, passage through electrodes with a highly developed surface, double layer formation and others). We highlight the properties leading to fractional dynamics, underlying modern theory of the anomalous diffusion. Electrochemical responses of supercapacitors are considered for various experimental techniques. The links between the kinetic models and equivalent circuits with fractional impedances are discussed. Keywords: Double electric layer, impedance spectroscopy, anomalous diffusion, tempered fractional derivative PACS: 61.43.Hv, 02.50.Ey, 82.20.Nk

1 Introduction Supercapacitors are high-capacity electrochemical devices with short charging time, very long cycle life, high specific power, low impedance and some others peculiarities, thanks to which they find practical application in various technical systems: wind turbines, photovoltaic devices, biomedical sensors, hybrid and electric cars, etc. [20]. Electrochemical supercapacitors (ES) differ in their charge storage mechanism, cell structure, electrolyte and electrode materials. There are distinguished three categories of ES, differing in mechanisms of charge storage [83]: 1. electric double-layer supercapacitors (EDLCs), whose capacitance arises due to the accumulation of charges at the electrode–electrolyte interface, 2. pseudocapacitors, where rapid reversible Faraday redox reactions take place at the electrodes, Acknowledgement: The authors thank the Russian Foundation for Basic Research (projects 18-5153018, 16-01-00556, 16-42-732113) and the Ministry of Education and Science of the Russian Federation (government programme 3.2111.2017/4.6) for financial support. Renat T. Sibatov, Ulyanovsk State University, Laboratory of Diffusion Processes, Ulyanovsk, Russia, e-mail: [email protected] Vladimir V. Uchaikin, Ulyanovsk State University, Department of Theoretical Physics, Ulyanovsk, Russia https://doi.org/10.1515/9783110571929-004

88 | R. T. Sibatov and V. V. Uchaikin 3.

hybrid electrochemical supercapacitors, which use EDL on one of the electrodes and Faraday capacitive mechanisms on the other.

In reality, any supercapacitor is characterized by both capacity of the EDL and pseudocapacity in various ratios. EDLCs with carbon electrodes have a weak pseudocapacity due to Faraday reaction of surface functional groups, and pseudocapacitors have some capacity of EDL proportional to the electrochemically effective surface area [20, 83]. High capacitances of ES with fixed sizes are achieved by increasing the effective area of the electrodes using materials with highly developed surface (carbon materials, conductive polymers, metal oxides, composite and nanostructured systems) [20]. Current approaches to increasing the energy density of ES are based on the hybridization of electrodes by adding electrochemically active additives or by complete replacing of carbon materials with electrochemically active materials [76]. Electrolytes used in supercapacitors are of three types: aqueous electrolytes, organic electrolytes and ionic liquids. A high ion concentration, low resistance, easy control of the processes and conditions of preparation are considered as main advantages of aqueous electrolytes. A significant disadvantage of aqueous electrolytes, however, is the low voltage of decomposition (1 ÷ 2 V). For this reason, organic electrolytes are often used. They are capable to provide a voltage window up to 3.5 V, but they are characterized by a higher resistance. One of the successful directions in phenomenological modeling of supercapacitors is based on the use of fractional calculus (see, e. g., [10, 18, 25, 59]). Usually, models are formulated in terms of equivalent circuits with one or more ‘constant phase element’, also known as a capacitor of fractional order [14, 33, 80]. The correspondent element with frequency dependent impedance Z(ω) ∝ (jω)−α is often called the fractal element [31, 80]. In special cases, when α is equal to = 1, 0 or −1, this element is associated with an ideal capacitor, a resistor or an inductor, respectively. When α = −1/2, it represents the Warburg diffusion impedance. Fractional differential models of supercapacitors were used to describe their operation in pulse power systems [16], to study the EDLC impedance in a limited frequency range for reducing number of model parameters [25, 50, 59], the temporal response of the EDLC [72, 78], to determine the parameters from transient characteristics [24, 30], for controlling systems of supercapacitors using fractional state models [25], or fractional nonlinear systems [10, 11]. Bertrand and Sabatier et al. [10, 61] have shown how fractional-order models can be used to capture the dynamical behavior of electrochemical devices, such as batteries or supercapacitors, and to build ‘state of charge’ or ‘state of health’ estimators. At the same time, they noted that this treatment does not reveal the nature of non-integer systems and not explain the physical interpretation of the models. In this chapter, we are going to fill the gap.

Fractional kinetics of charge carriers in supercapacitors | 89

2 Distribution of relaxation rates A typical ES consists of two porous electrodes separated by an ion-permeable separator, and an electrolyte ionically connecting electrodes (Figure 1(a)). Approximately, the porous electrode can be modeled by an array of individual pores (Figure 1(b)) arranged in parallel (Figure 1(c)). The engineering instruction for EDLC Panasonic [58] represents the non-exponential decay of discharging current by means of the equivalent scheme presented in Figure 1(c). The parallel arrangement of individual pores is responsible for distribution of relaxation times associated with ‘elementary capacitors’. Each elementary component discharges according to the simplest relaxation equation: f ̇(t) = −μf (t) (μ > 0), having a solution in the form of an exponential function. Considering a dichotomic stochastic process in which the relaxation between two states is given not by a single rate μ, but by a distribution ρ(μ), Glöckle and Nonnenmacher [34] represented the relaxation function by means of superposition ∞

f (t) = ∫ ρ(μ) exp(−μt) dμ. 0

Further, considering a scaling form of this distribution, they arrive at the equation of fractional differential relaxation Dαt f (t) = −μ0 f (t) + δα (t)f0 ,

δα (t) = t+−α /Γ(1 − α),

whose solution is expressed in terms of the Mittag-Leffler function, f (t) = f0 Eα (−μt α ),

zk . Γ(1 + αk) k=0 ∞

Eα (z) = ∑

Figure 1: Schematic of EDLC (a), elementary component (b) and the simplest equivalent circuit of EDLC (c).

(1)

90 | R. T. Sibatov and V. V. Uchaikin Without denying the methodological usefulness of this approach, it should be noted that even relaxation of an elementary electrolytic capacitor is apparently nonexponential. The charging–discharging kinetics of an EDLC is largely determined by the diffusion formation of the EDL, which arises as a result of charge accumulation at the electrode–electrolyte interface. The model for the formation of EDL was firstly proposed by Helmholtz (1853) and developed later. The Guy–Chapman (1910) model takes into account the thermal motion of adsorbed ions and their diffusion in solution, so that the surface charge of the electrode is balanced by a diffuse layer of oppositely charged ions. Stern (1924) modified the Guy–Chapman model taking into account the finite size of the ions and introduced a division into a compact layer of adsorbed ions closely adjacent to the electrode surface (Helmholtz layer) and diffuse layer in solution. This separation is important in the case of an electrolyte with a high concentration of ions. The model was refined by Graham (1947) taking into account the influence of solvent molecules and the difference in the sizes of cationic and anionic complexes. To date, the model of EDL continues to improve [35].

3 Discharging of a supercapacitor and ψ-derivative with respect to power function Ten years ago we (together with D. Uchaikin) numerically investigated the response to a voltage step for a ‘fractional’ capacitor at small deviations of the derivative order from the unit, with a view following up the transition from a Markovian (ν = 1) to a non-Markovian (ν ≠ 1) process [75]. We saw the quite understandable results: the exponential function in the first case and its “fractional” analogue, expressed through the Mittag-Leffler function, in the second one. When ν = 1, both curves coincided, but as soon as ν became a little less, a “power tail” appeared in the analogue, which was not present in a pure exponential function. Further studies have shown that this effect is a consequence of non-uniform convergence, due to which both curves almost coincide up to some critical time, and diverge. Because for ν = 1 the solution does not depend on the prehistory (“the memory is absent”), but for ν < 1 does (“the memory is present”), such behavior of the discharge curve looks like a memory regeneration. Let us consider this process in more detail. The charging EMF is turned on at t = −θ < 0, staying constant, with value ℰ0 , up to t = 0, and turned off at t = 0, when relaxation is activated. Writing the solution of the equation for the voltage Dνt U(t) + τ−ν U(t) = ℰ (t) via the Green function in the form of the Mittag-Leffler function Eν (t), and representing the latter in the power series form, we obtained [75] the following equation for the ratio U(t)/U(0), t > 0: U(t)/U(0) =

Eν (−t ν /τν ) − Eν [−(t + θ)ν /τν ] . 1 − Eν [−θν /τν ]

(2)

Fractional kinetics of charge carriers in supercapacitors | 91

Figure 2: Theoretical discharging curves of a ‘fractional’ capacitor for different charging times θ (ν = 0.998, τ = 0.5 s).

Each value of ν produces a family of curves with relation to different charging times θ, and only one value ν (ν = 1) produces a single curve independent on θ. At large times t ≫ θ the difference and the charging time θ in (2) can be interpreted as differential. Then it follows that U(t) ∼ θℰ0 t ν−1 Eν,ν [−(t/τ)ν ],

t ≫ 0.

This function has a power-tail: fν (t) ∝ t −ν−1 , as t → ∞. Theoretical discharging curves for different θ are shown in Figure 2. For ν ≈ 1, the voltage falls down accordingly to the Debye exponential law for a long time, but eventually we observe a split of the curves with different values of θ and the transition to a non-Debye power law. It looks like the memory returns into the system after some interval of time. Such a behavior was named a memory regeneration effect [75]. For ν = 1, relaxation follows the Debye law independently on θ: memory is absent. Interesting behavior of discharging current for Panasonic supercapacitors (0.22, 0.47 and 1 F) was revealed in [73]. This behavior conforms neither of known models based on equivalent circuit because the response is nonhomogeneous in time. Figure 3 demonstrates charging and discharging current curves obtained for the electrolytic capacitor Rubicon (1600 μF, 16 V) and the Panasonic EDLC (0.22 F). They differ by the type of current decay at the initial stage (exponential for Rubicon and stretched exponential for Panasonic EDLC) and by a flatter charging current decay for EDLC compared to Rubicon. In [72] the authors analyzed charging–discharging currents in Panasonic Gold capacitors in a wide range of time, from 10−1 to 104 s. At small times, relaxation is close to a stretched exponential law J(t) ∝ exp(−t β /τβ ) with β parameter values ranging from 0.5 to 0.7. For large times, the behavior of the current is asymptotically a power law J(t) ∝ t −α , with α values ranging from 0.8 to 1.2. Increase in temperature leads to decrease in internal resistance, which affects the change in the index of the power law and the tendency to pass-through conduction, as well as it reduces the scale parameter of the stretched exponential law.

92 | R. T. Sibatov and V. V. Uchaikin

Figure 3: Charging and discharging currents in electrolytic capacitor Rubicon (1600 μF, 16 V) (left) and in Panasonic Gold supercapacitor (0.22 F) for different charging times θ.

Figure 4: (a) Charging current in the Panasonic Gold capacitor (0.22 F). Points are experimental data, green dashed lines correspond to approximations by n-component superposition of exponential functions, the blue solid line is an approximation by the stretched fractional exponent (5). (b) Charging–discharging currents 0.22 F. Lines correspond to experimental data, points to an approximation by stretched fractional exponent.

In [72] and [73], it was established that the reaction of Panasonic Gold EDLCs (0.22, 0.47 and 1 F) to variations of the charging regime do not agree with the model of a homogeneous linear response (that is, a linear response depending only on the time difference t − t 󸀠 ). Approximation of the experimental data on charging–discharging of Panasonic Gold supercapacitors (0.22 F, 0.47 F, 1.0 F, 1.8 F) by a finite superposition of exponential functions converges nonuniformly (Figure 4(a)) to an observable curve. Such a behavior was not described by any known approximation, and a new representation of this process was suggested in [74] using the stretched fractional exponent obtained from the fractional exponent by the formal substitution t → t ν . This function is a solution of the linear integro-differential equation with the memory kernel non-invariant under a time shift.

Fractional kinetics of charge carriers in supercapacitors | 93

The stretched fractional exponential function has the form (−z αν )n , Γ(αn + 1) n=0 ∞

f (z) = expα,ν (−z) = ∑

α, ν > 0.

(3)

For α = ν = 1 this function coincides with the usual exponential function; for α = 1, ν < 1, it is a stretched exponential, for α ≠ 1, ν = 1, it is a fractional exponential expressed via the Mittag-Leffler function. The function (3) satisfies the following equation: C α 0 Dψ(t) f (t)

+ μf (t) = 0,

(4)

with the initial condition f (0) = 1. Here, C0 Dαψ(t) is the ψ-Caputo fractional derivative, that is, a Caputo-type operator of a function with respect to another function [2], C α 0 Dψ(t) f (t)

t

1 f 󸀠 (τ) = ψ󸀠 (τ) dτ, ∫ Γ(1 − α) (ψ(t) − ψ(τ))α

α < 1.

0

In [73, 74], the choice ψ(t) = t ν allowed us to adjust the model curves with experimental data on charging–discharging of a supercapacitor under consideration. The corresponding fractional operator is sometimes referred to as the Katugampola derivative (in the Caputo sense) [39]. Introducing the ψ-function could be explained by the time dependence subdiffusion coefficient Dα (t) similarly to the case considered in Section 4. Figure 4 demonstrates good agreement between the experimental data on charging of EDLC and the function J(t)/J(0) = expα,ν (−μt)

(5)

with parameters μ = 0.068, α = 0.97 and ν = 0.7. Additional parameters are indicated in Figure 4(a). During the data processing, we observe a violation of the linear homogeneous response model. The charging curve was approximated by the special response function (3). The same function was used to describe the discharging curves (Figure 4(b)), which provide excellent agreement.

4 Diffusion-limited charge transfer Due to the fact that EDLCs with carbon electrodes have a weak pseudocapacity due to the Faraday reaction of surface functional groups we consider the general case of diffusion-limited charge transfer at the electrode surface. Oldham notes in [55] that electrochemistry “was not the first scientific discipline to benefit from the fractional

94 | R. T. Sibatov and V. V. Uchaikin calculus, but it was certainly one of the first to reap a sustained harvest of useful concepts and methodologies.” In the simplest model, the transport of ions occurs due to 1D diffusion from the planar electrode in the semi-infinite electrolytic medium, 1 𝜕 𝜕2 c(x, t) = 2 c(x, t). D 𝜕t 𝜕x

(6)

This equation is usually solved (see, e. g., [55]) for the following initial and boundary conditions: c(x, 0) = cb ,

c(x → ∞, t) = cb ,

c(0, t) = cs (t),

where cb and cs (t) are concentrations of ions in the bulk electrolyte and near the electrode surface, respectively. The electric current is proportional to the flux j(0, t) of ions arriving at the electrode, J(t) = −nFAj(0, t),

(7)

where F is the Faraday constant, A is a surface area of the electrode. In terms of the Laplace transformation on time, a solution of the diffusion equation (6) under the indicated conditions has the form ̃ s) = c(x,

c cb s + [c̃0 (s) − b ] exp{−x√ }. s s D

(8)

The expression for the current density j(0, t) at the surface ∞

j(0, t) = ∫ 0

𝜕c(x, t) dx, 𝜕t

(9)

follows from the continuity equation 𝜕c/𝜕t + 𝜕j/𝜕x = 0. We use (9) instead of Fick’s law because relation (9) is more general and better applicable in the case of anomalous diffusion. Applying a Laplace transformation to the latter expression and substituting solution (8), we arrive at the expression ∞



c s ̃ s) = ∫ [sc(x, ̃ s) − cb ] dx = s[c̃0 (s) − b ] ∫ exp{−x√ } dx j(0, s D 0

c = √Ds [c̃0 (s) − b ]. s

0

1/2

(10)

Inserting it into equation (7) yields the fractional relationship between current and concentration of ions at the surface, J(t) = nFA√D D1/2 t [cb − cs (t)].

(11)

Fractional kinetics of charge carriers in supercapacitors | 95

If cs (t) = 0 for the case of the current response to a potential step without Faradaic reaction at the surface [21], we have J(t) = nFAcb √

D . πt

(12)

The expression for the concentration through the current contains a fractional integral: cb − cs (t) = BIt1/2 J(t),

(13)

where B = (nFA√D)−1 . The expression exp{−x √s/D} in equation (8) can be interpreted as the transform ̃ of the first passage time density of the Brownian motion starting from point p(s|x) x ≠ 0 and with the boundary located at surface x = 0, p(t|x) =

x2 x −3/2 ). t exp(− 4Dt 2√πD

(14)

The Lévy–Smirnov density (14) satisfies the fractional equation √ (D1/2 t + D

𝜕 )p(t|x) = 0, 𝜕x

p(t|0) = δ(t),

which could be derived by factoring the diffusion equation [55]. If we substitute an arbitrary transform of first passage time density instead of exp{−x √s/D} into (10), we obtain t

J(t) = nFA



d ∫ dτ{ ∫ p(t − τ|x) dx}[cb − cs (τ)]. dt 0

(15)

0

In the case of the time-dependent diffusion coefficient D(t), the diffusion equation (6) could be reduced to the ordinary form by the change of variables t

τ = ∫ D(t) dt = ψ(t). 0

In this case we arrive at generalizations of (11) and (13) that can be written as J(t) = nFA 0 D1/2 [c − cs (t)], ψ(t) b

1/2 cb − cs (t) = βIψ(t) J(t).

(16)

1/2 are ψ-Riemann–Liouville fractional derivative and integral, and 0 Iψ(t) Here, 0 D1/2 ψ(t) which are Riemann–Liouville-type operators of a function with respect to another function [2]. An application of these operators to experimenatl data on EDLC charging is shown in the previous section.

96 | R. T. Sibatov and V. V. Uchaikin Note that the first passage time pdf (14) is valid in the case of a plane homogeneous electrode and one-dimensional diffusion (6). Introducing instead of (14) a wide class of stable densities with an exponent 0 < κ < 1 it is easy to generalize result (13). The ̃ one-sided Lévy stable density has the Laplace transform p(s|x) = exp(−xsκ /b) (b is a scale constant). Then we obtain ∞

̃ s) = [sc̃ (s) − c ] ∫ exp{−xsκ /b} dx = bs1−κ [c̃ (s) − cb ]. j(0, 0 b 0 s

(17)

0

As a result, we arrive at the following relationship between current and concentration: J(t) = nFAb D1−κ t [cb − cs (t)],

(18)

which can be written in the form cb − cs (t) = aIt1−κ J(t),

(19)

where a = (nFAb)−1 . If cs (t) = 0 for the case of the current response to a potential step without Faradaic reaction at the surface [21], we have J(t) = nFAbcb

t −1+κ . Γ(κ)

(20)

It is easy to show that, in the case κ ∈ (0, 1/2), formula (17) corresponds to the subdiffusion equation with subdiffusion exponent α = 2κ ∈ (0, 1) and coefficient Kα = b2 , 1 C α 𝜕2 Dt c(x, t) = 2 c(x, t). Kα 𝜕x

(21)

5 Fractal electrode Characteristics of electrochemical devices and processes in them are defined generally by the properties of a metallic electrode surface in contact with a liquid or a solid electrolyte. In classical theory the influence of a boundary on an alternative current transmission through the system is described by a boundary capacitance inserted in parallel with the ohmic resistance of the electrolyte. At the same time the real part of the impedance does not depend on the frequency, and the imaginary part is inversely proportional to it. Many experiments, however, show a different behavior in most cases: at least in a restricted range the frequency dependence of the impedance is characterized by the additional power type term A(iω)−η with η ∈ (0, 1). This term is called the constant phase element (CPE). The presence of a CPE is associated with the surface roughness: the smoother the surface is, the closer η to the unity is. It is shown

Fractional kinetics of charge carriers in supercapacitors | 97

Figure 5: Equivalent scheme of rough electrode’s surface.

in [22] that the imitation of a rough surface via pores leads to a power dependence, but with a definite exponent η = 1/2, only if one does not assume a particular space structure of the resistance and capacitance distributions. As a ground for such structures it serves the investigations of surfaces using electronic microscopes. The absence of a natural scale of length, causing different zooming to show almost similar pictures of inhomogeneities distributions, makes it possible to apply the concept of fractals. Figure 5 shows an equivalent scheme often used to describe rough surface responses. The input impedance of this chain can be written in the form of an infinite chain fraction: Z(iω) = R +

iωC +

1 aR+

2

1

.

iωC+ 2 2 a R+⋅⋅⋅

Practical calculations are performed for finite chain fraction with the recursion number n and the solution behavior is investigated against the increase of n. At low frequencies the real part of impedance gets a plateau whose height rises a/2 times with every recursion. At high frequencies the impedance takes its extremal value R. At intermediate frequencies the system possess the property of CPE with η = 1 − df , where df = ln 2/ ln a is the fractal dimension of the Cantor set. The imaginary part Z(iω) is inversely proportional to the frequency at high and low frequencies and corresponds to CPE in the intermediate region. Thus the frequency representation of the generalized Ohm law for a rough surface has the form −1 ̃ ̃ ̃ ̃ = A−1 (iω)η U(iω), J(ω) = [Z(iω)] U(iω)

and in time representation it contains a fractional derivative: η

J(t) = A−1 Dt U(t). As is stated in [49], the power dependence of impedance on frequency and therefore the fractional order of the derivative are connected with the competition of resistive and capacitive currents. At low frequency the signal propagates farther through

98 | R. T. Sibatov and V. V. Uchaikin

Figure 6: SEM images of nanostructured electrodes for possible electrochemical applications. (a) Pt3 Co-SiC powder. Credit: Dan Liu et al. [48]; (b) Rh nanowires electrodeposited in aqueous solution. Credit: Liqiu Zhang et al. [82]; (c) MnPO4 ⋅H2 O nanowires. Credit: Bo Yan et al. [81]. (d) Cross-sectional images of hitosan/gelatin scaffolds. Credit: J.C. Forero et al. [29]. (e) Sample with deposited electroless Ni at 70°C for 75 s. Credit: Shu Huei Hsieh et al. [36]. (f) CH3 NH3 PbI3 /PbI2 /polyvinylpyrrolidone composite fibers. Credit: Li-Min Chao et al. [17]. CC BY 4.0 license.

a chain, before it can escape through surface capacity; therefore the low frequency impedance is larger. Of course, real surfaces are self-similar only in a finite frequency range, ultimately defining the frequency interval, where phase constancy (loss angle) takes place. A variety of electrodes proposed for EDLC are demonstrated in Figure 6. They are characterized by different types of porosity and disorder. Electrochemical responses to various signals contain important information as regards diffusion-controlled processes occurring at the electrode surface. Cottrell showed that the diffusion response to the potential step applied to a flat electrode leads to the power law decay of the current J(t) ∝ 1/√t. De Gennes predicted a similar power law behavior J(t) ∝ t −ν for current, flowing at fractal electrodes with dimension df . Moreover, ν is related to df by relation ν=

df − 1 2

.

(22)

The power law decay of the current under Cottrell conditions was confirmed in [5, 42]. Relation (22) was experimentally examined by Pajkossy and Nyikos [56] for specially prepared fractal electrodes with well-defined fractal dimensions. Le Mehaute and Crepy [42] proposed the generalized transfer equation of fractional order that describes the flow of an extensive quantity through a fractal medium in the linear ap-

Fractional kinetics of charge carriers in supercapacitors | 99

proximation of the thermodynamics of irreversible processes. Dassas and Duby used the fractional equation (21) with α = 3 − df to explain the electrochemical responses of fractal electrodes. This equation could be derived within different transport models such as continuous time random walks [62], multiple trapping [13], and the comb model [3]. In all these models we deal with asymptotical power law distributions of waiting times in some localized states. But, particularly for the fractal electrodes used in experiments [56, 57] it is not easy to substantiate reasons for such a localization. Fa and Lenzi [27] solved the equation for diffusion on fractals with spatially dependent coefficient at the presence of absorbing boundaries and derived the first passage time distribution, p(t|x) =

exp{−x 2+θ /(2 + θ)2 K1 t} x1+θ , tΓ((1 + θ)/(2 + θ)) [(2 + θ)2 K1 t](1+θ)/(2+θ)

t > 0.

(23)

In a homogeneous medium, when θ = 0, this density becomes the Lévy–Smirnov density. Substituting pdf (23) into expression (15), we also obtain a fractional relation between current and surface ion concentration, [cb − cs (t)]. J(t) = nFA b(θ, K1 ) D(1+θ)/(2+θ) t

(24)

Thus, anomalous transport of ions described by the fractional subdiffusion equation and diffusion with a variable diffusion coefficient both lead to the fractional relation (19) between current and concentration.

6 Trapping and percolation At present, both aqueous and organic electrolytes are used in supercapacitors. Organic electrolytes often have higher decomposition voltage (3 V against 1 V in aqueous solutions of acids and alkalis), allowing to produce supercapacitors with high operating voltage. However, the high resistance of electrolytes based on complex organic compounds compared with aqueous ones leads to a decrease in the specific power of the device. Some solid electrolytes used in electrochemical devices have polycrystalline structure with grain boundaries (for example, garnet with composition Li7 La3 Zr2 O12 ). As is known, the grain boundaries can increase or suppress the rate of ionic diffusion in polycrystalline oxides. Many approaches to the description of grain-boundary diffusion are based on the classical model proposed by Fisher in 1951, and use the system of normal-diffusion equations in an inhomogeneous medium. A direct relationship between the fractional diffusion approach and the Fisher model and its modifications without additional assumptions is demonstrated in Ref. [68]. Structural disorder peculiar to solid organic systems leads to the presence of traps with distributed localization energy. The dynamics of charge carriers in organic solids

100 | R. T. Sibatov and V. V. Uchaikin is often described by the multiple trapping (MT) model [54], which can be formulated in terms of the fractional diffusion equation [13, 69]. In the closely adjacent model, the so-called random activation energy model, it is assumed that: – the jump rate of a particle hopping over an energy barrier ΔE has the usual quasiclassical form W = Ae−ΔE/kT ; –

the conditional waiting time distribution corresponding to a given activation energy ΔE = ε is exponential P{T > t|ε} = e−W(ε)t ;



the activation energy is a random variable with the Boltzmann distribution density, p(ε) = ε0−1 e−ε/ε0 .

Averaging over the activation energy leads to the power law distribution ∞



0

0

P{T > t} = ∫ P{T > t|ε}p(ε) dε = ∫ exp[−(Ae−ε/kT )t] d(e−ε/ε0 ) = νΓ(ν)(At)−ν , with ν = kT/ε0 . Here, ε0 is the characteristic energy defining the width of localized density of states (DoS). For kT < ε0 , thermalization dominates and the transport becomes dispersive. For kT > ε0 , the carriers remain concentrated near the mobility edge and the charge transit exhibits non-dispersive behavior. Consequently, the physical meaning of ν is that of a representative of disorder: the smaller its value, the more dispersive the transport. Sibatov and Morozova [65] derived the following equation for the linearized MTmodel: t

𝜕p(x, t) 𝜕p(x, τ) 𝜕p(x, t) 𝜕2 p(x, t) = 0, + ∫ Q(t − τ) dτ + μE −D 𝜕t 𝜕τ 𝜕x 𝜕x 2

(25)

−∞

where ∞

Q(t) = ∫ λε exp{−λε t 0

Nf

Nt

e−ε/kT }ρ(ε) dε,

(26)

λε is a localization rate, Nf and Nt concentrations of free and localized states, ρ(ε) DoS of traps (see details in [65]).

Fractional kinetics of charge carriers in supercapacitors | 101

In the case of a weak dependence of the capture rate on energy ωε ≈ ω0 for exponential DoS, ρ(ε) = ε0−1 exp(−ε/ε0 ), we have Nf −ε/kT ω e } exp(−ε/ε0 ) dε. Q(t) = 0 ∫ exp{−ω0 t ε0 Nt ∞

0

Making the change of variable ξ = ω0 t (Nf /Nt ) e−ε/kT , we obtain the power law kernel ω0 α Q(t) = (ω0 tNf /Nt )α

ω0 t⋅Nf /Nt



e−ξ ξ α−1 dξ ∼

0

ω0 αΓ(α) −α t , (ω0 Nf /Nt )α

t → ∞,

α=

kT . ε0

The asymptotics of large times (t → ∞) implies that during time t a charge carrier undergoes a large number of localization–delocalization events t ≫ ω−1 0 Nt /Nf . As a result we arrive at the fractional equation 𝜕p(x, t) 𝜕2 p(x, t) 𝜕p(x, t) + ω0 cα−α C Dαt p(x, t) + μE −D = 0. 𝜕t 𝜕x 𝜕x 2

(27)

In electrodes based on nanoporous and nanostructured materials, there are many alternative ways for the current flow. Charge carrier transport is percolative and occurs through conducting channels, which are sequences of contact grains with metallic properties, and by tunneling or hopping mechanisms through dielectric layers. To study diffusion in percolation clusters, the comb model [4, 7, 79] is often used. In the simplest version of this model, a percolation cluster above criticality can be represented as a conductive axis, the x-axis (backbone of a cluster) with perpendicularly attached branches modeling ‘dead-ends’ (parallel to the y-axis). The advection– diffusion in the comb model can be described in the framework of the continuous time random walk (CTRW) concept [52, 71]. In the CTRW model [62], the Laplace transform ̃ K(s) of the kernel is associated with the transform of the probability P{θ > t} = Ψ(t) and the density ψ(t) of waiting times θ in traps by formula ̃ ̃ ̃ K(s) = Ψ(s)/ ψ(s).

(28)

The authors of [65] considered multiple trapping on a comb. If l is a random number of visited sites of a dead branch during one sojourn of the branch, localized states are in each site and corresponding waiting times are independent and identically distributed random variables with pdf ψ(t), then the distribution of the sojourn time in dead ends (averaged over l) is ∞

ψbr (t) = ∑ ψ∗l (t)pl . l=0

102 | R. T. Sibatov and V. V. Uchaikin Here, pl is a distribution of l. After the Laplace transformation, the convolution transforms into the product of densities ∞



l=0

0

l ̃ ̃ ̃ ̃ ln ψ(s)). ψ̃ br (s) = ∑ [ψ(s)] pl ∼ ∫ exp[l ln ψ(s)]p(l) dl = p(−

For the transform of the integral kernel of the generalized FP-equation, we have ̃ ̃ ln ψ(s)) 1 − p(− ̃ K(s) = . ̃ ̃ ln ψ(s)) sp(− ̃ In the case of power law distributions of waiting times in localized states ψ(s) ∼ α β ̃ ∼ 1 − (ηs) , the kernel transform has the form 1 − (τs) and ‘dead end’ lengths p(s) ̃ K(s) ∼ ηβ ταβ sαβ−1 . Substituting its original K(t) ∼ ηβ (t/τ)−αβ /Γ(1 − αβ) into equation (29), we arrive at the fractional diffusion equation of order αβ, that is, a product of energy and structural disorder exponents. In [65], the dispersive transport occurring due to several independent delocalization mechanisms is described within one-dimensional CTRW with a mixture distribution of waiting times. The corresponding kernel has the Laplace transform 1 − ∑k wk ψ̃ k (s) ̃ . K(s) = s ∑k wk ψ̃ k (s)

Here, ψk and wk are waiting time pdf and contribution ratio for the ith delocalization mechanism. In the quite general case of linear non-Fickian diffusion, when the anomalous behavior is not associated with a wide distribution of free path lengths, we can write the dispersive diffusion equation in the form t

∫ 0

𝜕 𝜕 𝜕c(x, τ) K(t − τ) dτ + {μEc(x, t) − D c(x, t)} = 0, 𝜕τ 𝜕x 𝜕x

(29)

where K(t) is a memory kernel defined by transport mechanism. Assuming a constant diffusion coefficient and zero macroscopic electric field, E = 0, due to screening by mobile charge carriers, and applying a Laplace transformation, we have the equation 𝜕2 ̃ c(x, ̃ ̃ s) − D 2 c(x, ̃ s) = c0 (x)K(s) sK(s) 𝜕x and the generalized Fick law

(30)

−1 𝜕 ̃ s) = −D [K(s)] ̃ ̃ s). J(x, c(x, (31) 𝜕x Discussing electrochemical responses, we have in view three forms of kernel K, having Laplace transforms listed in Table 1. Case A refers to pure subdiffusion, case B to tempered subdiffusion, and case C to distributed-order subdiffusion.

Fractional kinetics of charge carriers in supercapacitors | 103 ̃ Table 1: Important cases of K(s). No.

Case

K̃ (s)

A B C

Subdiffusion Tempered subdiffusion Distributed-order subdiffusion

̃ K(s) = τ α sα−1 , 0 < α ≤ 1 ̃ K(s) = τ α s−1 [(s + γ)α − γ α ] α ̃ K(s) = ∑i τi i sαi −1 , 0 < αi ≤ 1

7 Poisson–Nernst–Planck model and its fractional generalization The theory of electrochemical transport in bulk solutions is well developed [53, 60]. Its application to supercapacitor dynamics is a nontrivial task, because ions move through complex porous electrodes with internal surface charge. Microscopic models of electrolytes usually exploit the Nernst–Planck system of equations for the ionic fluxes (see, e. g., [63, 77]). Substituting these fluxes into the continuity equation, one can obtain the system of diffusion equations in the form Fz 𝜕ci = ∇[Di ci (∇ ln(γi ci ) + i ∇ϕ)], 𝜕t RT

(32)

where ϕ stands for the local electric potential, while ci and Di are the molar concentration and diffusion coefficient of ion species i in the electrolyte solution; F and R are the Faraday constant (F = 96485.3 sA/mol) and the universal gas constant (R = 8.314 J/(K⋅mol)). The reactivity coefficient γi can depend on ci , and the system is nonlinear in the general case. Equation (32) describes diffusion and electromigration in the electric field, produced by the ionic charge density, which could be self-consistently determined via Poisson’s equation, ∇2 ϕ = −

F N ∑z c . ε i=1 i i

(33)

The resulting model is called the Poisson–Nernst–Planck model, or briefly the PNPmodel. To expand the applicability of the PNP-model to porous media, recently Schmuck and Bazant [63] applied a formal homogenization procedure. The resulting macroscopic equations have a similar form to the microscopic PNP equations, except for two distinctions: parameters (diffusion coefficients, mobilities, permittivity) become tensors and the total surface charge per volume appears additionally in the macroscopic Poisson equation [63]. Difficulties of the application of microscopic models to supercapacitors are related to the need to account for the electron distribution in porous electrodes with complex morphology, the presence of surface charges and the anomalous character of electromigration in a percolative medium. Anomalous diffusion of ions in porous media is

104 | R. T. Sibatov and V. V. Uchaikin often ignored (see, e. g., [23, 63]). Existing fractional generalization of the PNP-model accounting for anomalous diffusion is developed in [26, 44] only for bulk electrolytic cell with planar electrodes. The approach was applied to granular organic materials [8, 9], but its modification for supercapacitor electrodes with complex morphology is a difficult problem waiting for a solution. One can say definitely today that any microscopic model of a supercapacitor is far from providing satisfactory ‘state of charge’ or ‘state of health’ estimators. Calculating the impedance spectra of an electrolytic cell, Lenzi et al. [44] consider a domain of length d along the z-axis with planar blocking electrodes at z = ±d/2: j± (±d/2, t) = 0. Neglecting recombination, i. e. assuming the conservation of the number of ions in the two component electrolytic system, d/2

d/2

∫ n+ (z, t) dz = ∫ n− (z, t) dz = Nd, −d/2

−d/2

they rewrite the Nernst–Planck transfer equation (32) in the linearized form (γi are constant) 𝜕c± (z, t) 𝜕2 Nq 𝜕2 ϕ = −D[ 2 c± (z, t) ± ]. 𝜕t kT 𝜕z 2 𝜕z

(34)

The 1D Poisson equation 𝜕2 ϕ(z, t) q = − [c+ (z, t) − c− (z, t)], ε 𝜕z 2

(35)

couples the equations for c+ and c− . Considering ϕ(z, t) = φ(z)ejωt , the authors of [6] obtain an expression for the impedance spectrum of the cell in the form Z = −j

βd 1 jωd 2 { }, tanh( ) + 2 2D ωεSβ2 λD2 β

(36)

where β=

1 √ jω 1 + λD2 λD D

and λD = √εk/(2Nq2 ) is Debye’s screening length. Lenzi et al. [44, 45] considered fractional generalization of the first time derivative in the Planck–Nernst equations due to anomalous diffusion of ions 1

𝜕 c (z, t) → ∫ dα p(α) C Dαt c± (z, t). 𝜕t ± 0

(37)

Fractional kinetics of charge carriers in supercapacitors | 105

The authors of [44, 45] see as an important motivation to introduce the generalization (37) the “possibility of incorporating effects which are not suitably described in terms of the usual approach. These effects are generally present in the low frequency limit where surface effects and diffusion have a pronounced role on the electrical response.” Using equation (29), one can write similarly the hereditary Nernst–Planck equation in the form t

∫ 0

𝜕c± (x, τ) 𝜕2 Nq 𝜕2 ϕ K± (t − τ) dτ = −D[ 2 c± (z, t) ± ], 𝜕τ kT 𝜕z 2 𝜕z

(38)

where kernels K+ and K− can differ. For coinciding kernels K+ = K− = K, based on the calculations of [44, 45], we can write the impedance in the form Z = −i

̃ βd jωK(jω)d 1 2 }, { 2 tanh( ) + 2 2 2D ωεSβ λD β

(39)

where β=

λD2 1 √ ̃ , 1 + jωK(jω) λD D

and transform K̃ could be chosen particularly from Table 1. The case of tempered sub̃ diffusion jωK(jω) = τα [(jω + γ)α − γ α ] is of particular interest because it has not been considered yet and inherent transient behavior could help to interpret a number of experimental data. In [45], the approach is extended to the systems characterized by Ohmic electrodes and is applied to the description of impedance spectra of electrolytic cells with Milli–Q water and a weak electrolytic solution of KCl. The authors of [46], studying the diffusion of ions in Milli-Q water, weak electrolytic solution KClO3 and some liquidcrystalline materials, demonstrate the presence of anomalous-diffusion regimes. Basu et al. [8] report on the study of gelatin, glycerol, formaldehyde films with various fractions of the LiClO4 salt. In particular, the morphology is studied using SEM and ionic conductivity by means of impedance spectroscopy. Assuming that the fractional PNP-model is applicable for a solid electrolyte, Basu et al. [8] successfully describe impedance data for gelatin-LiClO4 films. In spite of the success of the fractional PNP-model for an electrolytic cell, the approach cannot be directly applied to supercapacitors. Furthermore, despite attempts to modify the classic PNP-model for porous media [63] there are no congruent models based on microscopic approaches for supercapacitors. Therefore, people are often in favor of phenomenological models.

106 | R. T. Sibatov and V. V. Uchaikin

Figure 7: RC transmission line modeling single pore (a) and simple equivalent circuit containing CPE (b).

8 Impedance models and equivalent circuits According to the De Levie model, each pore is represented by a set of repeated RC components arranging an RC transmission line as shown in Figure 7(a). For this representation, the impedance tends to infinity at low frequency. This approach successfully describes the behavior of a range of materials with pores of similar (uniform) size, such as nanocrystalline TiO2 films in dye-sensitized solar cells [12]. The semi-infinite RC transmission line is characterized by the half-integer impedance Z = √R/jωC. A more general form used over a wide range of frequencies is represented by a constant phase element (CPE) with Z(ω) =

1 , Cν (jω)ν

0 ≤ ν ≤ 1.

(40)

A series coupling of resistor R0 and CPE (Figure 7(b)) represents a simple supercapacitor model [41], Z(s) = r +

1 , Cν s ν

(41)

used in [50, 59] to describe the EDLC impedance, and in [30] to model the transient response to a voltage-step input signal. Here, s is the Laplace variable. Connecting the element (41) in series with resistor R and applying the input voltage V(t), one could model the transient response of a supercapacitor. For the Laplace transform of the current in this circuit, one could write ν−1 ̃ V(s) ̃ = Cν (r + R)s J(s) s . ν 1 + (r + R)Cν s r + R

The inverse Laplace transformation leads to a convolution of the first derivative of the voltage V(t) with the one-parameter Mittag-Leffler function, t

J(t) = (r + R) ∫ V 󸀠 (τ)Eν (− −1

0

(t − τ)ν ) dτ. (r + R)Cν

(42)

Fractional kinetics of charge carriers in supercapacitors | 107

For the step voltage V 󸀠 (t) = V0 δ(t), the current decays as J(t) = (r + R)−1 V0 Eν (−

tν ). (r + R)Cν

In the survey of [31] of fractional-order modeling of supercapacitors, batteries and fuel cells, the authors select only three impedance models for supercapacitors. The first of them has already been mentioned (Figure 7(b)). Another model selected in [31] represents a combination of a resistor and three constant phase elements, so the impedance is Z(s) = R +

1 1 1 + . + Cα sα Cβ sβ C3 sα+β

(43)

It was used in [51] to describe the behavior of an HE0120C-0027A 120F supercapacitor in the frequency range from 1 mHz to 1 kHz. The third impedance mentioned in the survey, Z(s) = R + k

(1 + s/ω0 )α , sβ

(44)

was proposed in [59]. Using it, Quintana et al. [59] successfully described EPCOS 5 F and determined the fitting parameters (α = 0.5190, β = 0.9765, k = 0.3440 Ω/sβ ) from experimental data. In formulas (43) and (44), the exponents α and β are assumed to belong to interval [0, 1]. Figure 8 demonstrates some other simplified equivalent circuits of EDLC. The simple model presented in Figure 7(b) and used particularly in [1, 30] is a particular case of another model (Figure 8(b)), which is able to explain open circuit voltage decay, capacitance loss at high frequency, and voltammetric distortion at high scan rate for

Figure 8: The voltage distribution inside of the capacitor and its simplified equivalent circuit (taken from wiki/Supercapacitor, CC0 license) (a). Right panels demonstrate equivalent circuits of supercapacitors containing fractal capacitance (b)–(d).

108 | R. T. Sibatov and V. V. Uchaikin

Figure 9: Nyquist (left) and Bode (right) plots of a model circuit consisting of a fractance (CPE) in series with an RC-parallel network for different values of ν.

carbon-based supercapacitors [28]. Typical Nyquist and Bode plots for the model (Figure 8(b)) are demonstrated in Figure 9 for different values of ν. However, Fletcher et al. [28] note that finite ladder networks are more realistic than infinite ladder networks; this is represented by CPE in Figure 8(b). The generalized Warburg impedance obtained from the dispersive diffusion equation (29) with E = 0 takes the form Z(jω) =

̃ ϵ √ K(jω) . jω nF √D

(45)

For a limited domain with the second blocking electrode at x = L, we have Z(ω) = ζ √

̃ L K(jω) ̃ jωK(jω)], coth[ √D jω

(46)

where ζ = ϵ/nF √D is a constant. In the case of the tempered subdiffusion of ions, ZF

(α,γ)

(ω) =

√(jω + γ)α − γ α , Cα jω

0 < α ≤ 1.

(47)

Nyquist plots for a circuit including tempered fractional impedance (47) are demonstrated in Figure 10. The generalized fractional impedance, i. e. the element denoted by symbol ‘F’, represents generalization of the constant phase element and it is char̃ acterized by the impedance (46) with K(s) from Table 1. We can see a transition between two curves corresponding to circuits with a Warburg element (lower curve, low frequency) and with CPE (ν = 0.75, upper curve, high frequency). In the non-tempered case γ = 0, and ZF(α) (ω) = Cα−1 (jω)−(1−α/2)

(48)

Fractional kinetics of charge carriers in supercapacitors | 109

Figure 10: Nyquist plots for an equivalent circuit with a tempered fractional impedance.

Figure 11: Schematic diagram of PMSC: (a) top-view; and (b) cross-sectional view. Credit: Yun-Ting Chen et al. [19].

represents impedance of a CPE with exponent ν related to the subdiffusion parameter α, ν = 1 − α/2. For the normal diffusive case (α = 1), it becomes the Warburg impedance ∝ 1/√jω. If α = 0, it is an ideal capacitor with ZF(0) (ω) = (jωC0 )−1 . Recently, several works reported on supercapacitors based on planar configuration utilizing graphene, carbon nanotubes (CNT) and conducting polymers with different forms and architecture (see review in this issue [37]). Planar micro-supercapacitors (PMSC) have sizes about 1 cm or even several millimeters and can be placed directly on a chip and integrated with other microelectronic devices. Figure 11 depicts the typical geometry of PMSC. The structure provides high power density because ions can diffuse laterally between closely-spaced electrodes. There is no need in any binder or separator. PMSCs can be fabricated using photolithographic technique, laser patterning, screen-printing on a flexible substrate and some other methods [37]. Authors of Ref. [43] used a simple method of electrophoretic deposition of composite materials based on CNT arrays to produce electrodes for planar supercapacitors. In Figure 12 the simple equivalent scheme model of PMSC with CNT array based electrodes is depicted (it is a generalization of the scheme proposed in [40]). Two constant phase elements reflect subdiffusive behavior of ions in CNT array and in interelectrode space. In [40], it was observed that PMSC capacity C grows with height h of carbon nanotubes in electrodes according to a sublinear law, so the specific capacity and ratio C/h decreases. The authors explained such behavior by changing morphology of

110 | R. T. Sibatov and V. V. Uchaikin

Figure 12: The equivalent scheme model of planar PMSC with electrodes utilizing vertically aligned CNT arrays.

CNT forest for increased heights. These changes reduce the subdiffusion parameter. Fractional calculus approach looks as a very promising tool to understand the behavior of PMSC. To derive impedance models for hybrid ES, it is useful to attract well developed models of lithium-ion batteries. One of the popular hybrid type supercapacitors is the so-called lithium-ion capacitor (LIC), in which the anode consists of carbon material pre-doped with lithium ions. LICs are safer, have higher power densities than batteries and a higher output voltage than EDLC. For description of lithium-ion battery, the single particle model (SP-model) is often used [15]. In this model, the segment [0, L] on the x axis is divided into three sections corresponding to two electrodes and a separator, it is assumed that the electrodes consist of spherical particles with radii rp and rn for the cathode and the anode, respectively. The key assumption of the SP-model is that the current distribution is assumed to be uniform across the thickness of porous electrode. The SP-model has been developing since 2000 and is actively used in modern research of lithium ion batteries. Fractional equations in modifications of the SPmodel were used by authors of Refs. [61, 64, 66]. Assuming subdiffusion of lithium ions intercalated into spherical electrode particles, we consider the Fourier transform of the linearized subdiffusion equation in the form (see details in [66]): (iω)α c̃s (r, ω) = Ks

1 𝜕 2 𝜕c̃s (r, ω) (r ), 𝜕r r 2 𝜕r

0 < α ≤ 1,

0 < r < rn .

(49)

Using the standard procedure (see, e. g. [47]), one can derive the generalized expression for the electrode particle impedance from (49) under relevant boundary conditions and accounting for the generalized Fick law [66]: Zs = Rct +

Rs −1/2 1−α/2 coth {r Ks−1/2 (iω)α/2 } rKs (iω)

−1

.

(50)

This impedance corresponds to series connection of resistance to charge transfer Rct and the impedance of subdiffusion in solid phase, Rct =

RT , FJ0

Rs =

rn RT . F 2 Ks cssurf (1 − kcssurf )

Here cssurf = cs (rn , t) is a lithium concentration near the particle surface, J0 = max 0.5 kcs ce , k is a reaction rate constant, the transmission coefficient of the anode charge

is equal to 0.5, R and T are the gas constant and temperature.

Fractional kinetics of charge carriers in supercapacitors | 111

Figure 13: (a) Generalized Randle’s equivalent circuit (a circuit analogy of local interface impedance) for the LIC anode. (b) The electrochemical impedance spectra of the assembled high-purity vein graphite/Li half-cell. Credit: Xiaoyu Gao et al. [32].

With the account for subdiffusion (of order β) of ions in the electrolyte, the electric double layer impedance is found by linearizing the transport equation using the Fourier transform: (iω)β c̃e (r, ω) = Ke

1 𝜕 2 𝜕c̃e (r, ω) (r ), 𝜕r r 2 𝜕r

r > rn .

Solving this equation, by analogy with [38], we arrive at the subdiffusion impedance of a double layer on a spherical particle ZDL =

rn RTF −2 Ke−1

1 + rn Ke−1/2 (iω)1−β/2

.

The last expression can be considered as an impedance of a constant phase element connected in parallel with a resistor CPE ZDL =

RT

F 2 Ke1/2

(iω)−(1−β/2) ,

RDL =

rRT . F 2 Ke

Assuming an independent parallel arrangement of the double layer formation current and the lithium intercalation current on the surface of spherical electrode particles according to Randle and Graham’s reasoning, we arrive at the generalized circuit analogy of local interface impedance (Figure 13(a)). Figure 13(b) shows the electrochemical impedance spectra of the assembled high-purity vein graphite/Li half-cell (measured and presented in Ref. [32]). As mentioned in [32], these spectra are described by the equivalent scheme shown in the inset. This scheme contains the simplified version of the generalized Randle’s equivalent circuit.

112 | R. T. Sibatov and V. V. Uchaikin

9 Conclusion Equivalent circuit models including fractional impedances provide a reasonable description of the electrical behavior of supercapacitors. The main advantage of the approach is that the resulting models are simple, they are able to capture the dynamical behavior, and they can be used to build ‘state of charge’ or ‘state of health’ estimators [61]. Sometimes authors (see, e. g., [23]) reasonably note that the approach have no direct physical meaning. Often, the model parameterization is only applicable to certain operating conditions, and deviation from these condition reduces the applicability of the model. On the other hand, in spite of the success of microscopic approaches describing electromigration and diffusion (such as the Poisson–Nernst–Planck model) for electrolytic cells, the applications of the microscopic approach to supercapacitors have not met with great success in our opinion. Anomalous diffusion of ions in porous media is often ignored (see, e. g., [23, 63]). Existing fractional generalization of the PNP-model accounting for anomalous diffusion is developed in [26, 44] only for bulk electrolytic cell with plane electrodes. The approach was applied to granular organic materials [8, 9], but its modification for supercapacitor electrodes with complex morphology is a difficult problem waiting for solution. Here, we tried to demonstrate the relations between fractional kinetics produced by several physical mechanisms and the corresponding electrochemical responses of the supercapacitors. The diffusionlimited charge transfer at the electrodes is interpreted in terms of first passage time density. The obtained relations allowed us to extend the impedance models of EDLC by introducing tempered ‘fractance’ and a distributed-order fractional element. These elements refer to different types of anomalous transport of ions. Subdiffusion of species could be caused by localization of ions into states with a random activation energy, grain-boundary diffusion along aligned channels, and percolation on a comb structure. The tempered subdiffusion implying smooth truncation of the power law distribution of localization times could be caused by non-exponential distribution of activation energies [70], by finite size of grains in the model of grain-boundary diffusion [68] or teeth in the comb model, or due to competition between two independent delocalization processes. Distributed-order dynamics can be produced by competition between several independent transport mechanisms acting in parallel or by traps characterized by a mixture of power law waiting time distributions. Studies of memory effect in charging–discharging kinetics of EDLC indicated the behavior described by equation with the ψ-fractional derivative [2] with respect to power function ψ(t) = t ν . This behavior could be associated with fractional subdiffusion with time-dependent diffusion coefficient or multiple trapping on a percolation cluster. The latter model accounts for the joint action of energetic disorder and percolative pathways in porous media [65, 67].

Fractional kinetics of charge carriers in supercapacitors | 113

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Gabriel Bengochea, Manuel Ortigueira, Luis Verde-Star, and António M. Lopes

Recursive-operational method for fractional systems System theory without Laplace transform

Abstract: In this chapter we present a recursive-operational method for studying fractional continuous-time linear systems. The approach that we follow is an algebraic version of the usual convolution product. With it, we are able to compute the output of fractional linear systems. The method is recursive in the sense that we can add or remove (pseudo-) poles or zeros individually. The performance and accuracy of the method are illustrated by numerical examples. The procedure can be used also in nonlinear systems. To illustrate this feature we solve the fractional version of the logistic equation. Keywords: Operational calculus, fractional derivative, fractional systems MSC 2010: 44A40, 26A33

1 Introduction The importance of linear systems in science and engineering requires the development of simple but robust algorithms for computing their responses. This is valid in general and it is of primordial importance in fractional systems. These have begun to be applied in various areas of science, as can be seen in the accompanying chapters. In this chapter, we present an operational method, which allows input/output computations without using integral transforms. We will show how to compute the Acknowledgement: The research reported in this chapter was partially supported by the grant SEPCONACYT 220603. The first author was supported by SEP through the PRODEP under the project UAMPTC-630. The second author was supported by the Portuguese National Funds through the FCT Foundation for Science and Technology under the project PEst-UID/EEA/00066/2013. Gabriel Bengochea, Luis Verde-Star, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, Apartado 55-534, Ciudad de México, México, e-mails: [email protected], [email protected] Manuel Ortigueira, CTS-UNINOVA/Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Campus da FCT da UNL, Quinta da Torre 2825-149 Monte da Caparica, Portugal, e-mail: [email protected] António M. Lopes, UISPA-LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, e-mail: [email protected] https://doi.org/10.1515/9783110571929-005

120 | G. Bengochea et al. output, given the input, in the two usual situations: steady-state and transient response. The algorithm is an extension of the work developed in [1], where an algebraic version of the usual convolution was used; its generalization is described here. The initial conditions can be incorporated in the algorithm without any trick. Another property of the operational method presented here is that it is recursive in the sense that we can add and remove pseudo-poles and pseudo-zeros of a system without solving the whole system again. In terms of performance, we show that the algorithm turns out to be very efficient. The chapter is organized as follows. Section 2 contains the algebraic rules to be used in the chapter. In Section 3 we solve fractional linear systems by means of algebraic operations, as defined in Section 2, and some examples are presented. In Section 4 the initial conditions are introduced and some examples are presented. In Section 5 it is shown how it is possible to add and remove pseudo-poles and pseudo-zeros recursively. Section 6 contains a reformulation of the work presented in Section 2, with t nα which we show that the method does not depend on the objects Γ(nα+1) . Finally, in Section 7 we solve the fractional logistic equation using our operational method and present some simulations. Notation. – ( ⋅ )n∗ = (⋅) ∗ (⋅) ∗ ⋅ ⋅ ⋅ ∗ (⋅) ∗ (⋅). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ –

n times

We will omit the identity operator I when confusion is unlikely. For example, we will write (Dα − θ)y instead of (Dα − θI)y.

2 Algebraic framework The approach described herein is based on the operational calculus developed in [1]. This calculus is considered as a purely algebraic version of Mikusiński’s calculus [4]. Let u(t) be theHeaviside unit step function and let δ(t) be the Dirac delta distribution. Definition 2.1. We define a sequence of base functions ϕn (t), for 0 < α ≤ 1, by ϕn (t) = with the convention that

t −n Γ(−n+1)

t nα , Γ(nα + 1)

n ∈ ℤ,

(2.1)

= δ(n−1) (t), see [2].

This will be used to span F , the set of all the formal Laurent series with the format ∞

F = { ∑ ci ϕi (t) : k ∈ ℕ0 , ci ∈ ℂ}. i=−k

(2.2)

Recursive-operational method for fractional systems | 121

In this set we introduce a formal generalized convolution that will be our basic operation. Definition 2.2. We define a generalized convolution product ∗ as follows: ϕn (t) ∗ ϕk (t) = ϕn+k (t).

(2.3)

From this definition it is easy to verify that the generalized convolution ∗ has the following properties: – Identity element: From (2.1) and (2.3) we conclude that ϕ0 (t) = u(t) is the identity element. t −nα – Inverse element: For n ∈ ℤ, ϕ−n (t) = Γ(−nα+1) u(t) is the multiplicative inverse of nα

– –

t u(t). ϕn (t) = Γ(nα+1) Commutativity: This is obvious from (2.3). Associativity:

[ϕn (t) ∗ ϕm (t)] ∗ ϕl (t) = ϕn (t) ∗ [ϕm (t) ∗ ϕl (t)] = ϕn+m+l (t).



(2.4)

Distributivity: [ϕn (t) + ϕm (t)] ∗ ϕl (t) = ϕn (t) ∗ ϕl (t) + ϕm (t) ∗ ϕl (t) = ϕn+l (t) + ϕm+l (t).

(2.5)

We extend the generalized convolution in F as follows. If c = ∑∞ i=−kc ci ϕi (t) and d = ∞ ∑i=−kd di ϕi (t) are elements of F , then c∗d=f =





n=−kc −kd

fn ϕn (t),

(2.6)

where the coefficients fn are defined by the well-known discrete-time generalized convolution fn =

∑ −kc ≤i≤n+kd

ci dn−i .

(2.7)

Under this multiplication F is a field (there is no divisor of zero). Let us define ∞

Mγ,m (t) = ∑ ( i=m+1

i − 1 i−m−1 )γ ϕi (t), m

γ ∈ ℂ, m ≥ 0.

(2.8)

Observe that Mγ,0 (t) is the Mittag-Leffler function. Some properties of the function Mγ,m (t) are presented now. We have

122 | G. Bengochea et al. – Dkγ k!



Mγ,0 (t) = Mγ,k (t),

k ≥ 0,

(2.9)

where Dγ means derivation with respect to γ.

Mγ1 ,0 (t) ∗ Mγ2 ,0 (t) =

Mγ1 ,0 (t) − Mγ2 ,0 (t)

γ1 − γ2

,

γ1 ≠ γ2 .

(2.10)

– Mγ,m (t) ∗ Mγ,n (t) = Mγ,m+n+1 (t).



(2.11)

This formula is the analog to the usual generalized auto-convolution of a causal exponential. Partial mode decomposition )(−1)i (n+i i

m

Mγ1 ,m (t) ∗ Mγ2 ,n (t) = ∑

i=0

(γ1 − γ2 )1+n+i n

+∑

j=0

Mγ1 ,m−i (t)

(m+j )(−1)j j

(γ2 − γ1 )1+m+j

Mγ2 ,n−j (t),

(2.12)

with γ1 ≠ γ2 . This result is the analog of the one obtained from the partial fraction decomposition and the Laplace transform inversion. Let Dα be any fractional derivative [5] such that Diα

t (n−i)α t nα u(t) = u(t), Γ(nα + 1) Γ((n − i)α + 1)

(2.13)

then the following proposition is true. Proposition 2.3. Let Dα be an operator satisfying (2.13). Then Diα

t −iα t nα t nα u(t) = u(t) ∗ u(t) Γ(nα + 1) Γ(−iα + 1) Γ(nα + 1) =

t (n−i)α u(t) Γ((n − i)α + 1)

(2.14)

for all n, i ∈ ℕ. From the previous proposition we observe that the derivative operation is essentially a convolution. Let f (t) and g(t) be elements of F . From Proposition 2.3 and the

Recursive-operational method for fractional systems | 123

associativity of ∗ we obtain t −α u(t) ∗ (f (t) ∗ g(t)) Γ(−α + 1) t −α u(t) ∗ f (t)) ∗ g(t) =( Γ(−α + 1)

Dα (f (t) ∗ g(t)) =

= Dα f (t) ∗ g(t).

(2.15)

By a similar argument we deduce that Dα (f (t) ∗ g(t)) = f (t) ∗ Dα g(t). Then Dα (f (t) ∗ g(t)) = Dα f (t) ∗ g(t) = f (t) ∗ Dα g(t). This is a classic result of the derivative with the usual convolution.

3 Fractional continuous-time linear systems Linear systems defined by constant coefficient fractional differential equations are very useful tools in science and engineering [3, 8]. Their mathematical manageability and simplicity make them important for studying or modeling many natural or artificial systems. During the last 20 years the fractional systems called the attention of many application fields and invaded them. In the following, we shall be concerned with the study of such systems. The ones most interesting and easy to study are the commensurate systems that are written in the general form N

M

k=0

k=0

∑ ak Dkα y(t) = ∑ bk Dkα x(t),

(3.1)

with α ∈ ℝ+ and aN = 1. In current applications we assume that M ≤ N for stability reasons. We do not advance any particular fractional derivative definition. We will only suppose that Dα satisfies (2.13) with 0 < α ≤ 1.

3.1 Roots of the characteristic equation and poles The characteristic equation corresponding to the system described in (3.1) has N different roots allowing us to write it in the format A(z) = ∏Nk=1 (z − γk ). If | arg(γk )| < πα, k = 1, 2, . . . , N, then γk is a pole of the system [7]; therefore, we can have systems without poles. This leads to a decomposition of the system into a sum of integer and fractional components [7]. Anyway, this decomposition is not easy to obtain in the present context. To avoid a great change in the designations we will use the name “pseudopole” to refer to any characteristic equation root.

124 | G. Bengochea et al.

3.2 General case Suppose that A(z) in (3.1) can be factorized as A(z) = (z − γ0 )m0 +1 (z − γ1 )m1 +1 ⋅ ⋅ ⋅ (z − γr )mr +1 ,

(3.2)

where N = ∑ri=0 mi + 1, γ0 , γ1 , . . . , γr are the different roots of the polynomial A(z) (pseudo-poles) and m0 + 1, m1 + 1, . . . , mr + 1 are their multiplicities. Then system (3.1) can be written as r

mi +1

∏(Dα − γi ) i=0

M

y(t) = ∑ bj Djα x(t). j=0

(3.3)

Due to Proposition 2.3 and the fact that u(t) is the neutral element for ∗, the previous system can be written as r

∏( i=0

m +1

M i t −α t −jα y(t) = ( ∑ bj u(t) − γi u(t)) ) ∗ x(t), Γ(−α + 1) Γ(−jα + 1) ∗ j=0

(3.4)

where m +1

(

i t −α u(t) − γi u(t)) Γ(−α + 1) ∗

=(

t −α u(t) − γi u(t)) Γ(−α + 1)

∗ ⋅⋅⋅ ∗ (

t −α u(t) − γi u(t)), Γ(−α + 1)

(3.5)

mi + 1 times. Define γi Pmi +1

m +1

=(

i t −α , u(t) − γi u(t)) Γ(−α + 1) ∗

(3.6)

with i = 0, 1, . . . , r. On the other hand, from the definition and properties of the generalized convolution we deduce that ∞

u(t) ∗ Mγ,0 = ∑ γ i i=0

t (i+1)α u(t) Γ((i + 1)α + 1)

(3.7)

and ∞ tα t (i+2)α u(t) ∗ Mγ,0 = ∑ γ i u(t). Γ(α + 1) Γ((i + 2)α + 1) i=0

(3.8)

It follows that (u(t) − γ

tα tα u(t)) ∗ Mγ,0 (t) = u(t). Γ(α + 1) Γ(α + 1)

(3.9)

Recursive-operational method for fractional systems | 125

Thus (

t −α u(t) − γu(t)) ∗ Mγ,0 (t) = u(t). Γ(−α + 1)

(3.10)

This result can be generalized to m +1

(

i t −α u(t) − γu(t)) Γ(−α + 1) ∗

i +1 ∗ (Mγ,0 )m = u(t). ∗

(3.11)

The inverse of (3.6) is −1 γi Pmi +1

i +1 = (Mγi ,0 )m , ∗

(3.12)

which can be written as −1 γi Pmi +1

= Mγi ,mi .

(3.13)

From (2.9) we verify that m

Mγi ,mi =

Dγi i

mi !

Mγi ,0 .

(3.14)

It follows that the inverse operator of γ0 Pm0 +1 ∗ ⋅ ⋅ ⋅ ∗ γr Pmr +1 is m

m

Dγ00

⋅⋅⋅

m0 !

Dγr r

mr !

(Mγ0 ,0 ∗ Mγ1 ,0 ∗ ⋅ ⋅ ⋅ ∗ Mγr ,0 ).

(3.15)

From (2.10) it follows that Mγ0 ,0 ∗ Mγ1 ,0 ∗ ⋅ ⋅ ⋅ ∗ Mγr ,0 = d0 Mγ0 ,0 + d1 Mγ1 ,0 + ⋅ ⋅ ⋅ + dr Mγr ,0 ,

(3.16)

where di are constants to be determined. These constants only depend on the pseudopoles γ0 , γ1 , . . . , γr . Therefore the solution is given by M

y(t) = ∑ bj Djα (d0 Mγ0 ,m0 + d1 Mγ1 ,m1 + ⋅ ⋅ ⋅ + dr Mγr ,mr ) ∗ x(t) j=0 M

r



j=0

k=0

i=mk +1

= ∑ bj ∑ (dk ∑ (

i−m −1

k t (i−j)α i − 1 γk u(t)) ∗ x(t). ) mk Γ((i − j)α + 1)

(3.17)

When x(t) = u(t) we get the step response, which is given by M

r



j=0

k=0

i=mk +1

ru (t) = ∑ bj ∑ (dk ∑ (

i − 1 i−mk −1 t (i−j)α )γk )u(t). mk Γ((i − j)α + 1)

(3.18)

Observe that (3.17) and (3.18) can be computed only by means of the pseudo-poles of t (i−j)α the system and their multiplicities. In other words, we need not to work with Γ((i−j)α+1) as we will show at the end of the chapter. Below we study some particular cases.

126 | G. Bengochea et al.

3.3 Particular cases –

One pseudo-pole A generic pseudo-pole, γ0 , is a solution of the characteristic equation and originates in the differential equation by a term with the form k

[Dα − γ0 ] , where k is the multiplicity. The present algorithm makes a substitution of the above term by a symbolic operator. To see how this works, consider the system (Dα − γ0 )y(t) = x(t),

(3.19)

which can be rewritten as (

t −α u(t) − γ0 u(t)) ∗ y(t) = x(t). Γ(−α + 1)

(3.20)

t −α tα u(t) ∗ (u(t) − γ0 u(t)), Γ(−α + 1) Γ(α + 1)

(3.21)

In this case P1 =

and its inverse operator is P1−1 = Mγ0 ,0 (t).

(3.22)

Therefore, the solution of (3.19) is symbolically expressed by y(t) = Mγ0 ,0 (t) ∗ x(t).

(3.23)

This result allows us to solve all the problems involving a pseudo-pole with multiplicity one. When x(t) = u(t) we get the step response ru (t) = Mγ0 ,0 (t). –

(3.24)

Pseudo-pole with multiplicity two Now retake (3.1) with M = 0 and b0 = 1. Then the output of the system can be seen as the solution of the equation 2

(Dα − γ0 ) y(t) = x(t).

(3.25)

Suppose that the solution can be written as y(t) = ∑ yi i≥0

t (i+1)α u(t), Γ((i + 1)α + 1)

yi ∈ ℂ.

Recursive-operational method for fractional systems | 127

Proceeding as in Subsection 3.2, expression (3.25) can be written as 2

(

t −α u(t) − γ0 u(t)) ∗ y(t) = x(t). Γ(−α + 1) ∗

(3.26)

In this case P2 = (

2

t −α u(t) − γ0 u(t)) , Γ(−α + 1) ∗

(3.27)

and its multiplicative inverse is γ0 P2−1 = Mγ0 ,1 (t). Therefore, the solution is given by y(t) = Mγ0 ,1 (t) ∗ x(t).

(3.28)

When x(t) = u(t), we obtain the step response ru (t) = Mγ0 ,1 (t). –

(3.29)

Two simple pseudo-poles and one pseudo-zero Now we consider a system with two different pseudo-poles, but we join a pseudozero (Dα − γ0 )(Dα − γ1 )y(t) = (Dα − θ0 )x(t).

(3.30)

Then (3.30) can be written as (

t −α t −α u(t) − γ0 u(t)) ∗ ( u(t) − γ1 u(t)) ∗ y(t) Γ(−α + 1) Γ(−α + 1) t −α =( u(t) − θ0 u(t)) ∗ x(t). Γ(−α + 1)

(3.31)

Following the previous theory we can write γi P1

=(

t −α u(t) − γi u(t)), Γ(−α + 1)

(3.32)

with i = 0, 1. The inverse operator of γi P1 is given by −1 γi P1

:= Mγi ,0 (t),

i = 0, 1.

(3.33)

Therefore, the multiplicative inverse of γ0 P1 ∗ γ1 P1 is given by −1 γ0 P1

∗ γ1 P1−1 = Mγ0 ,0 (t) ∗ Mγ1 ,0 (t) =

Mγ0 ,0 (t) − Mγ1 ,0 (t)

γ0 − γ1

.

(3.34)

128 | G. Bengochea et al. On the other hand, observe that t −α u(t) − θ0 u(t)) ∗ x(t) = (Dα − θ0 )x(t). Γ(−α + 1)

(3.35)

Mγ ,0 (t) − Mγ1 ,0 (t) t −α ) ∗ x(t), u(t) − θ0 u(t)) ∗ ( 0 Γ(−α + 1) γ0 − γ1

(3.36)

( Hence, we obtain y(t) = (

which can be simplified to y(t) = −

θ0 ((1 + γ0 )Mγ0 ,0 (t) − (1 + γ1 )Mγ1 ,0 (t)) ∗ x(t), γ0 − γ1

(3.37)

which expresses symbolically the output of the system. As in the previous cases, the step response is given by ru (t) = −

θ0 ((1 + γ0 )Mγ0 ,0 (t) − (1 + γ1 )Mγ1 ,0 (t)). γ0 − γ1

(3.38)

Remark 3.1. In the case when γ0 and γ1 are complex conjugate we obtain ru (t) = −

θ0 ((1 + γ0 )Mγ0 ,0 (t) − (1 + γ1 )Mγ1 ,0 (t)), 2ℑ(γ0 )

(3.39)

where ℑ means the imaginary part of a complex number.

3.4 Examples Example 3.2. Consider the equation D2α y(t) + Dα y(t) − 2y(t) = u(t).

(3.40)

(Dα + 2)(Dα − 1)y(t) = u(t).

(3.41)

It can be written as

The multiplicative inverses of the operators (Dα +2) and (Dα −1) are M−2,0 (t) and M1,0 (t), respectively. Hence, we obtain y(t) = M−2,0 (t) ∗ M1,0 (t) ∗ u(t).

(3.42)

Since u(t) is the multiplicative neutral of the pseudo-convolution ∗ and by the property (2.10) we get 1 y(t) = − (M−2,0 (t) − M1,0 (t)). 3

(3.43)

Recursive-operational method for fractional systems | 129

Figure 1: Behavior of the solution for different values of arg(a).

Example 3.3. Consider the more general equation (Dα − a)(Dα − b)y(t) = u(t),

(3.44)

where a, b are complex numbers. The multiplicative inverses of the operators (Dα − a) and (Dα − b) are Ma,0 (t) and Mb,0 (t), respectively. Hence, we obtain y(t) = Ma,0 (t) ∗ Mb,0 (t) ∗ u(t).

(3.45)

Since u(t) is the multiplicative neutral of the pseudo-convolution ∗ and by the property (2.10), we get for a ≠ b y(t) =

1 (Ma,0 (t) − Mb,0 (t)), a−b

(3.46)

and for a = b y(t) = Ma,1 (t).

(3.47)

Figure 1 shows the behavior of our solution for α = 3/4, b = a∗ , ||a|| = √2 and arg(a) = 30∘ , 45∘ , 60∘ , 90∘ , 120∘ , 180∘ . Example 3.4. In the study of viscoelastic materials, the Voigt model is used to explain the creep behavior of polymers which is given by a0 Dy(t) + a1 y(t) = b0 x(t),

(3.48)

a0 Dα y(t) + a1 y(t) = b0 x(t).

(3.49)

and its generalization is

130 | G. Bengochea et al.

Figure 2: Behavior of the solution for some values of α.

In this case, we suppose a0 = 1 and Dα is the fractional Grünwald–Letnikov derivative. Also suppose that x(t) is in the image of the operator (Dα + a1 ). Hence (Dα + a1 )y(t) = b0 x(t).

(3.50)

The multiplicative inverse of the operator (Dα + a1 ) is the Mittag-Leffler function M−a1 ,0 (t). Then y(t) = M−a1 ,0 (t) ∗ b0 x(t).

(3.51)

y(t) = b0 M−a1 ,0 (t).

(3.52)

The step response is given by

Figure 2 shows the behavior of our solution for a1 = −1/2 and α = 1/2, 1, 3/2, 7/4, 2.

4 Initial conditions We consider the problem of initial conditions in the instant t0 = 0. For another instant t0 , it is sufficient to change u(t) by u(t − t0 ) in (2.1). Suppose that we have a system of the form N

M

k=0

k=0

∑ ak Dkα y(t) = ∑ bk Dkα x(t),

α ∈ ℝ+ ,

(4.1)

Recursive-operational method for fractional systems | 131

with initial conditions Dkα y(0) = yk , 0 ≤ k ≤ N − 1, output ∞

y(t) = ∑ yi i=0

t iα u(t), Γ(iα + 1)

(4.2)

t iα u(t). Γ(iα + 1)

(4.3)

and input ∞

x(t) = ∑ xi i=0

Observe that, when k ≥ 1, Dkα y(t) is a series beginning in the index −k, we need to rewrite Dkα y(t) by k−1

Dkα (y(t) − ∑ yi i=0

t iα u(t)). Γ(iα + 1)

(4.4)

Then the system is rewritten as N

k−1

k=0

i=0

M t iα u(t)) = ∑ bk Dkα x(t). Γ(iα + 1) k=0

∑ ak Dkα (y(t) − ∑ yi

(4.5)

After algebraic manipulations we obtain N

N−1

∑ ak Dkα (y(t) − ∑ yi i=0

k=0

N−1 N−1−j

= − ∑ ∑ aj yi+j j=0 i=0

t iα u(t)) Γ(iα + 1)

M t iα u(t) + ∑ bk Dkα x(t). Γ(iα + 1) k=0

(4.6)

By a similar argument, we can incorporate the initial conditions related to the input x(t), Dkα x(0) = xk , 0 ≤ k ≤ M − 1. That is, we rewrite, for k ≥ 1, Dkα x(t) by k−1

Dkα (x(t) − ∑ xi i=0

t iα u(t)). Γ(iα + 1)

(4.7)

The final system is given by N

N−1

∑ ak Dkα (y(t) − ∑ yi

k=0

i=0

N−1 N−1−j

= − ∑ ∑ aj yi+j j=0 i=0

t iα u(t)) Γ(iα + 1)

t iα u(t) Γ(iα + 1)

M

k−1

k=0

i=0

+ ∑ bk Dkα (x(t) − ∑ xi

t iα u(t)). Γ(iα + 1)

(4.8)

132 | G. Bengochea et al. Finally, the system can be solved by means of the operational method presented in the previous sections. As a particular case, we suppose that x(t) = 0 when t > 0 and x(iα) (0) = xi . Hence, the system (4.8) acquires the form N

N−1

∑ ak Dkα (y(t) − ∑ yi

k=0

i=0

N−1 N−1−j

= − ∑ ∑ aj yi+j j=0 i=0 M

k−1

k=0

i=0

+ ∑ bk ( ∑ xi

t iα u(t)) Γ(iα + 1)

t iα u(t) Γ(iα + 1)

t (i−k)α u(t)). Γ((i − k)α + 1)

(4.9)

Solving the system, we get N−1

y(t) = ∑ yi i=0

t iα u(t) − (d0 Mγ0 ,0 + d1 Mγ1 ,0 + ⋅ ⋅ ⋅ + dr Mγr ,0 ) Γ(iα + 1) N−1 N−1−j

∗ (− ∑ ∑ aj yi+j j=0 i=0

M k−1 t iα t (i−k)α u(t) − ∑ bk ( ∑ xi u(t))), Γ(iα + 1) Γ((i − k)α + 1) i=0 k=0

(4.10)

where γ0 , γ1 , . . . , γr are the pseudo-poles of the system and d0 , d1 , . . . , dr are constants to be determined. Next, we present two examples in which the initial conditions are introduced. Example 4.1. Consider the system (3.49) Dα y(t) − y(t) = 0,

(4.11)

with a0 = 1, a1 = −1, x(t) = 0 and initial condition y0 = y(0) = 1. Because Dα y(t) is a series beginning in the index −1, we need to rewrite Dα y(t) by Dα (y(t) − u(t)). Then we obtain the system Dα (y(t) − u(t)) − y(t) = 0.

(4.12)

(Dα − 1)(y(t) − u(t)) = u(t).

(4.13)

Factorizing we get

It follows that y(t) = u(t) + M1,0 (t) ∗ u(t) t iα u(t). Γ(iα + 1) i=0 ∞

=∑ If α = 1, then y(t) = et u(t).

(4.14)

Recursive-operational method for fractional systems | 133

Example 4.2. Consider the system of Example 3.2 D2α y(t) + Dα y(t) − 2y(t) = u(t),

(4.15)

with initial conditions y(0) = y0 and y(α) (0) = y1 . Suppose that the solution has the format ∞

∑ yi

i=0

t iα u(t). Γ(iα + 1)

(4.16)

Because D2α y(t) is a series beginning in the index −2, we need to rewrite D2α y(t) by tα ). By a similar argument, we need to rewrite Dα y(t) by Dα (y(t)− D2α (y(t)−y0 u(t)−y1 Γ(α+1) y0 u(t)). Then the system (3.40) can be rewritten as (D2α + Dα − 2)(y(t) − y0 u(t) − y1 = (1 + 2y0 − y1 )u(t) + 2y1

tα u(t)) Γ(α + 1)

tα u(t). Γ(α + 1)

(4.17)

Solving the system by the operational method we obtain y(t) = y0 u(t) + y1

t 2α tα u(t) + (1 − y1 + 2y0 ) u(t) Γ(α + 1) Γ(2α + 1)

+ (−1 − 2y0 + 3y1 )

t 3α u(t) Γ(3α + 1)

t 4α u(t) + ⋅ ⋅ ⋅ . Γ(4α + 1)

(4.18)

t 2α t 3α t 4α u(t) − 3 u(t) + 9 u(t) + ⋅ ⋅ ⋅ . Γ(2α + 1) Γ(3α + 1) Γ(4α + 1)

(4.19)

+ (3 + 6y0 − 5y1 ) Suppose that y0 = 1 and y1 = 0. Then y(t) = u(t) + 3

5 Recursive-operational solution to fractional linear systems 5.1 Forward Suppose that system (3.1) can be written as N

M

i=1

i=1

∏(Dα − γi )y(t) = ∏(Dα − θi )x(t),

(5.1)

134 | G. Bengochea et al. where θi , i = 1, . . . , M, are the pseudo-zeros and γi , i = 1, . . . , N, are the pseudo-poles of the system. Assume that all the pseudo-zeroes are different from all the pseudo-poles. This system can be written symbolically as N

∏( i=1

M t −α t −α u(t) − γi u(t)) ∗ y(t) = ∏( u(t) − θi u(t)) ∗ x(t), Γ(−α + 1) Γ(−α + 1) ∗ ∗ i=1

(5.2)

where ∏(⋅)∗ := (⋅) ∗ ⋅ ⋅ ⋅ ∗ (⋅). In practical applications the number of pseudo-zeros cannot be greater than the number of pseudo-poles and it is frequently less. So, we can take M pseudo-pole-zero pairs and N − M pseudo-poles. From (3.37), with γ1 = 0, it is easy to deduce that to each pseudo-pole-zero we can attach an inverse operator defined by Pi−1 (t) := u(t) + (γi − θi )Mγi ,0 ,

0 ≤ i ≤ M,

(5.3)

while for a simple pseudo-pole, see (3.22), the inverse operator is Pi−1 (t) := Mγ0 ,0 (t),

M + 1 ≤ i ≤ N.

(5.4)

Hence, the solution y(t) can be constructed recursively through successive convolutions. In Subsection 5.2 we describe this procedure.

5.2 The algorithm Suppose that we have already made the computation of the inverse operator corresponding to r pseudo-poles-zeroes of the system, that is, after m steps in the recursion we have found the partial step response m1

mr

0

i=0

i=0

i=−s

hm (t) = ∑ d1,i Mγ1 ,m1 −i (t) + ⋅ ⋅ ⋅ ∑ dr,i Mγr ,mr −i (t) + ∑ ci

t iα u(t), Γ(iα + 1)

(5.5)

where dj,i , ci and s are constants determined in the recursive process. Now, if we want to continue the recursion adding a pseudo-pole-zero we need to convolve the previous partial step response with (5.3), while if we want to add a simple pseudo-pole we need to convolve with (5.4). So, we need to pay attention to the convolution of a partial step response and a function of the kind Mγ0 ,0 (t). The convolution of an element with the t iα , i ≤ 0, and Mγ0 ,0 (t) can easily be computed (see (2.14)). Then we have two form Γ(iα+1) important cases. – When the pseudo-pole is new, γ0 ≠ γi , i = 1, . . . , r. From (2.12) we get Mγ0 ,0 (t) ∗ Mγi ,mi −k (t) =

1 M (t) (γ0 − γi )1+n γ0 ,0 mi −k

+ ∑

j=0

for each 1 ≤ i ≤ r and 0 ≤ k ≤ mi .

(−1)j Mγi ,mi −k−j (t), (γi − γ0 )1+j

(5.6)

Recursive-operational method for fractional systems | 135



When the pseudo-pole is not new, γ0 = γi for some fixed i in [1, r]. From (2.11) we get (5.7)

Mγi ,0 (t) ∗ Mγi ,n (t) = Mγi ,n+1 (t),

while that for γ0 ≠ γk , k ≠ i we can proceed as in the previous case. Next, we solve several examples in order to show the simplicity of the algorithm. Example 5.1. Consider the system defined by the transfer function H(s) =

sα − 1 . (sα + 2)(sα + 1)

(5.8)

The computation of the step response comes in two steps. The first step consists in calculating the step response only for a pseudo-pole-zero pair (for example (sα −1)/(sα + 1)). From (5.3) we obtain h1 (t) = u(t) − 2M−1,0 (t).

(5.9)

Now, from (5.4) we join the other simple pseudo-pole convolving by M−2,0 (t). So we get h(t) = u(t) ∗ M−2,0 (t) − 2M−1,0 (t) ∗ M−2,0 (t) = 3M−2,0 (t) − 2M−1,0 (t).

(5.10)

Example 5.2 (Adding a pseudo-pole). Consider the system defined by the transfer function H(s) =

sα − 1 . (sα + 2)2 (sα + 1)

(5.11)

The system has the same pseudo-poles as the previous example, but one is of double multiplicity. Here, we need to convolve the solution (5.10) by M−2,0 and then apply (2.11). Observe that really our algorithm is recursive. We have h(t) = 3M−2,0 (t) ∗ M−2,0 (t) − 2M−1,0 (t) ∗ M−2,0 (t) = 3M−2,1 (t) + 2M−2,0 (t) − 2M−1,0 (t).

(5.12)

5.3 Removing a pseudo-pole/adding a pseudo-zero In the above subsection we introduced a recursive algorithm for adding pseudo-poles. Here, we will show how to remove a given pseudo-pole. For this purpose, we only need t −α u(t) − γ0 u(t)). to note that the inverse of Mγ0 ,0 (t), under the product ∗, is ( Γ(−α+1)

136 | G. Bengochea et al. Example 5.3 (Remove a pseudo-pole). Consider the Example 5.2 and suppose that we want to remove one of the double pseudo-pole. For this purpose, we need to convolve t −α (5.12) by ( Γ(−α+1) u(t) + 2u(t)). So, we obtain h(t) = (3M−2,1 (t) + 2M−2,0 (t) − 2M−1,0 (t)) ∗ (

t −α u(t) + 2u(t)). Γ(−α + 1)

(5.13)

From (2.14) and the fact that u(t) is the multiplicative neutral of ∗ we get h(t) = 3M−2,0 (t) − 2M−1,0 (t).

(5.14)

It is not difficult to see that the action of removing a pseudo-pole is equivalent to the addition of a pseudo-zero. So, if we want to join a pseudo-zero (θ0 ) to the system, t −α it is enough to convolve the pre-existing step response h(t) with ( Γ(−α+1) u(t) − θ0 u(t)). Example 5.4 (Add a pseudo-zero). Consider the system introduced in Example 5.1 and suppose that we want to add the pseudo-zero θ0 = −2. Then h(t) = (3M−2,0 (t) − 2M−1,0 (t)) ∗ ( = u(t) − 2M−1,0 (t).

t −α u(t) + 2u(t)) Γ(−α + 1)

(5.15)

5.4 Removing pseudo-zeroes From the above considerations, it is clear that removing a pseudo-zero (θ0 ) is equivalent to joining a pseudo-pole, so we need to convolve the pre-existing step response h(t) by Mθ0 ,0 . Example 5.5 (Remove a pseudo-zero). Consider the system introduced in Example 5.1 and suppose that we want to remove the pseudo-zero θ0 = 1, then h(t) = (3M−2,0 (t) − 2M−1,0 (t)) ∗ M1,0 (t) = −M−2,0 + M−1,0 (t).

(5.16)

We present Table 1, which summarizes the previous process.

6 Reformulating the algorithm to avoid dealing with base functions In this section, we present a reformulation of the algorithm presented in the previous sections. We change the power series by doubly infinite sequence of complex numbers.

Recursive-operational method for fractional systems | 137 Table 1: Summary of the process. Situation

Convolve the pre-existing step response h(t) by

Adding a pseudo-pole γ0 Adding a pseudo-zero θ0 Remove a pseudo-pole γ0 Remove a pseudo-zero θ0

Mγ0 ,0 (t)

t u(t) − θ0 u(t)) ( Γ(−α+1) −α

t ( Γ(−α+1) u(t) − γ0 u(t)) −α

Mθ0 ,0 (t)



t ; The main advantage is in the avoidance of working with the power functions, Γ(iα+1) we only work with the coefficients ai , which simplify the operations in the algorithm. iα

t To each series a = ∑i≥n1 ai Γ(iα+1) ∈ F , we associate a doubly infinite sequence of complex number given by

0, i < n1 , an1 ,i = { ai , i ≥ n1 ,

n1 ∈ ℤ.

(6.1)

Denote by S the set of all these sequences. The operation of sum and product by a scalar in S are the usual between sequences. It is easy to see that, for all n1 ∈ ℤ, the sequence

ϕn1 ,i

0, i < n1 , { { { = {1, i = n1 , { { {0, i > n1 ,

(6.2)

is an element of S . Note that (6.2) is equivalent to (2.1). Definition 6.1. We define a convolution product ⬦ as follows: ϕn1 ,i ⬦ ϕn2 ,i = ϕn1 +n2 ,i .

(6.3)

The product ⬦ is a reformulation of the product ∗ introduced in (2.3). From this definition it is easy to verify that the convolution ⬦ has the following properties: – Identity element: From (6.3) it is easy to check that ϕ0,i is the identity element. – Inverse element: For n ∈ ℤ, ϕ−n,i is the multiplicative inverse of ϕn,i . – Commutativity: It is obvious from (6.3). – Associativity:



[ϕn,i ⬦ ϕm,i ] ⬦ ϕl,i = ϕn,i ⬦ [ϕm,i ⬦ ϕl,i ] = ϕn+m+l,i .

(6.4)

[ϕn,i + ϕm,i ] ⬦ ϕl,i = ϕn,i ⬦ ϕl,i + ϕm,i ⬦ ϕl,i = ϕn+l,i + ϕm+l,i .

(6.5)

Distributivity:

138 | G. Bengochea et al. We extend the convolution ⬦ to S as follows. For a−n1 ,i and a−n2 ,i in S , we define ⬦ as a−n1 ,i ⬦ a−n2 ,i = a−n3 ,i ,

n3 = n1 + n2 ,

(6.6)

where a−n3 ,i =

∑ −n1 ≤j≤i+n2

a−n1 ,j a−n2 ,i−j .

(6.7)

This convolution ⬦ is associative and commutative. Under this multiplication S is a field (there is no divisor of zero). Define γ Sm,i

0, i < m + 1, ={ i−m−1 i−1 , i ≥ m + 1, ( m )γ

γ ∈ ℂ, m ≥ 0.

(6.8)

Some properties of the function γ Sm,i and the convolution ⬦ are presented now: – Dkγ

k! γ



S0,i = γ Sk,i ,

(6.9)

where Dγ means derivation with respect to γ. γ1 S0,i



γ2 S0,i

– γ Sm,i



k ≥ 0,

=

1 ( S − S ). γ1 − γ2 γ1 0,i γ2 0,i

(6.10)

(6.11)

⬦ γ Sn,i = γ Sm+n+1,i .

Partial mode decomposition: γ1 Sm,i ⬦

(n+j )(−1)j j

m

γ2 Sn,i = ∑

j=0

(γ1 − γ2 )1+n+j n

+∑

j=0

γ1 Sm−j,i

(m+j )(−1)j j

(γ2 − γ1 )1+m+j

γ2 Sn−j,i ,

(6.12)

with γ1 ≠ γ2 . Remark 6.2. It is easy to verify that b S0,i

⬦ (ϕ−1,i − bϕ0,i ) = ϕ0,i .

(6.13)

Remark 6.3. It was shown previously that the action of Dα − b on a function is equivat −α − bu(t). Now, the action of such an operator is equivlent to convolving it with Γ(−α+1) alent to convolving, using ⬦ given by (6.3), with ϕ−1,i − bϕ0,i .

Recursive-operational method for fractional systems | 139

Next, we solve two examples. Example 6.4. Suppose that we want to solve the system (Dα + 1)(Dα − 1)y(t) = u(t).

(6.14)

From Remark 6.3, we associate to the operator Dα + 1 the sequence ϕ−1,i + ϕ0,i and to Dα − 1 the sequence ϕ−1,i − ϕ0,i . So, the previous equation can be rewritten as (ϕ−1,i + ϕ0,i ) ⬦ (ϕ−1,i − ϕ0,i ) ⬦ yn,i = ϕ0,i .

(6.15)

From Remark 6.2, we get y1,i = 1 S0,i ⬦ −1 S0,i ⬦ ϕ0,i 1 = (1 S0,i − −1 S0,i ) 2 1 = ((1)i−1 − (−1)i−1 ). 2

(6.16)

Finally, the solution is given by y(t) =

t iα 1 ∞ u(t) ∑((1)i−1 − (−1)i−1 ) 2 i=1 Γ(iα + 1) ∞

=∑ i=1

t 2iα u(t). Γ(2iα + 1)

(6.17)

Example 6.5. Suppose that we want to solve the system 2

(Dα + 1) y(t) = S−1,0 (t),

(6.18)

where S−1,0 (t) is defined in (6.8). In the same way as in Example 6.4, the equation can be written as (ϕ−1,i + ϕ0,i ) ⬦ (ϕ−1,i + ϕ0,i ) ⬦ yn,i = −1 S0,i .

(6.19)

From Remark 6.2 we have y3,i = −1 S0,i ⬦ −1 S0,i ⬦ −1 S0,i = −1 S2,i .

(6.20)

Therefore the solution is given by ∞

y(t) = ∑(−1)i+1 ( i=3

i2 3i t iα − + 1) u(t). 2 2 Γ(iα + 1)

(6.21)

In this reformulation it is possible to apply the recursive algorithm introduced in Subsection 5.2. We present the corresponding table and some examples.

140 | G. Bengochea et al. Table 2: Summary of process. Situation

Convolve the pre-existing step response h(t) by

Adding a pseudo-pole γ0

γ0 S0,i

Adding a pseudo-zero θ0

(ϕ−1,i − θ0 ϕ0,i )

Remove a pseudo-pole γ0

(ϕ−1,i − γ0 ϕ0,i )

Remove a pseudo-zero θ0

θ0 S0,i

Example 6.6 (Remove a pseudo-pole). Suppose that we want to remove the pseudopole 1 in (6.14). Following Table 2 we need to convolve (⬦) (6.16) by the sequence (ϕ−1,i − ϕ0,i ), so we get y1,i = 1 S0,i ⬦ −1 S0,i ⬦ ϕ0,i ⬦ (ϕ−1,i − ϕ0,i ) = −1 S0,i ⬦ ϕ0,i

(6.22)

= −1 S0,i . Rewriting y1,i in terms of powers, the solution is given by ∞

y(t) = ∑(−1)i−1 i=1

t iα u(t), Γ(iα + 1)

(6.23)

which corresponds to the system (Dα + 1)y(t) = u(t). Example 6.7 (Add a pseudo-pole). Now, we will add the pseudo-pole −1 to the system (Dα + 1)y(t) = u(t). Following Table 2 we need to convolve (⬦) (6.22) by the sequence −1 S0,i . Then we obtain y2,i = −1 S0,i ⬦ −1 S0,i = −1 S1,i .

(6.24)

The last equality follows from (6.11). In terms of powers we get the solution ∞

y(t) = ∑(−1)i (i − 1) i=2

t iα u(t), Γ(iα + 1)

(6.25)

which corresponds to the system (Dα + 1)2 y(t) = u(t). Example 6.8 (Add a pseudo-zero). In this example we will add the pseudo-zeros 1 and −1 to the system of previous example (Dα + 1)2 y(t) = u(t). Following Table 2 we need to convolve (⬦) (6.24) by (ϕ−1,i + ϕ0,i ) and (ϕ−1,i − ϕ0,i ). Since (ϕ−1,i + ϕ0,i ) is the inverse of −1 S0,i we see that the solution is given by y0,i = (ϕ−1,i ⬦ −1 S0,i ) − −1 S0,i

0, i < 0, { { { = {1, i = 0, { { i {2(−1) , i > 0.

(6.26)

Recursive-operational method for fractional systems | 141

In terms of powers, we get the solution ∞

y(t) = u(t) + ∑ 2(−1)i i=1

t iα u(t), Γ(iα + 1)

(6.27)

which corresponds to the system (Dα + 1)2 y(t) = (D2α − 1)u(t).

7 Application to the fractional logistic equation The fractional logistic equation is defined as y(α) (t) = βα y(t)(1 − y(t)),

t ≥ 0.

(7.1)

Although the fractional logistic equation is nonlinear, we can apply our operational method to solve it. We have to rewrite the nonlinear term y2 (t) by a power series and then apply the method.

7.1 Operational solution Assume that there exists a solution for equation (7.1) with fractional Taylor series ∞

y(t) = ∑ yn n=0

t nα u(t), Γ(nα + 1)

t > 0.

(7.2)

We suppose that β = 1. For y2 (t), we have 2

∞ t nα t nα u(t)) = ∑ bn u(t), y (t) = ( ∑ yn Γ(nα + 1) Γ(nα + 1) n=0 n=0 ∞

2

(7.3)

with n

bn = ∑ ( k=0

nα )y y ; kα k n−k

(7.4)

see [6]. Accordingly, (7.2) can be represented as (

∞ t −α t nα u(t) − u(t)) ∗ y(t) = − ∑ bn u(t). Γ(−α + 1) Γ(nα + 1) n=0

(7.5)

Following the theory established in Section 3, we incorporate the initial condition as follows: (

t −α u(t) − u(t)) ∗ (y(t) − y0 u(t)) Γ(−α + 1) ∞

= y0 u(t) − ∑ bn n=0

t nα u(t). Γ(nα + 1)

(7.6)

142 | G. Bengochea et al. So, we have ∞

y(t) = y0 u(t) + M1,0 (t) ∗ (y0 u(t) − ∑ bn = y0 u(t) + (y0 − b0 )

n=0

t nα u(t)) Γ(nα + 1)

t 2α t u(t) + (y0 − b0 − b1 ) u(t) Γ(α + 1) Γ(2α + 1)

+ (y0 − b0 − b1 − b2 )

α

t 3α u(t) + ⋅ ⋅ ⋅ . Γ(3α + 1)

(7.7)

Equating coefficients, we obtain y0 = y0 ,

y1 = y0 − b0 ,

y2 = y0 − b0 − b1 ,

y3 = y0 − b0 − b1 − b2 , .. .

.. .

(7.8)

and finally we have the recursive relation yk+1 = yk − bk ,

k = 0, 1, 2, . . . ,

(7.9)

where y0 is the initial condition y(0).

7.2 Implementation and numerical results In this section, we solve several examples to verify the effectiveness of the proposed approach. Theoretically our solution is exact, but when trying to implement it numerically two problems appear. t nα 1. Computation of the successive terms: yn (t) = yn Γ(nα+1) . For a fixed t, yn (t), first an increasing function of n < np , then becomes decreasing. The peak value is attained for kp around 200. However, for such values, both t nα and Γ(nα + 1) become ∞ in the machine representation. 2. Incrementing of the partial sum: Sn (t) = Sn−1 (t) + yn (t). Very frequently the nth term is very large. The same may happen with the partial sum. Adding two large numbers may lead to an unpredictable result and a convergent series becomes divergent. To avoid such problems, we use a Padé approximation for the solution series. For this t nα purpose, we truncate our solution series to the first 40 terms y(t) = ∑40 n=0 yn Γ(nα+1) u(t), n

z and then we make the variable change z = t α , so y(z) = ∑40 n=0 yn Γ(nα+1) u(z). We apply the [15/15] Padé approximation to y(z), getting a rational function P(z)/Q(z). Finally, returning to the original variable we obtain the approximation P(t α )/Q(t α ).

Recursive-operational method for fractional systems | 143

Example 7.1. Consider (7.1) with α = 1, β = 1, and initial condition y0 = y(0) = 2/3. It i

follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi ti! u(t), with y0 = 2/3,

y1 = y0 − b0 = 2/9,

y2 = y1 − b1 = −2/27,

y3 = y2 − b2 = −2/27,

y4 = y3 − b3 = 10/81,

y5 = y4 − b4 = 14/243, .. .

.. .

(7.10)

which constitutes the well-known Taylor series around zero of the order 1 solution [9] 2 . 2 + 1 exp(−t)

(7.11)

Example 7.2. Return to (7.1) with α = 0.5, β = 1, and initial condition y0 = y(0) = 2/3. 0.5i

t u(t), with It follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi Γ(0.5i+1)

y0 = 2/3,

y1 = y0 − b0 = 0.2222,

y2 = y1 − b1 = −0.0740,

y3 = y2 − b2 = −0.0381,

y4 = y3 − b3 = −0.0621,

y5 = y4 − b4 = −0.0028, .. .

.. .

(7.12)

Example 7.3. As in the previous example let α = 0.1, β = 1, and y0 = y(0) = 2/3. It 0.1i

t follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi Γ(0.1i+1) u(t), with

y0 = 2/3,

y1 = y0 − b0 = 0.2222,

y2 = y1 − b1 = −0.0740,

y3 = y2 − b2 = −0.0254,

y4 = y3 − b3 = −0.0422,

y5 = y4 − b4 = −0.0081, .. .

.. .

(7.13)

144 | G. Bengochea et al.

Figure 3: y0 = 2/3, β = 1 and (−−) α = 1, (⋅ ⋅ ⋅) α = 0.5, (—) α = 0.1.

The [15/15] Padé approximation using the first 40 terms of the solution in Examples 7.1, 7.2, and 7.3 is shown in Figure 3. Example 7.4. Consider (7.1) with α = 1, β = 1, and initial condition y0 = y(0) = 4/3. It i

follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi ti! u(t), with y0 = 4/3,

y1 = y0 − b0 = −4/9,

y2 = y1 − b1 = 20/27,

y3 = y2 − b2 = −44/27,

y4 = y3 − b3 = 380/81,

y5 = y4 − b4 = −4108/243, .. .

.. .

(7.14)

Example 7.5. Consider (7.1) with α = 0.5, β = 1, and initial condition y0 = y(0) = 4/3. 0.5i

t It follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi Γ(0.5i+1) u(t), with

y0 = 4/3,

y1 = y0 − b0 = −0.4444,

y2 = y1 − b1 = 0.7407,

y3 = y2 − b2 = −1.4860,

y4 = y3 − b3 = 3.4644,

y5 = y4 − b4 = −9.1139, .. .

.. .

(7.15)

Recursive-operational method for fractional systems | 145

Figure 4: y0 = 4/3, β = 1 and (−−) α = 1, (⋅ ⋅ ⋅) α = 0.5, (—) α = 0.1.

Example 7.6. As in the previous example let α = 0.1, β = 1, and y0 = y(0) = 4/3. It 0.1i

t follows from (7.9) that the solution is given by y(t) = ∑i≥0 yi Γ(0.1i+1) u(t), with

y0 = 4/3,

y1 = y0 − b0 = −0.4444,

y2 = y1 − b1 = 0.7407,

y3 = y2 − b2 = −1.4349,

y4 = y3 − b3 = 3.0681,

y5 = y4 − b4 = −7.0164, .. .

.. .

(7.16)

The [15/15] Padé approximation using the first 40 terms of the solution in the Examples 7.4, 7.5 and 7.6 is shown in Figure 4.

7.3 About the meaning of the fractional logistic equation The above results exhibit some degree of similarity with the integer order case and we wonder what the meaning is of the fractional logistic equation and its solution. The classic equation is a model for the growth of a given population, describing the population behavior of showing an increase and saturation. As our simulation shows it seems that the fractional solution is similar, but the increase is slower. This is the result of the long memory exhibited by the fractional derivatives. While the integer order derivatives are local operators in the sense that only recent values influence the output at a given instant, the fractional derivatives are nonlocal, because they involve the whole past in the computation. This increments the influence of the past on the

146 | G. Bengochea et al.

Figure 5: (−−) α = 0.7, (—) α = 0.5 and (⋅ ⋅ ⋅) α = 1.

future. Attending to these considerations and in the absence of experimental results we conjecture the possibility that the fractional order of the equation is a translation of the environment’s characteristics. Consider the development of two identical populations evolving in similar environments but having nutrients with different quality. The characteristics of the nutrients will surely influence the way how the population will grow. We conjecture that these environment conditions influencing the population development are stated in the order of the equation. Poor conditions mean a slow population increase and a delayed saturation. To illustrate this assertion we made a new simulation. We started with an initial value equal to 0.5 and a derivative order equal to 0.7. After a time t = 1 we retained the value of the output that we used as initial value for two new computations with orders 0.5 and 1. In Figure 5 we depict the result from which we verify that an increase of the order makes the growth of the solution faster.

Bibliography [1] G. Bengochea and L. Verde-Star, Linear algebraic foundations of the operational calculi, Advances in Applied Mathematics, 47 (2011), 330–351. [2] I. Gel’fand and G. Shilov, Generalized Functions, Academic Press, 1964. [3] E. Kamen and B. Heck, Fundamentals of Signals and Systems: Using the Web and Matlab, 2nd ed., Prentice Hall, Upper Saddle River, NJ, USA, 2000. [4] J. Mikusiński, Operational Calculus, Pergamon Press, 1959. [5] M. Ortigueira, Fractional Calculus for Scientists and Engineers, vol. 84, Springer, 2011. [6] M. Ortigueira and G. Bengochea, A new look at the fractionalization of the logistic equation, Physica A: Statistical Mechanics and Its Applications, 467 (2017), 554–561. [7] M. Ortigueira, T. Machado, M. Rivero, and J. Trujillo, Integer/fractional decomposition of the impulse response of fractional linear systems, Signal Processing, 114 (2015), 85–88.

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[8] M. J. Roberts, Signals and Systems: Analysis Using Transform Methods and Matlab, McGraw-Hill, 2003. [9] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Hachette UK, 2014.

Manuel D. Ortigueira, J. Tenreiro Machado, Fernando J. V. Coito, and Gabriel Bengochea

Discrete-time fractional signals and systems

Abstract: In this chapter, we formulate a coherent theory for discrete-time signals and systems taking two derivatives, namely the nabla (forward) and delta (backward), as basis. The eigenfunctions of such derivatives are the nabla and delta exponentials. With these eigenfunctions, two discrete-time Laplace transforms are introduced and their properties studied. These transforms are used to study the discrete-time linear systems defined by differential equations. The notions of impulse response and transfer function are introduced and discussed. Moreover, the Fourier transform and the frequency response are also considered. The framework is compatible with classic discrete-time signals and systems and allow for a uniform approximation of continuous systems when the sampling interval is reduced to zero. Keywords: Discrete-time, fractional, time scale, nabla Laplace transform, delta Laplace transform MSC 2010: 26A33, 34A08, 93C55, 65T50

1 Introduction The discrete-time (integer-order) signal processing [27, 4, 15, 40, 41] is a well-established scientific area being responsible for important realizations in our daily life. It has a formulation based on difference equations and uses as important tools the Z transform, mainly suited for system study, and the Fourier transform, very useful for signal analyis. With the Z transform we compute easily the transfer function and, from it, the impulse response. The discrete-time linear systems described by difference equations [16, 18, 6] have the discrete-time exponential, z n , n ∈ ℤ, z ∈ ℂ, as eigenfunction and the eigenvalue is exactly the transfer function. For these systems the transfer function is a rational function. All the tools developed in the context of the discretetime signal processing are based on stable coherent mathematical topics and lead to reliable calculation methods.

Manuel D. Ortigueira, Fernando J. V. Coito, CTS–UNINOVA/Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Campus da FCT da UNL, Quinta da Torre 2825-149 Monte da Caparica, Portugal, e-mails: [email protected], [email protected] J. Tenreiro Machado, Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal, e-mail: [email protected] Gabriel Bengochea, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, Apartado 55-534, Ciudad de México, Mexico, e-mail: [email protected] https://doi.org/10.1515/9783110571929-006

150 | M. D. Ortigueira et al. Originally, discrete-time signal processing was merely a set of numerical techniques to solve differential equations. It is of primordial importance that we can refer to the incremental ratio used to approximate the derivative that is currently known as the Euler method [6]. However, with small mathematical manipulation the substitution of these incremental ratia in the differential equations led to difference equations that are easy to treat, particularly when thinking in the computer implementation of the signal processing techniques. Therefore, the topic of difference equations gained the relevance exhibited previously by the differential equations. However, in the classical form, they are not suitable for introducing fractional behavior. This forced researchers to turn back and work with the incremental ratia that will be called discretetime derivatives in the sequel. In some applications the delta system form has been (t) used. This consists in substituting the derivative by the incremental ratio f (t+T)−f folT lowed by a sampling with interval T. The delta systems were applied in approximating continuous-time systems for filter implementation and control [26, 39, 25, 38, 12], as well as for modeling [10, 43, 44, 9]. Although there is not a formal introduction of the delta systems they were studied and their stability criteria have been formulated [7, 8]. The modern approach to differential discrete equations dates back to Hilger’s work of looking for a continuous/discrete unification theory [13], and nowadays called calculus on time scales [13]. His methodology consisted in defining a general domain that can be continuous, discrete or mixed (time scales or, more generally, measure chains) [3, 1, 14, 2]. Hilger defined two derivatives, delta and nabla, which are the incremental ratios or their limit to zero when not consisting of one isolated point. With these derivative definitions we can devise the differential equations representing linear systems. Using the standard nomenclature we will call them nabla and delta systems, in agreement with the two derivatives. The nabla derivative is causal, while the delta is anti-causal. In the follow-up we will study them in parallel and derive general formulas valid for any real order (the generalization to complex order is straightforward). When we try to extend the application of difference equations into the fractional domain we have some problems due to two main reasons: – When fractionalizing the discrete-time systems using the classic tools we obtain infinite dimension integer-order systems that are difficult to manipulate. – While the continuous-time fractional systems have long memory, the integerorder discrete-time systems currently used to approximate them have short memory, since their impulse responses tend to zero exponentially. As an attempt to overcome such problems, the fractional delay difference systems were proposed in [29]. However, a bridge between these systems and the continuous-time fractional systems was not found. Suitable alternatives, adopted here, are the discrete derivative-based systems that mimic the continuous time that will emerge from the discrete time as a limit when the

Discrete-time fractional signals and systems | 151

sample rate increases without bound (the inter-sample interval, also called graininess, goes to zero). We start from the nabla and delta derivatives. Each one has an exponential as eigenfunction [33, 32]. With such exponentials we define two discrete-time Laplace transforms that will be used for studying the general fractional discrete-time linear systems. These transforms are backwardly compatible with the classic two-sided Laplace transform and with the Z transform. From these transforms and with suitable variable changes we will arrive at the discrete-time Fourier transform [34, 32]. In this chapter, we will begin by introducing the nabla and delta derivatives and by computing the discrete-time exponentials that are their eigenfunctions. With these exponentials, we define two discrete-time Laplace transforms that are useful in the study of the discrete-time linear systems defined by discrete-time differential equations.

2 Derivatives and inverses 2.1 Fractional nabla and delta derivatives In the following we will consider our domain of work to be the measure chain (time scale) 𝕋 = (hℤ) = {. . . , −3h, −2h, −h, 0, h, 2h, 3h, . . . }, with h ∈ ℝ+ . The results we will obtain are readily generalized to the case where this chain is shifted by a given value, a < h [3, 2, 14]. We define the nabla derivative by f∇󸀠 (t) =

f (t) − f (t − h) h

(1)

fΔ󸀠 (t) =

f (t + h) − f (t) . h

(2)

and the delta derivative by

As can be seen the first is causal, while the second is anti-causal. The repeated application of these derivatives allows us to obtain the Nth order derivatives and from them the general fractional formulations: f∇(α) (t) =

n α ∑∞ n=0 (−1) (n)f (t − nh)



α∈ℝ

(3)

and fΔ(α) (t) = e−iαπ

n α ∑∞ n=0 (−1) (n)f (t + nh)



α ∈ ℝ,

(4)

152 | M. D. Ortigueira et al. obtained from the generalized Grünwald–Letnikov derivative [5, 5, 35]. As in [30] we will call these derivatives, respectively, forward and backward due to the “time flow”, from past to future and the reverse. This terminology is the reverse of the one used in some mathematical literature. Some mathematical manipulation allows us to write alternative formulations for the derivatives. Essentially, we have to give another form to the binomial coefficients and change the summation variable. In fact, these derivatives are expressed by a convolution that is commutative. We have (−α)n Γ(α + 1) Γ(−α + n) = = . n! Γ(α − n + 1)n! Γ(−α)n! We can write the above derivatives as f∇(α) (t) =

n h−α Γ(−α + n − k) f (kh) ∑ Γ(−α) k=−∞ (n − k)!

(5)

fΔ(α) (t) =

(−h)−α ∞ Γ(−α + n − k) f (kh), ∑ Γ(−α) k=n (n − k)!

(6)

and

with t = nh. To use these formulations for the fractional derivatives (FDs) some caution must be taken since they are dangerous for positive integer values of α, since the gamma function has poles at the negative integers. They state different forms of expressing the FD and should be compared with those we find in the current literature (see Bastos [2]). To compare the above formulas with the continuous-time derivatives, we consider k = τ/h and n = t/h, which yield Γ(−α + n − k) Γ(−α + (t − τ)/h) = ≈ (t − τ)/h−α−1 , (n − k)! Γ(((t − τ)/h) + 1) as h → 0. Treating the nabla case, we have lim

h→0

f∇(α) (t)

t

1 = ∫ (t − τ)−α−1 f (τ)dτ, Γ(−α) −∞

which is the forward Liouville derivative in agreement with [28]. For the delta case, the procedure is identical and leads to the backward Liouville derivative [28]. 2.1.1 Examples –

Consider the Heaviside unit step: 1 ε(nh) = { 0

n ≥ 0, n < 0.

(7)

Discrete-time fractional signals and systems | 153

It is straightforward to show that the nabla derivative of the unit step is 1

D∇ ε(nh) = { h 0

n = 0, n ≠ 0.

(8)

The anti-causal unit step is given by ε(−nh). Using (2) we verify that DΔ ε(−nh) = {

− h1 0

n = 0, n ≠ 0.

(9)

Expressions (7)–(9) lead us to introduce the discrete delta (impulse) function by δ(nh) = D∇ ε(nh).



(10)

These results are important when dealing with the linear systems obtained with these derivatives. The Nth order derivative of the discrete delta We have to insert the impulse into (5) and put α = N to obtain N DN∇ δ(nh) = h−N−1 (−1)n ( )ε(n) n −N−1 (−N)n ε(n). =h n! The term ε(n) indicates that the function is null for negative arguments (causal function). For the delta derivative, we have N DNΔ δ(nh) = (−h)−N−1 (−1)n ( )ε(−n). −n



Nth order derivatives of the causal functions We are going to obtain the derivative of a causal function. For N = 1 and remarking that ε(t − h) = ε(t) − hδ(t) we obtain D∇ [f (t)ε(t)] = f∇󸀠 (t)ε(t) + f (−h)δ(t)

(11)

f∇󸀠 (t)ε(t) = D∇ [f (t)ε(t)] − f (−h)δ(t),

(12)

and

which shows how the initial conditions appear. We are relating the causal part of the derivative with the derivative of the causal part, and the first one depends on the past. Repeating the process yields [f∇󸀠 (t)ε(t)]∇ = D2∇ [f (t)ε(t)] − f (−h)δ∇󸀠 (t). 󸀠

154 | M. D. Ortigueira et al. Using (11) we arrive at f∇󸀠󸀠 (t)ε(t) = D2∇ [f (t)ε(t)] − f (−h)δ∇󸀠 (t) − f∇󸀠 (−h)δ(t). This expression can be generalized for any positive integer order to N−1

f∇(N) (t)ε(t) = DN∇ [f (t)ε(t)] − ∑ f∇(k) (−h)δ∇(N−1−k) (t), k=0

(13)

which is similar to the well-known jump formula [11].

2.2 Main properties We are going to outline the main properties of the derivatives presented previously. The proofs are essentially similar to those of the corresponding properties of the generalized version of the Grünwald–Letnikov derivatives (see [30]). – Linearity The linearity property of the fractional derivative is evident from the above formulas. – Causality The causality property was already mentioned and can also be obtained easily. We only have to use (3). Assuming that f (t) = 0, for t < 0, we conclude from (3) that the derivative is also zero for t < 0. For the anti-causal case (4), the reasoning is similar. – Time reversal The substitution t ⇒ −t, converts the forward (nabla) derivative into the backward (delta) and vice versa. – Time shift The derivative operators are shift invariant as is clear from (3) and (4). – Additivity and commutativity of the orders Dα [Dβ f (t)] = Dβ [Dα f (t)] = Dα+β f (t). – Neutral element The existence of neutral element is a consequence of the last property. Putting β = −α, Dα [D−α f (t)] = D0 f (t) = f (t). This property is very important because it states the existence of an inverse. – Inverse element From the last result we conclude that there is always an inverse element: for every α order derivative, there is always a −α order derivative given by the same formula and so it is not needed to join any primitivation constant. – Associativity of the orders Dγ [Dα Dβ ]f (t) = Dγ+α+β f (t) = Dα+β+γ f (t) = Dα [Dβ+γ ]f (t). It is a consequence of the additivity.

Discrete-time fractional signals and systems | 155

Figure 1: Fractional derivatives of the impulse for α = {0.25, 0.5, . . . , 5}.



Fractional derivatives of the impulses It is straightforward to show that the derivative of any order of the impulse is essentially given by the binomial coefficients. In fact, from (3) and (4) we get Dα∇ δ(n) = h−α−1

(−α)n ε(nh) n!

(14)

(−α)−n ε(−nh). (−n)!

(15)

and DαΔ δ(n) = (−h)−α−1



In Figures 1 and 2 some plots of derivatives of the impulse are presented. Fractional derivatives of the unit steps According to the above properties, it is easy to obtain the fractional derivative of the step functions. We only have to substitute

156 | M. D. Ortigueira et al.

Figure 2: Fractional derivatives of the impulse for α = {−0.25, −0.5, . . . , −5}.

α − 1 for α and divide by h, so that Dα∇ ε(nh) = h−α and DαΔ ε(nh) = (−h)−α –

(−α + 1)n ε(nh) n! (−α + 1)−n ε(−nh). (−n)!

Fractional derivatives of the power functions For negative values of α these expressions can be considered the definitions of fractional “powers”. Their derivatives are given by β

D∇ [

(a − β)n (a)n ε(nh)] = h−β+1 ε(nh), n! n!

(16)

which states the analog of the derivative of the power function. Similarly we obtain for the delta derivative β

DΔ [

(a)−n (a − β)n ε(−nh)] = (−h)−β+1 ε(−nh). (−n)! (−n)!

(17)

Discrete-time fractional signals and systems | 157

2.3 On the ARMA models and their application to modeling fractional derivatives The constant coefficient ordinary difference equations are referred usually as ARMA (Autoregressive Moving Average) models and they are written in the general form [15, 40] N

M

k=0

m=0

∑ Ak y(n − k) = ∑ Bm x(n − m)

(18)

where n, M, N ∈ ℤ, and the coefficients Bk , k = 0, 1, . . . , N, and Bm , m = 0, 1, . . . , M, are real constants. It is possible to consider fractional delays [29], but we will not address that case here. In the regular case, the response of these systems to a sinusoid is also a sinusoid with the same frequency. This property leads to the introduction of the frequency response as another way of describing the system. In a general formulation we can say that the exponentials, z n , n ∈ ℤ, z ∈ ℂ, are the eigenfunctions of these systems. Let h(n) be the impulse response of the causal system defined by (18). We can recover the ARMA parameters from h(n) by means of the following relation: N Bj ∑ Ak h(j − k) = { 0 k=0

j = 0, 1, . . . , M,

j < 0 ∧ j > M,

(19)

with A0 = 1. Ak , k = 0, 1, . . . , N and Bm , m = 0, 1, . . . , M, are, respectively, the AR and MA parameters. To compute such parameters we can use the double Levinson recursion. The algorithm consists of the recursive solution of the system obtained from (19) with j = M to j = N +M for the AR coefficients followed by the use of the first M +1 equations to obtain the MA parameters. This algorithm gives the possibility of determining the orders of the systems when looking at the pattern formed by a sequence of coefficients. The procedure we describe here is based on a Schur-like algorithm. To begin with, we consider (19) and introduce a function fMN (j), j = 0, 1, . . ., given by N

fMN (j) = ∑ ANM (i)h(j + M − i), i=0

(20)

where we enhance the recursion orders, N and M. According to (19) this function has gaps (nulls) for j = 1, 2, . . . , N. For N = 0, we have fM0 (j) = h(j + M).

(21)

The algorithm uses an adjoint function defined by N

N N (i)h(N − j + M − i), gM (j) = ∑ GM i=0

(22)

158 | M. D. Ortigueira et al. with 0 gM (j) = h(−j + M).

(23)

N It is clear that gM (j) has also gaps for j = 1, 2, . . . , N. The solution of (20) is recursively constructed for successive values of N from N = 1 to N = N0 , where N0 is a positive integer. To accomplish this, assume that we constructed the (N − 1)th order functions N fMN (j) and gM (j) for j = 0, 1, 2, . . .. We construct the Nth order functions by the recursions N N−1 gM (N − j) fMN (j) = fMN−1 (j) + KM

(24)

N N−1 N N−1 gM (j) = gM (j) + HM fM (N − j),

(25)

and

N N where KM and HM are derived by forcing both functions to have a gap at j = N. We obtain N =− KM

fMN−1 (N) N−1 (0) gM

,

N =− HM

N−1 gM (N)

fMN−1 (0)

.

(26)

We have also N N N gM (0) = fMN−1 (0)[1 − KM HM ].

(27)

If the system with impulse response, hn , is really an ARMA(N0 , M0 ), then we will have N

Bj = fM 0 (j − M0 ) j = 0, 1, . . . , M0 . 0

(28)

For the AR coefficients we use the previous relations to obtain the so-called double Levinson recursion: N N−1 ANM (j) = AN−1 M (j) + KM GM (N − j),

N N−1 N N−1 GM (j) = GM (j) + HM GM (N − j),

(29)

N N with j = 0, 1, . . . , N, and where KM and HM represent the generalized reflection coefficients [27]. If the system was truly an ARMA model there would exist N and M for N N which KM HM+1 = 1, which is an useful relation for detecting suitable orders. For the application in ARMA modeling of FD, we consider the impulse response (14) as the signal to be modeled using the above algorithm. The model is only an approximation—the reason for calling it “pseudo-fractional ARMA modeling” [36]. In many applications, the MA model has been mostly used. It has been obtained by truncating the impulse response h(n), n = 0, 1, . . . , M. This and other alternatives can be found in [19–21, 23, 22, 42].

Discrete-time fractional signals and systems | 159

3 Discrete-time Laplace transforms 3.1 The nabla and delta general exponentials We introduce the nabla generalized exponential defined by e∇ (t, s) = [1 − sh]−t/h ,

(30)

where s ∈ ℂ. Following a similar procedure, we can obtain the delta generalized exponential: eΔ (t, s) = [1 + sh]t/h .

(31)

The corresponding complex sinusoids are obtained when s is on the right and left Hilger circles, respectively [1 − sh] = 1, s ∈ ℂ and [1 + sh] = 1, s ∈ ℂ [13, 2]. These exponentials have the following properties: 1. Relation between the nabla and delta exponentials: eΔ (t, s) = 1/e∇ (t, −s) = eΔ (−t, −s). 2. 3.

(32)

As h → 0 both exponentials converge to est . Attending to the way how both exponentials were obtained, we can conclude that Dα∇ e∇ (t, s) = sα e∇ (t, s)

(33)

DαΔ eΔ (t, s) = sα eΔ (t, s).

(34)

and

4. Behavior for s ∈ ℂ It is important to know how these exponentials increase/decrease for s ∈ ℂ. We are going to consider only the nabla case, since the delta is similar. Concerning the nabla exponential, (30), we can say that [13, 2]: – It is real for real s. – It is positive for s = x < h1 , x ∈ ℝ. – It oscillates for s = x > h1 , x ∈ ℝ. – For values of s inside the right Hilger circle it is bounded and goes to zero as s → h1 . – On the Hilger circle it has absolute value equal to 1 and it degenerates into a complex exponential. – Outside the Hilger circle its absolute value increases as |s| increases and goes to infinite as |s| → ∞.

160 | M. D. Ortigueira et al. 5.

Delayed exponentials Let n0 ∈ ℤ+ . The delayed exponentials verify the following relations: e∇ (t ∓ n0 h, s) = e∇ (t, s) ⋅ eΔ (±n0 h, −s)

(35)

eΔ (t ∓ n0 h, s) = eΔ (t, s) ⋅ e∇ (±n0 h, −s).

(36)

and

6.

Product of exponentials – Different types – same times s+v s+v e∇ (t, s) ⋅ eΔ (t, v) = e∇ (t, 1+vh ) = eΔ (t, 1−sh ),



with v ∈ ℂ. Different types – different times e∇ (t, s) ⋅ eΔ (τ, −s) = e∇ (t − τ, s), e∇ (t, −s) ⋅ eΔ (τ, s) = eΔ (τ − t, s),



(37)

(38)

with τ ∈ 𝕋. Same type e∇ (t, s) ⋅ e∇ (t, v) = e∇ (t, s + v − svh) and eΔ (t, s) ⋅ eΔ (t, v) = eΔ (t, s + v + svh).

7.

Cross derivatives DΔ e∇ (t, s) = sh ⋅ e∇ (t + h, s), D∇ eΔ (t, s) = sh ⋅ eΔ (t − h, s).

3.2 Nabla Laplace transform In Sub-section 3.1 we computed two exponentials that are eigenfunctions of the forward and backward derivatives. It is natural that we use them to define transforms [34, 32]. Let us start from the inverse transform, that is, from the synthesis formula. Assume that we have a signal, f (nh), and that the time flow is from past to future. Assume also that the signal has a given transform (undefined for now), F∇ (s). Its inverse transform would serve to synthesize it from a continuous set of elemental exponentials: f (nh) = −

1 ∮ F∇ (s)e∇ ((n + 1)h, s)ds, 2πi γ

(39)

Discrete-time fractional signals and systems | 161

where the integration path γ is any simple close contour in a region of analyticity of the integrand and including the point s = h1 . The simplest case is a circle with center at s = h1 . Let us start with a simple example: F(s) = 1. We are expecting to obtain an impulse as inverse transformed function. In fact, we obtain NLT−1 [1] = δ(nh), where NLT stands for nabla Laplace transform. The relation between the two exponentials expressed by (32) suggests that we define the NLT by +∞

F∇ (s) = h ∑ f (nh)eΔ (nh, −s). n=−∞

(40)

The region in ℂ where the series converges is called “region of convergence” (ROC). Attending to the properties of the exponential that we stated before, the limit as h → 0 in (40) leads to the usual two-sided Laplace transform.

3.3 Main properties of the NLT This transform enjoys the following properties [32]. We assume that s is inside the region of convergence: – Linearity This is an obvious property. – Transform of the derivative Attending to the previous results and to (33), we deduce that NLT[f∇(α) (nh)] = sα F∇ (s),



(41)

restating a well-known result in the context of the Laplace transform. The ROC is the disk inside the Hilger circle. It is worth to remark here that we are dealing with a two-sided transform. So the “initial values” are not needed. As we can see from (33), sα is, aside the 1/h factor, the NLT of the causal sequence formed by binomial coefficients. It is the transfer function of the system (differentiator) defined by the FD. The ROC is the interior of the Hilger cirle. Time shift The NLT of f (nh − n0 h) with n0 ∈ ℤ is given by NLT[f (nh − n0 h)] = eΔ (n0 h, −s)F∇ (s). This result comes from the definition of transform (37).

(42)

162 | M. D. Ortigueira et al. –

Convolution in time +∞

NLT[h ∑ f (kh)g(nh − kh)] = F∇ (s) ⋅ G∇ (s). k=−∞

(43)

To prove this relation we insert F∇ (s) ⋅ G∇ (s) into (39), substitute there F(s) by its expression given by (40) and use (39) again.

3.4 Examples We are going to compute the NLT of some relevant functions. – Causal and anti-causal exponentials Let f (t) = ε(t). A nabla causal exponential is defined by ec∇ (t, p) = e∇ (t, p) ⋅ ε(t). Its NLT is given by +∞

NLT[e∇ (t, p).ε(t)] = h ∑ e∇ (nh, p) ⋅ eΔ (nh, −s) n=0

p−s Using (37) e∇ (t, p) ⋅ eΔ (t, −s) = eΔ (t, 1−ph ). Summing the geometric series, we obtain

NLT[e∇ (t, p) ⋅ ε(t)] =

1 − ph . s−p

Similarly, let us consider the corresponding anti-causal exponential e∇ (t, p)⋅ε(−t− h). The transform is −1

NLT[e∇ (t, p) ⋅ ε(−t − h)] = h ∑ eΔ (nh, n=−∞

p−s ), 1 − ph

which leads to ∞

NLT[e∇ (t, p).ε(−t − h)] = h ∑ eΔ (−nh, n=1

1 − ph p−s )=− . 1 − ph s−p

These relations suggest a division of both members by 1 − ph to give NLT[e∇ (t + h, p) ⋅ ε(t)] =

1 s−p

(44)

and NLT[−e∇ (t + h, p) ⋅ ε(−t − h)] =

1 . s−p

(45)

Discrete-time fractional signals and systems | 163

1−sh To find the corresponding ROC we must test the fraction 1−ph . In the first case its absolute value must be smaller than 1. Furthermore, we must have

󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨s − 󵄨󵄨󵄨 < 󵄨󵄨󵄨p − 󵄨󵄨󵄨. 󵄨󵄨 h 󵄨󵄨 󵄨󵄨 h 󵄨󵄨



We conclude that the ROC is defined by all the points distancing from 1/h less than |p − h1 |. We can obtain an easier criterion by imposing the requirement that |s − h1 | < 1 and |p − h1 | > 1. The pole must stay outside the Hilger circle. For the anti-causal case, it is the inverse situation. We can generalize the results above for multiple poles by computing successive integer-order derivatives relatively to p. Unit steps Letting p = 0 in the above expressions we obtain immediately the NLT of the unit step: 1 (46) NLT[ε(t)] = . s The ROC is the disk delimited by the right Hilger circle |1 − sh| = 1. For the anticausal step, we obtain 1 NLT[−ε(−t − h)] = . s



Now the ROC is the region outside the right Hilger circle |1 − sh| > 1 Causal powers As seen above the NLT of the unit step is given by (46). Using the property of the derivative with negative α and using the results of Section 3.6, we obtain NLT[hα

(α + 1)n 1 ε(nh)] = α+1 . n! s

(47)

The ROC is the disk inside the Hilger circle. For the anti-causal case the situation is similar, but the ROC is outside the Hilger circle.

3.5 The delta transform. The correlation We studied the transform suitable for dealing with causal systems. However, we can formulate another transform taking as base the delta exponential as explained in the following. As before each function f (nh) can be synthesized by f (nh) =

1 ∮ FΔ (s)eΔ ((n − 1)h, s)ds, 2πi

(48)

where the integration path is any simple close contour in a region of analyticity of the integrand and including the point s = − h1 .

164 | M. D. Ortigueira et al. The relation between the two exponentials expressed by (32) suggests that we define the direct transform: +∞

FΔ (s) = h ∑ f (nh)e∇ (nh, −s). n=−∞

(49)

Equations (48) and (49) define the synthesis and analysis expressions of the delta Laplace transform (DLT). As stated before, the limit for h → 0 in both sides in (48) and in (49) leads again to the standard two-sided Laplace transform. The properties of this transform are similar to those of the NLT we studied above. The DLT is useful in the computation of the transform of the correlation. Let f (t) and g(t) be two functions with DLT and NLT, respectively. According to the well-known relation between the convolution and the correlation, we introduce this by +∞

rfg (nh) = h ∑ f (kh)g(nh + kh). k=−∞

(50)

We compute its NLT, using the shift property of the nabla transform. We have +∞

NLT[rfg (nh)] = h ∑ f (kh)eΔ (−kh, −s)G∇ (s). k=−∞

From expression (32) and due to (49) we can write NLT[rfg (nh)] = FΔ (−s)G∇ (s),

(51)

yielding a striking result that can be important when generalizing this theory for other time scales. We can show that in the continuous-time LT case, the right hand side is F(−s)G(s) and in the Z transform case it is F(z −1 )G(z).

3.6 Uniqueness and back compatibility Let us assume that we are working on a given time scale and let g(t) be a function defined on it. We can show that the NLT given by (40) is the unique function inside its region of convergence. This means that one function defined on a given time scale has a unique NLT. Since we can choose infinite time scales, we can have an infinite number of NLT. Concerning the inverse NLT the situation is not exactly the same. Even if we fix the time scale we can have several inverse transforms according to the ROC that we chose. In the rational transform case we can have three possibilities [40]: – one causal inverse, – one anti-causal inverse, – one or several non-causal inverses (we can use the term acausal).

Discrete-time fractional signals and systems | 165

Figure 3: Evolution of the Hilger circle as h decreases to zero. Both circles degenerate, in the limit, into the imaginary axis.

According to what we said above, when we perform the limit h → 0 we recover the −1 classic continuous-time formulations. Performing the Euler transformation s = 1−zh we obtain the current discrete-time difference-based equations. In Figure 3 we can see the evolution of the Hilger circle as h decreases to zero. – Therefore, when h → 0: – The two derivatives degenerate into the generalized fractional order derivatives. The negative integer order gives the well-known formulation of the Riemann integral. – The exponentials degenerate into one: est . – The nabla and delta Laplace transforms degenerate also into the classic twosided Laplace transform. – Setting s = iω, we obtain the Fourier transform. – Difference formulations In this case, instead of a limit computation we consider two distinct variable trans−1 formations, namely s = 1−zh in the nabla case and s = z−1 in the delta case. h – With the above transformations both exponentials degenerate into the current exponential z n (see Figure 4). – Both transforms recover the classic two-sided Z transform. – Putting s = eiω , we obtain the discrete-time Fourier transform.

166 | M. D. Ortigueira et al.

Figure 4: Transformations from the Hilger circles to the unit circle.

4 The differential discrete-time linear systems 4.1 Steady-state response We are going to consider systems with the general form [32] N

M

α

β

∑ ak D∇k y(t) = ∑ bk D∇k x(t) k=0

k=0

(52)

with aN = 1. The operator D is the nabla derivative defined above. The orders N and M are any positive integers. The αk and βk sequences of orders are strictly increasing and positive real numbers. It is interesting to remark that, when α = 1, expression (52) can be transformed into a difference equation as (18). Let g(t) be the impulse response of the system defined by (52): x(t) = δ(nh). The output is the convolution (43) of the input and the impulse response y(t) = g(t) ∗ x(t).

(53)

If x(t) = e∇ (nh, s), then the output is given by ∞

y(t) = e∇ (nh, s)[h ∑ g(nh)eΔ (nh, −s)]. n=−∞

The summation expression will be called transfer function as usually. We have ∞

G∇ (s) = h ∑ g(nh)eΔ (nh, −s), n=−∞

(54)

Discrete-time fractional signals and systems | 167

showing that the transfer function is the nabla Laplace transform [32] of the impulse response. With these results we can write the transfer function as G∇ (s) =

βk ∑M k=0 bk s

∑Nk=0 ak sαk

.

(55)

We conclude that: – The exponentials are the eigenfunctions of the linear systems (52). – The eigenvalues are the transfer function values. iθ

we obtain the frequency response. With the transformation s = 1−e h Let us consider the following example. Example 4.1. Let h = 1 and consider the differential equation y󸀠󸀠󸀠 (t) + y󸀠󸀠 (t) − 4y󸀠 (t) + 2y(t) = x(t). If x(n) = 2−n , that corresponds to s = −1, then the solution is given by y(n) =

1 1 2−n = 2−n . 6 (−1)3 + (−1)2 + 4 + 2

5 The Fourier transform and the frequency response In Sub-sections 3.2 and 3.5 we defined the nabla and delta Laplace transforms based on the nabla and delta derivatives. Those transforms used the exponentials obtained in Sub-section 3.1. As in the continuous-time case, the exponential degenerates into a complex sinusoid when its absolute value is equal to 1. To obtain here a suitable Fourier transform we look for the cases that make the exponentials have absolute value equal to 1. From (30) we obtain |1 − sh| = 1, which defines a circle centered at 1/h and has radius equal to 1/h. This is the right hand Hilger circle. Similarly we obtain the left Hilger circle by taking |1 + sh| = 1 from (31). With the change of variable −iωh s = 1−eh in (30), (39), and (40), as can be seen in Figure 4, we obtain e∇ (nh, ω) = eiωhn ,

n ∈ ℤ,

π/h

f (nh) =

1 ∫ F(eiω )eiωhn dω, 2π −π/h ∞

F(eiω ) = ∑ f (nh)e−iωhn . −∞

With the substitution ωh = Ω we obtain expressions that are independent of the sampling interval (graininess) h e∇ (n, Ω) = eiΩn ,

168 | M. D. Ortigueira et al. π

f (nh) =

1 ∫ F(eiΩ )eiΩn dω, 2π

(56)

−π

and ∞

F(eiΩ ) = h ∑ f (nh)e−iΩn . −∞

(57)

This means that we can use these expressions for any discrete-time signal independently of the underlying time scale. Expressions (56) and (57) define the discrete-time Fourier transform pair (DTFT), namely the inverse and direct transforms, respectively [15]. Turning back and considering the delta case we arrive at the same expressions iωh for the Fourier transform provided that we perform the substitution s = e h−1 . Now we are in conditions of defining the frequency response of a linear system. We −iωh only have to make s = 1−eh into (55). This involve the transformation of the parameters of the transfer function using the binomial coefficients. From these considerations we arrive at the conclusion that the transfer function (60) gives rise to the following frequency response: G(eiω ) =

M

0 Bk e−iωk ∑k=0

N

,

(58)

(−αl)k , k!

(59)

0 Ak e−iωk ∑k=0

where the coefficients Bk are given by M

Bk = ∑ bl h−αl−1 l=1

for k = 0, 1, 2, . . . , M0 . For the Ak parameters, the computation is similar. Computing the inverse Fourier transform of G(eiω ) we obtain a difference equation equivalent to the differential equation (52). As observed, we have to truncate the sequence. To confirm this claim we made some simulations according to the scheme: 1. Randomly generate a normalized bandwidth (uniform in (0, 21 )) of a FIR (finite impulse response) integer-order filter with a pre-specified order. 2. Generate a set of FIR coefficients corresponding to such bandwidth. 3. Compute the transformed coefficients using (59). 4. Compute the number of number of significative coefficients. Nonetheless, there is some difficulty in being concrete, in this case. We can use for example the number at which the absolute value of the coefficient becomes smaller than 10−3 . Several numerical simulations showed that, for α < 1, the number of such coefficients is less than twice the length of the FIR coefficient sequence and frequently less than this one.

Discrete-time fractional signals and systems | 169

6 Transient responses In general the systems described by (52) or (55) are infinite impulse response (IIR) systems that pose some difficulties due to the problems in computing the poles and zeros. Only when N = 0 and all the derivative orders are positive integers we have finite impulse response (FIR) systems [41]. Therefore, we will assume that all the orders are multiples of a given α, that we will assume to be real. The transfer function is then G∇ (s) =

αk ∑M k=0 bk s

∑Nk=0 ak sαk

.

(60)

In this case (60) can be decomposed into a sum of a polynomial (only zeros) plus a proper fraction (pole–zero). We will study separately the two cases.

6.1 Polynomial case Let us consider a transfer function with the form M

G∇ (s) = ∑ bk sαk . k=0

(61)

To invert this expression, we recall the previous statement on sα , that is, the transfer function of the differentiator. This is a system with impulse response given by the binomial coefficients in agreement with (14). The impulse response corresponding to (61) is M

g(nh) = b0 δ(nh) + ∑ bk h−α−1 k=1

(−αk)n ε(nh). n!

(62)

In Figure 5 we depict the impulse responses of this kind of system for α = {0.1, 0.2, . . . , 1}, where we set h = 1. In the computations it, we used as bk , k = 0, 1, . . . , M, the impulse response of a FIR system, as can be verified in the last strip (α = 1). Although they are theoretically IIR systems the impulse responses go to zero in a very fast way having an FIR behavior.

6.2 Proper fraction case Herein, we consider the case where we have M zeros and N poles given by (60) with k N > M. For the sake of simplicity we assume that the polynomial ∑M k=0 ak w has only simple roots. In this case we can write G(s) as N

Ak , α−p s k k=1

G∇ (s) = ∑

(63)

170 | M. D. Ortigueira et al.

Figure 5: Impulse responses of polynomial systems for α = {0.1, 0.2, . . . , 1}.

where the Ak and pz , k = 1, 2, . . . , N, are the residues and poles obtained by substituting w for sα in (60). This representation allows us to obtain the impulse response through the inversion of a combination of partial fractions with the form F∇ (s) =

A . sα − p

(64)

To invert (64) we can insert it into the inversion integral, (39). The residue theorem tells us that the inversion is obtained by computing the (n + 1)th derivative of F(s) and substituting 1/h for s. A simplest alternative is to use the properties of the geometric series. We have two possibilities corresponding to the regions:

Discrete-time fractional signals and systems | 171

Figure 6: Impulse response corresponding to

1.

1 sα −p

with p = 2 for α = {0.1, 0.2, . . . , 1}. 1

Intersection of the Hilger circle with the disk |s| < |p| α In this case we have F∇ (s) = −

∞ A A 1 = − [1 + p−k sαk ]. ∑ α p 1− s p k=1

(65)

p

From the above expression, it results that the impulse response f (nh), corresponding to a partial fraction of order one, is given by ∞

f (nh) = −A[ ∑ p−k−1 h−αk−1 k=0

(−αk)n ε(nh)]. n!

(66)

In Figure 6 we illustrate the procedure by showing the impulse responses of a onepole system for α = {0.1, 0.2, . . . , 1} and h = 1. We verify that the response goes to zero quickly resembling a FIR system. However, it will not be an interesting system in applications, since the values of p must be high for short h.

172 | M. D. Ortigueira et al.

Figure 7: Impulse response corresponding to system with a complex conjugate pair of poles for α = {0.1, 0.2, . . . , 1}.

Practical applications often present conjugate complex poles. This means that we must deal with terms of the form V∇ (s) =



A A∗ + α . − p s − p∗

(67)

Using (66) we obtain A v(nh) = −2 ⋅ Re{ }δ(nh) p ∞

− 2 ∑ Re{ k=1

2.

A

pk+1

}h−kα−1

(−αk)n ε(nh). n!

(68)

In Figure 7 we show the impulse responses of a complex conjugate pair of poles system for α = {0.1, 0.2, . . . , 1} and p = 1.001(1 + eiπ/2 ). We conclude that only with order α = 1 we can obtain a sinusoid. In the other cases, the system goes from the damped oscillation to the undamped with decreasing order. 1 Intersection of the Hilger circle with the disk |s| > |p| α For this case, we can write F∇ (s) = A

∞ s−α = A[ ∑ pk−1 s−αk .] −α 1 − ps k=1

(69)

Discrete-time fractional signals and systems | 173

Figure 8: Computation of the discrete-time Mittag-Leffler function for different values of α = {0.5, 0.7, 0.9, 1.1, 1.3} with h = 0.5. For comparison, we plotted also the continuous-time exponential e−t , t ∈ [0, 2.5].

The corresponding impulse response f (nh) is given by ∞

f (nh) = A[ ∑ pk−1 hαk−1 k=1

(αk)n ε(nh)]. n!

(70)

This expression is the discrete-time version of the α-exponential [17] we find in continuous-time systems and that is related to the Mittag-Leffler function, see Figure 8. Therefore, this case is interesting in practice, since it allows us to approach a continuous-time system as near as we want even with short values of the pole p (67). To obtain the expressions for the complex conjugate pair we proceed as above. Using (70) we obtain ∞

v(nh) = 2 ∑ Re{Apk−1 }hαk−1 k=1

(αk)n ε(nh). n!

(71)

that we illustrate in Figure 9.

6.3 Some stability issues The properties of the nabla exponential and the sequence of operations we followed to compute the NLT of the causal transform showed that if the poles are outside the right Hilger circle, then the system is stable. Additionally, the partial fraction inversions that we computed above showed that the series defining the time functions were convergent if |p|hα > 1, in the first case (65), and |p|hα < 1, in the second (69). This means

174 | M. D. Ortigueira et al.

Figure 9: Impulse response corresponding to system with a complex conjugate pair of poles for α = {0.1, 0.2, . . . , 1}.

that, if the p is outside the Hilger circle, then the system is stable, else the system can be stable even with the pole inside the Hilger circle provided that it is outside the circle 1 |s| = |p| α . This represents the discrete counterpart of the stability of continuous-time systems [24]. For the integer-order systems we can study the pole distribution by a Routh–Hurwitz like criterion [7].

6.4 On the initial conditions As seen in Sub-section 3.2, the NLT emerges naturally from the exponentials that are defined for −∞ < n < +∞, in Sub-section 3.1. This is very important when dealing with steady-state response of the systems. However, in a large number of problems we are interested in transient responses obtained for instants after the initial one, which usually we assume to be t = 0. This may lead to errors, because we may be forgetting information from the past. In the current literature the one-sided Laplace and Z transforms are used. In the fractional case the results may be incorrect due to the unsuitability of the used initial conditions (see [31, 30]). To start with, consider the simple problem: compute the output of a first order system under a suitable initial condition with zero input: y󸀠 (t) + ay(t) = x(t).

Discrete-time fractional signals and systems | 175

Consider x(t) ≠ 0 for t < 0. It is clear that y(t) is a two-sided function. If we multiply it by the unit step, ε(t), then we obtain a causal signal that is not solution of the above system, unless we remove the effect of multiplying y(t) by the step when computing the derivative. Therefore, the equation for the causal y(t) will be 󸀠

[y(t) − y(0)ε(t)] + ay(t) = 0, which leads to y󸀠 (t) − y(0)δ(t) + ay(t) = 0.

(72)

In the general integer-order case, we have from (13) N−1

g∇(N) (t)ε(t) = DN∇ [g(t)ε(t)] − ∑ g∇(k) (−h)δ∇(N−1−k) (t) k=0

N−1

= DN∇ [g(t)ε(t)] − ∑ g∇(k) (−h)ε(N−k) (t). k=0

Substituting Nα for N and kα for k we obtain N−1

(kα) (N−k)α g∇(Nα) (t)ε(t) = DNα (t). ∇ [g(t)ε(t)] − ∑ g∇ (−h)ε k=0

(73)

We can verify the coherence with the results presented in [31]. In agreement with the correspondence principle [37], we rewrite the above relation in the form N−1

(kα) −(N−k)α g∇(Nα) (t)ε(t) = DNα ∇ [g(t)ε(t)] − ∑ g∇ (−h)h k=0

(−(N − k)α + 1)n ε(nh). n!

Considering (13) and applying the NLT to both members, we obtain N−1

NLT[g∇(Nα) (t)ε(t)] = sNα NLT[f (t)ε(t)] − ∑ g∇(kα) (−h)s(N−k)α−1 . k=0

(74)

This result was obtained without defining a new transform. To solve the general integer-order initial condition problem, we only have to substitute the input and output transforms for the corresponding expressions given by (13). Therefore, with the above expression we can insert the initial conditions in any system as proposed in [31] and [30]. We must pay attention to an important detail that is sometimes forgotten: the initial conditions must be taken at instants before t = 0; in the continuous-time case they must be taken at t = 0− , as we conclude from (74).

176 | M. D. Ortigueira et al.

7 Conclusions This chapter presented a general approach to fractional discrete-time signal processing based on the discrete-time nabla and delta derivatives and obtained their eigenfunctions, called nabla and delta exponentials. With these exponentials the nabla and delta discrete-time Laplace transforms were defined. With suitable mathematical transformations it was possible to derive also the corresponding expressions for the Fourier transform. The nabla derivative and transform were used for the study of the general fractional causal discrete-time linear systems. The computation was discussed of the impulse response, transfer function and frequency response, and the stability and initial conditions problems analyzed.

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Tomas Skovranek and Vladimir Despotovic

Signal prediction using fractional derivative models Abstract: In this chapter the linear prediction (LP) and its generalisation to fractional linear prediction (FLP) is described with the possible applications to one-dimensional (1D) and two-dimensional (2D) signals. Standard test signals, such as the sine wave, the square wave, and the sawtooth wave, as well as the real-data signals, such as speech, electrocardiogram and electroencephalogram are used for the numerical experiments for the 1D case, and greyscale images for the 2D case. The 1D FLP model is proposed to have a similar construction as the LP model, i. e. it uses a linear combination of fractional derivatives with different values of the fractional order. The 2D FLP model uses a linear combination of the fractional derivatives in two directions, horizontal and vertical. The scheme for the computation of the optimal predictor coefficients for both 1D and 2D FLP models is also provided. The performance of the proposed FLP models is compared to the performance of the LP models, confirming that the proposed FLP can be successfully applied in processing of 1D and 2D signals, giving comparable or better performance using the same or even a smaller number of parameters. Keywords: Fractional calculus, linear prediction, Grünwald–Letnikov derivative, optimal predictor design, signal processing MSC 2010: 26A33, 62M20, 93C05, 94A08, 94A12

1 Introduction There are many redundant signals in nature, which contain information which is already present in previous instances of the signal (previous signal samples). For example, successive speech samples are usually similar, as well as successive pixels in images. There is no need to transmit the redundant information in a telecommunication system, as it can be successfully reconstructed at the receiver end, enabling signal Acknowledgement: This work was supported in part by the Slovak Research and Development Agency under the contract No.: SK-SRB-2016-0030, APVV-14-0892, SK-AT-2017-0015; by the Slovak Grant Agency for Science under grants VEGA: 1/0908/15; and under the framework of the COST Action CA15225. Tomas Skovranek, Technical University of Kosice, BERG Faculty, B. Nemcovej 3, 04200 Kosice, Slovakia, e-mail: [email protected] Vladimir Despotovic, University of Belgrade, Technical Faculty in Bor, Vojske Jugoslavije 12, 19210 Bor, Serbia, e-mail: [email protected] https://doi.org/10.1515/9783110571929-007

180 | T. Skovranek and V. Despotovic compression and substantial savings in resources necessary for signal transmission. Let us consider the sine signal as an example. It is not necessary to transmit every signal sample, as the sine is completely defined using only three parameters: amplitude, phase and frequency. Obviously, redundancy is huge in this simple example. However, this is not always the case; e. g. white Gaussian noise is uncorrelated, meaning that it does not contain any redundant information; hence compression is impossible in this case. In general, redundancy is usually considered to be directly proportional to correlation (although correlation is not the only source of redundancy) [27]. When the signal is correlated, it is possible to predict the current signal sample knowing its values in previous instances (signal history). Linear prediction (LP) is based on this principle: the current signal sample is represented as the linear combination of previous signal samples. Hence, there is no need to transmit all signal samples in the telecommunication system, only the coefficients of the linear function that is used to predict the current sample. Prediction in signal processing is a mathematical model where future values of a discrete-time signal are estimated based on the previous signal values. In linear prediction it is equivalent to finding an output of a linear time-invariant (LTI) filter by observing only previous output samples. A linear predictor is said to use p steps if the filter is using p previous instances for the prediction of the current signal sample. The task of linear prediction is to determine a set of filter coefficients which best describe the behaviour of an LTI system. Linear prediction is a fundamental tool in many diverse areas where processing of a one-dimensional or a two-dimensional signal is necessary. The idea of using the signal history is fundamentally rooted in fractional calculus. An important advantage of the fractional-order operators is that they are defined by infinite series, meaning that they have, implicitly, a “memory” of all past events [41]. Fractional derivatives were successfully used for modelling different physical processes instead of commonly used models involving integer-order derivatives. Although the foundation of the fractional calculus goes back to seventeenth century, it has been only in the last decades that the topic has been rediscovered in mathematics [32, 41, 59], physics [18], scientific and engineering applications, such as thermal engineering [1], control theory [8, 34, 57], viscoelasticity [30], diffusion modelling [31, 42, 48], and signal processing [11, 37–39, 44, 46, 52–54]. An extension of LP using fractional-order models, namely the fractional linear prediction (FLP), has attracted attention in different fields of signal processing, as it allows one to use the combination of the history of the signal with lower number of predictor coefficients. It was recently applied in speech analysis [2, 12, 13], detection and discrimination of premature ventricular contractions in electrocardiogram (ECG) signals [51], and in electroencephalogram (EEG) signal modelling [22]. In this study we present the one-dimensional FLP, and apply it to various test signals (sine, square and sawtooth waves), as well as to real-data one-dimensional signals, such as speech, electrocardiogram (ECG) and electroencephalogram (EEG) signal. Contrary to the previous studies on this topic [2, 22, 51], we present details of

Signal prediction using fractional derivative models | 181

the optimal fractional linear predictor design, with derivation of the optimal predictor coefficients. Furthermore, we generalise FLP for two-dimensional signals, hence we propose a two-dimensional FLP model and apply it to greyscale images. Finally, we compare the results with one-dimensional and two-dimensional LP.

2 One-dimensional linear prediction One-dimensional (1D) linear prediction (LP) is extensively studied and applied in different areas of signal processing, such as spectral estimation [5], adaptive filtering and channel equalisation [17, 43], system identification [5], audio signal processing [15, 56], electrocardiogram (ECG) signal processing [49], electroencephalographic (EEG) signal processing [40], seismic signal processing [6], object detection and object tracking [61], identification of an electromagnetic wave direction in radar, sensor networks and array processing [55]. However, the most common application is speech analysis, synthesis and coding [3]. 1D LP is an integral part of one of the most widely used speech coding algorithms nowadays, namely Code-Excited Linear Prediction (CELP), which is a basis of several important standards, such as ITU-T G.728 [19] used in video, cellular and satellite applications, ITU-T G.729.1 [20] used in wireless and multimedia network applications, ITU-T G.723.1 [21] used in speech coding for audio and video conferencing over public telephone networks, and many others.

2.1 Optimal linear predictor design in 1D Let the signal x(t) be a linear and stationary stochastic process such that x[n] = x(nT) is the nth signal sample at any arbitrary time t, and T is the small time interval between successive samples (sampling period). The signal x(t) can be represented as the output of the linear filter, such that at any time instance t = nT it can be represented by the linear combination of the previous signal samples: x[n] = a1 x[n − 1] + a2 x[n − 2] + a3 x[n − 3] + ⋅ ⋅ ⋅ .

(1)

In the case when only the finite subset of p previous samples is known, the nth signal sample can be approximated by p

̂ x[n] = ∑ ai x[n − i], i=1

(2)

̂ is the predicted signal sample and ai are the linear predictor coefficients. where x[n]

182 | T. Skovranek and V. Despotovic The prediction error (ep ) represents the difference between the original signal x and the predicted signal x:̂ p

̂ ep [n] = x[n] − x[n] = x[n] − ∑ ai x[n − i]. i=1

(3)

Applying the z-transform, the previous equation can be rewritten as p

p

i=1

i=1

Ep [z] = X[z] − ∑ ai X[z]z −i = X[z](1 − ∑ ai z −i ),

(4)

or A[z] =

Ep [z] X[z]

p

= 1 − ∑ ai z −i .

(5)

i=1

The prediction error signal (also known as residual signal) can be obtained as the output of the filter with the transfer function A[z]. At the receiver end of the telecommunication system the original signal can be reconstructed by filtering the residual signal using the filter with the transfer function H(z), which is inverse to the transfer function A[z]: H[z] =

1 1 . = A[z] 1 − ∑pi=1 ai z −i

(6)

Let us further define the mean-squared prediction error: 2

p

2

J = E[e [n]] = E[x[n] − ∑ ai x[n − i]] , i=1

(7)

where the operator E denotes the mathematical expectation. In an optimal predictor design the coefficients ai are chosen to satisfy a cost function (e. g. the mean-squared error); hence they can be determined by finding the first derivative of J with respect to ai and equating to zero: p

𝜕J = −2E[x[n − i](x[n] − ∑ ak x[n − k])] = 0, 𝜕ai k=1

i = 1, 2, . . . , p.

(8)

Solving (8) leads to the system of equations for determining the unknown linear predictor coefficients ak , k = 1, 2, . . . , p: p

E[ ∑ ak x[n − k]x[n − i]] = E[x[n]x[n − i]], k=1

i = 1, 2, . . . , p.

(9)

Signal prediction using fractional derivative models | 183

Since ak are constant with regard to the mathematical expectation, by replacing the positions of the sum and the mathematical expectation one obtains p

∑ ak E[x[n − k]x[n − i]] = E[x[n]x[n − i]],

i = 1, 2, . . . , p.

k=1

(10)

The stationary signals do not change when shifted in time; hence n can be replaced by n + i: p

∑ ak E[x[n]x[n + i − k]] = E[x[n]x[n + i]],

i = 1, 2, . . . , p.

k=1

(11)

Defining the autocorrelation function as Rxx (λ) = E[x[n]x[n + λ]],

(12)

equation (11) becomes p

∑ ak Rxx (i − k) = Rxx (i),

i = 1, 2, . . . , p.

k=1

(13)

Equation (13) is known as the Yule–Walker equation [55] and can be rewritten in the matrix form as Rxx ⋅ a = rxx ,

(14)

where Rxx

Rxx (0) [ R (1) [ xx =[ .. [ [ . R (p [ xx − 1)

Rxx (1) Rxx (2) .. . Rxx (p − 2)

a = [a1 rxx = [Rxx (1)

a2

Rxx (2) Rxx (3) .. . Rxx (p − 3) a3

⋅⋅⋅

Rxx (2) Rxx (3)

... ... .. . ...

ap ]T ,

⋅⋅⋅

Rxx (p − 1) Rxx (p − 2)] ] ], .. ] ] . 0 ] T

Rxx (p)] .

Finally, the unknown linear predictor coefficients a can be determined by solving the matrix equation: a = Rxx −1 ⋅ rxx .

(15)

2.2 Optimal fractional linear predictor design in 1D FLP is a generalisation of the notion of LP using the fractional-order derivatives. Using the analogy from LP the nth signal sample can be represented as the linear combination of its fractional derivatives: q

̂ x[n] = ∑ ai Dαi x[n − 1], i=1

(16)

184 | T. Skovranek and V. Despotovic ̂ where x[n] is the estimate of the nth signal sample, q is the number of “fractional terms” used for the prediction, ai are the FLP coefficients, and Dα x[n − 1] are the fractional derivatives of order αi , αi ∈ ℝ of the time-delayed signal. The time-delay is used to ensure that only the past samples are used for the estimation of the predicted signal sample, without including the current sample. The Grünwald–Letnikov (GL) definition is commonly used for the numerical evaluation of the fractional derivative of a function x(t) at time instant t [41]: α a Dt x(t)

⌊ t−a ⌋

1 h α = lim α ∑ (−1)j ( )x(t − jh), h→0 h j j=0

(17)

where ⌊⋅⌋ denotes the integer part of the fraction (t − a)/h, a and t are lower and upper terminals of differentiation, respectively, h is the sampling period and α ∈ ℝ is the order of fractional derivative. Note that in fractional derivatives limits must be considered, they vanish only in integer-order derivatives. Note, furthermore, that the upper limit of summation tends to infinity. Taking into account only the recent history of the signal, i. e. approximating the fractional derivative with the lower limit a by the fractional derivative with the moving lower limit t − L, where L is the memory length, the “short memory” principle [41] is employed. Due to this approximation, the number of addends in (17) is not greater than k = ⌊L/h⌋. For t = nh (17) becomes α 1 k ∑ (−1)j ( )x((n − j)h). h→0 hα j j=0

Dα x(nh) = lim

(18)

Denoting the nth sample of the signal x(t) as x[n], i. e. x[n] = x(nh), (18) can be rewritten as 1 k α ∑ (−1)j ( )x[n − j]. h→0 hα j j=0

Dα x[n] = lim

(19)

The binomial coefficients in (19), j α ω(α) j = (−1) ( j ),

j = 0, 1, 2, . . . ,

(20)

can be estimated using the recurrent relationship: ω(α) 0 = 1;

ω(α) j = (1 −

α + 1 (α) )ωj−1 , j

j = 1, 2, . . . .

(21)

Shifting in time by one sample backwards (19) becomes k α Dα x[n − 1] = h−α ∑ (−1)j ( ) x[n − 1 − j], j j=0

(22)

Signal prediction using fractional derivative models | 185

where k refers to the signal history, which is further used in the proposed FLP prediction model defined by (16). Let us rewrite the model given in (16) in the matrix form assuming N signal samples as ̂ x[1] Dα1 x[0] α [ x[2] ] [ [ ̂ ] [ D 1 x[1] [ . ]=[ .. [ . ] [ [ . ] [ . α1 ̂ [x[N]] [D x[N − 1]

Dα2 x[0] Dα2 x[1] .. . α2 D x[N − 1]

... ... .. . ...

Dαq x[0] a1 αq [ ] D x[1] ] [ a2 ] ] ] ⋅ [ . ], .. ] [.] ] [.] . αq D x[N − 1]] [aq ]

(23)

or in simpler form x̂ = Da,

(24)

where ̂ ̂ x̂ = [x[1] x[2] a = [a1

Dα1 x[0] [ Dα1 x[1] [ D=[ .. [ [ . α1 D x[N − 1] [

a2

T

⋅⋅⋅

̂ x[N]] ,

⋅⋅⋅

aq ]T ,

Dα2 x[0] Dα2 x[1] .. . Dα2 x[N − 1]

... ... .. . ...

Dαq x[0] Dαq x[1] ] ] ]. .. ] ] . αq D x[N − 1]]

The prediction error can be defined in the matrix form as ep = x − x̂ = x − Da,

(25)

where x = [x[1] x[2] ⋅ ⋅ ⋅ x[N]]T is the vector of the original signal samples and ep = [e[1] e[2] ⋅ ⋅ ⋅ e[N]]T is the vector of the prediction errors. Minimising the mean-squared prediction error defined as J = eTp ep = (x − Da)T ⋅ (x − Da)

= xT x − xT Da − aT DT x + aT DT Da = xT x − 2aT DT x + aT DT Da,

(26)

i. e. finding the first derivative of J with respect to a and equating to zero: 𝜕J = −2DT x + 2DT Da = 0, 𝜕a

(27)

DT Da = DT x.

(28)

we get normal equations:

186 | T. Skovranek and V. Despotovic Solving this for a the optimal FLP coefficients can be determined as a = (DT D) DT x, −1

(29)

provided that the inverse of DT D exists. As D is N × q matrix, the condition q ≤ N must hold, i. e. the number of FLP coefficients has to be smaller than or equal to the number of samples. In practice, this condition is always fulfilled.

2.3 Numerical results and discussion on 1D FLP The proposed fractional linear prediction can be applied in different areas of signal processing, such as speech or biomedical signal coding (e. g. EEG or ECG). In this section the experimental analysis and the results obtained, using three different test signals (sine, square and sawtooth waves), as well as the real-data signals (speech, ECG and EEG signals), are presented. Fractional-order models with the linear combination of two and three fractional derivatives, i. e. q = 2 and q = 3 in (16), were considered in this study, and compared to corresponding LP models with the same number of parameters, i. e. p = 2 and p = 3 in (2).1 Also it should be noted here that the fractional orders αi were chosen to be fixed throughout the whole signal duration, with the values that were heuristically proven to provide the best average performance for a wide variety of different signals. The prediction gain (PG) and the mean-squared error (MSE) are used as the measures of the predictor performance. PG is defined as the ratio between the variance of the input signal and the variance of the prediction error measured in decibels: PG [dB] = 10 log10

σx 2 . σep 2

(30)

A better predictor is capable of generating lower error, leading to higher gain [10]. MSE of a predictor measures the average of the squares of the errors, i. e. the differences between the predicted and the observed values: MSE =

1 N ∑ (x̂ − xi )2 . N i=1 i

(31)

MSE is always non-negative, with values closer to zero defining a better predictor. 1 The MATLAB implementation of 1D FLP is available from: http://www.mathworks.com/ matlabcentral/fileexchange/67867

Signal prediction using fractional derivative models | 187

2.3.1 Experimental analysis using sine wave The sine wave signal with the amplitude A = 1 and frequency f = 1 Hz given in the form x(t) = A sin (2πft)

(32)

was used for testing the performance of the proposed FLP defined in (16). The signal was sampled with the sampling frequency equal to 40 Hz. A sine wave is the representation of a single frequency with no harmonics, and it can be considered as the most fundamental building block of the sound (a pure tone). The estimated (predicted) signal obtained using two-step LP and FLP that uses two “fractional terms”, i. e. the linear combination of two fractional derivatives with fractional orders α1 = 0.15 and α2 = 0.30, is shown in Figure 1(a). In this case, the LP estimation better follows the original sine wave compared to the proposed FLP with the same number of coefficients. FLP that uses the linear combination of three “fractional terms” with fractional orders α1 = 0.15, α2 = 0.30 and α3 = 0.45 has improved prediction, giving almost perfect signal reconstruction, comparable to three-step LP, as shown in Figure 1(b). The results listed in Table 1 show a significant increase of PG (over 6 dB) and decrease in MSE in the case when FLP is used with more “fractional terms” (q = 3), while the LP provides no change with increasing the complexity from p = 2 to p = 3. The prediction gain as a function of the sampling frequency is displayed in Figure 2, showing that for lower sampling frequencies FLP using the linear combination of three fractional derivatives enables one to obtain approximately the same PG as LP model, while the performance of FLP using two fractional derivatives is significantly

Figure 1: Estimated sine wave test signal.

188 | T. Skovranek and V. Despotovic Table 1: Prediction gains for sine wave test signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

33.82 33.82 27.48 33.67

2.06 × 10−4 2.06 × 10−4 8.86 × 10−4 2.13 × 10−4

Figure 2: Prediction gain vs. sampling rate for sine wave test signal using two-step LP, FLP with α1 = 0.15 and α2 = 0.30, and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

lower. The result for three-step LP is not shown in Figure 2 for better visibility, as it overlaps two-step LP for all analysed sampling frequencies.

2.3.2 Experimental analysis using square wave Although the square wave has many equivalent definitions in mathematics, let us define it as the signum function of a sinusoid with the amplitude A = 1 and frequency f = 1 Hz: x(t) = A sgn(sin(2πft)).

(33)

It is equal to A when the sinusoid is positive, and equal to −A when it is negative. The signal was sampled with the sampling frequency equal to 40 Hz. Contrary to the sine wave, the square wave contains odd harmonics, besides the fundamental frequency. Square waves are used in digital circuits as clock and timing control signals, but also in sound synthesis to emulate the music instruments; e. g., clarinets.

Signal prediction using fractional derivative models | 189

Figure 3: Estimated square wave test signal. Table 2: Prediction gains for square wave test signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

7.05 7.06 7.31 7.35

1.97 × 10−1 1.97 × 10−1 1.86 × 10−1 1.84 × 10−1

The estimated (predicted) signal obtained using the two-step LP and FLP that uses the linear combination of two “fractional terms”, i. e. two fractional derivatives with fractional orders α1 = 0.15 and α2 = 0.30 is shown in Figure 3(a), from which it can be seen that FLP estimation follows the original input signal better than LP with the same number of coefficients, especially at sudden signal changes. A similar behaviour can be observed in Figure 3(b), where three-step LP and FLP that uses the linear combination of three fractional derivatives with fractional orders α1 = 0.15, α2 = 0.30 and α3 = 0.45 are shown. The results listed in Table 2 confirm higher prediction gain of both FLP models (when using two and three “fractional terms”) compared to LP with the same number of coefficients by approximately 0.3 dB. The improvement of PG is especially visible at the lower sampling frequencies, where PG is almost 1 dB higher using FLP with two fractional derivatives and almost 2 dB higher using FLP with three fractional derivatives, in comparison to two-step LP, as shown in Figure 4. The result for three-step LP is not shown in Figure 4 for better visibility, as it overlaps two-step LP for all analysed sampling frequencies.

190 | T. Skovranek and V. Despotovic

Figure 4: Prediction gain vs. sampling rate for square wave test signal using two-step LP, FLP with α1 = 0.15 and α2 = 0.30, and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

2.3.3 Experimental analysis using sawtooth wave The sawtooth wave signal can be constructed using the relation x(t) =

A A ∞ sin(2πkft) − ∑ , 2 π k=1 k

(34)

where A = 1 is the amplitude and f = 1 Hz is the used frequency. The signal was sampled with sampling frequency equal to 40 Hz. The sawtooth is a linearly increasing ramp, followed by an abrupt drop, forming the shape of teeth of a saw. Unlike a square wave which contains only odd harmonics, the sawtooth wave contains both even and odd harmonics of the fundamental frequency. It is known for its application in music, where it is used to generate sounds with music synthesisers, but it is also used to generate a raster on the CRT-based displays. The estimated (predicted) signal obtained using the two-step LP and FLP that uses the linear combination of two “fractional terms” with fractional orders α1 = 0.15 and α2 = 0.30 is shown in Figure 5(a). It is obvious that FLP estimation is able to follow the original input signal better than LP with the same number of coefficients. Similar behaviour can be observed in Figure 5(b), where LP with three-step prediction and FLP that uses the linear combination of three fractional derivatives with fractional orders α1 = 0.15, α2 = 0.30 and α3 = 0.45 are shown. The results listed in Table 3 confirm approximately 0.2 dB higher prediction gain of FLP with two fractional derivatives compared to LP with the same number of coefficients. Analysing the PG dependency on the sampling frequencies, a constant improvement of PG of the FLP models can be observed at all sampling frequencies, in comparison to the two-step LP, as shown in Figure 6. The result for three-step LP is not shown in Figure 6 for better visibility, as it overlaps two-step LP for all analysed sampling frequencies.

Signal prediction using fractional derivative models | 191

Figure 5: Estimated sawtooth wave test signal.

Table 3: Prediction gains for sawtooth wave test signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

5.34 5.35 5.55 5.56

9.91 × 10−2 9.89 × 10−2 9.46 × 10−2 9.43 × 10−2

Figure 6: Prediction gain vs. sampling rate for sawtooth wave test signal using two-step LP, FLP with α1 = 0.15 and α2 = 0.30, and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

192 | T. Skovranek and V. Despotovic 2.3.4 Experimental analysis using speech signal The “real-data” experiments were performed using the speech signal containing clean voice recordings, sampled at the frequency of 16 kHz, with the total duration of 67 s. As speech is a non-stationary process, with the properties changing rapidly over the time, it is usually processed in short intervals denoted as frames, where it can be considered approximately stationary. If the frames are too short, there is not enough time to accurately assess the speech properties, but on the other hand, if the intervals are too long, the signal properties vary significantly; hence, this stationarity is lost [33]. Frames with the duration of 160 samples were used in our experiments. The prediction gain obtained using the two-step and the three-step LP, as well as FLP with the linear combination of two and three “fractional terms,” is listed in Table 4. The prediction gain only slowly increases with the increase of LP steps, contrary to FLP, where the improvement of PG is more significant with the increase of the number of “fractional terms”. Finally, FLP with three fractional derivatives has slightly better performance compared to LP with the same number of coefficients. The improved performance of FLP using three fractional derivatives compared to the three-step LP can be observed in Figure 7 for two characteristic frames of voiced Table 4: Prediction gains for speech signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

14.02 14.32 13.04 14.38

6.40 × 10−5 5.99 × 10−5 7.81 × 10−5 5.97 × 10−5

Figure 7: Original speech signal and squared prediction error using LP (p = 3) and FLP (q = 3).

Signal prediction using fractional derivative models | 193

speech. The red dashed curve corresponds to the squared prediction error of FLP, whereas the black solid curve corresponds to the squared prediction error of LP. It is evident that the error is suppressed in the whole signal duration when FLP is used.

2.3.5 Experimental analysis using electrocardiogram signal An electrocardiogram (ECG) is the recording of the heart electrical activity over a period of time using the electrodes placed on the patient’s limbs and chest. The experiments were performed using ECG signals included in the MIT-BIH Arhythmia Database, obtained using long-term Holter recordings at the Beth Israel Hospital Arrhythmia Laboratory [35]. The database is a part of the PhysioNet collection of recorded physiologic signals [16]. Recordings of 10 patients were used, each of them containing slightly over 30-min long signals, and then concatenated to a single signal. The signal is a modified limb lead II (MLII), obtained by placing the electrodes on the chest, sampled at the frequency of 360 Hz [35]. The signal is framed with the frame duration of 160 samples. The prediction gain obtained using the two-step and the three-step LP, as well as FLP with the linear combination of two and three fractional derivatives is shown in Table 5. Similar to speech, in the case of using LP model the prediction gain only slightly increases with the increase of LP steps. For FLP this improvement is substantial, with over 3.3 dB higher PG when the number of “fractional terms” is increased from two to three. Finally, FLP with three fractional derivatives has 0.25 dB better performance compared to LP with the same number of coefficients. The improved performance of FLP using three fractional derivatives compared to the three-step LP can be observed in Figure 8 for six characteristic frames of ECG signal. The red dashed curve corresponds to the squared prediction error of FLP, whereas the black solid curve corresponds to the squared prediction error of LP. Again, it is evident that the error is suppressed in the whole signal duration when FLP is used. Table 5: Prediction gains for ECG signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

19.93 20.05 16.98 20.3

2.00 × 10−4 1.95 × 10−4 4.11 × 10−4 1.87 × 10−4

194 | T. Skovranek and V. Despotovic

Figure 8: Original ECG signal and squared prediction error using LP (p = 3) and FLP (q = 3).

2.3.6 Experimental analysis using electroencephalogram signal An electroencephalogram (EEG) is a recording of the electrical activity of the brain over a period of time using the electrodes placed on the scalp. The experiments were performed using EEG signals collected from 10 subjects upon rapid presentation of aerial images of London through the Rapid Serial Visual Presentation (RSVP) protocol at a speed of 10 Hz, and then concatenated to a single signal. EEG data have been filtered between 0.15–28 Hz and sampled at the frequency of 2048 Hz. The samples were measured in μV [29]. The dataset is a part of the PhysioNet collection of recorded physiologic signals [16]. The signal is framed with a frame duration of 160 samples. The prediction gain obtained using the two-step and the three-step LP, as well as FLP with the linear combination of two and three fractional derivatives is shown in Table 6. Similar to speech and ECG, in the case of using LP model the prediction gain only slightly increases with the increase of LP steps. For FLP this improvement is substantial, with almost 3 dB higher PG when the number of “fractional terms” is increased from two to three. Finally, FLP with three fractional derivatives has approximately 1 dB better performance compared to LP with the same number of coefficients.

Table 6: Prediction gains for EEG signal using two-step LP, three-step LP, FLP with α1 = 0.15 and α2 = 0.30 and FLP with α1 = 0.15, α2 = 0.30 and α3 = 0.45.

LP (p = 2) LP (p = 3) FLP (q = 2) FLP (q = 3)

PG [dB]

MSE

23.22 23.47 21.57 24.42

2.51 × 10−5 2.50 × 10−5 3.43 × 10−5 2.55 × 10−5

Signal prediction using fractional derivative models | 195

Figure 9: Original EEG signal and squared prediction error using LP (p = 3) and FLP (q = 3).

The improved performance of FLP using three fractional derivatives compared to the three-step LP can be observed in Figure 9 for four characteristic frames of EEG signal. The red dashed curve corresponds to the squared prediction error of FLP, whereas the black solid curve corresponds to the squared prediction error of LP. It is evident that the error is suppressed in the whole signal duration when FLP is used.

3 Two-dimensional linear prediction Assuming more than one random variable, the univariate linear prediction can be extended to the multivariate case [55]. Applications of the two-dimensional (2D) LP models can be found in image coding [14, 26, 36, 60], 2D object detection [50] and spectral estimation of 2D series [7, 24]. The Levinson–Durbin approach for 2D series is proposed in [62], while the 2D autoregressive parameter estimation algorithm based on the functional Schur coefficients is presented in [45]. An extension of the 1D lattice technique for LP parameter estimation to the 2D case is given in [23, 28]. 2D LP has also been adopted for the lossless compression in the JPEG-LS standard [4, 58]. An extension of the solution for 2D Yule–Walker equations, used for LP parameter estimation, to the three-dimensional (3D) case is proposed in [25] and [9]. However, the increasing computational complexity limits the potential use of the multivariate LP models. Generally, for a “causal” prediction (see Figure 10), the predicted signal sample ̂ x[m, n] is estimated using only “previous” samples, i. e. samples that are located above and left from it (x[m − i, n], x[m, n − i], x[m − i, n − i]).

196 | T. Skovranek and V. Despotovic

Figure 10: “Causal” 2D prediction scheme.

To the best of our knowledge there are no reported results on two-dimensional fractional linear prediction (2D FLP) in the literature. Therefore, motivated by the use of the fractional derivative operator, and the approach described in the previous section, a two-dimensional fractional linear prediction model (2D FLP) is proposed here, and the computation of the optimal predictor coefficients is provided. The classical 2D linear prediction is also defined, which serves as the baseline for comparison. Optimal predictor coefficients estimation for the case of using four previous samples for the 2D LP model is also provided.

3.1 Optimal linear predictor design in 2D Let us assume a 2D sequence x[k, l] such that k = {0, 1, . . . , K − 1} and l = {0, 1, . . . , L − 1}. The 2D LP, which predicts signal x at the position [m, n] in 2D plane can be defined as: ̂ x[m, n] = ∑ ∑ ai,j x[m − i, n − j], i∈Π j∈Π

(35)

̂ where x[m, n] is the predicted value of the signal sample x[m, n], ai,j are the predictor coefficients, and Π defines the support region of the predictor. Assuming quarterplane support region Q × Q, Π = {(i, j) : 0 ≤ i, j ≤ Q − 1 and (i, j) ≠ (0, 0)}, the number of samples used for the prediction is defined as P = Q2 − 1. This support region in 2D system theory is known as recursively computable [47]. Inspired by the definition (35) let us propose a 2D LP model that for the current sample estimation uses previous samples in the horizontal and the vertical direction,

Signal prediction using fractional derivative models | 197

and that can be defined as p

p

i=1

i=1

̂ x[m, n] = ∑ a0,i x[m, n − i] + ∑ ai,0 x[m − i, n].

(36)

Taking into account two previous samples (p = 2) in each direction, horizontal and vertical, (36) can be written as ̂ x[m, n] = a0,1 x[m, n − 1] + a0,2 x[m, n − 2] + a1,0 x[m − 1, n] + a2,0 x[m − 2, n].

(37)

̂ The prediction error is then defined as ep [m, n] = x[m, n] − x[m, n]. The optimal predictor coefficients are determined by minimising the mean-squared prediction error, i. e. by finding the partial derivatives of E[ep2 [m, n]] with respect to the predictor coefficients and equating to zero (E denotes the mathematical expectation). The optimal predictor is then given by the solution of the 2D Yule–Walker equation, in the matrix form Rxx ⋅ a = rxx ,

(38)

or equivalently Rxx (0, 0) [ R (0, 1) [ xx [ [Rxx (1, −1) [Rxx (2, −1)

Rxx (0, 1) Rxx (0, 0) Rxx (1, −2) Rxx (2, −2)

Rxx (1, −1) Rxx (1, −2) Rxx (0, 0) Rxx (1, 0)

Rxx (2, −1) a0,1 Rxx (0, 1) [a ] [R (0, 2)] Rxx (2, −2)] ] [ 0,2 ] [ xx ] ][ ]=[ ], Rxx (1, 0) ] [a1,0 ] [Rxx (1, 0)] Rxx (0, 0) ] [a2,0 ] [Rxx (2, 0)]

(39)

and where Rxx (i, j) = E[x[m, n]x[m − i, n − j]]

(40)

is the autocorrelation function. The vector of the optimal predictor coefficients can be computed the same way as in the 1D case (14): a = Rxx −1 ⋅ rxx .

(41)

3.2 Optimal fractional linear predictor design in 2D A two-dimensional fractional linear prediction model can be constructed using the fractional derivatives of the input signal (in 2D case the input signal is a matrix) in both directions, the horizontal and the vertical. Following the notion given in Section 2.2, let us first define the fractional derivative of the horizontal time-delayed signal, with

198 | T. Skovranek and V. Despotovic the delay in the horizontal direction, as k α Dα x[m, n − 1] = h−α ∑ (−1)j ( ) x[m, n − 1 − j], j j=0

(42)

of the fractional order α, where α ∈ ℝ, and the fractional derivative in the vertical direction of order β, where β ∈ ℝ, of the time-delayed signal, with the delay in the vertical direction: k β Dβ x[m − 1, n] = h−β ∑ (−1)j ( ) x[m − 1 − j, n]. j j=0

(43)

Since we deal here with a 2D input signal, the fractional derivatives for all rows and columns have to be computed first. Let us arrange these derivatives (1D signals) in the form of matrices Dα and Dβ , where the matrix Dα represents the “horizontal” fractional derivatives Dα x[m, n − 1], which are arranged in the rows of Dα , for all the rows m = 1, 2, . . . , M: Dα x[1, 0] [ Dα x[2, 0] [ Dα = [ .. [ [ . α [D x[M, 0]

Dα x[1, 1] Dα x[2, 1] .. . α D x[M, 1]

... ... .. . ...

Dα x[1, N − 1] Dα x[2, N − 1] ] ] ], .. ] ] . α D x[M, N − 1]]

and Dβ is the matrix of the “vertical” fractional derivatives Dβ x[m − 1, n], which are arranged in the columns of Dβ , for all the columns n = 1, 2, . . . , N: Dβ x[0, 1] [ Dβ1 x[1, 1] [ Dβ = [ .. [ [ .

β [D x[M − 1, 1]

Dβ x[0, 2] Dβ x[1, 2] .. . Dβ x[M − 1, 2]

... ... .. . ...

Dβ x[0, N] Dβ x[1, N] ] ] ]. .. ] ] . Dβ x[M − 1, N]]

Assuming a square 2D input signal, i. e. a M × N matrix (where M = N), the 2D FLP model that uses two “fractional terms”, one in each direction (the horizontal and the vertical), can be given in the matrix form: ̂ x[m, n] = a1 Dα [m, n] + a2 Dβ [m, n],

(44)

where m = 1, 2, . . . , M represents the rows, n = 1, 2, . . . , N the columns of the 2D signal ̂ (see Figure 10), x[m, n] is the predicted signal sample located in the mth row and the nth column of the prediction matrix X,̂ and ai are the predictor coefficients, optimisation of which can be done using the relation (29).

Signal prediction using fractional derivative models | 199

It should be noted here that the model (44) can be generalised for any number of “fractional terms” having vectors of the fractional derivatives α = [α1 , α2 , . . . , αv ] and/or β = [β1 , β2 , . . . , βw ], obtaining the model in the form v

w

i=1

i=1

̂ x[m, n] = ∑ ai Dαi [m, n] + ∑ bi Dβi [m, n],

(45)

with v matrices Dαi , and/or w matrices Dβi , and with corresponding vectors of predictor coefficients a and b, but this was not done in this work for readability.

3.3 Numerical results and discussion on 2D FLP The proposed 2D FLP can, in general, be applied to any two-dimensional signal. The numerical experiments were performed using standard test images in the greyscale format (a representation of the 2D signal), with the resolution 256 × 256 and 512 × 512 pixels. In the case of using images in RGB format, first the conversion to eight-bit greyscale was realised by eliminating the hue and saturation information, while retaining the luminance. The luminance of a pixel in a greyscale image ranges from 0 to 255, with 0 denoting black, 255 denoting white, and values in-between denoting shades of grey. The proposed 2D FLP model was applied to the test images and compared to the proposed 2D LP model, which was used as the baseline. The prediction error (Ep ) and the prediction gain (PG) were used as the measures of the predictor performance. The prediction error can in the matrix form be defined as: Ep = X − X,̂

(46)

where X̂ is the matrix of the predicted 2D signal, and X is the original 2D signal defined as: x[1, 1] [ x[2, 1] [ X=[ [ .. [ . [x[M, 1]

x[1, 2] x[2, 2] .. . x[M, 2]

... ... .. . ...

x[1, N] x[2, N] ] ] .. ] ], . ] x[M, N]]

with x[m, n] being the signal sample located in the mth row and nth column of the 2D signal, and M, N being the number of rows and columns in the matrix. Similarly to the 1D case, PG defined as the ratio between the variance of the input signal (now a 2D signal represented by the matrix X) and the variance of the 2D prediction error (represented by the matrix Ep ) measured in decibels is defined as: PG [dB] = 10 log10

σX 2 . σEp 2

(47)

200 | T. Skovranek and V. Despotovic Table 7: Prediction gains using 2D LP with four predictor coefficients, and 2D FLP with two “fractional terms” and α = β = 0.01.

Cameraman Baboon Kiss Peppers

256 × 256 512 × 512 512 × 512 512 × 512

2D LP PG [dB]

2D FLP PG [dB]

12.2604 6.9583 11.9417 15.7545

12.4044 7.1093 12.5713 15.9191

While 2D LP, which was used to perform the experiments, has four predictor coefficients, two in each direction, i. e. p = 2 in (36), the 2D FLP assumes only two predictor coefficients (a1 , a2 ). However, also the fractional orders α and β have to be taken into account (see model (44)), resulting in four parameters used by the 2D FLP in total. This makes the number of used parameters in LP and FLP equal, leading to a fair comparison. Based on the experiments with various 2D input signals, different combinations of the fractional orders α and β of the FLP model were analysed, with 0 < α, β < 1, resulting in the conclusion that the orders can be chosen to be constant values that are equal, i. e. α = β, making the 2D FLP model using one parameter less than the corresponding 2D LP model (three parameters in total, two predictor coefficients and one parameter value for the fractional orders α = β). The total number of parameters is important, since they have to be quantised and transmitted to the receiver end of the communication system, and thus, influence the compression rate. Analysing the selected input images (“Cameraman”, “Baboon”, “Kiss”, “Peppers”) the fractional orders of the 2D FLP were assumed fixed α = β = 0.01 instead of optimally determined fractional orders, which in these cases did not lead to significant loss in PG. The results listed in Table 7 confirm higher prediction gain of the 2D FLP model with two “fractional terms” compared to 2D LP that uses four predictor coefficients (two in each direction). The difference between the performances of the 2D LP and 2D FLP model for the tested images (2D input signals) ranges from 0.14 dB to 0.63 dB. The prediction error results for the four selected images were also analysed, and the errors are shown in Figure 11, where the first column represents the original image, the second column is the prediction error of the 2D LP model and the third column is the prediction error of the 2D FLP model. The results confirm that the proposed 2D FLP model is fully competitive with the 2D LP model, as it removes most of the redundancy, leaving only the information along contours in the used test images, i. e. the regions where luminance suddenly changes. The improvement in terms of lower prediction error is especially notable in the “Kiss” image.

Signal prediction using fractional derivative models | 201

Figure 11: 2D prediction error results: 1st column—original image / 2nd column—prediction error for 2D LP with four predictor coefficients / 3rd column—prediction error for 2D FLP with two “fractional terms” with α = β = 0.01.

4 Conclusion In this chapter the linear prediction and its generalisation to fractional linear prediction is described with the possible applications to 1D and 2D signals.

202 | T. Skovranek and V. Despotovic The numerical experiments in 1D are performed using standard test signals, such as the sine wave, the square wave, and the sawtooth wave signals, as well as using the real-data signals, such as speech, ECG and EEG signals. The 1D fractional linear prediction model is proposed to have a similar construction as the linear prediction model, i. e. it is defined as a linear combination of q “fractional terms,” meaning that it uses linear combination of q fractional derivatives with different values of the fractional order. The scheme for the optimal predictor coefficients computation of the FLP model is also provided. The 1D FLP model is compared to the two-step and three-step linear predictor, i. e. a linear prediction model that for the estimation of the predicted signal sample uses a linear combination of two and three previous samples (p = 2 and p = 3 in the 1D LP model). The fractional orders for FLP models used in the experiments were fixed (empirically determined, as a good trade-off for various signals) making the models comparable, using same number of predictor coefficients. The results show that already the FLP model with three “fractional terms” is fully competitive with the linear prediction that uses three predictor coefficients, and even outperforms it, measured by PG. Analysing the results using test signals it is evident that LP has better performance only when the signal slowly changes over time (e. g. sine wave), while it is not able to track abrupt signal transitions, as in the case of the square and sawtooth wave. FLP performs better in that cases. The performance of FLP for tested real-data signals is always better than the performance of the LP model, with up to 1 dB higher PG in the case of the EEG signal. The one-dimensional FLP model is further generalised to two-dimensional case using the linear combination of the fractional derivatives, one in each direction, horizontal and vertical, with fractional orders α and β, having four parameters in total (two coefficients and two fractional orders). The results are compared to 2D LP that uses the same number of parameters, i. e. four predictor coefficients, two in each direction, horizontal and vertical. Based on the numerical experiments using standard greyscale test images as the input signal, an interesting observation can be made: the fractional orders used in the 2D FLP model can be assumed to be equal, making the 2D FLP model use one parameter less. The obtained results confirm that the proposed 2D FLP model is fully competitive with the 2D linear prediction, and even outperforms it in terms of prediction gain, while using one parameter less. The experiments undertaken in this study prove that FLP can be successfully applied in processing of both 1D and 2D signals, giving comparable or better performance using the same or even a smaller number of parameters. This might be important in any parametric coding scheme, where the parameters have to be quantised and transmitted to the receiver end of the communication system, influencing the transmission and/or compression rate.

Signal prediction using fractional derivative models | 203

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E. H. Doha, M. A. Zaky, and M. A. Abdelkawy

Spectral methods within fractional calculus Abstract: Less than a decade ago, the formulation and implementation of robust, accurate and efficient spectral methods for the solution of fractional differential models were considered as an extremely challenging task. This was partly due to the nonlocality of the fractional differential operators, and the nonnegativity of the adjoint of a fractional differential operator used to describe the dynamics. In this chapter, we review the current state of spectral methods within fractional calculus and emphasize recent developments in the area of fractional differential models. We also include a discussion of the formulation of fractional spectral tau methods for multi-term time-fractional diffusion equations and conclude with examples on the application of collocation methods for solving the nonlinear variable-order fractional advection– diffusion equation. Keywords: 76M22, 35R11, 41A10, 41A30 MSC 2010: Spectral methods, fractional calculus, fractional Jacobi functions, fractional approximations

1 Introduction The past two decades have witnessed an astounding growth in the field of fractional calculus. The key idea behind this calculus has a history coinciding with that of classical calculus and the topic has been promoted by several prominent mathematicians who contributed fundamentally to the development of classical calculus, including L’Hospital, Leibniz, Riemann, Liouville, Grünwald, and Letnikov. This progress has been unleashed with the ever-increasing applications of generalized differentiation and integration in practically all imaginable fields of science [4, 50, 82]. The advances in the field require the development of methods for solving differential models with fractional derivatives and/or fractional integrals [15, 17, 89]. A very important challenge in many applications of fractional calculus is the reliable approximation of target models. Unfortunately, in many scenarios such an approximation cannot be obtained analytically and one has to resort to numerical methods. Filling this gap has become the key to understanding fractional calculus and solve E. H. Doha, Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt, e-mail: [email protected] M. A. Zaky, Department of Mathematics, National Research Centre, Giza 12622, Dokki, Egypt, e-mail: [email protected] M. A. Abdelkawy, Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia, e-mail: [email protected] https://doi.org/10.1515/9783110571929-008

208 | E. H. Doha et al. FDEs of increasing interest in recent years; see [32]. However, the development of numerical methods in this area does not have a long history and has undergone a fast development in the last few years. The literature on this issue can be broadly divided into two major categories: (i) FDEs with smooth solutions: Most of the known methods that have been developed for FDEs impose smoothness conditions on the solution. The fractional operators in these equations are discretized by the L1 method and its modification [61, 99], the weighted fractional central difference methods [33], the shifted Grünwald– Letnikov formula [46], the weighted shifted Grünwald–Letnikov formulas [44], the fractional linear multi-step methods [64, 98], the spectral approximations [19, 86, 87], the finite-element method [29], and some others [1, 71, 88]. (ii) FDEs with non-smooth solutions: Unlike the integer-order differential equations, solutions to FDEs are generally not smooth even for smooth data. Up to now, a few numerical schemes have been developed for these equations, such as using nonuniform grids in the discretization of the fractional operators [69, 72], and the non-polynomial basis functions [23, 45, 66, 91, 96, 100] to capture the singularity of the solutions to FDEs. Numerical methods for FDEs can also be classified into local and global methods. The finite difference, finite volume and finite-element methods are based on local interpolation strategies and depend on a mesh for local approximation, whereas the spectral methods are global in character, which should be better suited for non-local problems. In practice, finite-element methods are particularly well suited to problems in complex geometries, whereas spectral methods can provide superior accuracy, at the expense of domain flexibility. We emphasize that there are many numerical approaches, such as the hp finite-element and spectral-element methods, which combine the advantages of the global and the local methods. However, in this chapter, we shall restrict our attention to the global spectral methods. Spectral methods are promising candidates for solving FDEs since their global nature fits well with the non-local definition of fractional operators [16, 18, 90, 92]. They can be thought of as a development of the so-called method of weighted residuals. According to this method, the solution to the FDE is expressed in terms of a finite series of known functions, which are global in the sense that they are defined over the entire domain; they are called trial/basis functions. After substituting this series in the FDE, an inner product of the resulting equation with the so-called test functions is formed, which is used in order to guarantee that the equation is satisfied as closely as possible by the truncated series expansion. This is accomplished by minimizing the error in the differential equation produced by using the truncated series expansion instead of the exact solution, with respect to a suitable inner product. As in the traditional spectral methods for integer-order differential equations (see, e. g., [21]), it is important to choose appropriate approximation bases in the spectral methods for FDEs. When the solution is smooth enough, the classical Jacobi polynomials (typically Legendre or Chebyshev polynomials) can be used

Spectral methods within fractional calculus | 209

as an approximation basis. Unlike for the classical spectral polynomials, Zayernouri and Karniadakis [95] and Khosravian-Arab et al. [57] proposed two new classes of orthogonal systems in bounded and unbounded domains, respectively, which were the eigenfunctions of the fractional Sturm–Liouville eigenvalue problems. Based on the choice of trial/test functions, four major classes of spectral methods can be distinguished. (a) Galerkin spectral method: In this method the trial and test functions should be the same. The trial functions must individually satisfy the boundary conditions and need not necessarily be orthonormal to each other, although orthonormality reduces the complexity since no matrix inversion is required. Ervin and Roop [41] proved the well-posedness of a Galerkin weak formulation to a conservative space-fractional diffusion equation with a constant diffusivity coefficient and homogeneous Dirichlet boundary conditions. The fact that the conservative Caputo and Riemann–Liouville space-fractional diffusion equations coincide in the context of homogeneous Dirichlet boundary conditions is used heavily in the work [41]. Of the same flavor is the work of [60], where a space–time spectral Galerkin method with smooth basis functions (Legendre polynomials) for a time-fractional diffusion equation was considered. To the best of our knowledge, Li and Xu [60] were the first to achieve exponential convergence in their numerical tests in agreement with their error analysis. Nie et al. [70] proposed a numerical method based on the Galerkin spectral method for the Riemann–Liouville fractional derivative in space and a finite difference scheme in time for solving a spatial-fractional diffusion equation. Mao and Shen [66] developed this work to solve multi-dimensional fractional elliptic equations with variable coefficients in conserved form as well as in non-conserved form. Zheng et al. [101] considered the time-fractional Fokker– Planck equation. The proposed Galerkin method was obtained by employing the Jacobi polynomials to the temporal discretization and Fourier-like basis functions to the spatial discretization. It was proved that the same approximation order was attained as in [60]. This technique was also studied in [102], in which the multiterm time-fractional diffusion equation was considered. Yang [84] proposed Jacobi spectral Galerkin methods for solving fractional integro-differential equations. Kamrani [53] proposed Galerkin method based on Jacobi polynomials to investigate the numerical solution of stochastic FDEs. Dehghan et al. [30] applied two high order numerical algorithms, namely a compact finite difference scheme and a Galerkin spectral method for solving the multi-term FDE. Guo et al. [47] proposed a numerical approach for the two-dimensional time-fractional nonlinear reaction-diffusion-wave equation. In this approach, the Crank–Nicolson scheme is implemented for the time stepping and the Legendre–Galerkin spectral method is applied to the space discretization. In [24], the authors derived efficient spectralGalerkin method using the generalized Laguerre functions for solving the tempered fractional diffusion equation on the half/whole line.

210 | E. H. Doha et al. (b) Petrov–Galerkin spectral method: The use of different test and trial functions distinguishes the Petrov–Galerkin method from the Galerkin one. Thanks to this flexibility, the Petrov–Galerkin method can be very useful for some nonself-adjoint problems. Zhang et al. [100] analyzed spectral Petrov–Galerkin and collocation methods for single-term Riemann–Liouville fractional ordinary differential equations with initial value on a finite interval. They also developed Laguerre spectral Petrov–Galerkin methods and collocation methods for FDEs on the half line. Mao et al. [66] developed Petrov–Galerkin algorithms for Riesz FDEs with homogeneous Dirichlet boundary conditions and fractional integral boundary conditions. The methods are based on a new class of generalized Jacobi functions which are tailored to Riesz fractional derivatives. They derived useful properties of these generalized Jacobi functions, and in particular, their optimal approximation results in non-uniformly weighted Sobolev spaces. Zayernouri and Karniadakis [96] developed exponentially accurate Petrov–Galerkin spectral methods for fractional initial and final-value problems subject to Dirichlet initial/final conditions. They employed the developed spectral theory in [95] for fractional Sturm–Liouville problems, which provided the corresponding basis and test functions utilized in their schemes. Samiee et al. [77] developed a unified Petrov–Galerkin spectral method for a class of fractional partial differential equations with constant coefficients in a (1 + d)-dimensional space–time hypercube, d = 1, 2, 3, etc., subject to homogeneous Dirichlet initial/boundary conditions. They employed Jacobi polyfractonomials, as temporal trial/test functions, and the Legendre polynomials as spatial trial/test functions. Khosravian-Arab et al. [57] introduced exponentially accurate Galerkin, Petrov–Galerkin and pseudospectral methods based on non-classical Lagrange basis functions [58] for FDEs on a semi-infinite interval. Kharazmi et al. [56] developed a Petrov–Galerkin spectral method for distributed-order FDEs. They introduced the distributed Sobolev spaces and their associated equivalent norms, and investigated the stability and error analysis of the scheme. Deng and Zhang [31] provided a variational framework and a Petrov–Galerkin method for the tempered PDEs describing the tempered anomalous diffusion. In [94], the authors derived a Petrov–Galerkin method for solving tempered fractional ODEs on a finite interval by using the eigenfunctions of tempered fractional Sturm–Liouville problems. In [49], the authors proposed a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial FDEs by using a class of generalized Laguerre functions. (c) Tau spectral method: The spectral tau approach can be viewed as a special case of the Petrov–Galerkin. The tau method was proposed by Lanczos in 1938 [21], and it differs from the Galerkin one for the treatment of boundary conditions. In particular, the test functions are not required to satisfy the boundary conditions, since the latter are enforced through a set of supplementary equations. To the best of our knowledge, Saadatmandi and Dehghan [74, 75] introduced the first

Spectral methods within fractional calculus | 211

attempt to reconstruct the spectral tau method for solving FDEs. The method was obtained by employing the Legendre polynomial together with its operational matrix of the Caputo fractional derivative. Based on their formulation, Lotfi et al. [63] proposed a numerical technique for solving fractional optimal control problems. Doha et al. [38] derived the Jacobi operational matrix of the left-sided Caputo fractional derivative, which was applied in conjunction with the spectral tau scheme for solving one-sided FDEs. The Chebyshev [37] and Legendre [74] operational matrices can be viewed as special cases from Jacobi operational matrix [38]. Bhrawy et al. [6] developed a direct solver for the multi-term FDEs using a generalized Laguerre spectral tau approximation. Bhrawy and Zaky [9] proposed and analyze an efficient operational formulation of spectral tau method for two-sided spaceand multi-term time-FDE with Dirichlet boundary conditions. Zaky [86] revisited the Legendre spectral tau method to handle the multi-term time-fractional diffusion equations and studied its convergence analysis. In [87] the author developed efficient algorithms based on the Legendre-tau approximation for one- and twodimensional fractional Rayleigh–Stokes problems for a generalized second-grade fluid. Ahmadian et al. [2] proposed a spectral tau technique based on the generalized fractional Legendre functions to solve the Kelvin–Voigt equation and nonNewtonian fluid behavior model with fuzzy parameters. Mokhtary et al. [68] developed an efficient method for solving linear multi-term FDEs using the spectral tau method based on Müntz–Legendre polynomials. Bhrawy and Zaky [11] prosed a fractional Jacobi tau method to solve systems of FDEs. Ito et al. [51] revisited the Legendre-tau method for two-point boundary value problems with a Caputo fractional derivative in the leading term, and established an L2 error estimate for smooth solutions. Further, they applied the method to the Sturm–Liouville problem. Mokhtary [67] developed and analyzed an operational tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. Bhrawy et al. [7] investigated fractional generalized Laguerre tau method for solving linear FDEs on the half line. The review article [8] contains a bibliography on operational matrices and spectral tau techniques for fractional calculus. Hanert and Piret [48] developed a Chebyshev spectral tau scheme to discretize the space–time fractional diffusion equation with exponential tempering in both space and time. (d) Collocation spectral method: The test functions in the collocation method are Dirac delta functions. The collocation method forces the residual to vanish pointwisely at a set of preassigned points. Galerkin and tau spectral methods are implemented in the frequency space and well suited for linear problems with constant (or polynomial) coefficients. However, their implementations are not convenient for problems with general variable coefficients. On the other hand, the collocation method is more flexible to deal with complicated problems, such as FDEs with variable coefficients [5, 36], multi-term FDEs [27, 42], and variable-order FDEs [13, 14], but it cannot always be reformulated as a suitable variational formulation (most

212 | E. H. Doha et al. convenient for error analysis). A combination of the Galerkin and collocation methods leads to the Galerkin method with numerical integration, or sometimes called the collocation method in the weak form. Rawashdeh [73] proposed a polynomial spline collocation method for fractional integro-differential equations. Lin and Xu [61] proposed an effective numerical method based on a finite difference scheme in time and Legendre spectral collocation method in space to solve the time-fractional diffusion equation with smooth enough solution. Amore et al. [3] derived a collocation scheme based on little sinc functions to solve the space-fractional Schrödinger equation on a uniform grid. Pedas and Tamme [71] discussed the numerical solution of linear multi-term FDEs by piecewise polynomial collocation methods. Esmaeili et al. [43] proposed a collocation method based on Müntz–Legendre polynomials for solving a general class of fractional ordinary differential equations. Saadatmandi et al. [76] proposed a collocation scheme based on the sinc functions in time and Legendre polynomials in space to solve the fractional convection-diffusion equation with variable coefficients. In [10, 26, 59], by using the Legendre, Chebyshev and Jacobi polynomials, the authors developed the spectral approximations to evaluate the fractional integral and the Caputo derivative numerically. Ma and Huang [65] proposed and analyzed a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. Yang et al. [83] developed a numerical technique based on orthogonal spline collocation method in space and a finite difference method in time for the solution of the two-dimensional fractional sub-diffusion equation. Zayernouri and Karniadakis [97] developed an exponentially accurate fractional spectral collocation method for solving linear/nonlinear variable-order fractional partial differential equations. Chen et al. [22] proposed a multi-domain spectral method for high-order integrations of the time-fractional partial differential equations. Jiao et al. [52] introduced new fractional collocation schemes using fractional Birkhoff interpolation basis functions. Ejlali and Hosseini [40] introduced a direct pseudo-spectral method based on fractional Lagrange interpolating functions to solve the fractional optimal control problem. Tang et al. [80] provided fractional pseudo-spectral methods for solving fractional optimal control problems. They developed differential and integral fractional pseudo-spectral methods and proved the equivalence between them from the distinctive perspective of Caputo fractional Birkhoff interpolation. Zaky [85] and Zaky and Tenreiro Machado [93] proposed Legendre- and Jacobi-collocation schemes for solving distributed-order fractional optimal control problems in oneand two-dimensions, respectively. Cai and Chen [20] developed a fractional-order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. Taheri et al. [79] proposed the Legendre spectral collocation method to solve stochastic fractional integro-differential equations.

Spectral methods within fractional calculus | 213

Based on the above review, we note that Jacobi polynomials (typically Legendre or Chebyshev polynomials) can be used as an approximation basis in the spectral methods for fractional differential models when the solution is smooth enough, but as we already stated solutions of fractional differential models usually have a weak singularity at the boundary. As a result, the polynomial approximation exhibits a very limited order of convergence. This chapter aims at providing a family of orthogonal systems of fractional functions. The orthogonal systems are based on Jacobi polynomials through a fractional coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of fractional differential models. Moreover, we derived the fractional Jacobi–Gauss integration formula. As an example of applications, an efficient spectral tau approximation based on these functions to multi-term time-fractional diffusion equations is presented. This approximation takes into account the potential irregularity of the solution. We also include a discussion of the formulation of the spectral collocation method for solving the nonlinear variable-order fractional advection–diffusion equation. We refer to [47, 56, 66, 100] for the detailed description of the Galerkin and Petrov–Galerkin spectral methods.

2 Fractional Jacobi functions Whereas the classical orthogonal polynomials perform well for numerical solution of integer-order differential equations [34, 35, 39], their application to the FDEs implies at least two difficulties in connection with the spectral methods. First, the solutions of FDEs may contain some fractional power terms and the classical orthogonal polynomials cannot match with them completely. In this case, the rate of convergence for the numerical approximations will not be reasonable. Second, in the spectral methods, the derivative of the trial functions can be expressed according to the same trial bases. However, the fractional derivatives of classical orthogonal polynomials are not polynomials, so another non-polynomial approximation is needed to present them in terms of trial functions [11, 54]. Consequently, we have to consider suitable basis functions in the proposed spectral methods. 1 For given λ ∈ (0, 1], let x = ( 1+y ) λ , which maps y ∈ (−1, 1) into x ∈ Λ ≡ (0, 1). Then 2 y = 2xλ − 1,

1 + y = 2xλ ,

1 − y = 2 − 2xλ ,

(2.1)

and dy 1+y = 2λxλ−1 = 2λ( ) dx 2

λ−1 λ

,

1 1 2 dx = x1−λ = ( ) dy 2λ 2λ 1 + y

Definition 2.1. The set of fractional Jacobi functions Pn

(α,β,λ)

n

(α,β,n) λk

Pn(α,β,λ) (x) = Pn(α,β) (2xλ − 1) = ∑ Ek k=0

x ,

λ−1 λ

.

(2.2)

(x); α, β > −1, is defined by

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

(2.3)

214 | E. H. Doha et al. where Pn

(α,β)

(y) is the classical Jacobi polynomial of order n and Ek

(α,β,n)

=

(−1)n−k Γ(1 + n + β)Γ(k + α + n + β + 1) . Γ(k + β + 1)k!Γ(n + α + β + 1)(n − k)!

(2.4)

The set of fractional Jacobi functions is the nth eigenfunction of the singular Sturm–Liouville problem d d −1 α α+1 (λ (1 − xλ ) xλβ+1 v(x)) + λγn(α,β) (1 − xλ ) xλβ+λ−1 v(x) = 0, dx dx

x ∈ Λ,

(2.5)

with the corresponding eigenvalue γn = γn = n(n + α + β + 1). It is worth recalling important special cases of the fractional Jacobi functions, namely: – Müntz–Legendre polynomials [43]: (α,β)

,λ) (0, 1−λ λ

L(λ) i (x) = Pi –

(x),

fractional Legendre functions [55]: Pi(λ) (x) = Pi(0,0,λ) (x).

Let χF (x) = λx λβ+λ−1 (1 − xλ )α . Then the fractional Jacobi functions form a complete 2 L (α,β,λ) (Λ) orthogonal system, that is, (α,β,λ)

χF

∫ Pi

(α,β,λ)

Λ

(α,β,λ)

(x)Pj

(α,β,λ)

(x)χF

(x) dx = hj

(α,β)

δij ,

i, j ≥ 0,

(2.6)

j ≥ 0.

(2.7)

where hj = hj

(α,β)

=

Γ(j + α + 1)Γ(j + 1 + β) , (2j + 1 + α + β)j!Γ(j + 1 + α + β) 1

) λ ). Then, by (2.1) and (2.2), we have For any v ∈ L2 (α,β,λ) (Λ), we set u(y) = v(( 1+y 2 χF

∫ u2 (y)χ (α,β) (y) dy = 2α+β+1 ∫ v2 (x)χF

(α,β,λ)

(x) dx < ∞.

Λ

I

Therefore, u ∈ L2χ(α,β) (−1, 1). According to the completeness of the set of Jacobi polynomials, we have ∞

u(y) = ∑ aj Pj j=0

(α,β)

(y).

Spectral methods within fractional calculus | 215

By the definition of Pn

(α,β,λ)

(x), we obtain ∞

u(x) = ∑ aj Pj

(α,β,λ)

j=0

This implies the completeness of the set of Pj

(α,β,λ)

(x).

(x) in L2 (α,β,λ) (Λ).

χF (α,β) (α,β) N {xZ,N,j , ϖZ,N,j }j=0 ,

For a given positive integer N. Suppose with Z = G, R, L the Jacobi–Gauss, Jacobi–Gauss–Radau and Jacobi–Gauss–Lobatto quadrature, are nodes and weights, respectively. Let 𝒫N denote the space of all polynomials of degree not exceeding N. Then, for any u, v ∈ C[−1, 1], the discrete inner product is defined by N

(u, v)N,χ(α,β) = ∑ u(xZ,N,j )v(xZ,N,j )ϖZ,N,j . (α,β)

(α,β)

(α,β)

j=0

It is straightforward to derive the fractional Jacobi–Gauss-type integration formulas, that is, 1

∫ ϕ(x)χF

(α,β,λ)

N

dx = ∑ ϕ(xZ,N,j )ϖZ,N,j + EN [ϕ], (α,β,λ)

(2.8)

(α,β,λ)

j=0

0

where {xZ,N,j , ϖZ,N,j }Nj=0 , with Z = G, R, L the fractional Jacobi–Gauss, the fractional Jacobi–Gauss–Radau and the fractional Jacobi–Gauss–Lobatto quadrature, are nodes and weights, respectively, and EN [ϕ] is the quadrature error. If EN [ϕ] = 0, we say (2.8) is exact for ϕ. (α,β,λ)

(α,β,λ)

Theorem 2.1 (Fractional Jacobi–Gauss quadrature). Let {xG,N,j , ϖG,N,j }Nj=0 be the Jac(α,β)

(α,β)

obi–Gauss quadrature nodes and weights, {xG,N,j }Nj=0 be the zeros of PN+1 (x) and (α,β,λ)

(α,β,λ)

ϖG,N,j =

GN

(α,β)

(α,β,λ)

=

PN

(α,β)

(xG,N,j )𝜕x PN+1 (xG,N,j ) (α,β)

(α,β)

(α,β)

(α,β) Ĝ N

(1 − (xG,N,j )2 )[𝜕x PN+1 (xG,N,j )]2 (α,β)

(α,β)

(α,β)

,

(2.9)

where GN

(α,β)

(α,β) Ĝ N

Γ(N + α + 1)(2N + α + β + 2)Γ(N + β + 1) , 2(N + 1)!Γ(N + α + β + 2) Γ(N + β + 2)Γ(N + α + 2) = . Γ(N + α + β + 2)(N + 1)!

=

Then the quadrature formula (2.8) is exact for any ϕ(x) ∈ ℱ2N+1 , where (α,β,λ)

(υ,ϑ,λ)

ℱN

= span{Pn(υ,ϑ,λ) : 0 ≤ n ≤ N}.

(2.10)

216 | E. H. Doha et al. 1

Proof. As pointed out in [78, 91], for any ϕ(x) ∈ ℱ2N+1 , we have ϕ(( x+1 ) λ ) ∈ 𝒫2N+1 , 2 (α,β,λ)

1 (α,β,λ) (x) dx ∫ ϕ(x)χF 0

α+β+1 1

1

x + 1 λ (α,β) ) )χ (x) dx ∫ ϕ(( 2

1 =( ) 2

−1

α+β+1 N

1 =( ) 2 N

xG,N,j + 1

1 λ

(α,β)

∑ ϖG,N,j ϕ(( (α,β)

2

j=0

) )

= ∑ ϖG,N,j ϕ(xG,N,j ), (α,β,λ)

(2.11)

(α,β,λ)

j=0

x

(α,β)

+1

1

) λ are zeros of PN+1 (x), 0 ≤ j ≤ N. They are arranged in dewhere xG,N,j = ( G,N,j 2 creasing order and the corresponding Christoffel numbers are given by (α,β,λ)

(α,β,λ)

(α,β,λ) ϖG,N,j

=

1

2−α−β−1

PN+1 (x) (α,β)

χ (α,β) (x) dx ∫ (α,β) (α,β) (α,β) 𝜕x PN+1 (xG,N,j ) −1 x − xG,N,j GN

(α,β)

= =

PN

(α,β)

(xG,N,j )𝜕x PN+1 (xG,N,j ) (α,β)

(α,β)

(α,β)

(α,β) Ĝ N

(1 − (xG,N,j )2 )[𝜕x PN+1 (xG,N,j )]2 (α,β)

(α,β)

(α,β)

(2.12)

.

The fractional Jacobi–Gauss–Radau and the fractional Jacobi–Gauss–Lobatto quadrature formulas can be given in the same way as in Theorem 2.1. For bounding some approximation errors of the fractional Jacobi functions, we need the following nonuniformly weighted Sobolev space: Hχm(α,β) ,∗ (−1, 1) := {v : 𝜕xk v ∈ L2χ(α+k,β+k) (−1, 1), 0 ≤ k ≤ m}, equipped with the inner product and the norm as m

(u, v)m,χ(α,β) ,∗ = ∑ (𝜕xk u, 𝜕xk v)χ(α+k,β+k) , k=0

1

2 ‖u‖m,χ(α,β) ,∗ = (u, u)m,χ (α,β) ,∗ .

1

) λ ) ∈ Hχm(α,β) ,∗ (−1, 1) for some m ≥ 1 and ϕ(x) = Theorem 2.2 (See [91]). Let u(x) = v(( 1+x 2 1 λ

φ(( 1+x ) ) ∈ ℱN . Then, for the fractional Jacobi–Gauss and the fractional Jacobi– 2 Gauss–Radau integration, we have (α,β,λ)

󵄨󵄨 󵄨 −m 󵄩 m 󵄩 󵄨󵄨(v, φ)χ(α,β,λ) − (v, φ)N,χ(α,β,λ) 󵄨󵄨󵄨 ≤ CN 󵄩󵄩󵄩𝜕x u󵄩󵄩󵄩χ(α+m,β+m) ‖ϕ‖χ(α,β) ,

(2.13)

and for the fractional Jacobi–Gauss–Lobatto integration, we have 󵄨 󵄨󵄨 −m 󵄩 m 󵄩 󵄨󵄨(v, φ)χ(α,β,λ) − (v, φ)N,χ(α,β,λ) 󵄨󵄨󵄨 ≤ CN 󵄩󵄩󵄩𝜕x u󵄩󵄩󵄩χ(α+m−1,β+m−1) ‖ϕ‖χ(α,β) , where C is a positive constant.

(2.14)

Spectral methods within fractional calculus | 217

3 Fractional tau method By the attempts to describe some real processes with the equations of the fractional order, several researches addressed the situation that the order of the fractional derivative from the corresponding model equations did not remain constant and changed say, in the interval from 0 to 1, from 1 to 2 or even from 0 to 2. To manage these phenomena, several approaches were suggested. To manage this phenomenon, Lorenzo and Hartley [62] developed the conception of variable- and distributed-order fractional operators. These operators open up new possibilities for robust mathematical modeling and simulation of diverse physical problems in science and engineering [10, 28]. The mathematical analysis of these operators was investigated by several researchers; see, e. g., [25]. Multi-term time-fractional diffusion equations can be considered as a particular case of the distributed-order time-fractional diffusion equation whose weight function is chosen as a linear combination of Dirac δ-functions. The multi-term time-fractional diffusion model is given by C

ℒμ1 ,...,μn ,μ (0 Dt )f (x, t) − 𝜕xx f (x, t) = q(x, t),

f (0, t) = g(t),

f (1, t) = h(t),

f (x, 0) = y(x),

(x, t) ∈ Λ × (0, 1], t ∈ (0, 1], x ∈ Λ,

(3.1)

where the differential operator ℒμ1 ,...,μn ,μ (C0 Dt ) is defined by C

n

μ

C

μ

ℒμ1 ,...,μn ,μ (𝜕t ) = 0 Dt + ∑ εr 0 Dt r , r=1

where 0 < μn ≤ ⋅ ⋅ ⋅ ≤ μ1 < μ < 1, εr ≥ 0, r = 1, 2, . . . , n. The fractional-order derivatives appearing in (3.1) is defined in the Caputo sense. Unlike Riemann–Liouville derivatives, Caputo derivatives are not singular on the domain boundaries. That feature makes them particularly appealing for non-local numerical methods, like the spectral methods, as most basis functions take a nonzero value on the boundary. This model was proposed to improve the modeling accuracy of the single-term model for describing anomalous diffusion. It is not only a useful tool for modeling the behavior of viscoelastic fluid and rheological material, but also often appears while discretizing the distributed-order derivative in approximating the distributed-order differential equations. Hence, studies on the multi-term time-fractional differential equations have become important and useful for different applications. In this section, we derive a space–time discretization of (3.1) based on the tau method with fractional Jacobi expansions in time and Jacobi expansions space. The integrated form of (3.1) is given by n

μ−μr

f (x, t) + ∑ εr 0 It r=1

2 μ 𝜕 f (x, t) 𝜕x 2

f (x, t) = 0 It

̃ t), + q(x,

(3.2)

218 | E. H. Doha et al. with the boundary conditions f (0, t) = g(t),

f (1, t) = h(t),

μ−μr

(3.3)

μ

t ̃ t) = (1 + ∑nr=1 εr Γ(μ−μ where q(x, )y(x) + 0 It q(x, t). The model solution is expressed in r +1) terms of a matrix of unknown coefficients Fij as follows: M N

f (x, t) ≃ fM,N (x, t) = ∑ ∑ Pi

(α,β,λ)

i=0 j=0

(t)Fij Pj

(α,β,1)

(x) = ΨTλ,M (t)FΨ1,N (x),

(3.4)

T

(3.5)

where Ψλ,M (t) = [P0

(α,β,λ)

(t), P1

(α,β,λ)

(t), . . . , PM

(α,β,λ)

(t)] .

̃ t) can be discretized as The boundary conditions (3.3) and the reaction function q(x, f (0, t) ≃ ΨTλ,M (t)GΨ1,N (0), f (1, t) ≃ ΨTλ,M (t)HΨ1,N (1),

̃ ̃ t) ≃ ΨTλ,M (t)QΨ q(x, 1,N (x),

(3.6)

where the entries of the matrices Q,̃ G and H are defined as 1 1

1 ̃ t)Pi(α,β,λ) (t)χF(α,β,λ) (t)Pj(α,β,1) (x)χF(α,β,1) (x) dx dt, Q̃ ij = ∫ ∫ q(x, hi hj 0 0

Gij = Hij =

δj,0 hi

δj,0 hi

1

∫ g(t)Pi

(α,β,λ)

0

(α,β,λ)

(t) dt,

(α,β,λ)

(t) dt.

(t)χF

1

∫ h(t)Pi

(α,β,λ)

0

(t)χF

(3.7)

All these integrals can be computed to essentially machine precision with the fractional Jacobi–Gauss quadrature: Q̃ ij =

1 M N (α,β,1) (α,β,λ) (α,β,λ) (α,β,λ) (α,β,λ) ̃ G,N,δ (tG,N,ϵ ) , tG,N,ϵ )ϖG,N,ϵ Pi ∑ ∑ q(x hi hj ϵ=0 δ=0 × ϖG,N,δ Pj

(α,β,1) (α,β,1)

Gij = Hij =

δj,0 hi

δj,0 hi

M

(α,β,1)

(xG,N,δ ),

∑ g(tG,N,ϵ )ϖG,N,ϵ Pi (α,β,λ)

(α,β,λ) (α,β,λ)

ϵ

M

∑ h(tG,N,ϵ )ϖG,N,ϵ Pi ϵ

(α,β,λ)

(α,β,λ) (α,β,λ)

(α,β,λ)

(tG,N,ϵ ), (α,β,λ)

(tG,N,ϵ ).

(3.8)

Spectral methods within fractional calculus | 219

Lemma 3.1 (see [39]). The first derivative of the Jacobi vector Ψ1,N (x) is given by d Ψ (x) = D(1) Ψ1,N (x), dx 1,N

(3.9)

with D(1) is the (N + 1) × (N + 1) operational matrix of derivative with entries σ(i, j), j < i, D(1) ij = { 0, otherwise, where σ(i, j) =

(α + β + i + 2)j (α + β + i + 1)(j + α + 2)i−j−1 Γ(j + α + β + 1) (i − j − 1)! Γ(α + β + 2j + 1)

× 3 F2 (

j − i + 1, j + α + 2,

i + j + α + β + 2, j + α + 1 ; 1) . 2j + α + β + 2

(3.10)

By using (3.9), it is clear that dn n Ψ (x) = (D(1) ) Ψ1,N (x) = D(n) Ψ1,N (x), dxn 1,N

(3.11)

where n ∈ ℕ and the superscript in D(1) denotes matrix powers. Lemma 3.2 (see [12]). The fractional integral of the fractional Jacobi vector ΨTλ,N (x) is given by μ T 0 It Ψλ,N (t)

≃ ΨTλ,N (t)I(μ) ,

(3.12)

where I(μ) is the (N + 1) × (N + 1) operational matrix of fractional integration of order μ whose entries are given by j

(−1)j+k Γ(j + β + 1) Γ(α + 1) i! Γ(j + k + α + β + 1) Γ(k ν + 1) Γ(k + β + 1) Γ(j + β + α + 1) (j − k)! k! Γ(kν + μ + 1) k=0

Iij = ∑ (μ)

i

×∑

s=0

(−1)i+s Γ(i + s + β + α + 1) Γ(s + k + β +

μ λ

+ 1) (2i + α + β + 1)

Γ(i + α + 1) Γ(s + β + 1)(i − s)! s! Γ(s + k + β + α +

μ λ

+ 2)

.

(3.13)

Using Lemmas 3.1 and 3.2 yields 2 μ 𝜕 f (x, t) 𝜕x 2

0 It

μ 0 It f (x, t)

≃ ΨTλ,M (t)I(μ) FD(2) Ψ1,N (x), ≃ ΨTλ,M (t)I(μ) FΨ1,N (x).

(3.14)

Next, by substituting (3.4), (3.6) and (3.14) back into (3.2) and re-arranging the terms we obtain n

̃ ΨTλ,M (t)(F + ∑ εr I(μ−μr ) F − I(μ) FD(2) − Q)Ψ 1,N (x) = 0. r=1

(3.15)

220 | E. H. Doha et al. Using again the L2 (α,β,λ) orthogonality of the Jacobi polynomials, we obtain (M + 1) × χF

(N − 1) algebraic equations in the coefficients, Fij , 0 ≤ i ≤ M, 0 ≤ j ≤ N: n

̃ (N+1,N−1) = 0, I(M+1,M+1) (F + ∑ εr I(μ−μr ) F − I(μ) FD(2) − Q)I r=1

(3.16)

where the matrix I(M+1,M+1) is such that I(M+1,M+1) = δij , 0 ≤ i ≤ M, 0 ≤ j ≤ N. ij Also, we obtain (2M + 2) equations from the boundary conditions (3.3) I(M+1,M+1) (F − G)Ψ1,N (0) = 0, I(M+1,M+1) (F − H)Ψ1,N (1) = 0.

(3.17)

This produces a system of linear algebraic equations: ̃ (N+1,N−1) = 0, I(M+1,M+1) (F + ∑nr=1 εr I(μ−μr ) F − I(μ) FD(2) − Q)I { { { (M+1,M+1) I (F − G)Ψ1,N (0) = 0, { { { (M+1,M+1) (F − H)Ψ1,N (1) = 0. {I

(3.18)

Now, we present two numerical examples to demonstrate the high accuracy and efficiency of the proposed fractional tau method. The most important feature of this method is that the convergence of the numerical solution is exponential with respect to the degree of the fractional function if f (x, t 1/λ ) is smooth where f (x, t) is the exact solution of problem (3.1). The numerical examples that follow will aim at confirming this feature. Example 3.1. Consider the one-term time-fractional diffusion equation: C

λ

𝜕2 f (x,t)

D f (x, t) = 𝜕x2 + q(x, t), { { {0 t f (x, 0) = 0, { { { {f (0, t) = f (1, t) = 0,

(3.19)

where q(x, t) = sin(πx)Eλ,1 (t λ ) + π 2 sin(πx)(Eλ,1 (t λ ) − 1). The exact solution is f (x, t) = sin(πx)(Eλ,1 (t λ ) − 1). In this example, the exact solution has limited regularity in time direction at t = 0. In order to investigate the exponential convergence, we plot the L∞ -error in the semi-log scale. The results obtained by using the fractional tau method, with λ = 1/3, 1/2 and 3/4, respectively, are plotted in Figure 1, showing exponential decay of the errors with respect to N and M for all employed values of λ. These results are in good agreement with the theoretical prediction given in Theorem 2.2.

Spectral methods within fractional calculus | 221

Figure 1: The L∞ -errors versus N = M at α = β = 0.

Example 3.2. Consider the three-term time-fractional diffusion equation [102] μ

μ

μ

C D f (x, t) + C0 Dt 1 f (x, t) + C0 Dt 2 f (x, t) = { { {0 t f (x, 0) = 0, { { { {f (0, t) = f (1, t) = 0,

where q(x, t) = π 2 t 3 sin πx + +

𝜕2 f (x,t) 𝜕x 2

+ q(x, t),

(3.20)

6 6 t 3−μ sin πx + t 3−μ1 sin πx Γ(4 − μ) Γ(4 − μ1 )

6 t 3−μ2 sin πx. Γ(4 − μ2 )

It is easily verified that the exact solution is f (x, t) = t 3 sin πx. We investigate the convergence behavior as for Example 3.1. The computed results for μ = 0.9, μ1 = 0.5, μ2 = 0.1 and λ = 1/2, 3/4, 1 are shown in Figure 2. It is observed from this figure that all the error curves are straight lines in the semi-log representation, which indicates that only algebraic accuracy is obtained. The accuracy of the classical spectral method based on classical polynomial approximations, i. e., λ = 1 can be significantly improved by using the fractional spectral method introduced in this chapter.

4 Jacobi–Gauss–Lobatto collocation method This section discusses a polynomial spectral collocation method for the following nonlinear variable-order fractional advection–diffusion equation: 𝜕f 𝜕μ(t) f 𝜕f − ν(x, t) + q(f , x, t), = κ(x, t) 𝜕t 𝜕x 𝜕|x|μ(t)

(x, t) ∈ Λ × (0, 1],

(4.1)

222 | E. H. Doha et al.

Figure 2: The L∞ -errors versus N = M at μ = 0.9, μ1 = 0.5, μ2 = 0.1 and α = β = 1/2.

subject to the initial-boundary conditions f (x, 0) = ξ (x),

x ∈ Λ,

f (0, t) = f (1, t) = 0,

(4.2)

t ∈ (0, 1].

(4.3)

This model can accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. The proposed method is based on the Jacobi–Gauss–Lobatto collocation method for spatial discretization and the Legendre–Gauss–Radau collocation method for temporal discretization. We assume that N

f (x, t) = ∑ ϕj (t)Pj

(α,β,1)

j=0

(4.4)

(x),

where ϕj (t) are given by ϕj (t) =

1 (α,β,1) (α,β,1) (x)χF (x) dx ∫ f (x, t)Pj hj Λ

=

1 N (α,β,1) (α,β,1) (α,β,1) (α,β,1) (xL,N,i )ϖL,N,i f (xL,N,i , t). ∑P hj i=0 j

(4.5)

Hence, the approximate solution takes the form N

N

1 (α,β,1) (α,β,1) (α,β,1) (α,β,1) (α,β,1) Pj (xL,N,i )ϖL,N,i Pj (x))f (xL,N,i , t). h j j=0

f (x, t) = ∑ ( ∑ i=0

(4.6)

The discretization of the left-sided Riemann–Liouville variable-order fractional derivative of f (x, t) at the (N − 1) interior Jacobi–Gauss–Lobatto points becomes 󵄨󵄨 RL μ(t) 0 Dx f (x, t)󵄨󵄨x=x (α,β,1) L,N,j

N

μ(t)

= ∑ L Dij f (xL,N,i , t), i=0

(α,β,1)

(4.7)

Spectral methods within fractional calculus | 223

where the differentiation matrix L Dμ(t) of the left-sided Riemann–Liouville fractional derivative is given by μ(t)

L Dij

N

1 (α,β,1) (α,β,1) (α,β,1) (α,β,1) (α,β,1) (xL,N,i )ϖL,N,i PL ,k (xL,N,j , t) P h k k=0 k

= ∑

(4.8)

and k

r! Er xr−μ(t) . Γ(r − μ(t) + 1) r=0 (α,β,k)

PL ,k (x, t) = ∑ (α,β,1)

(4.9)

By following the same procedure, the right-sided Riemann–Liouville variable-order fractional derivative of f (x, t) can be approximated as follows: 󵄨󵄨 RL μ(t) x D1 f (x, t)󵄨󵄨x=x(α,β,1) L,N,j

where the differentiation matrix tional derivative is given by μ(t)

R Dij

RD

μ(t)

N

μ(t)

= ∑ R Dij f (xL,N,i , t), (α,β,1)

i=0

(4.10)

of the right-sided Riemann–Liouville frac-

N

1 (α,β,1) (α,β,1) (α,β,1) (α,β,1) (α,β,1) (xL,N,i )ϖL,N,i PR,k (xL,N,j , t) Pk h k=0 k

= ∑

(4.11)

and k

(−1)r−k r! Er (x − 1)r−μ(t) . Γ(r − μ(t) + 1) r=0

PR,k (x, t) = ∑ (α,β,1)

(β,α,k)

(4.12)

The Riesz fractional derivative of f (x, t) can be approximated as a combination of the left- and right-sided Riemann–Liouville fractional derivatives as follows: N 𝜕μ(t) f (x, t) 󵄨󵄨󵄨󵄨 (α,β,1) μ(t) = Dij f (xL,N,i , t), ∑ 󵄨 (α,β,1) 𝜕|x|μ(t) 󵄨󵄨󵄨x=xL,N,j i=0

(4.13)

where μ(t)

Dij

=−

1 2 cos(

μ(t) [ D πμ(t) L ij ) 2

μ(t)

+ R Dij ],

j = 1, 2, . . . , N − 1.

(4.14)

The first-order partial derivative with respect to x, at the specific collocation nodes (α,β,1) xL,N,i , can be derived from (4.7) by setting μ(t) = 1: N 𝜕f (α,β,1) = ∑ L Dij f (xL,N,i , t), 𝜕x i=0

(4.15)

224 | E. H. Doha et al. where N

1 (α,β,1) (α,β,1) (α,β,1) (α+1,β+1,1) (α,β,1) (xL,N,j ). (xL,N,i )ϖL,N,i Pk−1 Pk h k=1 k

L Dij = ∑

(4.16)

Making use of equations (4.4)–(4.16) and the two-point boundary conditions (4.3) enables us to generate a system of (N − 1) ODEs in time: N−1

N−1

μ(t)

fṅ (t) = κn (t) ∑ Din fi (t) − νn (t) ∑ i=1

fn (t) = ξn ,

i=1

L Din fi (t)

+ qn (fn (t), t),

n = 1, . . . , N − 1,

(4.17)

where fn (t) = f (xL,N,n , t), (α,β,1)

κn (t) = κ(xL,N,n , t), (α,β,1)

νn (t) = ν(xL,N,n , t), (α,β,1)

(4.18)

and qn (fn (t), t) = q(fn (t), xL,N,n , t).

(4.19)

(α,β,1)

This system may be written in the following matrix form: ̇ = Q(t, FT (t)), F(t) F(0) = f,

(4.20)

where T

F(t) = [f1 (t), f2 (t), . . . , fN−1 (t)] ,

T

f = [ξ (xL,N,1 ), fξ (xL,N,2 ), . . . , ξ (xL,N,N−1 )] , (α,β,1)

(α,β,1)

(α,β,1)

T

Q(t, FT (t)) = [Q1 (t, FT (t)), Q2 , (t, FT (t)), . . . , QN−1 (t, FT (t))] ,

(4.21)

and N−1

N−1

μ(t)

Qn (t, FT (t)) = κn (t) ∑ Din fi (t) − νn (t) ∑ i=1

i=1

L Din fi (t)

+ qn (fn (t), t).

(4.22)

The system of ordinary differential equations (4.20) in time may be solved using the Legendre–Gauss–Radau collocation method [81]. We are going to see below that the Jacobi–Gauss–Lobatto collocation method proposed here is well adapted to such problems when the solution is smooth enough. Example 4.1. Consider the following variable-order fractional advection–diffusion equation with a Riesz fractional derivative: 𝜕μ(t) f (x, t) 𝜕f (x, t) 𝜕f (x, t) = κ(x, t) + q(f , x, t), − ν(x, t) 𝜕t 𝜕x 𝜕|x|μ(t) f (x, 0) = x2 (1 − x)2 , f (0, t) = 0,

x ∈ Λ,

f (1, t) = 0,

t ∈ (0, 1],

(4.23)

Spectral methods within fractional calculus | 225

We choose

π μ(t)+2 κ(x, t) = −2 cos( μ(t))Γ(5 − μ(t))(x − x2 ) , 2

ν(x, t) = et x3 (1 − x)3 ,

and q(f , x, t) = (1 − ρ(x, t) − σ(x, t))f + 2(1 − 2x)f 2 , where ρ(x, t) = 2x 2 (1 − x)2 + μ(t)[12x2 + (4 − μ(t))(−6x + 3 − μ(t))], σ(x, t) = 2(1 − x)2 x2 + μ(t)[12(1 − x)2 + (4 − μ(t))(6x − 3 − μ(t))]. This problem has the exact solution f (x, t) = et x2 (1 − x)2 . In Figures 3 and 4, we show the L∞ -errors of the Jacobi–Gauss–Lobatto collocation method for μ(t) = 1.8 and μ(t) = 21 (tanh(t) + 3), respectively. The numerical results demonstrate that the method converges rapidly, irrespective of α and β. They also indicate clearly the efficiency and accuracy of the suggested method for smooth solutions.

Figure 3: The L∞ -errors versus N at μ = 1.8.

Figure 4: The L∞ -errors versus N at μ(t) = 21 (tanh(t) + 3).

226 | E. H. Doha et al.

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Liping Chen and Ranchao Wu

Design and generation of fractional-order multi-scroll chaotic attractors Abstract: Generation and circuit implementation of three kinds of fractional-order (FO) multi-scroll (MS) attractors, one dimensional FOMS attractors, one dimensional hyper-FOMS attractors and multi-directional multi-scroll (MDMS) attractors, are discussed in the chapter. By extending the number of saddle equilibrium points with index 2, hysteresis series, stair nonlinear function series (SNFS) and a saturated nonlinear functions (SNLF) are used to generate FOMS attractors based on three FO linear linear autonomous system. Phase portraits, Poincaré map and maximum Lyapunov exponents are applied to verifying the chaotic behaviors of the generated multi-scroll chaotic attractors. Circuits for the these multi-scroll attractor are designed, which are in accordance with the numerical simulation. Keywords: Fractional order, multi-scroll, circuit implementation PACS: 05.45.Ac, 05.45.Gg, 02.30.Hq

1 Introduction This chapter focuses on the design and implementation of several fractional-order (FO) multi-scroll (MS) chaotic systems. In recent years, research on the frontier issues such as chaos and fractional calculus have shifted from pure mathematical theory to engineering applications. FOMS chaotic system has the characteristics of a simple structure, complicated dynamic, being of multi-scroll character and having high sensitivity. Also it has the advantages of having high space, high dynamic storage capacity, and anti-decipher and antiinterference properties when used as information encryption key. The FOMS chaotic system may have wide application in the field of information security. The contributions of this chapter are threefold. First, by employing a piecewiselinear function (PWL) to a linear autonomous system, one-dimensional FOMS attractors are generated. The chaotic behavior of the constructed system is confirmed through numerical tools such as largest Lyapunov exponent (LE) and Poincaré maps. The simulation result and physical experimental result agree, confirming the efficacy of our approach. Second, motivated by the existing work, we attempt to generate Liping Chen, School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, 230009, China, e-mail: [email protected] Ranchao Wu, School of Mathematics, Anhui University, Hefei, 230039, China, e-mail: [email protected] https://doi.org/10.1515/9783110571929-009

234 | L. Chen and R. Wu FO hyper-chaotic attractors based on a four-dimensional linear system via a hysteresis series. As is well known, there are essential differences between hyper-chaotic behavior and chaotic behavior, since a hyper-chaotic system possesses at least two positive Lyapunov exponents, while chaos has only one. With more than one positive Lyapunov exponent, a hyper-chaotic system diverges in more directions and hence has richer and more sophisticated dynamics. Therefore the generation of a hyper-chaotic system is much more significant. Third, two different kinds of PWL, namely, a stair nonlinear function series (SNFS) and saturated nonlinear functions (SNLFs), are combined to obtain multi-directional multi-scroll (MDMS) attractors in a three-dimensional FO system. It is worth mentioning that the SNFS and the SNLF are different kinds of functions with different forming mechanisms, which is quite different from those systems in [1, 4, 5, 12, 13]. The chaos generation mechanism of the system is elaborated through equilibrium stability theory, using the phase diagram, by Poincaré maps and by bifurcation diagrams. A circuit simulation is also carried out to validate the feasibility of our work.

2 Preliminary Fractional calculus is a generalization of integration and differentiation to noninteger-order fundamental operators. There are many different kinds of definitions for fractional derivatives, but the ones most commonly used are the Riemann–Liouville definition and the Caputo definition. Since the Caputo definition has a wider range of application in engineering, the Caputo fractional derivative operator Dα is adopted here. Definition 1 ([8]). The Caputo derivative of FO α of the function x(t) is defined as follows: α

t

D x(t) = D

−(n−α)

dn 1 x(t) = ∫(t − τ)n−α−1 x(n) (τ)dτ, dt n Γ(n − α) t0

where n − 1 ≤ α < n ∈ Z + . The stability of a fractional linear system is quite different from the integer case. In order to illustrate our works briefly, the following definitions and lemmas are presented firstly. Definition 2 ([4]). Consider a general n-dimensional fractional autonomous system Dα (X) = f (X);

(1)

the roots of the equation f (X) = 0 are called the equilibrium points of fractional differential system, where Dα (X) = (Dα x1 , Dα x2 , . . . , Dα xn )T , X = (x1 , x2 , . . . , xn )T ∈ Rn .

Design and generation of fractional-order multi-scroll chaotic attractors | 235

Lemma 1 ([8]). System (1) is asymptotically stable at the equilibrium point O if | arg(λi (J))| > απ/2, i = 1, 2, . . . , n. J denotes the Jacobi matrix of f (X), the λi are the eigenvalues of J. Lemma 2 ([3]). The equilibrium point O of system (1) is unstable if the order α satisfies the following condition for at least one eigenvalue: α>

2 | Im(λ)| arctan , π | Re(λ)|

where Im(λ) and Re(λ) denote the real and imaginary part of λ, respectively. Lemma 3 ([5]). For n = 3, if one of the eigenvalues λ1 < 0 and for the other two conjugate eigenvalues | arg(λ2 )| = | arg(λ2 )| < απ/2, then the equilibrium point O is called a saddle point with index 2; if one of the eigenvalues λ1 > 0 and the other two conjugate eigenvalues | arg(λ2 )| = | arg(λ3 )| > απ/2, then the equilibrium point O is called a saddle point with index 1. Lemma 4 ([7]). For n > 3, suppose there is one saddle point O with index 2. That is, the Jacobi matrix at O has two real eigenvalues γ1 and γ2 , and a pair of conjugate complex eigenvalues δ ± jω satisfying at least one of the conditions γi < 0, i = 1, 2, δ > 0, ω > 0 and | arctan(ω/δ)| < απ/2, where α is the order, then for each equilibrium point with index 2 a scroll emerges.

3 Generation and circuit implementation of one-directional FOMS chaotic attractors 3.1 Model description and dynamic analysis In this section, SNFS is introduced to a three-dimensional FO system to generate MS chaotic attractors. In 2000, Sprott proposed the following integer-order jerk system [10]: ẋ = y, { { { ẏ = z, { { { {ż = −x − y − βz + G(x),

(2)

where x, y, z are dimensionless variable, β is a suitable constant, G(x) is one of a number of elementary piecewise linear functions. When β = 0.5 and G(x) = sgn(x), the characteristic roots of the Jacobian matrix at equilibrium point (0, 0, 0) are −0.8038, 0.1519+1.1050i and 0.1519−1.1050i, so (0, 0, 0) is a saddle-focus point of index 2. When the initial conditions are chosen as (0, 1, 0), the Lyapunov exponents of system (2) are 0.152, 0 and −0.652 [11]. Chaotic attractors of 2-scroll type are displayed in Figure 1.

236 | L. Chen and R. Wu

Figure 1: Chaotic behaviors of system (2) (a) x–y, (b) x–z.

The following FO system is derived from system (2) with 0 < α < 1: Dα x = y, { { { α D y = z, { { { α D { z = −x − y − βz + F(x).

(3)

To guide system (3) to generating chaotic behavior, the function F(x) should be a nonlinear function. On the other hand, to generate an MS attractor, one should focus on extending the equilibrium points of system (3). Let the left side of system (3) equal 0; it is not hard to see that the equilibrium points of system (3) are (xi , 0, 0), where xi are the roots of the equation F(x) = x, i = 1, 2, . . . , n. The nonlinearity F(x) is modified by adding time-shifted versions of the signum function. Thus we form an SNFS as follows: N

F(x) = A2 sgn(x) + A2 ∑ [sgn(x − 2nA1 ) + 1] n=1

M

+ A2 ∑ [sgn(x + 2mA1 ) − 1] m=1

N

= (N − M)A2 + A2 sgn(x) + A2 ∑ sgn(x − 2nA1 ) n=1

M

+ A2 ∑ sgn(x + 2mA1 ),

(4)

m=1

where N, M = 1, 2, 3, . . . . If N = M and A1 = A2 = A, then N

N

n=1

m=1

F(x) = A sgn(x) + A ∑ sgn(x − 2nA) + A ∑ sgn(x + 2mA).

(5)

Design and generation of fractional-order multi-scroll chaotic attractors | 237

Figure 2: Distribution of equilibrium points with parameter A, (a) Even number case, (b) odd number case.

An even number of equilibrium points are formed utilizing function F(x) above in (5). As well, an odd number of equilibrium points can be obtained by recasting the function (4) as N

N

n=1

m=1

F(x) = A ∑ sgn[x − (2n − 1)A] + A ∑ sgn[x + (2m − 1)A].

(6)

When F(x) in (3) is taken as (5), the equilibrium points of system (3) are (xi , 0, 0) = (±(2i − 1)A, 0, 0), where i = 1, 2, . . . , (2N + 2), N = 0, 1, 2, . . . , so (2N + 2) attractors are generated. If F(x) in (4) is chosen as (6), the equilibrium points of system (3) are (xi , 0, 0) = (±2iA, 0, 0), where i = 1, 2, . . . , (2N + 1), N = 1, 2, 3, . . . , thus (2N + 1) attractors are formed. By changing the value of the parameter A, we could control the distance of two attractors. The distribution of equilibrium points in the constructed system is depicted in Figure 2. Now the stability of these equilibrium points will be discussed. Obviously, the Jacobi matrices of system (3) at all equilibrium points (xi , 0, 0) are identical, i. e., 0 J = (0 −1

1 0 −1

0 1 ), −β

(7)

λ3 + βλ2 + λ + 1 = 0.

(8)

the characteristic equation of (7) is given by

According to Lemma 3, the stability of the equilibrium of (3) is determined by the eigen2 3 1 2 31 values of (8). Denote a = 27 β − 31 β + 1, δ = 271 β3 − 108 β − 61 β + 108 , and one has β a a λ1 = − + √3 − + √δ + √3 − − √δ 3 2 2

238 | L. Chen and R. Wu and β 1 a a λ2,3 = − − (√3 − + √δ + √3 − − √δ) 3 2 2 2 √3 3 a a ± i(√− + √δ − √3 − − √δ) 2 2 2 = η ± γi, where η = Re(λ2,3 ), γ = Im(λ2,3 ). In system (3), every equilibrium point (xi , 0, 0) has the same eigenvalue, i. e., all equilibrium points (xi , 0, 0) are saddle-focus points of index 2. It is obvious that the number of the saddle-focus equilibrium point and the number of scrolls are equal. By employing the above method, any odd or even number of equilibrium points can be generated, i. e., any number of FOMS chaotic attractors can be created based on (3). When β = 0.3, α = 0.9, then λ1 = −0.7486, γ = 1.1338, η = 0.2243, | arctan(γ/η)| = 0.4378π < 0.45π = απ/2. It follows from Lemma 3 that every equilibrium point is the saddle-focus point of index 2. Figure 3(a) shows the maximum Lyapunov exponent with respect to the breaking parameter β of system (3). Figure 3(b) shows the Poincaré mapping of a 3-scroll attractor at the section y = 0. When β = 0.3, the largest Lyapunov exponent LE = 0.0312. By using an Adams–Bashforth–Moulton predictor–corrector algorithm [3], 2-, 3-, 4-, 5-, 6- and 7-scroll attractors are generated, and 7-scroll attractors are depicted in Figure 4. It should be pointed out that β is not unique, but selecting proper values can make the attractors clear and distinct. Parameters can be adjusted in a certain range in order to modify the size, shape and location of the attractors. Theoretically, a stair function series can be adopted to generate any number of attractors in system (3). However,

Figure 3: The maximum Lyapunov exponent with respect to the parameter β in 4(a), Poincaré mapping of 3-scroll attractor at the section y = 0 (b).

Design and generation of fractional-order multi-scroll chaotic attractors | 239

Figure 4: Attractors of 7-scroll type on the x–y plane obtained from a numerical simulation.

increasing the number of attractors leads to a higher computational load and to numerical errors. Therefore, the theoretical analysis will not agree with the simulation results and generating distinct attractors will be more difficult.

3.2 Circuit design and experimental result The circuit implementation is somewhat different from the numerical simulation due to hardware limitations. Since circuits cannot keep such a high operational precision as computers, some distortion will be introduced and the input signal cannot be well tracked under the condition of large input signals. Because chaotic systems are very sensitive to the initial values and the parameters of the system, it is difficult to control the value of the resistors and the capacitors very precisely in the circuit implementation. For an FO circuit, there is not a device to implement the FO calculations directly; we adopt a number of resistors and capacitors connecting in parallel, or in series, to approximate the transfer function of 1/sα (where α is the order), which often causes an increasing error. For convenience, we only discuss the case of α = 0.9, α = 0.8 and an even lower order circuit can be designed in a similar way. The operational amplifier in Figure 5(a) is the TL082. Moreover, all original devices in our circuit diagrams below are operational amplifiers of type TL082 with voltage supply ±15 V and saturated output voltage about ±13.5 V.

Figure 5: The circuit unit of the FO with order α = 0.9.

240 | L. Chen and R. Wu According to references [2], the linear transfer function approximation of fractional integrator of order 0.9 with maximum discrepancy y = 2 dB is 1

s0.9



2.2675(s + 1.292)(s + 215.4) . (s + 0.01292)(s + 2.154)(s + 359.4)

(9)

The circuit diagram shown in Figure 5 is utilized to implement the function in the Laplace domain. Then the transfer function H(s) between a and b in Figure 5 is H(s) =

C/C3 C/C2 C/C1 + + s + 1/R1 C1 s + 1/R2 C2 s + 1/R3 C3 C

0 1 ( C1 + = C0

C0 C2

+

C0 )[s2 C3

+

((C2 +C3 )/R1 +(C2 +C3 )/R2 +(C1 +C2 )/R3 )s+ C1 C2 +C2 C3 +C1 C3

(s + 1/R1 C1 )(s + 1/R2 C2 )(s + 1/R3 C3 )

R1 +R2 +R3 R1 R2 R3

]

,

(10)

1 . Comparing where C0 is a unit parameter. Let C0 = 1 μF, H(s) = F(s)/C0 and F(s) = s0.9 (9) with (10), the resistance and capacitance for the chain circuit unit shown in Figure 5 are R1 = 62.84 MΩ, C1 = 1.232 μF, R2 = 250 kΩ, C2 = 1.835 μF, R3 = 2.5 kΩ, C3 = 1.1 μF. Making A = 0.5 in the function (5) and A = 1 in the function (6), then the circuit units to realize equation (5) and (6) are designed as follows, respectively. In Figure 6(a), R4 = R5 = R6 = R7 = R8 = R9 = 13.5 kΩ, R10 = 1 kΩ, R11 = R12 = 10 kΩ. In Figure 6(b), R4 = R5 = R6 = R7 = R8 = 13.5 kΩ, R9 = 1 kΩ, R10 = 10 kΩ, R11 = 5 kΩ. The circuit shown in Figure 6(a) can generate an even number of scrolls, 2, 4, 6 scrolls and so on. One deals with it by turning on or turning off the switch S1 and S2 , respectively. An odd number of scrolls can be generated by the circuit diagram described in Figure 6(b). For simplicity, a circuit of generating 7-scroll attractors is presented as an instance to verify the efficacy of the design. In Figure 7, all values of resistance and capacitance are calculated and the β = 100 kΩ/333 kΩ = 0.3 simulation result is shown in Figure 8. In addition, the 3- and 5-scroll chaotic attractors are experimentally confirmed via the circuit design and oscilloscope observations, which are shown in Figure 9.

Figure 6: Odd (a) and even (b) number scrolls generated by circuit unit.

Design and generation of fractional-order multi-scroll chaotic attractors | 241

Figure 7: Complete circuit for creating 7-scroll attractors.

4 Design and implementation of FO hyper-chaotic (H) MS systems based on hysteresis series In this section, a FOHMS system is constructed from a four-dimensional FO linear system via a hysteresis function series.

242 | L. Chen and R. Wu

Figure 8: The circuit simulation result of MS attractors, 7-scroll attractors.

Figure 9: Hardware circuit implementation; (a) 3-scroll case, (b) 5-scroll case.

4.1 The four-dimensional FOHMS system Consider the following four-dimensional linear autonomous system: Dα x = y, { { { { { {Dα y = z, { { {Dα z = −ax − by − cz − dw, { { { α {D w = −y − w,

(11)

where 0 < α < 1 is order, a, b, c and d are real constants. System (11) has only one equilibrium point (0, 0, 0, 0). Since there is no nonlinear term on the right-hand side of the equations, system (11) cannot behave chaotically. Here, to generate FOHMS attractors, the basic hysteresis function 0 if x − mA < A, h(x − mA) = { A if x − mA > 0,

(12)

is used. Let A = 1, m = 0 in equation (12); if x reaches the threshold value 0 from the right side, then h(x) switches from 1 to 0. If x reaches the threshold value 1 from the

Design and generation of fractional-order multi-scroll chaotic attractors | 243

Figure 10: Mechanism of basic hysteresis function.

Figure 11: Phase portraits of (a) double-scroll attractors with k = 0 on x–y; (b) double-scroll attractors with k = 0 on x–w.

left side, then h(x) switches from 0 to 1. Figure 10 shows the work mechanism of the hysteresis function. Recasting system (11) by including the hysteresis function yields Dα x = y, { { { { { {Dα y = z, { {Dα z = −a[x − h(x)] − by − cz − dw, { { { { α {D w = −y − w + k ⋅ h(x),

(13)

where the parameter k is 0 or 1. When a = 5, b = 1, c = 3, d = 1, and k = 0, the two largest Lyapunov exponents of system (13) are λmax1 = 0.1051 and λmax2 = 0.0932, which confirms that (13) is a H system. The corresponding phase portraits are shown in Figure 11. When a = 5, b = 1, c = 3, d = 1, and k = 1, the two largest Lyapunov exponents of (13) are λmax1 = 0.1051 and λmax2 = 0.0932, which verify that system (13) is hyperchaotic. The phase portrait is shown in Figure 12. From Figure 11 and Figure 12, it is verified that a single hysteresis function can generate H double-scroll attractors in system (11). Stimulated by this result, several hysteresis functions are added to system (11) to constructed FOHMS attractors. Using

244 | L. Chen and R. Wu

Figure 12: Phase portraits of (a) double-scroll attractors with k = 1 on x–y; (b) double-scroll attractors with k = 1 on x–w.

the hysteresis function series p

q

i=1

i=1

h(x, p, q) = ∑ h−i (x) + ∑ hi (x),

(14)

where p and q are positive integer constants, hi (x) = h(x − i + 1) and h−i (x) = −hi (x). Inserting the hysteresis function series (14) into system (13) yields Dα x = y, { { { { { {Dα y = z, { { Dα z = −a[x − h(x, p, q)] − by − cz − dw, { { { { α {D w = −y − w + k ⋅ h(x, p, q).

(15)

System (15) will generate FOHMS attractors if the parameters a, b, c, and d are chosen adequately. Further discussion will be presented in Subsection 4.2.

4.2 Dynamic analysis and numerical simulation Let the right side of system (15) be zero. Hence, y = 0, { { { { { {z = 0, { { {−a[x − h(x, p, q)] − by − cz − dw = 0, { { { {−y − w + k ⋅ h(x, p, q) = 0.

(16)

When k = 0, the symbol xi∗ denotes the roots of h(x, p, q) − x = 0. Therefore, the equilibrium points of system (15) are (xi∗ , 0, 0, 0). The location of (xi∗ , h(xi∗ , p, q)) (marked

Design and generation of fractional-order multi-scroll chaotic attractors | 245

Figure 13: The distribution of (xi∗ , h(xi∗ , p, q)) among h(x, p, q).

by ’∙’) is depicted in Figure 13, from which we can infer that (xi∗ , h(xi∗ , p, q)) are actually located in the switching points of the hysteresis function series h(x, p, q) . Since there are (p+q+1) switching points in h(x, p, q), system (15) has (p+q+1) equilibrium points. These equilibrium points are distributed along the x-axis, so that x∗ = (−p, −p + 1, . . . , 0, . . . , q − 1, q). Therefore, parameters p and q control the number of equilibrium points in the negative and positive x-axis, respectively. Variable numbers of equilibrium points can be obtained by adjusting p and q. When k = 1, the symbol xi∗ denotes the roots of ax − a ⋅ h(x, p, q) + k ⋅ h(x, p, q) = 0, ∗ wi denotes the roots of h(x, p, q) − w = 0, and the equilibrium points of system (15) are [(a − k)u∗ /k, 0, 0, u∗ ], where u∗ = (−p, −p + 1, . . . , 0, q − 1, q). We observe that the parameter k adjusts the distribution of the equilibrium points in the x- and w-axes. Consider the property of the hysteresis function series, the Jacobi matrices of these equilibrium points are identical regardless of the value of k, which implies that as regards the stability these equilibrium points are also identical. The Jacobi matrix of equilibrium point (xi∗ , 0, 0, wi∗ ) is 0 0 J=( −a 0

1 0 −b −1

0 1 −c 0

0 0 ). −d −1

The characteristic equation is λ4 + (c + 1)λ3 + (b + c)λ2 + (a + b − d)λ + a = 0.

(17)

When α = 0.96, a = 3, b = 1, c = 2.3, d = 1, k = 1, p = 1, and q = 1, there are (p+q+1) = 3 equilibrium points. The corresponding characteristic equation is λ4 + 3.3λ3 + 3.3λ2 + 3λ + 3 = 0,

246 | L. Chen and R. Wu

Figure 14: Phase portraits of 3-scroll attractors: (a) on x–y; (b) on x–w;(c) on x–y–z; (d) Poincaré map with z = 0 on x–y.

and the characteristic roots are λ1 = −2.0645, λ2 = −1.4252, and λ3,4 = 0.0948 ± 1.0053i, with | arg(λ3 )| = | arg(λ4 )| = 1.4767 < απ/2 = 1.5079. According to Lemma 4, three equilibrium points have index 2, and attractors will be generated around them. The two largest Lyapunov exponents of (15) are λmax1 = 0.1106 and λmax2 = 0.1020, which confirm that the system is hyperchaotic. The phase portraits are presented in Figure 14(a)– (c). The Poincaré map is shown in Figure 14(d). Here we choose the four groups of parameters shown in Table 1 to simulate and test the chaotic behavior of the constructed system, the two largest Lyapunove exponents are calculated, and 7-scroll attractors are depicted in Figure 15.

4.3 Circuit simulation In this subsection, a circuit for system (15) is designed and circuit simulation results are shown to validate the theoretical formulation.

Design and generation of fractional-order multi-scroll chaotic attractors | 247 Table 1: Lyapunov exponents of system with different parameters. Portrait – – – Figure 14

Parameters p q

a

b

c

d

k

7 5 11 11

1 1 1 1

2.5 2 2.5 2.3

1 1 1 1

1 1 1 1

1 2 2 3

Lyapunov exponents

n-scroll attractors

0.2160, 0.2111 0.2076, 0.2034 0.3438, 0.3292 0.3890, 0.3633

4 5 6 7

2 2 3 3

Figure 15: Phase portraits of (a) 7-scroll attractors generate by (15) on x–y; (b) 7-scroll attractors generate by (15) on x–w.

First one is to realize the hysteresis function series. The corresponding circuit diagram is shown in Figure 16, where D is a Zener diode, R2 is a current-limiting resistance for the Zener diode, and R1 is a positive feedback resistor that can produce time delay effects in the circuit. Because of symmetry, only the case in Figure 16(a) is considered. Suppose that Vs is the regulated value of the Zener diode D and ±|Vsat | are the saturation voltages of the operational amplifier. The symbols VA and VB denote the input voltages of the comparators, and Evn and VH are the input and output voltages, respectively. When VA > VB , Vc = +|Vsat |, and VO = 0, the non-inverting terminal voltage VA is VA+ =

Evn R1 , R1 + R3

(18)

E R

that is, when VB < R vn+R1 , VO = 0. 1 3 When VA < VB , VC = −|Vsat |, and VO = −VS , we get VA− = Evn −

(Evn + Vs )R3 . R3 + R1

(19)

248 | L. Chen and R. Wu

Figure 16: Basic circuit diagram of hysteresis function series.

Thus, when VB > Evn −

(Evn +Vs )R3 , R1 +R3

and VO = −Vs , it follows from (18) and (19) that the

difference between the two thresholds of the hysteresis function is VA+ − VA− = According to equation (12), VA+ − VA− = 1 V, that is,

VS R3 . R1 +R3

Vs R3 = 1 V. R1 + R3 Let Vs = 7.5 V and R3 = 14 kΩ, then R1 = 91 kΩ. Since VH = − {0, VH = { R V F s , { Rv

if VB
Evn −

RF Vo , Rv

we get

Evn R3 , R1 +R3

where Rv = 7.5 kΩ and RF = 1 kΩ. If the hysteresis function series is constructed in system (15) by adding several hysteresis functions, then the right and left thresholds of the nth hysteresis function should satisfy the following rules: Evn R1 = n + 1, R1 + R3

Evn −

Evn R3 = n, R1 + R3

n = 1, 2, 3, . . . .

Thus, Evn = (n + 1)Vs R3 /R1 and, when n = 0, 1, 2, 3, Evn is given by {Ev0 , Ev1 , Ev2 , Ev3 } = {1.05, 2.19, 3.33, 4.20} V. The circuit diagram of the hysteresis function series for generating 3-scroll attractors is depicted in Figure 17, where R = 10 kΩ and E = 1.05 V. According to reference [6],

Design and generation of fractional-order multi-scroll chaotic attractors | 249

Figure 17: Circuit diagram of the hysteresis function series for generating 3-scroll attractors.

Figure 18: Circuit diagram of 0.96-order fractional differential unit.

the approximating fractional transfer function of order α = 0.96 is H(s) =

1.515(s + 1433)(s + 3.565) . (s + 1821)(s + 4.532)(s + 0.01127)

The circuit that implements the 0.96-order fractional differential is depicted in Figure 18. The values of the circuit components are calculated as R3 = 81.95 MΩ, R4 = 1.22 MΩ, R5 = 0.0039 MΩ, C1 = 0.232 μF, C2 = 0.18 μF, and C3 = 0.66 μF. The circuit to realize the 3-scroll FOHMS attractors is shown in Figure 19, where R1 = 10 kΩ, R2 = 5 kΩ, and R3 = 100 kΩ. The box ‘F’ stands for the basic circuit unit of 0.96-order fractional differential in Figure 18. Box ‘H(x)’ stands for the circuit unit shown in Figure 17, with k = 1 if the switch is on, and k = 0 if the switch is off. Under zero initial conditions, we can obtain from (15) the following equations: R2 Cs0.96 Ux (s) = Uy (s), { { { { { {R2 Cs0.96 Uy (s) = Uz (s), R R R { 0.96 { {R2 Cs Uz (s) = − R43 Ux (s) − R35 Uy (s) − R63 Uz (s) − { { { R3 R3 0.96 {R2 Cs Uw (s) = − R − R Uw (s) + k ⋅ H(s), 9

R3 U (s) R7 w

+

R3 H(s), R8

(20)

10

where Ux (s), Uy (s), Uz (s) and Uw (s) are the Laplace transform of ux , uy , uz , uw , respectively. The term s0.96 denotes the 0.96-order Laplace transform. Equation (20) can be

250 | L. Chen and R. Wu

Figure 19: Circuit diagram for generating 3-scroll attractors.

rewritten as { s0.96 Ux (s) = R1C Uy (s), { 2 { { { {s0.96 U (s) = 1 U (s), { y R2 C z { R R R { s0.96 Uz (s) = R1C [− R3 Ux (s) − R3 Uy (s) − R3 Uz (s) − { { 2 4 5 6 { { 0.96 R R { s Uw (s) = R1C [− R3 − R 3 Uw (s) + k ⋅ H(s)], 2 9 10 {

R3 U (s) R7 w

+

R3 H(s)], R8

(21)

where 1/R2 C is the integration constant. Comparing system (21) and (15), it is straightforward to obtain a = R3 /R4 = R3 /R8 , b = R3 /R5 , c = R3 /R6 , and d = R3 /R7 . The

resistance R3 is chosen as above, while the theoretical resistance values of R4 , R5 ,

R6 , R7 , R8 , R9 are easy to calculate. Since the chaotic system is very sensitive to the

initial state and the parameter values, during the experiment, the actual resistance values are more or less different from the ones calculated theoretically. For example,

for generating a 3-scroll attractor, the theoretical value of each resistance is chosen as R4 = R8 = 33.3 kΩ, R5 = R7 = 100 kΩ, and R6 = 50 kΩ, while during a simulation,

distinct scrolls emerge in the oscilloscope when R4 = R8 = 33.3 kΩ, R5 = R7 = 100 kΩ, and R6 = 60 kΩ. Simulation results are depicted in Figure 20.

Design and generation of fractional-order multi-scroll chaotic attractors | 251

Figure 20: Circuit simulation result of (a) 3-scroll on x–y; (b) 3-scroll on x–w; (c) 7-scroll on x–y; (d) 7-scroll on x–w cases.

5 Generating MDMS (multi-directional multi-scroll) attractors 5.1 Model and numerical simulations First we introduce a three-dimensional autonomous FO system, Dα x = y + z, { { { α D y = μz, { { { α {D z = −x − z,

(22)

where μ is constant, μ > 0, and α is the system order, 0 < α < 1. Since (22) is linear, nonlinear terms need to be brought in for this system to be chaotic. As is well known, one saddle point with index 2 will only generate one attractor at most. The generation of MDMS chaotic attractors will need a large number of

252 | L. Chen and R. Wu saddle points with index 2. To this end, system (22) is recast as follows: Dα x = y + z − f (y), { { { α D y = μz, { { { α {D z = −x − z + f (x),

(23)

where 0 < α < 1 is order and μ > 0. The functions f (x) and f (y) are PWLF. To generate MDMS chaotic attractors, two different kinds of PWLF, namely the SNFS and the SNLF, are chosen and added to system (22) to show the working principle. Take F1 (x) =

k=K

󵄨󵄨 p 󵄨󵄨󵄨󵄨 |k| 󵄨 {󵄨󵄨x − p(2k − ) + q󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2q k k=−K,k =0 ̸ ∑

󵄨󵄨 󵄨󵄨 |k| 󵄨 󵄨 − 󵄨󵄨󵄨x − p(2k − ) − q󵄨󵄨󵄨} 󵄨󵄨 󵄨󵄨 k and N

F2 (x) = A1 sgn(x) + ∑ sgn(x − 2nA2 ) M

n=1

+ ∑ sgn(x + 2mA3 ), m=1

where K, M, N, m, n are positive integers, k is an integer, A1 , A2 , A3 , p > 0 and q ∈ (0, p). Note that F1 (x) is an SNLF, the saturated slope is p/q and the delay time is (2k − |k|/k)p. The function F2 (x) is an SNFS. System (23) can generate (2K + 1) attractors in x-axis or y-axis when f (x) = F1 (x) or f (y) = F1 (x), respectively. Similarly, (m + n + 1) attractors will be created in system (23) by F2 in x-axis or y-axis when A1 = 0, and (n + m + 2) attractors will be generated when A1 ≠ 0. The parameters K, p, q, A1 , A2 , A3 are all adjustable. By changing these parameters, one can design the number of equilibrium points meeting necessary conditions to generate MS chaotic attractors. Let p = 1, q = 0.02p, K = 2, A1 = A2 = A3 = 1, m = n = 1, thus forming 5 × 4 attractors in x–y plane, corresponding to the type 1 attractors in Table 2. The attractors in x-axis are generated by SNLF and in y-axis by SNFS. SNFS and SNLF are different kinds of functions, which have different forming mechanisms. The topologies of attractors created by different combinations of those are not equivalent with each other. Table 2 gives the details of some parameters and the corresponding MDMS chaotic attractors generated with these parameters, where type 1, 2, 3, and 4 in Table 1 are used to identify these different attractors. That is to say, type 1, 2, 3 and 4 attractors are created by the combinations of SNLF and SNFS, SNLF and SNLF, SNFS and SNLF, SNFS and SNFS, respectively. Take the parameters p = 1, q = 0.02p in Table 2. Numerical simulations are depicted in Figures 21(a), (b), (c) and (d). The parameters in Table 2 are not unique and can be chosen within a certain range while the system still keeps being chaotic.

Design and generation of fractional-order multi-scroll chaotic attractors | 253 Table 2: Attractors generated with parameters μ, K, M, N, A1 , A2 , A3 . type 1 2 3 4

x-axis

y-axis

μ

K

M

N

A1

A2

A3

α

attractors

SNLF SNLF SNFS SNFS

SNFS SNLF SNLF SNFS

2.5 2.5 3 2.5

2 2 2 -

1 – 1 2

1 – 1 2

1 – 0 1

1 – 0.5 1

1 – 0.5 1

0.96 0.96 0.9 0.96

5×4 5×5 3×5 6×6

Figure 21: Phase portrait of four kinds of attractors: (a) type 1; (b) type 2; (c) type 3; (d) type 4.

5.2 Dynamic behavior analysis In this section, the type 2 MS chaotic attractors are taken as an example to analyze the dynamic behavior. The corresponding system of type 2 attractors is Dα x = y + z − F1 (y), { { { α D y = μz, { { { α {D z = −x − z + F1 (x).

(24)

254 | L. Chen and R. Wu

Figure 22: (a) 3-D phase portrait attractors of system (24). (b) Largest Lyapunov exponent of system (24) with respect to μ. (c) Poincaré map of system (24) on the section y = 0. (d) Largest Lyapunov exponent of system (24) with respect to α.

According to Table 2, one has the parameters μ = 2.5, K = 2, p = 1, q = 0.02p and α = 0.96. Figure 22(a) shows the 3-D phase portrait of 5 × 5 attractors of type 2. Figure 22(b) shows the largest Lyapunov exponent with respect to μ of system (24), Figure 22(c) represents the Poincaré map on the section y = 0 and Figure 22(d) shows the largest Lyapunov exponent with respect to α of system (24). Making Dα x = Dα y = Dα z = 0 yields y − F1 (y) = 0, z = 0, and −x + F1 (x) = 0. All the equilibrium points of system (24) are calculated and depicted in Figure 23, verifying that there are (2k + 1) × (2K + 1) + 2K × 2K = 41 equilibrium points. At equilibrium point E ∗ = (x∗ , y∗ , z ∗ ), the Jacobi matrix of system (24) is JE ∗ where F1󸀠 (y∗ ) =

dF1 (y) | ∗, dy y

0 =( 0 F1󸀠 (x∗ ) − 1

F1󸀠 (x∗ ) =

dF1 (x) | ∗. dx x

1 − F1󸀠 (y∗ ) 0 0

0 μ ), −1

(25)

Design and generation of fractional-order multi-scroll chaotic attractors | 255

Figure 23: Equilibrium point distribution of system (24).

The derivative of F1 (u) at point u∗ (u denotes x or y) is F1󸀠 (u∗ ) =

dF1 (u) 󵄨󵄨󵄨󵄨 󵄨 du 󵄨󵄨󵄨u∗ k=K

=

p |k| {sgn(u∗ − p(2k − ) + q) 2q k k=−K,k =0 ̸ ∑

− sgn(u∗ − p(2k −

|k| ) − q)}. k

If u∗ = 2kp, k = 0, ±1, ±2, . . . , ±K, which corresponds to the equilibrium points marked by ‘∘’ in Figure 23, then F1󸀠 (u∗ ) = 0. Alternatively, if u∗ = (2k − |k|/k)p, k = 0, ±1, ±2, . . . , ±K, which correspond to the equilibrium points marked by ‘∙’ in Figure 23, then F1󸀠 (u∗ ) = p/q. The characteristic equation of matrix (25) is f (λ) = λ3 + λ2 − (F1󸀠 (x∗ ) − 1)λ

+ μ(F1󸀠 (x ∗ ) − 1)(F1󸀠 (y∗ ) − 1) = 0.

(26)

For equilibrium points ‘∘’, as F1󸀠 (x∗ ) = F1󸀠 (y∗ ) = 0, the characteristic equation can be simplified as f (λ) = λ3 + λ2 + λ + μ = 0.

(27)

The roots of equation (27) are λ1 = −1.4732, λ2,3 = 0.2366 ± 1.2810i, yielding | arg(λ2,3 )| = 1.3882 < 1.5080 = απ/2. According to Lemma 3, points ‘∘’ are saddle points with index 2, and an attractor will form around each ‘∘’. For equilibrium points ‘∙’, F1󸀠 (x∗ ) = F1󸀠 (y∗ ) = p/q = 50 and the characteristic equation can be simplified to f (λ) = λ3 + λ2 + (1 − p/q)λ + (1 − p/q)2 μ = 0.

(28)

256 | L. Chen and R. Wu The roots of equation (28) are λ1 = −19.4269, λ2,3 = 9.2135 ± 14.9697i, yielding | arg(λ2,3 )| = 1.0191 < 1.5080 = απ/2. According to Lemma 3, the points ‘∙’ are saddle points with index 2. However, the numerical simulation in Figure 21(d) shows that only the points ‘∘’ can generate attractors. In fact, having a saddle point with index 2 is only a necessary condition, not a sufficient one for generating attractors. According to the homoclinic Ŝilnikov theorem [9], an existence condition is needed of a homoclinic orbit in the neighboring region of the equilibrium point to generate attractors. This is why we say that one saddle point with index 2 will generate only one attractor at most. The attractors of type 1, type 3 and type 4 can be investigated in a similar way.

5.3 Circuit implementation for MS attractors In this section, several circuit diagrams are designed to realize various MS chaotic attractors. Firstly, we investigate the circuit diagram which is the basic cell for designing the SNLF in Figure 24(a). The Ui –Uo relationship of the ith basic cell, depicted in Figure 24(b), is given by Vsat if Ui < (Ei − Vsat RR2 ), { { 1 { { { {− RR1 (Ui − Ei ) if (Ei − Vsat RR2 ) ≤ Ui 1 Uo = { 2 { ≤ (Ei + Vsat RR2 ), { { 1 { { if Ui > (Ei + Vsat RR2 ), {−Vsat 1

(29)

where Ui and Uo are the input and the output voltages (notice that the subscript i of Ui means ‘input’, it is not a counter i). Furthermore, Vsat denotes the saturated output voltage of the operational amplifier, so that Vsat = 13.5 V. The values (Ei − Vsat R2 /R1 ) and (Ei + Vsat R2 /R1 ) are switching points of the ith cell, and the slope is −R1 /R2 . Equation (29) can be rewritten as Uo = −

V R 󵄨󵄨󵄨 󵄨󵄨󵄨 V R 󵄨󵄨󵄨 R1 󵄨󵄨󵄨󵄨 {󵄨󵄨Ui − Ei + sat 2 󵄨󵄨󵄨 − 󵄨󵄨󵄨Ui − Ei − sat 2 󵄨󵄨󵄨}. 2R2 󵄨󵄨 R1 󵄨󵄨 󵄨󵄨 R1 󵄨󵄨

By connecting several basic cells in parallel, and adding an invert sum box circuit in series, as shown in Figure 24(c), we can easily deduce that f1 (u) =

k=K

R1 R4 󵄨󵄨󵄨󵄨 V R 󵄨󵄨󵄨 {󵄨󵄨u − Ei + sat 2 󵄨󵄨󵄨 2R2 R3 󵄨󵄨 R1 󵄨󵄨 k=−K,k =0 ̸ ∑

󵄨󵄨 V R 󵄨󵄨󵄨 󵄨 − 󵄨󵄨󵄨u − Ei − sat 2 󵄨󵄨󵄨}. 󵄨󵄨 R1 󵄨󵄨

(30)

Assume that E = 1 is the unit voltage and take Ei = (2k − |k|/k)E. Comparing equation (30) with the function F1 (x), we obtain q = Vsat R2 /R1 and p = E. Setting R1 = 200 kΩ,

Design and generation of fractional-order multi-scroll chaotic attractors | 257

Figure 24: The circuit diagram. (a) basic cell; (b) Ui –Uo relationship; (c) the realization of f1 (u); (d) diagram of 0.96-order differential unit.

since q = 0.02 and Vsat = 13.5 V, we have R2 = 300 Ω. While R1 R4 /2R2 R3 = p/2q, take R4 = 1 kΩ, and then R3 = 13.5 kΩ. Moreover, if resistor R1 → ∞ in Figure 24(c), that is, R1 is off, then −R1 /R2 → −∞. The switching points (Ei − Vsat R2 /R1 ) and (Ei + Vsat R2 /R1 ) both tend to Ei . Therefore, the circuit to realize the SNLF tends to the SNFS. Equation (29) can be recast as Vsat Uo = { −Vsat

if Ui ≤ Ei ,

if Ui > Ei .

(31)

Similarly, the SNFS function F2 (x) can be realized as follows: N

M

i=1

j=1

f2 (u) = A1 sgn(u) + ∑ sgn(u − Ei ) + ∑ sgn(u + Ej ), where Ei = 2iE1 , Ej = 2jE2 , E1 and E2 are the unit voltages, and E1 = A2 , E2 = A3 . The circuit diagram is omitted here for saving space.

258 | L. Chen and R. Wu

Figure 25: Circuit diagram to generate attractors of type 2.

For the circuit to realize the 0.96-order differentiation depicted in Figure 24(d) we calculated R3 = 81.95 MΩ, R4 = 1.22 MΩ, R5 = 0.0039 MΩ, C1 = 0.232 μF, C2 = 0.18 μF, and C3 = 0.66 μF, for α = 0.96. Based on all these rules, we choose the type 2 attractors as a vehicle to generate MDMS chaotic attractors of order 0.96 and a maximum 9 × 9 attractor is observed. The circuit diagram is represented in Figure 25. This circuit diagram includes four different parts. – Part 1: invert sum box N0 ; – Part 2: FO integrator box N1 ; – Part 3: invert box N2 ; – Part 4: saturated function generator SNLF. Box F stands for the 0.96-order calculation unit shown in Figure 24(d). Box SNLF stands for the saturated function unit in Figure 24(c). For all resistors R1 = 10 kΩ, R2 = 100 Ω. According to the Kirchhoff current law, and for the zero initial condition, we derive the following equations: { s0.96 Ux (s) = { { { 0.96 s Uy (s) = { { { { 0.96 s Uz (s) = {

1 [Uy (s) + Uz (s) − f (Uy (s))], R2 C 1 R1 U (s), R2 C R3 z 1 [−Ux (s) − Uz (s) + f (Ux (s))], R2 C

(32)

Design and generation of fractional-order multi-scroll chaotic attractors | 259

Figure 26: Circuit results: (a) 9 × 9 attractors, with x-axis using SNLF, 5 V/div, y-axis using SNLF, 5 V/div. (b) 5 × 4 attractors, with x-axis using SNLF, 2 V/div, and y-axis using SNLF, 2 V/div. (c) 5 × 5 attractors, with x-axis using SNLF, 2 V/div, and y-axis using SNLF, 2 V/div. (d) 5 × 4 attractors, with x-axis using SNFS, 2 V/div, and y-axis using SNFS, 2 V/div.

where 1/R2 C is the integrator constant of the circuit and the transformation factor of the time scale. The parameter μ = R1 /R3 is not unique and, in our system, we choose μ = 3, yielding R3 = 3.3 kΩ. The 9×9 attractors generated by SNLF in two directions are shown in Figure 26(a). Other kinds of attractors are addressed by adjusting the circuit diagram, and a few are listed in Figures 26(b), (c) and (d).

Bibliography [1] [2] [3]

W. Ahmad, Generation and control of multi-scroll chaotic attractors in fractional order systems, Chaos Solitons & Fractals, 25(3) (2005), 727–735. W. Ahmad and J. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos Solitons & Fractals, 16(2) (2003), 339–351. D. Cafagna and G. Grassi, Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos, International Journal of Bifurcation and Chaos, 18(3) (2008), 615–639.

260 | L. Chen and R. Wu

[4] [5] [6] [7]

[8] [9] [10] [11] [12] [13]

W. Deng and J. Lu, Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control, Chaos, 16(4) (2006), 043120. W. Deng and J. Lu, Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system, Physics Letters A, 369(5) (2007), 438–443. F. Min, S. Shao, W. Huang, and E. Wang, Circuit implementations, bifurcations and chaos of a novel fractional-order dynamical system, Chinese Physics Letters, 32 (2015), 3, 21. L. J. Ontanón García, E. Jiménez-López, E. Campos-Cantón, and M. Basin, A family of hyperchaotic multi-scroll attractors in R n , Applied Mathematics and Computation, 233 (2014), 522–533. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. P. S. Silva, Shil’nikov’s theorem-a tutorial, IEEE Transactions on Circuits and Systems. I, Fundamental Theory and Applications, 40(10) (1993), 675–682. J. Sprott, A new class of chaotic circuit, Physics Letters A, 266(1) (2000), 19–23. J. Sprott, Simple chaotic systems and circuits, American Journal of Physics, 68(8) (2000), 758–763. G. Sun, M. Wang, L. Huang, and L. Shen, Generating multi-scroll chaotic attractors via switched fractional systems, Circuits, Systems, and Signal Processing, 30(6) (2011), 1183–1195. F. Xu, Integer and fractional order multiwing chaotic attractors via the Chen system and the Lü system with switching controls, International Journal of Bifurcation and Chaos, 24(3) (2014), 1450029.

Guo-Cheng Wu, Dumitru Baleanu, and Yun-Ru Bai

Discrete fractional masks and their applications to image enhancement Abstract: Fractional differences for image enhancement are revisited and the general methodology is illustrated in this chapter. Several fractional differences are theoretically analyzed and numerically compared. The weight coefficients derived from the discrete fractional calculus are a set of conserved quantities and they are suitable for image processing. Then a discrete fractional mask is designed within the Caputo difference and the mask coefficients are given by use of the Gamma functions. In comparison with the Grünwald–Letnikov difference and Riemann–Liouville masks, the results show this novel mask’s efficiency and simplicity. Keywords: Discrete fractional calculus, image enhancement, fractional difference mask MSC 2010: 76M20, 94A08, 26A33

1 Introduction The Kirsch operator, the Sobel operator, the Laplace method and the difference method et al. [7] are the methods most often used in image enhancements. However, the operators used in these methods are local ones. Pictures and images in our life time often contain information of long interaction or connection. They exhibit a feature of non-locality. Hence, these methods are not efficient for image enhancement. Fractional derivatives hold non-locality. The fractional image enhancement was suggested. It has a very short history. To the best of our knowledge, the earliest idea was proposed by Pu et al. [16–18]. The fractional derivative’s memory effects were considered. For all pixels in images, they need numerical discretization of the continuous fractional derivatives like the Caputo derivative, Riemann–Liouville derivative and Grünwald–Letnikov (G–L) derivative so that the weight coefficients are obtained (see [8, 9]). The mask can fully use all the points’ information due to the fractional derivative’s memory effects. Later, Gao [10] and Liu et al. [22] proposed modified versions and improved the enhanced results by use of G–L difference and Riesz operator, Acknowledgement: This work was financially supported by Sichuan Science and Technology Program and China Postdoctoral Science Foundation (Grant No. 2016M602632). Guo-Cheng Wu, Yun-Ru Bai, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, P.R. China, e-mails: [email protected], [email protected] Dumitru Baleanu, Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey; and Institute of Space Sciences, Magurele–Bucharest, Romania, e-mail: [email protected] https://doi.org/10.1515/9783110571929-010

262 | G.-C. Wu et al. respectively. The Riesz derivative is a two-sided operator which can use both the left and the right pixels’ information. This new mask reduces the round number and improves the speed. However, the numerical discretization of continuous fractional calculus readily results in truncation errors. It causes much information lost or leads to tedious noise in image enhancement. Noise will damage the crucial and useful information. Fractional differences derived from numerical discretization cannot satisfy the need of high accuracy. In this study, by use of the fractional Caputo-like difference [1, 2], a novel fractional difference mask is designed. The difference operator is naturally a discrete one which is defined on an isolated time scale. We adopt this discrete fractional calculus to illustrate that it is a more straightforward tool for signal processing and the results demonstrate the merits.

2 Fractional difference masks Several authors [1–4, 12] defined discrete fractional calculus on the time scale and developed an efficient fractionalization tool for difference equations. Recently, rich results were obtained on this topic [5, 6, 11, 13–15, 19–21]. Denote by ℕa the isolated time scale {a, a + 1, a + 2, . . .}, a ∈ ℝ. Let us revisit the definition. Definition (See [2] and [1]). For 0 < ν, ν ∉ N, the fractional Caputo-like difference is defined as C ν a Δt u(t)

:=

1 Γ(m − ν)

t−(m−ν)

(m−ν−1) m

∑ (t − σ(s))

s=a

Δ u(s),

t ∈ Na+m−ν , m = [ν] + 1,

(1)

where ν is the difference order and t (ν) is defined as the falling function t (ν) =

Γ(t + 1) . Γ(t + 1 − ν)

For 0 < ν < 1, the fractional Caputo-like difference can be rewritten as C ν a Δt u(t)

j−1

=

1 Γ(j − i − ν) Δu(i) ∑ Γ(1 − ν) i=0 Γ(j − i)

= u(j) +

j−1

Γ(j − i + 1 − ν) Γ(j − i − ν) Γ(j − ν) 1 − )u(i) − u(0), ∑( Γ(1 − ν) i=1 Γ(j − i + 1) Γ(j − i) Γ(1 − ν)Γ(j)

0 < ν < 1.

(2)

It can be rewritten as C ν a Δt u(t)

= C0 u(j) + C1 u(j − 1) + ⋅ ⋅ ⋅ + Cj−i u(i) + ⋅ ⋅ ⋅ + Cj u(0)

(3)

Discrete fractional masks and their applications to image enhancement | 263 Table 1: Fractional difference masks 3 × 3. ∑i=0 Ci

j=3

RL α

D

GL α

C α

ν ν ν ν

= 0.8 = 0.6 = 0.4 = 0.2

0.09 0.23 0.43 0.69

0.08 0.22 0.41 0.67

−0.21 −0.5 −0.93 −1.43

D

D

C ν a Δt

0 0 0 0

Table 2: Fractional difference masks 5 × 5. ∑i=0 Ci

j=5

RL α

D

GL α

C α

ν ν ν ν

= 0.9 = 0.7 = 0.5 = 0.3

0.02 0.01 0.25 0.48

0.02 0.01 0.24 0.47

−0.05 −0.02 −0.53 −0.98

D

D

C ν a Δt

0 0 0 0

and the coefficients Cj are given from equation (2) by Ci = 1, i = 0, { { { { 1 Ci = Γ(1−ν) ( Γ(i+1−ν) − Γ(i−ν) ), Γ(i+1) Γ(i) { { { { Γ(j−ν) {Ci = − Γ(1−ν)Γ(j) , i = j.

1 ≤ i ≤ j − 1,

(4)

Along the eight directions, we can design the masks by Table 1 to Table 8. For example, for j = 4, the coefficients can be given by { C0 = 1, { { { { Γ(2−ν) Γ(1−ν) 1 { {C1 = Γ(1−ν) ( Γ(2) − Γ(1) ), { { { { 1 ( Γ(3−ν) − Γ(2−ν) ), C2 = Γ(1−ν) Γ(3) Γ(2) { { { { 1 { { ( Γ(4−ν) − Γ(3−ν) ), C3 = Γ(1−ν) { Γ(4) Γ(3) { { { { 1 Γ(4−ν) C = − Γ(1−ν) Γ(4) . { 4

(5)

Here we also list the mask coefficients derived from continuous fractional derivatives: Dα u(t) ≈ C0 un + C1 un−1 + ⋅ ⋅ ⋅ + Cn un where the step length is set to one. (a) Differences of the Caputo derivative C Dα u(t) [9]: −1 Ci = Γ(2−ν) , i = 0, { { { { 2(j−i)1−ν −(j−i+1)1−ν −(j−i−1)1−ν , Ci = { Γ(2−ν) { { { j1−ν −(j−1)1−ν {Ci = − Γ(2−ν) , i = j.

1 ≤ i ≤ j − 1,

(6)

264 | G.-C. Wu et al. Table 3: The coefficient Tx + along the direction 0°. .. . 0 C0 0 .. .

.. . 0 ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 Ci 0 .. .

.. . ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 Cj 0 .. .

Table 4: The coefficient Tx + y + along the direction 45°. .. . 0 0 0 C0

.. . 0 ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 Ci 0 .. .

0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

Cj 0 0 .. .

Table 5: The coefficient Ty + along the direction 90°. ⋅⋅⋅

0

⋅⋅⋅ ⋅⋅⋅

0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅

Cj .. . Ci .. . C0

0

⋅⋅⋅

0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

Table 6: The coefficient Tx − y + along the direction 135°. Cj 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 .. 0 0 0

.

.. .

.. .

.. .

0 Ci

0 0 ..

0 0

0 0

0

.

0 C0

Table 7: The coefficient Tx − along the direction 180°. .. . 0 Cj 0 .. .

.. . 0 ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 Ci 0 .. .

.. . ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 C0 0 .. .

Discrete fractional masks and their applications to image enhancement | 265 Table 8: The coefficient Tx − y − along the direction 225°. .. . 0 0 0 Cj

.. . 0 ⋅⋅⋅ ⋅⋅⋅ .. .

.. . 0 Ci 0 .. .

0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

C0 0 0 0 .. .

(b) Differences of the Riemann–Liouville derivative RL Dα u(t) [8]: Ci = { { { { Ci = { { { { { Ci =

1 , Γ(2−ν) 1−ν

i = 0,

−2i +(i−1)1−ν , Γ(2−ν) (1−ν)j−ν −j1−ν +(j−1)1−ν , Γ(2−ν)

(i+1)

1−ν

1 ≤ i ≤ j − 1,

(7)

i = j.

(c) Differences of the Grünwald–Letnikov derivative GL Dα u(t) [16–18]: {Ci = 1, i = 0, { (−1)i Γ(ν+1) {Ci = Γ(i+1)Γ(ν−i+1) ,

1 ≤ i ≤ j.

(8)

Fix the fractional order ν and pixel number j. We compare all the sums of the coefficients in the following tables for the mask matrix 3 × 3 and 5 × 5. We can observe that only the weight coefficients derived from the DFC are a set of conservative quantities and they do not generate any numerical errors if we neglect the aspect of the number of digits.

3 Algorithm and results We use the presented fractional difference mask in Section 2 and follow the general steps suggested by Pu [16–18]. Assume that f (x, y) is the gray value of the enhanced pixel (x, y), g(x, y) is the increment and F(x, y) is the enhanced result. Step I. Set n = 2m. Apply the Caputo fractional difference to the gray function f (x, y). Fully using the points around the enhanced pixel (x, y), one can calculate the g along the eight directions following the rules m

2m

gx+ = ∑ ∑ Tx+ (m + a, b)f (x + a, y + b); a=−m b=0 0

2m

gx+ y+ = ∑ ∑ Tx+ y+ (2m + a, b)f (x + a, y + b); a=−2m b=0

266 | G.-C. Wu et al. Table 9: The coefficient Ty − along the direction 270°. ⋅⋅⋅

0

⋅⋅⋅ ⋅⋅⋅

0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅

C0 .. . Ci .. . Cj

0

⋅⋅⋅

0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

Table 10: The coefficient Tx + y − along the direction 315°. C0 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 .. 0 0 0

.

.. .

.. .

.. .

0 Ci

0 0 ..

0 0

0 0

0

.

0 Cj m

0

∑ Ty+ (2m + a, m + b)f (x + a, y + b);

gy+ = ∑

a=−2m b=−m 0

0

∑ Tx− y+ (2m + a, 2m + b)f (x + a, y + b);

gx− y+ = ∑

a=−2m b=−2m 0

m

∑ Tx− (m + a, 2m + b)f (x + a, y + b);

gx− = ∑

a=−m b=−2m 2m

0

gx− y− = ∑ ∑ Tx− y− (a, 2m + b)f (x + a, y + b); a=0 b=−2m 2m

m

gy− = ∑ ∑ Ty− (a, m + b)f (x + a, y + b); a=0 b=−m 2m 2m

gx+ y− = ∑ ∑ Tx+ y− (a, b)f (x + a, y + b). a=0 b=0

Tables 3–10 are the corresponding fractional difference masks. Step II. Select the above maximal value as the fractional increment which is denoted Δαmax f (x, y), Δαmax f = max(gx+ , gx+ y+ , gy+ , gx− y+ , gx− , gx− y− , gy− , gx+ y− ).

(9)

Step III. The enhanced result is calculated by the equality F(x, y) = Δαmax f + f (x, y).

(10)

Discrete fractional masks and their applications to image enhancement | 267

Figure 1: Plan image (Figure 8a in [16]).

Figure 2: Discrete fractional mask in present study.

Figure 3: G–L difference mask used in [17].

Now we give the enhanced results of the remote sensing satellite image (Figure 1) used in [16] (see Figure 8a therein). Figure 2 is our result using the above steps, named the DFC method. Figure 3 and Figure 4 are existing results by use of the classical G–L difference and the Riemann–Liouville (R–L) derivative, respectively. In Figure 2–Figure 4, all the parameters are set to have the fractional difference order ν = 0.8 and m = 2. Figure 5 and Figure 6 are the results derived from the methods most often used, by operators called Kirsch and Sobel operators in Matlab.

268 | G.-C. Wu et al.

Figure 4: R–L mask used in [8].

Figure 5: Kirsch operator.

Figure 6: Sobel operator.

We can find that all the image edges are enhanced in Figure 2–Figure 6. The results in Figure 2, Figure 3 and Figure 4 are clearer when all figures are compared with Figure 1. Particularly, Figure 2 is the clearest one.

Discrete fractional masks and their applications to image enhancement | 269

4 Conclusions This study newly suggests a fractional difference tool (discrete fractional calculus) for image enhancement, and a novel fractional difference mask is derived via the weight coefficients of the definition. Furthermore, we theoretically compare the results derived by use of the continuous fractional calculus. For the same plain image and the same fractional order between zero and one, we can see that the DFC method leads to a better and enhanced result. It can be concluded that the DFC is suitable for discrete systems, since the tool is straightforwardly defined on an isolated time scale and holds a beautiful and conservative discrete memory effect.

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T. Abdeljawad, On Riemann and Caputo fractional differences, Computers & Mathematics with Applications, 62 (2011), 1602–1611. G. A. Anastassiou, About discrete fractional calculus with inequalities, Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library, 5 (2011), 575–585. F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, International Journal of Differential Equations, 2 (2007), 165–176. F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 137 (2009), 981–989. F. M. Atici and P. W. Eloe, Gronwall’s inequality on discrete fractional calculus, Computers & Mathematics with Applications, 64 (2012), 3193–3200. D. Baleanu, G. C. Wu, Y. R. Bai, and F. L. Chen, Stability analysis of Caputo-like discrete fractional systems, Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 520–530. T. Bräunl, S. Feyrer, W. Rapf, and M. Reinhardt, Parallel Image Processing, Springer-Verlag, Berlin, Heidelberg, 2001. Q. L. Chen, G. Huang, and Y. F. Pu, Riemann–Liouville fractional-order image enhancement algorithm and tis circuit realization, Journal of Shenyang University of Technology, 34 (2012), 550–555. Q. L. Chen, G. Huang, X. Q. Zhang, J. Yang, and W. Xiang, A Caputo fractional differential approach to image enhancement, Journal of Computer-Aided Design & Computer Graphics, 25 (2013), 519–525 (in Chinese). C. B. Gao, J. L. Zhou, J. R. Hu, and F. N. Lang, Edge detection of colour image based on quaternion fractional differential, IEEE Transactions on Image Processing, 5 (2011), 261–272. C. S. Goodrich, On a discrete fractional three-point boundary value problem, Journal of Difference Equations and Applications, 18 (2012), 397–415. M. T. Holm, The Theory of Discrete Fractional Calculus: Development and Application, Ph.D. Thesis, University of Nebraska-Lincoln, Lincoln, Nebraska, 2011. D. Mozyrska and E. Girejko, Overview of Fractional h-Difference Operators, Operator Theory: Advances and Applications, vol. 229, Springer, Basel, 2013. D. Mozyrska and P. Ostalczyk, Generalized fractional-order discrete-time integrator, Complexity, 2017 (2017), 3452409.

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[15] D. Mozyrska and M. Wyrwas, The Z-transform method and delta type fractional difference operators, Discrete Dynamics in Nature and Society, 2015 (2015), 852734. [16] Y. F. Pu and W. X. Wang, Fractional differential masks of digital image and their numerical implementation algorithms, Acta Automatica Sinica, 33 (2007), 1128–1135 (in Chinese). [17] Y. F. Pu, W. X. Wang, J. L. Zhou, Y. Y. Wang, and H. D. Jia, Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation, Science in China. Series F. Information Sciences, 51 (2008), 1319–1339. [18] Y. F. Pu, J. L. Zhou, and X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Transactions on Image Processing, 19 (2010), 491–511. [19] G. C. Wu and D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynamics, 75 (2014), 283–287. [20] G. C. Wu, D. Baleanu, and L. L. Huang, Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse, Applied Mathematics Letters, 82 (2018), 71–78. [21] G. C. Wu, D. Baleanu, and S. D. Zeng, Discrete chaos in fractional sine and standard maps, Physics Letters A, 378 (2014), 484–487. [22] Q. Yu, V. Vegh, F. Liu, and I. Turner, A variable order fractional differential-based texture enhancement algorithm with application in medical imaging, Plos One, 10 (2015), e0132952.

Bashir Ahmad, Sotiris K. Ntouyas, and Ahmed Alsaedi

Existence theory for fractional differential equations with nonlocal integro-multipoint boundary conditions with applications Abstract: We discuss the existence and uniqueness of solutions for a Caputo fractional differential equation involving a nonlinearity depending upon the unknown function together with its lower order fractional derivative supplemented with nonlocal integro-multipoint boundary conditions. We make use of standard fixed point theorems to derive the main results. Some applications of the obtained work are also indicated. Keywords: Caputo fractional derivative, nonlocal, multipoint, multi-strip, boundary conditions MSC 2010: 26A33, 34A08, 34B15

1 Introduction Fractional calculus has developed into an important area of research in view of its growing interest in the mathematical modeling of many real world phenomena occurring in several disciplines. Examples include fractional dynamics [1], bioengineering [2], mechanical manipulators [3], viscoelasticity [4], control theory [5], fractional calculus models [6], blood flow problems [7], fractional electrical circuits [8], and financial economics [9]. Fractional-order boundary value problems (FBVPs) received immense attraction in recent years. The literature on FBVPs is now much enriched and varies from theoretical aspects to analytic and numerical methods for the solution of these problems. The boundary data associated with fractional-order differential equations include nonlocal, multipoint, multi-strip, and fractional-order boundary conditions in addition to classical Sturm–Liouville conditions; see, for instance, [10–18]. Nonlocal multipoint boundary data find useful applications in certain problems of thermodynamics, elasticity and wave propagation when the controllers at the end points of the interval dissipate or add energy according to censors located at variable interior positions of the domain. On the other hand, integral boundary conditions provide the means to take into account an arbitrary shaped cross-section of blood vessels in comBashir Ahmad, Ahmed Alsaedi, Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Sotiris K. Ntouyas, Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece https://doi.org/10.1515/9783110571929-011

272 | B. Ahmad et al. putational fluid dynamics (CFD) studies of blood flow problems [19], and they regularize ill-posed parabolic backward problems in time partial differential equations, for example, mathematical models for bacterial self-regularization [20]. In this paper, we propose and study a new boundary value problem consisting of a Caputo fractional differential equation and nonlocal integro-multipoint boundary conditions. In precise terms, we consider the following Caputo fractional differential equation: c q

D x(t) = f (t, x(t), c Dβ x(t)),

0 < β < 1, 1 < q ≤ 2, t ∈ [0, 1],

(1)

equipped with boundary conditions of the form 1

1

m−2

m−2

∫ x(s)ds + μ1 = ∑ αi x(σi ),

∫ x󸀠 (s)ds + μ2 = ∑ δi x󸀠 (σi ),

0

0

i=1

i=1

(2)

where c Dq , c Dβ denote the Caputo fractional derivative of order q and β, respectively, 0 < σ1 < σ2 < ⋅ ⋅ ⋅ < σm−2 < 1, f is a given continuous function, and αi , δi (i = 1, 2, . . . , m− 2), μ1 , μ2 are real constants. One can interpret the first boundary condition in (2) as the difference of distribution of the unknown function over the given domain differs from the sum of finite many values of the unknown function at arbitrary positions located within the domain by a constant, and the same holds for the derivative of the unknown function in the second boundary condition of (2).

2 Preliminaries Let us begin with some basic definitions of fractional calculus [21]. Definition 2.1. The Riemann–Liouville fractional integral of order α > 0 for a locally integrable real-valued function f on 0 ≤ t < b < ∞ is defined as t

1 I y(t) = ∫(t − s)α−1 y(s)ds. Γ(α) α

0

Definition 2.2. If f ∈ C n [a, b], then the Caputo fractional derivative c Dαa of f of order α ∈ ℝ (n − 1 < α < n, n ∈ ℕ) is defined as c α Da f (t)

=

Ia1−α f (n) (t)

t

1 = ∫ (t − s)n−1−α f (n) (s)ds. Γ(n − α) a

Here we denote the Caputo fractional derivative c Dαa with a = 0 as c Dα . Now we prove an auxiliary lemma, which plays a key role in the study of the problem at hand.

Nonlocal fractional integro-multipoint boundary value problems | 273

Lemma 2.3. Let Λ1 , Λ2 ≠ 0. For g ∈ C([0, 1], ℝ), the solution of the linear fractional differential equation c q

D x(t) = g(t),

1 < q ≤ 2,

t ∈ [0, 1],

(3)

subject to the boundary conditions (2) is equivalent to the integral equation t

x(t) = ∫ 0

(t − s)q−1 g(s)ds Γ(q) σi

1 s

(Λ t − Λ3 ) m−2 (σ − s)q−2 (s − w)q−2 { ∑ δi ∫ i g(s)ds − ∫ ∫ g(w)dwds − μ2 } + 2 Λ1 Λ2 Γ(q − 1) Γ(q − 1) i=1 0

σi

+

0 0

1 s

(σ − s)q−1 1 m−2 (s − w)q−1 g(s)ds − ∫ ∫ g(w)dwds − μ1 }, { ∑ αi ∫ i Λ2 i=1 Γ(q) Γ(q)

(4)

0 0

0

where m−2

m−2

Λ2 = 1 − ∑ αi ,

Λ1 = 1 − ∑ δi ,

Λ3 =

i=1

i=1

1 m−2 − ∑ ασ. 2 i=1 i i

(5)

Proof. It is well known [21] that the solution of fractional differential equation (3) can be written as t

x(t) = ∫ 0

(t − s)q−1 g(s)ds + b0 + b1 t, Γ(q)

(6)

where b0 , b1 ∈ ℝ are arbitrary constants. From (6), we have t

x (t) = ∫ 󸀠

0

(t − s)q−2 g(s)ds + b1 . Γ(q − 1)

(7)

Using the boundary conditions (2) in (6) and (7), we find that σi

1 s

0

0 0

(σ − s)q−2 1 m−2 (s − w)q−2 b1 = { ∑ δi ∫ i g(s)ds − ∫ ∫ g(w)dwds − μ2 }, Λ1 i=1 Γ(q − 1) Γ(q − 1) σi

(8)

1 s

m−2 Λ (σ − s)q−2 (s − w)q−2 b0 = − 3 { ∑ δi ∫ i g(s)ds − ∫ ∫ g(w)dwds − μ2 } Λ1 Λ2 i=1 Γ(q − 1) Γ(q − 1)

+

m−2

0 σi

0 0

1 s

(σ − s)q−1 (s − w)q−1 1 g(s)ds − ∫ ∫ g(w)dwds − μ1 }, { ∑ αi ∫ i Λ2 i=1 Γ(q) Γ(q) 0

(9)

0 0

where Λk (k = 1, 2, 3) are given by (5). Substituting the values of b1 and b0 given by (8) and (9), respectively, in (6), we obtain the required integral equation (4). By direct computation, we can complete the converse. This completes the proof.

274 | B. Ahmad et al. In the sequel, denote by X = {x | x ∈ C([0, 1], ℝ) and c Dβ x ∈ C([0, 1], ℝ)} the space equipped with the norm ‖x‖X = supt∈[0,1] |x(t)|+supt∈[0,1] |c Dβ x(t)|. Notice that (X, ‖⋅‖X ) is a Banach space [22]. Introduce an operator F : X → X by t

F(x)(t) = ∫ 0

(t − s)q−1 f (s, x(s), c Dβ x(s))ds Γ(q) σi

(Λ t − Λ3 ) m−2 (σ − s)q−2 { ∑ δi ∫ i f (s, x(s), c Dβ x(s))ds + 2 Λ1 Λ2 Γ(q − 1) i=1 1 s

− ∫∫ 0 0

0

(s − w)q−2 f (w, x(w), c Dβ x(w))dwds − μ2 } Γ(q − 1) σi

(σ − s)q−1 1 m−2 + { ∑ αi ∫ i f (s, x(s), c Dβ x(s))ds Λ2 i=1 Γ(q) 1 s

− ∫∫ 0 0

(10)

0

(s − w)q−1 f (w, x(w), c Dβ x(w))dwds − μ1 } Γ(q)

and observe that the problem (1)–(2) has solutions if and only if the operator (10) has fixed points. Furthermore, as argued in [21], we have t

c β

D (Fx)(t) = ∫ 0

(t − s)q−β−1 f (s, x(s), c Dβ x(s))ds Γ(q − β) σi

m−2 (σ − s)q−2 t 1−β + { ∑ δi ∫ i f (s, x(s), c Dβ x(s))ds Λ1 Γ(2 − β) i=1 Γ(q − 1) 1 s

− ∫∫ 0 0

(11)

0

(s − w)q−2 f (w, x(w), c Dβ x(w))dwds − μ2 }. Γ(q − 1)

For convenience, we set the notations: σiq−1 |Λ2 t − Λ3 | m−2 1 1 + max { ∑ |δi | + } Ω= Γ(q + 1) t∈[0,1] |Λ1 Λ2 | Γ(q) Γ(q + 1) i=1 + Ω1 = ε1 =

σiq 1 m−2 1 { ∑ |αi | + }, |Λ2 | i=1 Γ(q + 1) Γ(q + 2)

m−2 σ q−1 1 1 1 + { ∑ |δi | i + }, Γ(q − β + 1) |Λ1 |Γ(2 − β) i=1 Γ(q) Γ(q + 1) 1−γ

1−γ 1 ( ) Γ(q) q − γ

|Λ2 t − Λ3 |ε3 t∈[0,1] |Λ2 |

+ max

(12) (13)

Nonlocal fractional integro-multipoint boundary value problems | 275 1−γ

+

1−γ 1 ( ) |Λ2 |Γ(q) q − γ

m−2

q−γ

{ ∑ |αi |σi i=1

1−γ

ε2 =

1−γ 1 ( ) Γ(q − β) q − β − γ

ε3 =

1−γ 1 ( ) |Λ1 |Γ(q − 1) q − γ − 1

+

+

q−1 }, q−γ+1

(14)

ε3 , Γ(2 − β)

1−γ

m−2

(15) q−γ−1

{ ∑ |δi |σi i=1

+

1 }. q−γ

(16)

3 Existence results In this section, we discuss the existence of solutions for the problem (1)–(2). Our first existence result is based on Schauder’s fixed point theorem. Theorem 3.1. Assume that f : [0, 1] × ℝ × ℝ → ℝ is a continuous function satisfying the assumption: 1

(A1 ) |f (t, x, y)| ≤ ϕ(t) + λ1 |x|ρ1 + λ2 |y|ρ2 , ∀(t, x, y) ∈ [0, 1] × ℝ × ℝ, ϕ ∈ L γ ([0, 1], ℝ+ ) with 1

1

‖ϕ‖ = (∫0 |ϕ(s)| γ ds)γ , γ ∈ (0, q − 1), λi ≥ 0, 0 ≤ ρi < 1, i = 1, 2.

Then the boundary value problem (1)–(2) has at least one solution on [0, 1]. Proof. Let us consider Br = {x ∈ X : ‖x‖X ≤ r} with r > 0 to be specified later. Observe that Br is a closed, bounded and convex subset of the Banach space X. It will be shown that there exists r > 0 such that the operator F maps Br into Br . For x ∈ Br , we have 󵄨 󵄨󵄨 󵄨󵄨(Fx)(t)󵄨󵄨󵄨 t

≤∫ 0

(t − s)q−1 󵄨ρ 󵄨 󵄨ρ 󵄨 [ϕ(s) + λ1 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(s)󵄨󵄨󵄨 2 ]ds Γ(q) σi

|Λ t − Λ3 | m−2 (σ − s)q−2 󵄨ρ 󵄨 󵄨ρ 󵄨 + max 2 [ ∑ |δi | ∫ i [ϕ(s) + λ1 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(s)󵄨󵄨󵄨 2 ]ds t∈[0,1] |Λ1 Λ2 | Γ(q − 1) i=1 1 s

+ ∫∫ 0 0

0

(s − u)q−2 󵄨ρ 󵄨 󵄨ρ 󵄨 [ϕ(u) + λ1 󵄨󵄨󵄨x(u)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(u)󵄨󵄨󵄨 2 ]duds + |μ2 |] Γ(q − 1) σi

(σ − s)q−1 1 m−2 󵄨ρ 󵄨 󵄨ρ 󵄨 + [ ∑ |αi | ∫ i [ϕ(s) + λ1 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(s)󵄨󵄨󵄨 2 ]ds |Λ2 | i=1 Γ(q) 1 s

+ ∫∫ 0 0

0

(s − u)q−1 󵄨ρ 󵄨 󵄨ρ 󵄨 [ϕ(u) + λ1 󵄨󵄨󵄨x(u)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(u)󵄨󵄨󵄨 2 ]duds + |μ1 |] Γ(q)

276 | B. Ahmad et al. 1−γ



‖ϕ‖ 1 − γ ( ) Γ(q) q − γ

λ1 r ρ1 + λ2 r ρ2 m−2 1 { ∑ |αi |σiq + + |Λ2 |} |Λ2 |Γ(q + 1) i=1 q+1

+

1−γ

|Λ2 t − Λ3 | ‖ϕ‖ 1−γ [ ( ) t∈[0,1] |Λ1 Λ2 | Γ(q − 1) q − γ − 1

+ max ρ1

m−2

q−γ−1

+

{ ∑ |δi |σi i=1

1 } q−γ

m−2

ρ2

+

λ1 r + λ2 r 1 { ∑ |δi |σiq−1 + } + |μ2 |] Γ(q) q i=1

+

‖ϕ‖ 1 − γ 1 [ ( ) |Λ2 | Γ(q) q − γ

1−γ

m−2

q−γ

{ ∑ |αi |σi i=1

1 } + |μ1 |] q−γ+1

+

and 󵄨 󵄨󵄨 c β 󵄨󵄨( D Fx)(t)󵄨󵄨󵄨 t

≤∫ 0

+

(t − s)q−β−1 󵄨ρ 󵄨 󵄨ρ 󵄨 [ϕ(s) + λ1 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(s)󵄨󵄨󵄨 2 ]ds Γ(q − β) σi

m−2 (σ − s)q−2 t 1−β 1 󵄨ρ 󵄨 󵄨ρ 󵄨 [ ∑ |δi | ∫ i [ϕ(s) + λ1 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(s)󵄨󵄨󵄨 2 ]ds |Λ1 | Γ(2 − β) i=1 Γ(q − 1) 1

s

+ ∫(∫ 0



0

0

q−2

(s − u) 󵄨ρ 󵄨 󵄨ρ 󵄨 [ϕ(u) + λ1 󵄨󵄨󵄨x(u)󵄨󵄨󵄨 1 + λ2 󵄨󵄨󵄨c Dβ x(u)󵄨󵄨󵄨 2 ]du)ds + |μ2 |] Γ(q − 1) 1−γ

‖ϕ‖ 1−γ ( ) Γ(q − β) q − β − γ + +

+

λ 1 r ρ1 + λ 2 r ρ2 Γ(q − β + 1)

1−γ

‖ϕ‖ 1−γ 1 1 [ ( ) |Λ1 | Γ(2 − β) Γ(q − 1) q − γ − 1 ρ1

m−2

q−γ−1

{ ∑ |δi |σi i=1

+

1 } q−γ

m−2

ρ2

λ1 r + λ2 r 1 { ∑ |δi |σiq−1 + } + |μ2 |]. Γ(q) q i=1

From the above inequalities, we obtain ‖Fx‖X ≤ L + M[λ1 r ρ1 + λ2 r ρ2 ], where L=

1−γ

‖ϕ‖ 1 − γ ( ) Γ(q) q − γ

1−γ

+

‖ϕ‖ 1−γ ( ) Γ(q − β) q − β − γ

1−γ

|Λ2 t − Λ3 | ‖ϕ‖ 1−γ [ ( ) t∈[0,1] |Λ1 Λ2 | Γ(q − 1) q − γ − 1

+ max

1−γ

+

‖ϕ‖ 1 − γ 1 [ ( ) |Λ2 | Γ(q) q − γ

m−2

q−γ

{ ∑ |αi |σi i=1

+

m−2

q−γ−1

{ ∑ |δi |σi i=1

+

1 } + |μ1 |] q−γ+1

1 } + |μ2 ] q−γ

Nonlocal fractional integro-multipoint boundary value problems | 277 1−γ

+

‖ϕ‖ 1−γ 1 1 [ ( ) |Λ1 | Γ(2 − β) Γ(q − 1) q − γ − 1

m−2

q−γ−1

{ ∑ |δi |σi i=1

+

1 } + |μ2 |] q−γ

and M=

m−2 1 1 1 { ∑ |αi |σiq + + |Λ2 |} + |Λ2 |Γ(q + 1) i=1 q+1 Γ(q − β + 1)

+

m−2 |Λ t − Λ3 | 1 1 1 ( max 2 + ){ ∑ |δi |σiq−1 + }. Γ(q)|Λ1 | t∈[0,1] Λ2 | Γ(2 − β) q i=1 1

1

Let r be a positive number such that r ≥ max{3L, (3Mλ1 ) 1−ρ1 , (3Mλ2 ) 1−ρ2 }. Then, for any x ∈ Br , it follows that ‖Fx‖X ≤ L + M[λ1 r ρ1 + λ2 r ρ2 ] ≤

r r r + + = r. 3 3 3

In view of the continuity of f , it is easy to verify that F is continuous. Next, for every bounded subset B̄ of X, we can show that the families F(B)̄ and c β D F(B)̄ are equicontinuous, where B̄ is any bounded subset of X. Since f is continuous, we can assume that |f (t, x(t), c Dβ x(t))| ≤ N for any x ∈ B̄ and t ∈ [0, 1]. Now, for 0 ≤ t1 < t2 ≤ 1, we have 󵄨 󵄨󵄨 󵄨󵄨(Fx)(t2 ) − (Fx)(t1 )󵄨󵄨󵄨

t1 󵄨󵄨 󵄨󵄨 1 󵄨 ≤ 󵄨󵄨 ∫[(t2 − s)q−1 − (t1 − s)q−1 ]f (s, x(s), c Dβ x(s))ds 󵄨󵄨 Γ(q) 󵄨 0 t2 󵄨󵄨 1 󵄨󵄨 q−1 c β + ∫(t2 − s) f (s, x(s), D x(t))ds󵄨󵄨󵄨 󵄨󵄨 Γ(q) 󵄨 t1

+

1 s

+ ∫∫ 0 0



σi

(σ − s)q−2 󵄨󵄨 |t2 − t1 | m−2 󵄨 c β [ ∑ |δi | ∫ i 󵄨󵄨f (s, x(s), D x(s))󵄨󵄨󵄨ds |Λ1 | Γ(q − 1) i=1 0

q−2

(s − u) 󵄨󵄨 󵄨 c β 󵄨f (s, x(s), D x(s))󵄨󵄨󵄨ds] Γ(q − 1) 󵄨

σ q−1 N|t2 − t1 | m−2 1 N 󵄨󵄨 q󵄨 q q [ ∑ |δi | i + ] 󵄨󵄨2(t2 − t1 ) + t1 − t2 󵄨󵄨󵄨 + Γ(q + 1) |ρ1 | Γ(q) Γ(q + 1) i=1

and N q−β 󵄨 q−β 󵄨 󵄨󵄨 c β 󵄨󵄨 c β q−β + t1 − t2 󵄨󵄨󵄨 󵄨󵄨( D Fx)(t2 ) − ( D Fx)(t1 )󵄨󵄨󵄨 ≤ 󵄨2(t − t ) Γ(q − β + 1) 󵄨 2 1 +

1−β 1−β σ q−1 N |t2 − t1 | m−2 1 [ ∑ |δi | i + ]. |Λ1 | Γ(2 − β) Γ(q) Γ(q + 1) i=1

278 | B. Ahmad et al. In consequence, we obtain 󵄨 󵄨 󵄨c β 󵄨󵄨 c β 󵄨󵄨(Fx)(t2 ) − (Fx)(t1 )󵄨󵄨󵄨 + 󵄨󵄨󵄨( D Fx)(t2 ) − ( D Fx)(t1 )󵄨󵄨󵄨 → 0

as t2 → t1 ,

independent of x ∈ B.̄ Therefore the operator F : Br → Br is equicontinuous and uniformly bounded. Hence, by the Arzelà–Ascoli theorem, it follows that F(Br ) is relatively compact in X. Therefore, Schauder’s fixed point theorem applies and consequently the problem (1)–(2) has at least one solution on [0, 1]. The proof is completed. Corollary 3.2. Let f : [0, 1] × ℝ × ℝ → ℝ be a continuous function such that |f (t, x, y)| ≤ ν(t), ∀(t, x, y) ∈ [0, 1] × ℝ × ℝ, where ν ∈ C([0, 1], ℝ+ ). Then the boundary value problem (1)–(2) has at least one solution on [0, 1]. For ρ1 = ρ2 = 1, Theorem 3.1 takes the following form. Corollary 3.3. Assume that f : [0, 1] × ℝ × ℝ → ℝ is a continuous function satisfying the 1

assumption: |f (t, x, y)| ≤ ϕ(t)+λ1 |x|+λ2 |y|, ∀(t, x, y) ∈ [0, 1]×ℝ×ℝ, and ϕ ∈ L γ ([0, 1], ℝ+ ), γ ∈ (0, q − 1), λi ≥ 0, i = 1, 2. Then the problem (1)–(2) has at least one solution on [0, 1]. Our next existence result is based on Krasnoselskii’s fixed point theorem.

Lemma 3.4 (Krasnoselskii’s fixed point theorem [23]). Let Y be a closed, bounded, convex and nonempty subset of Banach space X. Let A, B be the operators such that (i) Ay1 + By2 ∈ Y whenever y1 , y2 ∈ Y; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists y3 ∈ Y such that y3 = Ay3 + By3 . Theorem 3.5. Let f : [0, 1]×ℝ×ℝ → ℝ be a continuous function satisfying the following conditions: (A2 ) |f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ ψ(t)(|x1 − x2 | + |y1 − y2 |) for t ∈ [0, 1], xi , yi ∈ ℝ, i = 1, 2 and γ ∈ (0, q − 1) and the function ψ : [0, 1] → ℝ+ is Lebesgue integrable with power γ1 , 1

1

1

that is, ψ ∈ L γ ([0, 1], ℝ+ ) with ‖ψ‖ = (∫0 |m(s)| γ ds)γ .

1

(A3 ) |f (t, x, y)| ≤ ψ(t), ∀(t, x, y) ∈ [0, 1] × ℝ × ℝ, and ψ ∈ L γ ([0, 1], ℝ+ ), γ ∈ (0, q − 1). Then the problem (1)–(2) has at least one solution on [0, 1] provided that 1−γ

‖ψ‖(ε1 −

1−γ 1 ( ) Γ(q) q − γ

+ ε2 −

1−γ

1−γ 1 ( ) Γ(q − β) q − β − γ

) < 1,

(17)

where ε1 , ε2 are defined by (14) and (15), respectively. Proof. Selecting ρ > L, we define Bρ = {x ∈ 𝒞 : ‖x‖X ≤ ρ} and introduce the operators A and B on Bρ as follows: t

A(x)(t) = ∫ 0

(t − s)q−1 f (s, x(s), c Dβ x(s))ds, Γ(q)

Nonlocal fractional integro-multipoint boundary value problems | 279 σi

(Λ t − Λ3 ) m−2 (σ − s)q−2 B(x)(t) = 2 f (s, x(s), c Dβ x(s))ds [ ∑ δi ∫ i Λ1 Λ2 Γ(q − 1) i=1 1 s

− ∫∫ 0 0

0

(s − w)q−2 f (w, x(w), c Dβ x(w))dwds − μ2 } Γ(q − 1) σi

(σ − s)q−1 1 m−2 { ∑ αi ∫ i f (s, x(s), c Dβ x(s))ds + Λ2 i=1 Γ(q) 1 s

− ∫∫ 0 0

0

(s − w)q−1 f (w, x(w), c Dβ x(w))dwds − μ1 }. Γ(q)

For any x, y ∈ Bρ , as in the proof of Theorem 3.1, it can be shown that ‖Ax + By‖X ≤ L < ρ. This shows that Ax + By ∈ Bρ . The operator A is completely continuous as in Theorem 3.1. Using the assumption (A2 ), it can be shown that the operator B is a contraction with the aid of (17). Thus all the assumptions of Lemma 3.4 are satisfied. Hence the conclusion of Lemma 3.4 implies that the problem (1)–(2) has at least one solution on [0, 1].

4 Uniqueness results In the following result, we establish the uniqueness of solutions for the problem (1)– (2) by applying Banach’s fixed point theorem together with Hölder’s inequality. Theorem 4.1. Let f : [0, 1] × ℝ × ℝ → ℝ be a continuous function satisfying the assumption (A2 ). Then there exists a unique solution for the problem (1)–(2) on [0, 1] if ‖ψ‖(ε1 + ε2 ) < 1, where ε1 , ε2 are defined by (14) and (15), respectively. Proof. For x, y ∈ X and for each t ∈ [0, 1], by Hölder’s inequality, we have 󵄨 󵄨󵄨 󵄨󵄨(Fx)(t) − (Fy)(t)󵄨󵄨󵄨 t

≤∫ 0

|Λ t − Λ3 | (t − s)q−1 󵄨 󵄨 󵄨 󵄨 ψ(s)(󵄨󵄨󵄨x(s) − y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(s) − c Dβ y(s)󵄨󵄨󵄨)ds + max 2 t∈[0,1] |Λ1 Λ2 | Γ(q) m−2

σi

i=1

0

× [ ∑ |δi | ∫ 1

s

+ ∫(∫ 0

0

(σi − s)q−2 󵄨 󵄨 󵄨 󵄨 ψ(s)(󵄨󵄨󵄨x(s) − y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(s) − c Dβ y(s)󵄨󵄨󵄨)ds Γ(q − 1)

(s − u)q−1 󵄨 󵄨 󵄨 󵄨 ψ(u)(󵄨󵄨󵄨x(u) − y(u)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(u) − c Dβ y(u)󵄨󵄨󵄨)du)ds] Γ(q)

280 | B. Ahmad et al. σi

(σ − s)q−1 1 m−2 󵄨 󵄨 󵄨 󵄨 + [ ∑ |αi | ∫ i ψ(s)(󵄨󵄨󵄨x(s) − y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(s) − c Dβ y(s)󵄨󵄨󵄨)ds |Λ2 | i=1 Γ(q) 1 s

+ ∫∫ 0 0

≤[

0

(s − u)q−1 󵄨 󵄨 󵄨 󵄨 ψ(u)(󵄨󵄨󵄨x(u) − y(u)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(u) − c Dβ y(u)󵄨󵄨󵄨)duds] Γ(q) 1−γ

1−γ 1 ( ) Γ(q) q − γ

+ max

t∈[0,1]

1−γ

1−γ 1 ( ) |Λ2 |Γ(q) q − γ

+

|Λ2 t − Λ3 |ε3 |Λ2 |

m−2

q−γ

{ ∑ |αi |σi i=1

+

q−1 }]‖ψ‖‖x − y‖X q−γ+1

= ε1 ‖ψ‖‖x − y‖X . Similarly, we have 󵄨 󵄨󵄨 c β c β 󵄨󵄨( D Fx)(t) − ( D Fy)(t)󵄨󵄨󵄨 t

≤∫ 0

(t − s)q−β−1 1 t 1−β 󵄨 󵄨 󵄨 󵄨 ψ(s)(󵄨󵄨󵄨x(s) − y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(s) − c Dβ y(s)󵄨󵄨󵄨)ds + Γ(q − β) |Λ1 | Γ(2 − β) m−2

σi

i=1

0

× [ ∑ |δi | ∫ 1

s

+ ∫(∫ 0

≤[

0

(σi − s)q−2 󵄨 󵄨 󵄨 󵄨 ψ(s)(󵄨󵄨󵄨x(s) − y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(s) − c Dβ y(s)󵄨󵄨󵄨)ds Γ(q − 1)

(s − u)q−2 󵄨 󵄨 󵄨 󵄨 ψ(u)(󵄨󵄨󵄨x(u) − y(u)󵄨󵄨󵄨 + 󵄨󵄨󵄨c Dβ x(u) − c Dβ y(u)󵄨󵄨󵄨)du)ds] Γ(q − 1) 1−γ

1−γ 1 ( ) Γ(q − β) q − β − γ

+

ε3 ]‖ψ‖‖x − y‖X Γ(2 − β)

= ε2 ‖ψ‖‖x − y‖X . From the above inequalities, we deduce that ‖Fx − Fy‖X ≤ (ε1 + ε2 )‖ψ‖‖x − y‖X . In view of the given condition, ‖ψ‖(ε1 + ε2 ) < 1, it follows that the mapping F is a contraction. Hence, by the Banach fixed point theorem, F has a unique fixed point which is a unique solution of problem (1)–(2). This completes the proof. Corollary 4.2. Suppose that the continuous function f : [0, 1] × ℝ × ℝ → ℝ satisfies the following assumption: (A2 )󸀠 |f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ L(|x1 − x2 | + |y1 − y2 |), for t ∈ [0, 1], xi , yi ∈ ℝ, i = 1, 2 and L > 0 is a constant. If L(Ω + Ω1 ) < 1, where Ω, Ω1 are defined by (12) and (13), respectively, then the boundary value problem (1)–(2) has a unique solution on [0, 1].

Nonlocal fractional integro-multipoint boundary value problems | 281

5 Applications (i) Our results correspond to fractional-order Duffing equation if we take q = 2β (0 < β < 1) in (1) with f (t, x(t), c Dβ x(t)) = a0 x(t) − b0 x3 x(t) − k0 c Dβ x(t) + r0 cos(ω0 t),

where a0 , b0 , k0 , amplitude r0 , angular frequency ω0 and order β are parameters; for details, see [24].

(ii) Letting f (t, x(t), c Dβ x(t)) =

1 (e(t) − x(t) − RL c Dβ x(t)) in (1) with LC

q = 2β (0 < β < 1),

equation (1) corresponds to a fractional-order differential equation satisfied by the voltage function x(t) in RLC circuit with current e(t) [25].

1

(iii) Our results correspond to the conserved type conditions: ∫0 x(s)ds 1 ∫0 x󸀠 (s)ds 1 ∫0 x󸀠 (s)ds

= 0 (μ1 = 0 =

=

1 μ2 ) and non-conserved type conditions: ∫0 x(s)ds + μ1

0, = 0,

+ μ2 = 0 (μ1 ≠ 0 ≠ μ2 ) provided that there is no contribution from the

interior positions of the given domain.

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Index 2D prediction error 199 2D Yule–Walker equation 197 a memory regeneration effect 91 add a pseudo-pole 140 add a pseudo-zero 136, 140 adding a pseudo-pole 135 ARMA models 157 Autoregressive Moving Average 157 bifurcations 75 calculus on time scales 150 Caputo definition 66 Caputo fractional derivative 272 chaotification 233 circuits design 233 collocation spectral method 211 conserved type conditions 281 constant phase element 96 correlation 163, 164 delta derivative 151 – fractional 151 delta Laplace transform 163, 164 delta system 150 difference equations 149 discrete convolution 121 discrete fractional calculus 261, 262 discrete-time – differential linear systems 166 – steady state response 166 – transient response 169 – exponential 149 – linear systems 149 – Mittag-Leffler function 173 double Levinson recursion 158 electrocardiogram signal 193 electroencephalogram signal 194 energy harvesting 63 existence results 275 ferroelectric 64 finite impulse response 169 Fourier transform 149, 167 fractional derivative 7, 179 https://doi.org/10.1515/9783110571929-012

fractional derivatives of the impulses 155 fractional derivatives of the power functions 156 fractional derivatives of the unit steps 155 fractional difference mask 261 fractional Jacobi functions 213 fractional Jacobi–Gauss quadrature 215 fractional Legendre functions 214 fractional linear prediction 179 fractional logistic equation 141 fractional order see also constant phase element, 33 – filter 33 – bandpass 33 – notch 33 – fractance 33 – fractor 33 – α 33 – capacitive fractor 33 – CPZ 33 – inductive fractor 33 – phase band 33 – GIC 33 – GIC resistor 33 – ladder 33 – resonator 33 – Q factor 33 – sensitivity 33 fractional order Duffing equation 281 fractional systems 119, 123 fractional tau method 217 fractional viscoelasticity 70 fractional-order inductance 65 frequency response 168 Galerkin spectral method 209 generalized convolution 121, 124 greyscale image 199, 201 Grünwald–Letnikov definition 184 Grünwald–Letnikov derivative 130, 152 Harrod–Domar model 4, 5 Hilger circle 159 – stability 173 homoclinic orbit 75 hysteretic damping 63 image enhancement 261

284 | Index

impulse response 166 infinite impulse response 169 initial conditions 153, 174 jump formula 154 Krasnoselskii’s fixed point theorem 278 Laplace transform 151 – discrete-time 151, 159 Laurent series 120 linear prediction 179 long memory 2–7, 12, 15, 17, 18, 20, 23–26 macroeconomics 1, 4–12, 14–17, 20, 21, 23–26 mean-squared error 186 mechanical resonators 64 mechanical vibration 68 Mikusiński’s calculus 120 Mittag-Leffler function 121, 130 Müntz–Legendre polynomials 214 nabla and delta exponentials 159 nabla derivative 151 – examples 152 – fractional 151 – properties 154 nabla exponential 159 nabla Laplace transform 161 – examples 162 – properties 161 – uniqueness 164 non-conserved type conditions 281 nonlinear dynamics 233 nonlinear stiffnes 70

nonlocal integro-multipoint boundary conditions 272 one-dimensional signal 179, 181 operational calculus 120 optimal predictor design 181, 183, 196, 197 Petrov–Galerkin spectral method 210 piezo-electric 64 power-law memory 4, 7, 8, 10, 12, 14–16, 20, 21, 24–26 prediction gain 186, 199 recursive algorithm 135, 139 remove a pseudo-pole 136, 140 remove a pseudo-zero 136 Riemann–Liouville fractional integral 272 signal processing 149, 179 spectral methods 207 speech signal 192 stability 173 tau spectral method 210 technological growth rate 10–12 time scale 151 transfer function 149, 166 two-dimensional signal 179, 195 uniqueness results 279 Yule–Walker equation 183 Z transform 149