*198*
*56*
*27MB*

*English*
*Pages 302
[304]*
*Year 2023*

Table of contents :

Cover

Title

Copyright

End User License Agreement

Contents

Preface

List of Contributors

Modified Adaptive Synchronization and Anti-Synchronization Method for Fractional OrderChaotic Systems with Uncertain Parameters

Improved Generalized Differential TransformMethod for a Class of Linear NonhomogeneousOrdinary Fractional Differential Equations

Incomplete K2-function

Some Results on Incomplete HypergeometricFunctions

Transcendental Bernstein Series: Interpolation andApproximation

Some Sufficient Conditions for Uniform Convexityof Normalized 1F2 Function

From Abel Continuity Theorem to Paley-WienerTheorem

A New Class of Truncated Exponential-Gould-Hopper-based Genocchi Polynomials

Computational Preconditioned Gauss-Seidel viaHalf-Sweep Approximation to Caputo's Time-Fractional Differential Equations

Krasnoselskii-type Theorems for MonotoneOperators in Ordered Banach Algebra withApplications in Fractional Differential Equationsand Inclusion

General Fractional Order Quadratic FunctionalIntegral Equations: Existence, Properties ofSolutions, and Some of their Applications

Non-linear Set-Valued Delay Functional IntegralEquations of Volterra-Stieltjes Type: Existenceof Solutions, Continuous Dependence and Applications

Certain Saigo Fractional Derivatives of ExtendedHypergeometric Functions

Some Erdélyi-Kober Fractional Integrals of theExtended Hypergeometric Functions

On Solutions of the Kinetic Model by SumuduTransform

Subject Index

Back Cover

Cover

Title

Copyright

End User License Agreement

Contents

Preface

List of Contributors

Modified Adaptive Synchronization and Anti-Synchronization Method for Fractional OrderChaotic Systems with Uncertain Parameters

Improved Generalized Differential TransformMethod for a Class of Linear NonhomogeneousOrdinary Fractional Differential Equations

Incomplete K2-function

Some Results on Incomplete HypergeometricFunctions

Transcendental Bernstein Series: Interpolation andApproximation

Some Sufficient Conditions for Uniform Convexityof Normalized 1F2 Function

From Abel Continuity Theorem to Paley-WienerTheorem

A New Class of Truncated Exponential-Gould-Hopper-based Genocchi Polynomials

Computational Preconditioned Gauss-Seidel viaHalf-Sweep Approximation to Caputo's Time-Fractional Differential Equations

Krasnoselskii-type Theorems for MonotoneOperators in Ordered Banach Algebra withApplications in Fractional Differential Equationsand Inclusion

General Fractional Order Quadratic FunctionalIntegral Equations: Existence, Properties ofSolutions, and Some of their Applications

Non-linear Set-Valued Delay Functional IntegralEquations of Volterra-Stieltjes Type: Existenceof Solutions, Continuous Dependence and Applications

Certain Saigo Fractional Derivatives of ExtendedHypergeometric Functions

Some Erdélyi-Kober Fractional Integrals of theExtended Hypergeometric Functions

On Solutions of the Kinetic Model by SumuduTransform

Subject Index

Back Cover

- Author / Uploaded
- Praveen Agarwal
- Shilpi Jain

- Similar Topics
- Mathematics
- Analysis

Advances in Special Functions of Fractional Calculus: Special Functions in Fractional Calculus and Their Applications in Engineering Edited by

Praveen Agarwal Anand International College of Engineering, India

&

Shilpi Jain Poornima College of Engineering, India

Advances in Special Functions of Fractional Calculus: Special Functions in Fractional Calculus and Their Applications in Engineering Editors: Praveen Agarwal and Shilpi Jain ISBN (Online): 978-981-5079-33-3 ISBN (Print): 978-981-5079-34-0 ISBN (Paperback): 978-981-5079-35-7 © 2023, Bentham Books imprint. Published by Bentham Science Publishers Pte. Ltd. Singapore. All Rights Reserved. First published in 2023.

BSP-EB-PRO-9789815079333-TP-288-TC-15-PD-20230411

BENTHAM SCIENCE PUBLISHERS LTD. End User License Agreement (for non-institutional, personal use) This is an agreement between you and Bentham Science Publishers Ltd. Please read this License Agreement carefully before using the ebook/echapter/ejournal (“Work”). Your use of the Work constitutes your agreement to the terms and conditions set forth in this License Agreement. If you do not agree to these terms and conditions then you should not use the Work. Bentham Science Publishers agrees to grant you a non-exclusive, non-transferable limited license to use the Work subject to and in accordance with the following terms and conditions. This License Agreement is for non-library, personal use only. For a library / institutional / multi user license in respect of the Work, please contact: [email protected].

Usage Rules: 1. All rights reserved: The Work is the subject of copyright and Bentham Science Publishers either owns the Work (and the copyright in it) or is licensed to distribute the Work. You shall not copy, reproduce, modify, remove, delete, augment, add to, publish, transmit, sell, resell, create derivative works from, or in any way exploit the Work or make the Work available for others to do any of the same, in any form or by any means, in whole or in part, in each case without the prior written permission of Bentham Science Publishers, unless stated otherwise in this License Agreement. 2. You may download a copy of the Work on one occasion to one personal computer (including tablet, laptop, desktop, or other such devices). You may make one back-up copy of the Work to avoid losing it. 3. The unauthorised use or distribution of copyrighted or other proprietary content is illegal and could subject you to liability for substantial money damages. You will be liable for any damage resulting from your misuse of the Work or any violation of this License Agreement, including any infringement by you of copyrights or proprietary rights.

Disclaimer: Bentham Science Publishers does not guarantee that the information in the Work is error-free, or warrant that it will meet your requirements or that access to the Work will be uninterrupted or error-free. The Work is provided "as is" without warranty of any kind, either express or implied or statutory, including, without limitation, implied warranties of merchantability and fitness for a particular purpose. The entire risk as to the results and performance of the Work is assumed by you. No responsibility is assumed by Bentham Science Publishers, its staff, editors and/or authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products instruction, advertisements or ideas contained in the Work.

Limitation of Liability: In no event will Bentham Science Publishers, its staff, editors and/or authors, be liable for any damages, including, without limitation, special, incidental and/or consequential damages and/or damages for lost data and/or profits arising out of (whether directly or indirectly) the use or inability to use the Work. The entire liability of Bentham Science Publishers shall be limited to the amount actually paid by you for the Work.

General: 1. Any dispute or claim arising out of or in connection with this License Agreement or the Work (including non-contractual disputes or claims) will be governed by and construed in accordance with the laws of Singapore. Each party agrees that the courts of the state of Singapore shall have exclusive jurisdiction to settle any dispute or claim arising out of or in connection with this License Agreement or the Work (including non-contractual disputes or claims). 2. Your rights under this License Agreement will automatically terminate without notice and without the

need for a court order if at any point you breach any terms of this License Agreement. In no event will any delay or failure by Bentham Science Publishers in enforcing your compliance with this License Agreement constitute a waiver of any of its rights. 3. You acknowledge that you have read this License Agreement, and agree to be bound by its terms and conditions. To the extent that any other terms and conditions presented on any website of Bentham Science Publishers conflict with, or are inconsistent with, the terms and conditions set out in this License Agreement, you acknowledge that the terms and conditions set out in this License Agreement shall prevail. Bentham Science Publishers Pte. Ltd. 80 Robinson Road #02-00 Singapore 068898 Singapore Email: [email protected]

BSP-EB-PRO-9789815079333-TP-288-TC-15-PD-20230411

CONTENTS PREFACE ................................................................................................................................................... i LIST OF CONTRIBUTORS ...................................................................................................................... iv CHAPTER 1 MODIFIED ADAPTIVE SYNCHRONIZATION AND ANTI-SYNCHRONIZATION METHOD FOR FRACTIONAL ORDER CHAOTIC SYSTEMS WITH UNCERTAIN PARAMETERS ……………………………………………………………………………………………………………………… 1 S. K. Agrawa, Lalit Batra, V. Mishra and D. Datta 1. INTRODUCTION ......................................................................................................................... 1 2. PRELIMINARIES, THEOREM, PROBLEM DESCRIPTION AND CONTROL DESIGN . 5 2.1. Fractional Calculus ............................................................................................................. 5 2.2. Problem Description .......................................................................................................... 6 2.3. Modified Adaptive Synchronization Controller Des-Ign For Synchronization ................... 7 2.4. Modified Adaptive Synchronization Controller Design for Anti-Synchronization ............. 7 3. SYSTEM'S DESCRIPTION ........................................................................................................ 9 3.1. Fractional Order 3D Autonomous Chaotic System ............................................................. 9 3.2. Fractional Order Novel 3d Autonomous Chaotic System ................................................... 10 4. CIRCUIT IMPLEMENTATIONS OF THE CHAOTIC SYSTEM .......................................... 11 4.1. Numerical Simulation and Results ...................................................................................... 13 5. ANTI-SYNCHRONIZATION BETWEEN FRACTIONAL-ORDER 3D AUTONOMOUS AND NOVEL 3D AUTONOMOUS CHAOTIC SYSTEMS USING MODIFIED ADAPTIVE CONTROL METHOD………………………………………. 23 5.1. Numerical Simulation and Results……………………………………………………….. 24 CONCLUSION .................................................................................................................................. 33 CONSENT FOR PUBLICATION ................................................................................................... 34 CONFLICT OF INTEREST ............................................................................................................ 34 ACKNOWLEDGEMENT ................................................................................................................ 34 REFERENCES .................................................................................................................................. 34 CHAPTER 2 IMPROVED GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR A CLASS OF LINEAR NONHOMOGENEOUS ORDINARY FRACTIONAL DIFFERENTIAL EQUATIONS….……………………………………………………………………………………………………………………… 39 İ. Onur KIYMAZ, and Ayşegül ÇETİNKAYA 1. INTRODUCTION ......................................................................................................................... 39 2. IMPROVED GENERALIZED DIFFERENTIAL TRANSFORM METHOD ........................ 41 3. APPLICATION OF THE METHOD .......................................................................................... 43 CONCLUDING REMARKS AND OBSERVATIONS .................................................................. 47 CONSENT FOR PUBLICATION ................................................................................................... 49 CONFLICT OF INTEREST ............................................................................................................ 49 ACKNOWLEDGEMENT ................................................................................................................ 49 REFERENCES .................................................................................................................................. 49 CHAPTER 3 INCOMPLETE 𝑲𝟐-FUNCTION ………………………………………………………… Dharmendra Kumar Singh and Vijay Laxmi Verma 1. INTRODUCTION ......................................................................................................................... 2. SPECIAL CASES .......................................................................................................................... 3. RELATIONS WITH RIEMANN - LIOUVILLE FRACTIONAL CALCULUS OPERATORS ................................................................................................................................... CONCLUSION ..................................................................................................................................

52 52 56 58 60

CONSENT FOR PUBLICATION ................................................................................................... 61 CONFLICT OF INTEREST ............................................................................................................ 61 ACKNOWLEDGEMENT ................................................................................................................ 61 REFERENCES .................................................................................................................................. 61 CHAPTER 4 SOME RESULTS ON INCOMPLETE HYPERGEOMETRIC FUNCTIONS …….... 62 Dharmendra Kumar Singh and Geeta Yadav 1. INTRODUCTION ......................................................................................................................... 62 1.1. Incomplete Hypergeometric Function ................................................................................. 62 1.2 Incomplete Wright Function ................................................................................................ 63 1.3. Hypergeometric Function ................................................................................................... 64 2. THEOREMS .................................................................................................................................. 64 CONCLUSION .................................................................................................................................. 71 CONSENT FOR PUBLICATION ................................................................................................... 71 CONFLICT OF INTEREST ............................................................................................................ 72 ACKNOWLEDGEMENT ................................................................................................................ 72 REFERENCES .................................................................................................................................. 72 CHAPTER 5 TRANSCENDENTAL BERNSTEIN SERIES: INTERPOLATION AND APPROXIMATION…………………………………………………………………………………………………………...…… 73 Z. Avazzadeh, H. Hassani, J.A. Tenreiro Machado, P. Agarwal and E. Naraghirad 1. INTRODUCTION ......................................................................................................................... 73 2. BERNSTEIN POLYNOMIALS ................................................................................................... 74 2.1. Properties of BP .................................................................................................................. 75 3. TRANSCENDENTAL BERNSTEIN SERIES ............................................................................ 76 3.1 Properties of TBS................................................................................................................. 77 3.2. Convergence Analysis ........................................................................................................ 79 4. NUMERICAL EXPERIMENTS .................................................................................................. 83 CONCLUSION .................................................................................................................................. 90 CONSENT FOR PUBLICATION ................................................................................................... 91 CONFLICT OF INTEREST ............................................................................................................ 91 ACKNOWLEDGEMENT ................................................................................................................ 91 REFERENCES .................................................................................................................................. 91 CHAPTER 6 SOME SUFFICIENT CONDITIONS FOR UNIFORM CONVEXITY OF NORMALIZED 1𝑭𝟐 FUNCTION ………………………………………………………………………………………………………..… 94 Deepak Bansal and Shilpi Jain 1. INTRODUCTION ......................................................................................................................... 94 2. SUFFICIENCY CONDITIONS FOR UNIFORMLY CONVEXITY ....................................... 101 3. SUFFICIENCY CONDITIONS FOR

Re1F2 a,b,c; z1/2 ......................................... 107

CONSENT FOR PUBLICATION ................................................................................................... 109 CONFLICT OF INTEREST ............................................................................................................ 109 ACKNOWLEDGEMENT ................................................................................................................ 109 REFERENCES .................................................................................................................................. 109 CHAPTER 7 FROM ABEL CONTINUITY THEOREM TO PALEY-WIENER THEOREM ……… 112 S. Yu, P. Agarwal and S. Kanemitsu 1. INTRODUCTION ......................................................................................................................... 112 2. PALEY-WIENER THEOREM .................................................................................................... 118 CONSENT FOR PUBLICATION ................................................................................................... 119

CONFLICT OF INTEREST ............................................................................................................ 119 ACKNOWLEDGEMENT ................................................................................................................ 119 REFERENCES .................................................................................................................................. 119 CHAPTER 8 A NEW CLASS OF TRUNCATED EXPONENTIAL-GOULD-HOPPER-BASED GENOCCHI POLYNOMIALS …..…………………………………………………………………………………………… 121 Ghazala Yasmin and Hibah Islahi 1. INTRODUCTION ......................................................................................................................... 121 2. VARIABLE TRUNCATED EXPONENTIAL-GOULD-HOPPER BASED GENOCCHI POLYNOMIALS ............................................................................................................................... 124 3. OPERATIONAL AND INTEGRAL REPRESENTATIONS .................................................... 128 4. GRAPHICAL REPRESENTATION AND ROOTS................................................................... 131 CONSENT FOR PUBLICATION ................................................................................................... 134 CONFLICT OF INTEREST ............................................................................................................ .134 ACKNOWLEDGEMENT ................................................................................................................ 134 REFERENCES .................................................................................................................................. 134 CHAPTER 9 COMPUTATIONAL PRECONDITIONED GAUSS-SEIDEL VIA HALF-SWEEP APPROXIMATION TO CAPUTO'S TIME-FRACTIONAL DIFFERENTIAL EQUATIONS……. 136 Andang Sunarto, Jumat Sulaiman and Jackel Vui Lung Chew 1. INTRODUCTION ......................................................................................................................... 137 2. IMPLICIT APPROXIMATION WITH CAPUTO'S TIME-FRACTIONAL .......................... 138 3. STABILITY ANALYSIS .............................................................................................................. 141 4. HALF-SWEEP PRECONDITIONED GAUSS-SEIDEL ........................................................... 143 4.1. Algorithm 1: HSPGS Iterative Method ............................................................................... 144 5. NUMERICAL EXPERIMENT .................................................................................................... 145 CONCLUDING REMARKS ............................................................................................................ 154 CONSENT FOR PUBLICATION ................................................................................................... 154 CONFLICT OF INTEREST ............................................................................................................ 154 ACKNOWLEDGEMENT ................................................................................................................ 154 REFERENCES .................................................................................................................................. 154 CHAPTER 10 KRASNOSELSKII-TYPE THEOREMS FOR MONOTONE OPERATORS IN ORDERED BANACH ALGEBRA WITH APPLICATIONS IN FRACTIONAL DIFFERENTIAL EQUATIONS AND INCLUSION …,,………………………………………………………………………………………… 157 Nayyar Mehmood and Niaz Ahmad 1. INTRODUCTION ......................................................................................................................... 157 2. KRASNOSELSKII-TYPE RESULTS FOR TWO MONOTONE SELF OPERATORS ........ 162 3. FIXED POINT RESULTS FOR SET VALUED MONOTONE MAPPINGS .......................... 166 3.1. A Generalization of Darbo's Fixed Point Theorem for Monotone Operators ...................... 169 3.2. Krasnoselskii-type Results For The Sum Of Two Monotone Multivalued Operators......... 172 4. APPLICATIONS ........................................................................................................................... 173 CONCLUSION .................................................................................................................................. 182 CONSENT FOR PUBLICATION ................................................................................................... 182 CONFLICT OF INTEREST ............................................................................................................ 182 ACKNOWLEDGEMENT ................................................................................................................ 182 REFERENCES .................................................................................................................................. 182 CHAPTER 11 GENERAL FRACTIONAL ORDER QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS: EXISTENCE, PROPERTIES OF SOLUTIONS, AND SOME OF THEIR APPLICATIONS……………………………………………………………………………………………………………………. 185

Ahmed M.A. El-Sayed and Hind H.G. Hashem 1. INTRODUCTION ......................................................................................................................... 185 2. PRELIMINARIES ........................................................................................................................ 187 3. EXISTENCE OF SOLUTIONS ................................................................................................... 191 3.1. Carathéodory Theorem for the Quadratic Integral Equation (1) ......................................... 191 4. SPECIAL CASES AND REMARKS ........................................................................................... 196 5. PROPERTIES OF SOLUTIONS ................................................................................................. 198 5.1. Uniqueness of Solutions of QFIE (1) .................................................................................. 199 5.2. Maximal and Minimal Solutions ......................................................................................... 201 5.3. Comparison Theorem.......................................................................................................... 205 5.4. 𝜹−Approximate Solutions ................................................................................................... 207 6. APPLICATIONS ........................................................................................................................... 212 6.1. Hybrid Functional 𝝓−Differential Equation of Fractional Order ........................................ 212 6.2. Pantograph Functional 𝝓−Differential Equation of Fractional Order ................................. 213 CONCLUSION .................................................................................................................................. 214 CONSENT FOR PUBLICATION ................................................................................................... 214 CONFLICT OF INTEREST ............................................................................................................ 214 ACKNOWLEDGEMENT ................................................................................................................ 215 REFERENCES .................................................................................................................................. 215 CHAPTER 12 NON-LINEAR SET-VALUED DELAY FUNCTIONAL INTEGRAL EQUATIONS OF VOLTERRASTIELTJES TYPE: EXISTENCE OF SOLUTIONS, CONTINUOUS DEPENDENCE AND APPLICATIONS…………………………………………………………………………………………………………… 219 A. M. A. El-Sayed, Sh. M Al-Issa and Y. M. Y. Omar 1. INTRODUCTION ......................................................................................................................... 219 2. PRELIMINARIES ........................................................................................................................ 221 3. SINGLE-VALUED PROBLEM ................................................................................................... 223 3.1. Existence of Solutions I ...................................................................................................... 223 4. UNIQUENESS OF THE SOLUTION ......................................................................................... 227 4.1. Continuous Dependence of the Solution ............................................................................. 288 4.1.1. Continuous Dependence on the Delay Functions (𝒕) ........................................................ 288 4.1.2. Continuous Dependence on the Functions (𝒕,𝒔)................................................................ 230 4.2. Application 1 ...................................................................................................................... 232 5. EXISTENCE OF SOLUTIONS II ............................................................................................... 233 5.1. Application 2 ...................................................................................................................... 235 6. SET-VALUED PROBLEM .......................................................................................................... 235 6.1. Existence of Solution .......................................................................................................... 236 6.2. Continuous Dependence on the Set of Selection 𝑺𝑭𝟏 .......................................................... 237 6.3. Set-valued Chandrasekhar non-linear Quadratic Functional Integral Inclusion .................. 239 6.4. Volterra Integral Inclusion of Fractional Order................................................................... 239 6.4.1. Differential Inclusion ....................................................................................................... 241 CONCLUSION .................................................................................................................................. 241 CONSENT FOR PUBLICATION ................................................................................................... 241 CONFLICT OF INTEREST ............................................................................................................ 241 ACKNOWLEDGEMENT ................................................................................................................ 242 REFERENCES .................................................................................................................................. 242 CHAPTER 13 CERTAIN SAIGO FRACTIONAL DERIVATIVES OF EXTENDED HYPERGEOMETRIC FUNCTIONS…………………………………………………………………………………………………………… 244 S. Jain, R. Goyal, P. Agarwal and S. Momani 1. INTRODUCTION ......................................................................................................................... 244

1.1. Main Results ....................................................................................................................... 1.2. Differential Inclusion .......................................................................................................... CONCLUSION .................................................................................................................................. CONSENT FOR PUBLICATION ................................................................................................... CONFLICT OF INTEREST ............................................................................................................ ACKNOWLEDGEMENTS .............................................................................................................. REFERENCES .................................................................................................................................. CHAPTER 14 SOME ERDÉLYI-KOBER FRACTIONAL INTEGRALS OF THE EXTENDED HYPERGEOMETRIC FUNCTIONS .……………………………………………………………………………………… S. Jain, R. Goyal, P. Agarwal, Clemente Cesarano4 and Juan L.G. Guirao 1. INTRODUCTION AND PRELIMINARIES .............................................................................. 2. MAIN RESULTS ........................................................................................................................... 3. SOME SPECIAL CASES OF THE ABOVE FRACTIONAL INTEGRAL FORMULAS ..... CONCLUSION .................................................................................................................................. CONSENT FOR PUBLICATION ................................................................................................... CONFLICT OF INTEREST ............................................................................................................ ACKNOWLEDGEMENTS .............................................................................................................. REFERENCES ..................................................................................................................................

249 241 255 256 256 256 256

259 259 263 265 266 267 267 267 267

CHAPTER 15 ON SOLUTIONS OF THE KINETIC MODEL BY SUMUDU TRANSFORM ……… 270 Esra Karatas Akgül, Fethi Bin Muhammed Belgacem and Ali Akgül 1. INTRODUCTION ......................................................................................................................... 270 2. MATHEMATICAL BACKGROUND ......................................................................................... 271 3. MAIN RESULTS ........................................................................................................................... 274 3.1. Lewis Drying Kinetic Model in Fractional Cases ............................................................... 274 3.1.1. Caputo Fractional Derivative for Drying Kinetic ............................................................. 274 3.1.2. Caputo-Fabrizio Fractional Derivative for Drying Kinetic .............................................. 275 3.2. Atangan-Baleanu Fractional Derivative for Drying Kinetic ............................................... 276 3.2.1. Constant Proportional Caputo Fractional Derivative for Drying Kinetic ......................... 277 4. NUMERICAL RESULTS ............................................................................................................. 279 CONCLUSION .................................................................................................................................. 280 CONSENT FOR PUBLICATION ................................................................................................... 281 CONFLICT OF INTEREST ............................................................................................................ 281 ACKNOWLEDGEMENT ................................................................................................................ 281 REFERENCES .................................................................................................................................. 281 SUBJECT INDEX .................................................................................................................................... 2

i

PREFACE In recent years special functions have been developed and applied in a variety of fields, such as combinatory, astronomy, applied mathematics, physics, and engineering due mainly to their remarkable properties. The main purpose of this Special Issue is to be a forum of recently-developed theories and formulas of special functions with their possible applications to some other research areas. This Special Issue provides readers with an opportunity to develop an understanding of recent trends of special functions and the skills needed to apply advanced mathematical techniques to solve complex problems in the theory of partial differential equations. Subject matters are normally related to special functions involving mathematical analysis and its numerous applications, as well as to more abstract methods in the theory of partial differential equations. The main objective of this book is to highlight the importance of fundamental results and techniques of the theory of complex analysis for PDEs, and emphasize articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. In chapter 1, the authors investigated the Adaptive synchronization and Anti synchronization between fractional-order 3D autonomous chaotic systems and novel 3D autonomous chaotic system with quadratic exponential terms using modified adaptive control method with unknown parameters. In chapter 2, the authors improved the generalized differential transform method by using the generalized Taylor's formula. In chapter 3, the authors introduced an incomplete K2-Function.Incomplete hypergeometric function, incomplete hypergeometric function, incomplete confluent hypergeometric function, incomplete Mittag-Leffler function can be deduced as special cases of our findings. In chapter 4, the authors present some new results for the in-complete hypergeometric function. In chapter 5, the authors adopt the transcendental Bernstein series (TBS), a set of basis functions based on the Bernstein polynomials (BP), for approximating analytical functions.

ii

In chapter 6, the authors find sufficient conditions under which 1F2 (a; b, c; z) belongsto UCV (α, β) and Sp(α, β). Here, 1F2 (a; b, c; z) is a special case of generalized hypergeometric function for p = 1 and q = 2. In chapter 7, the authors reveal that the missing link among a few crucial results in analysis, Abel continuity theorem, convergence theorem on (generalized) Dirichlet series, Paley-Wiener theorem is the Laplace transform with Stieltjes integration.

In chapter 8, the authors introduce a hybrid family of truncated exponentialGould-Hopper based Genocchi polynomials by means of generating function and series definition. Some significant properties of these polynomials are established. In chapter 9, the authors derived a finite difference approximation equation from the discretization of the one-dimensional linear time-fractional diffusion equations with Caputo's time-fractional derivative. In chapter 10, the authors derived some important theorems like Krasnoselskii-type

Theorems for Monotone Operators in Ordered Banach Algebra with Applications in Fractional Differential Equations and Inclusion. In chapter 11, authors studied general fractional order quadratic functional integral equations: Existence, properties of solutions and some of their applications. In chapter 12, the authors consider a nonlinear set-valued delay functional integral equations of Volterra-Stieltjes type. In chapter 13, the authors establish Saigo fractional derivatives of extended hypergeometric functions. Some special cases of these integrals are also derived. In chapter 14, the authors establish some new formulas and new results related to the Erdelyi-Kober fractional integral operator which was applied to the extended hypergeometric functions.

iii

In 15 chapter, the authors investigated the kinetic model with four different fractional derivatives. They obtained the solutions of the models by Sumudu transform. They demonstrated results by some figures and prove the accuracy of the Sumudu transform by some theoretical results and applications. Praveen Agarwal Anand International College of Engineering India & Shilpi Jain Poornima College of Engineering India

iv

List of Contributors Ahmed M.A. El-Sayed

Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria 21526, Egypt

Ali Akgül

Department of Computer Science and MathematicsDepartment of MathematicsDepartment of Mathematics, Lebanese American UniversitySiirt University, Art and Science FacultyNear East University, Mathematics Research Center, Near East Boulevard, Beirut56100 SiirtPC: 99138, Nicosia /Mersin, LebanonTurkeyTurkey

Andang Sunarto

Tadris Matematika, Universitas Islam Negeri (UIN) Fatmawati Sukarno, Bengkulu, Indonesia

Ayşegül ÇETİNKAYA1

Deptartment of Mathematics, Ahi Evran University, Kırşehir, Turkey

Clemente Cesarano

Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy

D. Datta

SRM Institute of Science and Technology, Bharathi Salai, Ramapuram, Chennai, India

Deepak Bansal

Department of Mathematics, University College of Engineering and Technology, Bikaner Technical University, Bikaner, Rajasthan, India

Dharmendra Kumar Singh

Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India

E. Naraghirad

Department of Mathematics, Yasouj University, Yasouj, Iran

Esra Karatas Akgül

Department of Mathematics, Siirt University, Art and Science Faculty, 56100 Siirt, Turkey

Fethi Bin Muhammed Belgacem

Department of Mathematics, Faculty of Basic Education, PAAET, Al-Ardhiya, Kuwait

Geeta Yadav

Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India

Ghazala Yasmin

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India

H. Hassani

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India

Hibah Islahi

Institute of Applied Sciences, Mangalayatan University, Aligarh-202145, India

Hind H.G. Hashem

College of Science, Qassim University, P.O. Box 6644 , Buraidah 51452, Saudi Arabia

J.A. Tenreiro Machado3

Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, R. Dr. António Bernardino de Almeida, Porto 431, Portugal

Jackel Vui Lung Chew

Faculty of Computing and Informatics, Universiti Malaysia Sabah Labuan International Campus, Labuan, 87000, Malaysia

Jumat Sulaiman

Universiti Malaysia Sabah, Faculty of Science and Natural Resources, Kota Kinabalu, Malaysia

Lalit Batra

Dept. of Applied Sciences, Bharati Vidyapeeth’s College of Engineering, New Delhi, India

v Nayyar Mehmood

1Department of Mathematics, & Statistics International Islamic University H10 , Islamabad, Pakistan

Niaz Ahmad

1Department of Mathematics, & Statistics International Islamic University H10 , Islamabad, Pakistan

P. Agarwal

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India

R. Goyal

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, ndia

S. Yu

Taishan College, Shandong University, Jinan, 250110, P.R.China

S. Kanemitsu

KSCSTE-Kerala School of Mathematics, Kunnamangalam, Kozhikode, Kerala, India

S. Jain

Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India

S. Momani

Nonlinear Dynamics Research Center (NDRC)Department of Mathematics, Ajman UniversityFaculty of Science, The University of Jordan, AjmanAmman 11942, UAEJordan

S.K. Agrawal

Dept. of Applied Sciences, Bharati Vidyapeeth’s College of Engineering, New Delhi, India

Sh. M Al-Issa

Faculty of Arts and Sciences, Department of MathematicsFaculty of Arts and Sciences, Department of Mathematics, Lebanese International UniversityThe International University of Beirut, SaidaBeirut, LebanonLebanon

Shilpi Jain

Department of Mathematics, Poornima College of Engineering, Jaipur, India

V. Mishra

Dept. of Mathematics, Thakur College of Engineering and Technology, Mumbai, India

Vijay Laxmi Verma

Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India

Y. M. Y. Omar

Faculty of Science, Omar Al-Mukhtar University, Libya

Z. Avazzadeh

Department of Mathematical Sciences, University of South Africa, Florida, South Africa

İ. Onur KIYMAZ

Deptartment of Mathematics, Ahi Evran University, Kırşehir, Turkey

Advances in Special Functions of Fractional Calculus, 2023, 01-38

1

CHAPTER 1

Modified Adaptive Synchronization and AntiSynchronization Method for Fractional Order Chaotic Systems with Uncertain Parameters S. K. Agrawal1, Lalit Batra1, V. Mishra2,* and D. Datta3 1

Department of Applied Sciences, Bharati Vidyapeeth’s College of Engineering, New Delhi, India

2

Department of Mathematics, Thakur College of Engineering and Technology, Mumbai, India

3

SRM Institute of Science and Technology, Bharathi Salai, Ramapuram, Chennai, India Abstract: In the present article, we have investigated the Adaptive synchronization and Anti-synchronization between fractional order 3D autonomous chaotic system and novel 3D autonomous chaotic system with quadratic exponential term using Modified adaptive control method with unknown parameters. The modified adaptive control method is very affective and more convenient in comparison to the existing method for the synchronization of the fractional order chaotic systems. The chaotic attractors and synchronization of the systems are found for fractional order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams-Boshforth-Moulton method show that the method is reliable and effective for synchronization and anti-synchronization of autonomous chaotic systems.

Keywords: Modified Adaptive control method, Synchronization and AntiSynchronization; Fractional derivative, 3D autonomous chaotic systems, Unknown parameters. 1. INTRODUCTION Nowadays, fractional order derivative has become a popular field of research since fractional order system response ultimately converges to the integer order system. For high accuracy, fractional derivatives are used to describe the dynamics of systems. The attribute of fractional order systems for which they have gained *Corresponding

author V. Mishra: Deptartment of Mathematics, Thakur College of Engineering and Technology, Mumbai, India; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

2

Advances in Special Functions of Fractional Calculus

Agrawal et al.

popularity in the investigation of dynamical systems is that they allow a greater degree of flexibility in the model. An integer order differential operator is a local operator. Whereas the fractional order differential operator is non-local in that it considers that the future state not only depends upon the present state but also upon all of the history of its previous states. For this realistic property, the usage of fractional order systems is becoming popular. Fractional differential equations have garnered a lot of attention and appreciation recently due to their ability to provide an exact description of different nonlinear phenomena. The advantage of fractional order systems is that they allow greater flexibility in the model. Another advantage of fractional order systems is that they possess memory and display much more sophisticated dynamics compared to its integral order counterpart, which is of great significance in secure communication and control processes. The applications of fractional calculus are growing rapidly. During the last few years, the applications can be found in the fields of science and engineering, including Fluid Mechanics [1, 2], Quantum Mechanics [3], Material Science [4], Viscoelasticity [5], Bioengineering [6], Medicine [7], Biological models [8, 9], Cardiac Tissues [10], etc. Analysis of fractional order dynamical systems involving Riemann-Liouville as well as Caputo derivatives have been found in the study [11-13]. The field of chaos in nonlinear dynamics has grabbed the attention of researchers, and this contributes to a significant amount of ongoing research these days. Synchronization of chaos is a naturally occurring phenomenon where one chaotic dynamical system mimics the dynamical behavior of another chaotic system. This phenomenon can be used in a chaotic communication system as a mechanism for information decoding of the dynamical system. The application of nonlinear dynamical systems has nowadays spread to a wide spectrum of disciplines, including science, engineering, biology, sociology, etc. In nonlinear systems, a small change in a parameter in system parameters can lead to sudden and dramatic changes in both the qualitative and quantitative behavior. The idea of synchronizing chaotic systems was introduced by Pecora and Carroll [14] in 1990. They showed that it was possible to synchronize several chaotic systems through a simple coupling. Synchronization of chaotic dynamical systems has been extensively studied by many researchers [15-17] due to its important applications in an ecological system [18], physical system [19], chemical system [20], modeling brain activity, system identification, pattern recognition phenomena and secure communications [21, 22] and so on. In recent years, several different types of synchronization schemes have been proposed. These include a nonlinear time-delay feedback approach [23], adaptive control [24-26], active control [27-29], sliding mode control [30, 31] and so on. The

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

3

concept of synchronization can be extended to complete synchronization [32, 33], phase synchronization [34], projective synchronization [35, 36] and function projective synchronization [37, 38]. The synchronization of chaotic systems is a difficult problem due to their extremely sensitive dependence on initial conditions. Any initial correlations present between identical and non-identical systems, starting from very close initial conditions, exponentially decrease to zero with time. Thus, for all practical purposes, any initial synchronization between the systems is bound to disappear rapidly. The important feature of the study of synchronization is where the difference of states of chaotic systems converge to zero for a long time. This phenomenon is known as complete synchronization. Mathematically, the synchronization is achieved when lim x1 (t ) x2 (t ) 0, where x1 (t ) and x2 (t ) are the state vectors t

of the drive and response systems, respectively. The phenomenon of antisynchronization is also observed in periodic, chaotic systems. This is a phenomenon in which the state variables of synchronized systems with different initial values have the same absolute but opposite signs. The sum of the two signals is expected to converge to zero when anti-synchronization occurs. Mathematically, the antisynchronization is achieved when lim x1 (t ) x2 (t ) 0 . t

Adaptive Control methods are the control scheme used by a controller which must adapt to a controlled system with parameters that vary from time to time. In practical situations, these parameters may be unknown or initially uncertain. Thus the derivation of adaptive controller for the synchronization of chaotic systems in the presence of system parameter uncertainty is an important problem. This technique is used when the system parameters are unknown. In an adaptive method, a control law and a parameter update rule for unknown parameters are designed in such a way that the chaotic drive system controls the chaotic response system. Most of the studies in synchronization/anti-synchronization involve two identical/nonidentical systems under the hypotheses that all the parameters of the master and slave systems are known prior, a controller is constructed with the known parameters and systems are free from external perturbations. But in practical situations, the uncertainties like parameter mismatch and external disturbances may destroy the synchronization and even break it. Therefore, it is necessary to design an adaptive controller and parameter update law for control and synchronization of chaotic systems consisting of unknown parameters to get rid of internal and external noises. In the presence of model uncertainties and external disturbances, an

4

Advances in Special Functions of Fractional Calculus

Agrawal et al.

appropriate adaptive control scheme is applied to stabilize a group of chaotic systems. Recently, many authors studied adaptive synchronization for chaotic systems; Chen et al. [26] proposed a novel parameter identification and synchronization method for synchronizing non-identical chaotic systems with unknown parameters based on the adaptive control method in 2002. In 2006, Zang et al. [39] proposed a method for the adaptive synchronization of two chaotic systems with different structures and unknown parameters. In the letter, based on Lyapunov stability theory, an adaptive synchronization controller is designed and analytic expression of the controller and the adaptive laws of parameters are developed for the synchronization of two different chaotic systems in the presence of unknown parameters, which are very much useful in real-life applications. In 2008, Salarieh and Shahrokhi [40] investigated the adaptive synchronization of two different chaotic systems with time-varying unknown parameters. Mossa et al. [41] investigated adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters in 2010. Li et al. [42] proposed Complete (anti-) synchronization of chaotic systems with fully uncertain parameters by adaptive control, in 2011. Yu et al. [43] studied anti-synchronization of a novel hyperchaotic system with parameter mismatch and external disturbances. After this, in 2013, Agrawal and Das [44] modified the adaptive synchronization and parameter identification method with unknown parameters for use in fractional-order chaotic systems and designed the appropriate adaptive synchronization controller and parameter identification which are very much useful in real-life situations. From the literature survey, it is seen that with the development of nonlinear control theory, nowadays, adaptive synchronization & anti-synchronization method has become very much effective to control and synchronize & anti-synchronize chaotic systems with uncertain parameters. In the present article, the adaptive synchronization & anti-synchronization and parameter identification method with unknown parameters have been introduced for use in fractional order chaotic systems and designed the appropriate adaptive synchronization & anti-synchronization controller and parameter identification which are based on the Lyapunov stability method. The synchronization & antisynchronization between two different pairs of fractional order chaotic systems with different fractional order time derivatives using the modified adaptive control method has been achieved. Using the Adams–Bashforth–Moulton method [45, 46], numerical simulations are carried out for different fractional order derivatives, which are depicted through figures for different particular cases.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

5

2. PRELIMINARIES, THEOREM, PROBLEM DESCRIPTION AND CONTROL DESIGN FOR SYNCHRONIZATION 2.1. Fractional Calculus Fractional calculus is a generalization of integration and differentiation to a non-

q t

integer order integro-differential operator a D , which is defined by:

dq , dt q q a Dt 1, t (d ) q a

R(q) 0, R(q) 0, R(q) 0, (1)

Where q is the fractional order, which can be a complex number, R(q) denotes the real part of q and a t , a is the fixed lower terminal & t is the moving upper terminal. There are some definitions for fractional derivatives. The commonly used definition is the Riemann-Liouville definition, defined by:

d n nq q jt x(t ), a Dt x (t ) dt n

q 0,

(2)

n q i.e., n is the first integer that is not less than q, jt is the -order RiemannLiouville integral operator, which is described as follows: 1 ( ) jt (t ) d , ( ) 0 (t )1

t

where 0 1 and (.) is the gamma function. The Caputo differential operator of fractional order q is defined as

(3)

6 c

Advances in Special Functions of Fractional Calculus

Dtq x(t ) jt

n q

where n q .

q 0,

x ( n) (t ),

Agrawal et al.

Lemma 1: In Riemann Liouville derivatives, if p q 0 , m and n are integers such that 0 m 1 p m , 0 n 1 q n , then we obtain [47]. a

Dtp

a

Dt q f (t ) a Dtp q f (t ) .

(4)

Lemma 2: In Riemann Liouville derivatives, if p, q 0 , m and n are integers such that 0 m 1 p m , 0 n 1 q n , then we obtain [47].

a

Dt

p

a

D f (t ) a Dt q t

pq

n

f (t ) a D j 1

q j t

f (t ) t a

(t a) p j . (1 p j )

(5)

2.2. Problem Description We consider the master (derive) chaotic system as Dtq x f ( x) F ( x) ,

(6)

and the slave (response) system as Dtq y g ( y ) G ( y ) u.

(7)

where x, y R n are the state vectors, and R m are the unknown parameter vectors of the systems, f ( x) and g ( y) are n 1 matrices, F ( x) and G( y) are the n m matrices, the elements Fij (x ) in the matrix F (x) and Gij ( y ) in matrix G( y) satisfy Fij ( x) L , x R n and Gij ( y) L , y R n respectively. Let, e y x is the error vector for synchronization and e y x is the error vector for Anti-synchronization. Our goal is to design a controller u such that the trajectory of the response system (7) with initial condition y 0 asymptotically approaches the drive system (6) with initial conditions x0, and finally, we get required for synchronization lim e lim y (t , y0 ) x(t , x0 ) 0, and for Antit

t

Synchronization lim e lim y(t , y 0 ) x(t , x0 ) 0, where t

norm.

t

. is the Euclidean

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

7

2.3. Modified Adaptive Synchronization Controller Des-Ign For Synchronization Theorem 1: If a nonlinear control function is selected in the system (7) as [44] (t ) ( q 1)1 u f ( x) F ( x) g ( y) G( y) Dtq 1 F ( x)e G( y)e ( Dtq 1e(t )) ek , ((q 1))

(8)

and adaptive laws of parameters are taken as

([F ( x)]T e e ) , ([G( y)]T e e ) , Where, e ( ) and e ( )

(9)

then the response system (7) can synchronize the derived system (6) globally and asymptotically, and satisfies lim ( ) lim ( ) 0 , where k 0 is a t

t

constant, q [0,1] is the order of derivative, and , are the estimated parameters of and . 2.4. Modified Adaptive Synchronization Controller Design for Anti-Synchronization Theorem 1: If a nonlinear control function is selected in the system (7) as

u f ( x) F ( x) g ( y ) G( y ) (t ) ( q 1)1 Dtq 1 F ( x)e G( y )e ( Dtq 1e(t )) ek , ((q 1))

(10)

and adaptive laws of parameters are taken as

([F ( x)]T e e ) , ([G( y)]T e e ) ,

(11)

where e ( ) and e ( ) . then the response system (7) can Anti-synchronize the derive system (6) globally and asymptotically, and satisfies lim ( ) lim ( ) 0 ,where k 0 is a t

t

8

Advances in Special Functions of Fractional Calculus

Agrawal et al.

constant, q [0,1] is the order of derivative and , are the estimated parameters of and . Proof: From equations (6), (7) and (16) we get

(t ) ( q 1)1 Dtq e(t ) Dtq 1 F ( x)e G( y)e ( Dtq 1e(t )) ek , ((q 1))

(12)

let us choose the Lyapunov function as: V( e, e , e )

1 T 1 T T e e e e e e e T e ( ) T ( ) ( ) T ( ) , (13) 2 2

where V( e, e , e ) R n , and the time derivative of V( e, e , e ) along the trajectory of the error dynamics system (13) is as follows:

( e, e , e ) [e T e ( ) T ( ) T ] , V

(14)

Using Lemma2 in equation (20) we get ( q 1) 1 T T T V( e, e , e ) D1q D q e D q 1e(t ) (t ) , (15) e ( ) ( ) t t t ((q 1))

From equations (11) and (14), we get T

(t ) ( q 1)1 (t ) ( q 1)1 V ( e, e , e ) Dt1q Dtq 1 F ( x)e G( y)e ( Dtq 1e(t )) ek Dtq 1e(t ) e ((q 1)) ((q 1))

( )T ( )T ,

(16)

Since q [0,1] , (1 q) 0 and (q 1) 0 . Now using Lemma1 and equation (17), the equation (16) reduces to T

(t ) ( q 1) 1 (t ) ( q 1) 1 V ( e, e , e ) F ( x)e G ( y )e ( Dtq 1e(t )) ek Dtq 1e(t ) e ((q 1)) ((q 1))

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

9

( )T ([F ( x)]T e e ) ( )T ([G( y)]T e e ) ,

[e F ( x)T e G( y) T keT ]e e [ F ( x)]T e e e e [G( y)]T e e e T

T

T

T

keT e e e e e 0 . T

T

T

T

(17)

Which is show that the response system (7) is anti-synchronized to the drive system (6) globally and asymptotically based on the Lyapunov stability theory [48, 49]. It is also seen that the Anti-synchronization error e and the parameters’ estimation errors e , e decay to zero as time becomes large. 3. SYSTEM'S DESCRIPTION 3.1. Fractional Order 3D Autonomous Chaotic System In 2009, Lui et al. [50, 51] investigated a three-dimensional autonomous system that relies on two multipliers and one quadratic term to introduce the nonlinearity, necessary for folding trajectories. The given chaotic system (18) is a new chaotic system whose chaotic attractor is similar to Lorenz's chaotic attractor. This is described by the following nonlinear fractional order differential equation.

d q1 x1 2 (a1 x1 d1 y1 ) , q1 dt d q2 y1 (b1 y1 k1 x1 z1 ) , dt q2

(18)

d q3 z1 (c1 z1 m1 x1 y1 ) . dt q3 Fig. (1) describes the chaotic attractor of the system (18) for the parameters a1 1, b1 2.5, c1 5, d1 1, k1 4, m1 2 with initial conditions (0.2, 0, 0.5).

10

Advances in Special Functions of Fractional Calculus

Agrawal et al.

2 1.5

1.5

1

1 0.5

0

z(t)

z(t)

0.5

0

-0.5 -0.5

-1 -1

-1.5 -2 4

2

0

-2

-4

-3

-2

-1

1

1 0 2 0

-1 -3

y(t)

1(a) Fig. (1). Phase portrait of the system (18) in x-y-z spaces: 1(a) for

i 1, 2,3.

-2

-2 -4

x(t)

y(t)

0

-1.5 4

x(t)

1(b)

q i 1 & 1(b) for qi 0.95 ,

3.2. Fractional Order Novel 3d Autonomous Chaotic System Fractional order novel 3D autonomous chaotic system is expressed as [52, 53].

d q1 x2 a2 ( y 2 x2 ) , dt q1 d q2 y 2 (b2 x2 c2 x2 z 2 ) , dt q2

(19)

d q3 z 2 (e x 2 y 2 d 2 z 2 ) . dt q3 where a2 , b2 , c2 , d 2 are parameters and x, y, z are the state variables. It is a threedimensional autonomous system with six terms on the right-hand side, but it mainly relies on two quadratic nonlinearities, one is a quadratic exponential nonlinear term, and the other is a quadratic cross-product term to introduce the nonlinearity necessary for the chaotic attractor.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

11

System (19) generates the chaotic attractor for the parameters a2 10, b2 40, c2 2, d 2 2.5 with the initial conditions (2.2, 2.4, 28). From Fig. (2), we can easily see that the chaotic attractor is similar to the Lorenz attractor but is different from that of the Lorenz system or any existing chaotic systems.

60

50

50

40

40

z(t)

z(t)

30

30 20

20 10

10 0 5

0 5

4 2

0

0

4 2

0

0

-2

y(t)

-5

-4

-2

x(t)

y(t)

-5

2(a) Fig. (2). Phase portrait of the system (19) in x-y-z spaces: 2(a) for

i 1, 2,3.

-4

x(t)

2(b)

q i 1 & 2(b) for qi 0.95 ,

4. SYNCHRONIZATION BETWEEN FRACTIONAL-ORDER 3D AUTONOMOUS AND NOVEL 3D AUTONOMOUS CHAOTIC SYSTEMS USING A MODIFIED ADAPTIVE CONTROL METHOD This section studies the synchronization behavior between the fractional order 3D autonomous and novel 3D autonomous chaotic systems. It is assumed that 3D autonomous system drives the novel 3D autonomous chaotic system. Now the drive system (18) and the response system (19) are defined with control parameters u [u1 , u 2 , u 3 ]T as:

d q1 x1 2 (a1 x1 d1 y1 ) , q1 dt d q2 y1 (b1 y1 k1 x1 z1 ) , dt q2

.

(20)

12

Advances in Special Functions of Fractional Calculus

Agrawal et al.

d q3 z1 (c1 z1 m1 x1 y1 ) . dt q3 and

d q1 x2 a 2 ( y 2 x2 ) u1 , dt q1 d q2 y 2 (b2 x2 c2 x2 z 2 ) u 2 , dt q2

… (21)

d q3 z 2 (e x 2 y 2 d 2 z 2 ) u 3 . q3 dt Then the error function can be defined as:

e1 x 2 x1 e2 y 2 y1 , e z z 2 1 3

(22)

So, according to the theorem 1, the controller function is taken as u1 (a1 x1 d1 y1 ) a 2 ( y 2 x 2 ) 2

D

q1 1

, (t ) ( q1 1) 1 2 q1 1 e1 k1 ea1 x1 ed1 y1 ea2 y 2 ( Dt e1 (t )) ((q1 1))

(23)

u 2 (b1 y1 k1 x1 z1 ) (b2 x2 c2 x2 z 2 ) , (24) (t ) ( q2 1)1 D q2 1 eb1 y1 ek1 x1 z1 eb2 x2 ec2 x 2 z 2 ( Dtq2 1e2 (t )) e2 k 2 ((q 2 1)) u 3 (c1 z1 m1 x1 y1 ) (e x2 y2 d 2 z 2 ) , (t ) ( q3 1) 1 D q3 1 ec1 z1 em1 x1 y1 ed 2 z 2 ( Dtq3 1e3 (t )) e3 k 3 ((q3 1))

(25)

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

13

where k1 , k 2 , k 3 are the real positive constants, and ea1 (a1 a1 ) , eb1 (b1 b1 ) ,

ec1 (c1 c1 ) , ed1 (d1 d1 ) , ek1 (k1 k1 ) , em1 ( m1 m1 ) , ea2 (a 2 a 2 ) , eb2 (b2 b2 ) , ec2 (c 2 c 2 ) , ed2 (d 2 d 2 ) . The estimated parameters are calculated as:

a1 ( x1e1 ea1 ), b1 ( y1e2 eb1 ), c1 ( z1e3 ec1 ), 2 (26) d1 ( y1 e1 ed1 ), k1 ( x1 z1e2 ek1 ), m1 ( x1 y1e3 em1 ), a2 (( y 2 x2 )e1 ea2 ), b2 ( x2 e2 eb2 ), c2 ( x2 z 2 e2 ec2 ), d 2 ( z 2 e3 ed 2 ). 4.1. Numerical Simulation and Results In numerical simulations for the adaptive synchronization of drive and response systems, the unknown parameter vectors of the drive and response systems are taken the same as in Section 3, and the initial values of the estimated unknown parameter vectors of drive and response system are taken as (a1 0, b1 0,

c1 0, d1 0, k1 0, m1 0) and (a2 0, b2 0, c2 0, d 2 0) respectively. The initial values of a drive system (18) and response system (19) of the state vectors are similar to Section 3, and the initial value of the error vector and constant k are e [2,2.4,27.5]T and k [1,1,1]T , respectively. The response system (18) is synchronized with the drive system (19), as shown in Figs. (3-5) for the order of derivatives q [1.0,1.0,1.0]T , q [0.98,0.98,0.98]T and q [0.95,0.95,0.95]T , respectively, and it is seen that the synchronization error vectors converge asymptotically to zero. (Figs. 3e & f), (Figs. 4e & f) and (Figs. 5e & f) show the variations of estimated parameter vectors which converge to the original parameter vectors for both the drive and response systems at the order of derivatives q [1.0,1.0,1.0]T , q [0.98,0.98,0.98]T and q [0.95,0.95,0.95]T , respectively.

14

Advances in Special Functions of Fractional Calculus

Agrawal et al.

8

x1(t) 6

x2(t)

x1(t), x2(t)

4 2 0 -2 -4 -6

0

1

2

3

4

5

6

7

t

Fig. (3a). State trajectory between state vectors x1 and x2. 25 y1(t) y (t)

20

2

y1(t),y2(t)

15

10

5

0

-5

0

1

2

3

4

t

Fig. (3b). State trajectory between state vectors y1 and y2.

5

6

7

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

30 z1(t) 25

z (t) 2

z1(t), z2(t)

20 15 10 5 0 -5

0

1

2

3

4

5

6

7

t

Fig. (3c). State trajectory between state vectors z1 and z2. 30 e1(t) e (t)

20

2

e (t)

e1(t), e2(t), e3(t)

3

10

0

-10

-20

-30

0

1

2

3

4

t

Fig. (3d) State trajectory between error vectors

e1 , e2 , e3 and e4 .

5

6

7

15

16

Advances in Special Functions of Fractional Calculus

Fig. (3e). State trajectories of adaptive parameters

Fig. (3f) State trajectories of adaptive parameters

Agrawal et al.

a1 , b1 , c1 , d1 , k1 and m1.

a2 , b2 , c2 , and d 2 .

Fig. (3) State trajectories of the state vectors, error system and estimated parameters of a drive system (20) and response system (21) for q [1.0,1.0,1.0]T .

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

17

8 x1(t) 6

x (t) 2

x1(t), x2(t)

4 2 0 -2 -4 -6

0

1

2

3

4

5

6

7

t Fig. (4a). State trajectory between state vectors x1 and x1.

25 y 1(t) y 2(t)

20

y1(t), y2(t)

15

10

5

0

-5

0

1

2

3

4

t Fig. (4b). State trajectory between state vectors y1 and y2.

5

6

7

18

Advances in Special Functions of Fractional Calculus

Agrawal et al.

30 z1(t) 25

z (t) 2

z1(t), z2(t)

20 15 10 5 0 -5

0

1

2

3

4

5

6

7

t Fig. (4c). State trajectory between state vectors z1 and z2. 30 e1(t) e (t)

20

2

e (t)

e1(t), e2(t), e3(t)

3

10

0

-10

-20

-30

0

1

2

3

4

t

Fig. (4d). State trajectory between error vectors

e1 , e2 , e3 and e4 .

5

6

7

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

Fig. (4e). State trajectories of adaptive parameters

a1 , b1 , c1 , d1 , k1 and m1.

Fig. (4f). State trajectories of adaptive parameters

a2 , b2 , c2 , and d 2 .

19

Fig. (4). State trajectories of the state vectors, error system and estimated parameters of a drive system (20) and response system (21) for q [0.98, 0.98, 0.98]T .

20

Advances in Special Functions of Fractional Calculus

Agrawal et al.

8 x1(t) 6

x (t) 2

x1(t), x2(t)

4 2 0 -2 -4 -6

0

1

2

3

4

5

6

7

t

Fig. (5a). State trajectory between state vectors x1 and x1. 25 y1(t) y (t)

20

2

y1(t), y2(t)

15

10

5

0

-5

0

1

2

3

4

t

Fig. (5b). State trajectory between state vectors y1 and y2.

5

6

7

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

30 z1(t) 25

z (t) 2

z1(t), z2(t)

20 15 10 5 0 -5

0

1

2

3

4

5

6

7

t

Fig. (5c). State trajectory between state vectors z1 and z2. 30 e1(t) e (t)

20

2

e (t)

e1(t), e2(t), e3(t)

3

10

0

-10

-20

-30

0

1

2

3

4

5

t

Fig. (5d). State trajectory between error vectors

e1 , e2 , e3 and e4 .

6

7

21

22

Advances in Special Functions of Fractional Calculus

Fig. (5e). State trajectories of adaptive parameters

a1 , b1 , c1 , d1 , k1 and m1.

Fig. (5f). State trajectories of adaptive parameters

a2 , b2 , c2 , and d 2 .

Agrawal et al.

Fig. (5) State trajectories of the state vectors, error system and estimated parameters of a drive system (20) and response system (21) for q [0.95, 0.95, 0.95]T .

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

23

5. ANTI-SYNCHRONIZATION BETWEEN FRACTIONAL-ORDER 3D AUTONOMOUS AND NOVEL 3D AUTONOMOUS CHAOTIC SYSTEMS USING MODIFIED ADAPTIVE CONTROL METHOD The present section shows the anti-synchronization behavior between the fractional order 3D autonomous and novel 3D autonomous chaotic systems. Like the above section, we assumed that 3D autonomous system drives the novel 3D autonomous chaotic system. Now taking equation (20) as a drive system and equation (21) as a response system, where u [u1 , u 2 , u 3 ]T are the control parameters Then, the error system can be defined for the anti-synchronization as

e1 x2 x1 e2 y 2 y1 , e z z 2 1 3

(22)

So, according to Theorem 2, the controller function for anti-synchronization is taken as u1 (a1 x1 d1 y1 ) a 2 ( y 2 x2 ) 2

, (t ) ( q1 1)1 2 D q1 1 ea1 x1 ed1 y1 ea2 y 2 ( Dtq1 1e1 (t )) e1k1 ((q1 1))

u 2 (b1 y1 k1 x1 z1 ) (b2 x2 c2 x2 z 2 ) , (t ) ( q2 1)1 D q2 1 eb1 y1 ek1 x1 z1 eb2 x2 ec2 x2 z 2 ( Dtq2 1e2 (t )) e2 k 2 ((q2 1)) u3 (c1 z1 m1 x1 y1 ) (e x2 y2 d 2 z 2 ) D

q3 1

. (t ) ( q3 1)1 q3 1 e3 k 3 ec1 z1 em1 x1 y1 ed 2 z 2 ( Dt e3 (t )) ((q3 1))

where k1 , k 2 , k 3 are again the real positive constants, and the estimated parameters error functions e a1 , eb , ec1 , e d1 , ek1 , e m1 , e a 2 , eb , ec2 , e d 2 are taken same as the 1

Section 4.

2

24

Advances in Special Functions of Fractional Calculus

Agrawal et al.

The estimated parameters are calculated as: a1 ( x1e1 ea1 ), b1 ( y1e2 eb1 ), c1 ( z1e3 ec1 ), 2 d1 ( y1 e1 ed1 ), k1 ( x1 z1e2 ek1 ), m1 ( x1 y1e3 em1 ), a 2 (( y 2 x2 )e1 ea2 ), b2 ( x2 e2 eb2 ), c2 ( x2 z 2 e2 ec2 ), d 2 ( z 2 e3 ed 2 ).

5.1. Numerical Simulation and Results In numerical simulations for the adaptive anti-synchronization of drive and response systems, the unknown parameter vectors of the drive and response systems and initial values of the estimated unknown parameter vectors are taken the same as in previous sections. The initial values of a drive system (18) and response system (19) of the state vectors are also taken, similar to Section 3, and the initial value of the error vector for anti-synchronization and constant k are taken as e [2.4,2.4,28.5]T and k [1,1,1]T , respectively. The response system (18) is antisynchronized with the drive system (19), as shown in Figs. (6-8) for the order of derivatives q [1.0,1.0,1.0]T , q [0.98,0.98,0.98]T and q [0.95,0.95,0.95]T , respectively, and it is seen that the anti-synchronization error vectors converge asymptotically to zero. (Figs. 6e & 6f), (Figs.7e & 7f) and (Figs. 8e & 8f) show the variations of estimated parameter vectors which converge to the original parameter vectors for both the drive and response systems at the order of derivatives q [1.0,1.0,1.0]T , q [0.98,0.98,0.98]T and q [0.95,0.95,0.95]T , respectively. 8 x1(t) 6

x (t) 2

x1(t), x2(t)

4 2 0

-2 -4 -6

0

5

10

t

Fig. (6a). State trajectory between state vectors x1 and x1.

15

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

25 y1(t) y (t)

20

2

y1(t),y2(t)

15

10

5

0

-5

0

5

10

15

t Fig. (6b). State trajectory between state vectors y1 and y2. 30 z1(t)

25

z2(t)

z1(t), z2(t)

20 15 10 5 0 -5

0

5

10

t Fig. (6c). State trajectory between state vectors z1 and z2.

15

25

26

Advances in Special Functions of Fractional Calculus

Agrawal et al.

30 e1(t)

e1(t), e2(t), e3(t)

25

e2(t)

20

e (t) 3

15 10 5 0 -5

-10

0

1

2

3

4

5

6

7

8

t

Fig. (6d). State trajectory between error vectors

e1 , e2 , e3 and e4 .

Fig. (6e). State trajectories of adaptive parameters

a1 , b1 , c1 , d1 , k1 and m1.

9

10

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

Fig. (6f). State trajectories of adaptive parameters

27

a2 , b2 , c2 , and d 2 .

Fig. (6) State trajectories of the state vectors, error system and estimated parameters of the drive system (20) and response system (21). q [1.0,1.0,1.0]T . 8 x1(t) 6

x (t) 2

x1(t), x2(t)

4 2 0

-2 -4 -6

0

1

2

3

4

5

t Fig. (7a). State trajectory between state vectors x1 and x1.

6

7

8

9

10

28

Advances in Special Functions of Fractional Calculus

Agrawal et al.

25 y1(t) y (t)

20

2

y1(t),y2(t)

15

10

5

0

-5

0

1

2

3

4

5

6

7

8

9

10

t

Fig. (7b). State trajectory between state vectors y1 and y2. 30 z1(t)

25

z2(t)

z1(t), z2(t)

20 15 10 5 0 -5

0

1

2

3

4

5

t Fig. (7c). State trajectory between state vectors z1 and z2.

6

7

8

9

10

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

30 e1(t)

25

e2(t)

e1(t), e2(t), e3(t)

20

e (t) 3

15 10 5 0 -5

-10

0

1

2

3

4

5

6

7

8

t

Fig. (7d). State trajectory between error vectors

e1 , e2 , e3 and e4 .

Fig. (7e). State trajectories of adaptive parameters

a1 , b1 , c1 , d1 , k1 and m1.

9

10

29

30

Advances in Special Functions of Fractional Calculus

Fig. (7f). State trajectories of adaptive parameters

Agrawal et al.

a2 , b2 , c2 , and d 2 .

Fig. (7) State trajectories of the state vectors, error system and estimated parameters of drive system (20) and response system (21). q [0.98, 0.98, 0.98]T . 8 x1(t) 6

x (t) 2

x1(t), x2(t)

4 2 0

-2 -4 -6

0

1

2

3

4

5

t

Fig. (8a). State trajectory between state vectors x1 and x1.

6

7

8

9

10

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

25 y1(t) y (t)

20

2

y1(t),y2(t)

15

10

5

0

-5

0

1

2

3

4

5

6

7

8

9

10

t Fig. (8b). State trajectory between state vectors y1 and y2. 30 z1(t)

25

z2(t)

z1(t), z2(t)

20 15 10 5 0 -5

0

1

2

3

4

5

t

Fig. (8c). State trajectory between state vectors z1 and z2.

6

7

8

9

10

31

32

Advances in Special Functions of Fractional Calculus

Agrawal et al.

30 e1(t)

25

e1(t), e2(t), e3(t)

e2(t) e (t)

20

3

15 10 5 0 -5

0

1

2

3

4

5

6

7

8

t

Fig. (8d). State trajectory between error vectors

e1 , e2 , e3 and e4 .

Fig. (8e). State trajectories of adaptive parameters a1 , b1 , c1 , d1 , k1 and m1.

9

10

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

33

Fig. (8f). State trajectories of adaptive parameters a2 , b2 , c2 , and d 2 .

Fig. (8) State trajectories of the state vectors, error system, and estimated parameters of a drive system (20) and response system (21). q [0.95, 0.95, 0.95]T . CONCLUSION The present investigation has attained accomplishment in two significant capacities. Firstly it successfully studied a modified adaptive synchronization & antisynchronization method proposed for application in fractional-order chaotic systems. The Adaptive controller and parameters update law are designed properly to synchronize/anti-synchronizes two different pairs of chaotic systems based on the Lyapunov direct stability theory. The controller and identification parameters law is designed so that the components of the error systems and parameter estimation error systems decay towards zero as time increases. The second one is the numerical simulation, which is carried out using the Adams–Bashforth– Moulton method, which calls for appreciation to show that the method is reliable and effective for modified adaptive synchronization & anti-synchronization of nonlinear dynamical systems. This proposed modified method clearly exhibits its simplicity and suitability for a large class of fractional order as well as integer-order chaotic systems. This type of modified adaptive synchronization with uncertainties is highly applicable in secure communication.

34

Advances in Special Functions of Fractional Calculus

Agrawal et al.

CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1]

[2]

[3]

[4]

[5] [6]

[7] [8]

[9]

V.V. Kulish, and J´.L. Lage, "Application of Fractional Calculus to Fluid Mechanics", J. Fluids Eng., vol. 124, no. 3, pp. 803-806, 2002. http://dx.doi.org/10.1115/1.1478062 D. Tripathi, S.K. Pandey, and S. Das, "Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel", Appl. Math. Comput., vol. 215, no. 10, pp. 3645-3654, 2010. http://dx.doi.org/10.1016/j.amc.2009.11.002 A. Yildirim, "An Algorithm for Solving the Fractional Nonlinear Schrödinger Equation by Means of the Homotopy Perturbation Method", Int. J. Nonlinear Sci. Numer. Simul., vol. 10, pp. 445-451, 2011. A. Carpinteri, P. Cornetti, and K.M. Kolwankar, "Calculation of the tensile and flexural strength of disordered materials using fractional calculus", Chaos Solitons Fractals, vol. 21, no. 3, pp. 623-632, 2004. http://dx.doi.org/10.1016/j.chaos.2003.12.081 E. Soczkiewicz, "Application of fractional calculus in the theory of viscoelasticity", Mol. Qua. Acou., vol. 23, pp. 397-404, 2002. R.L. Magin, "Fractional calculus in bioengineering, part 3", Crit. Rev. Biomed. Eng., vol. 32, no. 3-4, pp. 195-377, 2004. http://dx.doi.org/10.1615/CritRevBiomedEng.v32.i34.10 PMID: 15651636 W.G. Glockle, T. Mattfeld, T.F. Nonnenmacher, and E.R. Weibel, Fractals in Biology and Medicine., vol. 2, Birkhauser Basel, 1998. R.L. Magin, "Fractional calculus models of complex dynamics in biological tissues", Comput. Math. Appl., vol. 59, no. 5, pp. 1586-1593, 2010. http://dx.doi.org/10.1016/j.camwa.2009.08.039 A. Gökdoğan, M. Merdan, and A. Yildirim, "A multistage differential transformation method for approximate solution of Hantavirus infection model", Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 1, pp. 1-8, 2012. http://dx.doi.org/10.1016/j.cnsns.2011.05.023

Uncertain Parameters

[10]

[11] [12] [13]

[14]

[15]

[16]

[17]

[18]

[19] [20]

[21]

[22] [23]

[24]

Advances in Special Functions of Fractional Calculus

35

R.L. Magin, and M. Ovadia, "Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus", J. Vib. Control, vol. 14, no. 9-10, pp. 1431-1442, 2008. http://dx.doi.org/10.1177/1077546307087439 S.G. Samko, A.A. Kilbas, and O.I. Maricev, Fractional Integrals and Derivatives, Theory and Applications., Gordon and Breach, 1993. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Dierential Equations., Elsevier Science: Amsterdam, 2006. D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus Models and Numerical Methods., Series on Complexity, Nonlinearity and Chaos, World Scientic, 2012. http://dx.doi.org/10.1142/8180 L.M. Pecora, and T.L. Carroll, "Synchronization in chaotic systems", Phys. Rev. Lett., vol. 64, no. 8, pp. 821-824, 1990. http://dx.doi.org/10.1103/PhysRevLett.64.821 PMID: 10042089 E. Ott, C. Grebogi, and J.A. Yorke, "Controlling chaos", Phys. Rev. Lett., vol. 64, no. 11, pp. 1196-1199, 1990. http://dx.doi.org/10.1103/PhysRevLett.64.1196 PMID: 10041332 C.C. Fuh, and P.C. Tung, "Controlling chaos using differential geometric method", Phys. Rev. Lett., vol. 75, no. 16, pp. 2952-2955, 1995. http://dx.doi.org/10.1103/PhysRevLett.75.2952 PMID: 10059451 G. Chen, and X. Dong, "On feedback control of chaotic continuous-time systems", IEEE Trans. Circ. Syst. I Fundam. Theory Appl., vol. 40, no. 9, pp. 591-601, 1993. http://dx.doi.org/10.1109/81.244908 B. Blasius, A. Huppert, and L. Stone, "Complex dynamics and phase synchronization in spatially extended ecological systems", Nature, vol. 399, no. 6734, pp. 354-359, 1999. http://dx.doi.org/10.1038/20676 PMID: 10360572 M.K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization., World Scientific: Singapore, 1996. Han, C. Kerrer, and Y. Kuramoto, “D-phasing and bursting in coupled neural Oscillators”, Phys. Rev. Lett., vol. 75, pp. 3190-3193, 1995. http://dx.doi.org/10.1103/PhysRevLett.75.3190 PMID: 10059517 K.M. Cuomo, and A.V. Oppenheim, "Circuit implementation of synchronized chaos with applications to communications", Phys. Rev. Lett., vol. 71, no. 1, pp. 65-68, 1993. http://dx.doi.org/10.1103/PhysRevLett.71.65 PMID: 10054374 K. Murali, and M. Lakshmanan, "Secure communication using a compound signal using sampled-data feedback", Appl. Math. Mech., vol. 11, pp. 1309-1315, 2003. J.H. Park, and O.M. Kwon, "A novel criterion for delayed feedback control of time-delay chaotic systems", Chaos Solitons Fractals, vol. 23, no. 2, pp. 495-501, 2005. http://dx.doi.org/10.1016/j.chaos.2004.05.023 R. Guo, "A simple adaptive controller for chaos and hyperchaos synchronization", Phys. Lett. A, vol. 372, no. 34, pp. 5593-5597, 2008. http://dx.doi.org/10.1016/j.physleta.2008.07.016

36

[25]

Advances in Special Functions of Fractional Calculus

Agrawal et al.

J. Huang, "Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters", Phys. Lett. A, vol. 372, no. 27-28, pp. 4799-4804, 2008. http://dx.doi.org/10.1016/j.physleta.2008.05.025 [26] S. Chen, and J. Lü, "Parameters identification and synchronization of chaotic systems based upon adaptive control", Phys. Lett. A, vol. 299, no. 4, pp. 353-358, 2002. http://dx.doi.org/10.1016/S0375-9601(02)00522-4 [27] M.T. Yassen, "Chaos synchronization between two different chaotic systems using active control", Chaos Solitons Fractals, vol. 23, no. 1, pp. 131-140, 2005. http://dx.doi.org/10.1016/j.chaos.2004.03.038 [28] S.K. Agrawal, M. Srivastava, and S. Das, "Synchronization of fractional order chaotic systems using active control method", Chaos Solitons Fractals, vol. 45, no. 6, pp. 737-752, 2012. http://dx.doi.org/10.1016/j.chaos.2012.02.004 [29] S.K. Agrawal, M. Srivastava, and S. Das, "Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems", Nonlinear Dyn., vol. 69, no. 4, pp. 2277-2288, 2012. http://dx.doi.org/10.1007/s11071-012-0426-y [30] D.M. Senejohnny, and H. Delavari, "Active sliding observer scheme based fractional chaos synchronization", Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 11, pp. 4373-4383, 2012. http://dx.doi.org/10.1016/j.cnsns.2012.03.004 [31] Y. Chai, and L.Q. Chen, "Projective lag synchronization of spatiotemporal chaos via active sliding mode control", Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 8, pp. 3390-3398, 2012. http://dx.doi.org/10.1016/j.cnsns.2011.12.011 [32] H. Yu, and Y. Liu, "Chaotic synchronization based on stability criterion of linear systems", Phys. Lett. A, vol. 314, no. 4, pp. 292-298, 2003. http://dx.doi.org/10.1016/S0375-9601(03)00908-3 [33] M. Srivastava, S.K. Agrawal, and S. Das, “Adaptive projective synchronization between different chaotic systems with parametric uncertainties and external disturbances”, Pramana-journal of physics, vol. 81, pp. 417-437, 2013. http://dx.doi.org/10.1007/s12043-013-0580-x [34] G.H. Erjaee, and S. Momani, "Phase synchronization in fractional differential chaotic systems", Phys. Lett. A, vol. 372, no. 14, pp. 2350-2354, 2008. http://dx.doi.org/10.1016/j.physleta.2007.11.065 [35] K. Zhang, H. Wang, and H. Fang, "Feedback control and hybrid projective synchronization of a fractional-order Newton–Leipnik system", Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 1, pp. 317-328, 2012. http://dx.doi.org/10.1016/j.cnsns.2011.04.003 [36] S.K. Agrawal, and S. Das, "Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique", Math. Methods Appl. Sci., vol. 37, no. 17, 2013.

Uncertain Parameters

[37]

[38] [39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47] [48] [49]

Advances in Special Functions of Fractional Calculus

37

K.S. Sudheer, and M. Sabir, "Function projective synchronization in chaotic and hyperchaotic systems through open-plus-closed-loop coupling", Chaos, vol. 20, no. 1, 2010.013115. http://dx.doi.org/10.1063/1.3309019 PMID: 20370270 H. Dua, Q. Zeng, C. Wang, and M. Ling, "Function projective synchronization in coupled chaotic systems. Nonlin", Analy: Real World Appli., vol. 11, pp. 705-712, 2010. H. Zhang, W. Huang, Z. Wang, and T. Chai, "Adaptive synchronization between two different chaotic systems with unknown parameters", Phys. Lett. A, vol. 350, no. 5-6, pp. 363-366, 2006. http://dx.doi.org/10.1016/j.physleta.2005.10.033 H. Salarieh, and M. Shahrokhi, "Adaptive synchronization of two different chaotic systems with time varying unknown parameters", Chaos Solitons Fractals, vol. 37, no. 1, pp. 125136, 2008. http://dx.doi.org/10.1016/j.chaos.2006.08.038 M.M. Al-Sawalha, and M.S.M. Noorani, "Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters", Commun. Nonlinear Sci. Numer. Simul., vol. 15, no. 4, pp. 1036-1047, 2010. http://dx.doi.org/10.1016/j.cnsns.2009.05.037 X.F. Li, A.C.S. Leung, X.P. Han, X.J. Liu, and Y.D. Chu, "Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control", Nonlinear Dyn., vol. 63, no. 1-2, pp. 263-275, 2011. http://dx.doi.org/10.1007/s11071-010-9802-7 F. Yu, C.H. Wang, Y. Hu, and J.W. Yin, "Antisynchronization of a novel hyperchaotic system with parameter mismatch and external disturbances", Pramana, vol. 79, no. 1, pp. 8193, 2012. http://dx.doi.org/10.1007/s12043-012-0285-6 S.K. Agrawal, and S. Das, "A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters", Nonlinear Dyn., vol. 73, no. 1-2, pp. 907-919, 2013. http://dx.doi.org/10.1007/s11071-013-0842-7 K. Diethelm, N.J. Ford, and A.D. Freed, "Detailed error analysis for a fractional Adams method", Numer. Algorithms, vol. 36, no. 1, pp. 31-52, 2004. http://dx.doi.org/10.1023/B:NUMA.0000027736.85078.be K. Diethelm, and N.J. Ford, "Multi-order fractional differential equations and their numerical solution", Appl. Math. Comput., vol. 154, no. 3, pp. 621-640, 2004. http://dx.doi.org/10.1016/S0096-3003(03)00739-2 I. Podlubny, Fractional Differential Equations., Academic Press: New York, 1999. W. Hahn, The Stability of Motion., Springer: New York, 1967. http://dx.doi.org/10.1007/978-3-642-50085-5 A.M. Liapunov, Stability of Motion., Elservier, Academic Press New York and London, 1966.

38

[50]

[51] [52]

[53]

Advances in Special Functions of Fractional Calculus

Agrawal et al.

C. Liu, L. Liu, and T. Liu, "A novel three-dimensional autonomous chaos system", Chaos Solitons Fractals, vol. 39, no. 4, pp. 1950-1958, 2009. http://dx.doi.org/10.1016/j.chaos.2007.06.079 H. Wang, and J. Ma, "Chaos Control and Synchronization of a Fractional-order Autonomous System", WSEAS Trans. Math., vol. 11, no. 8, pp. 704-711, 2012. C. Liu, T. Liu, L. Liu, and K. Liu, "A new chaotic attractor", Chaos Solitons Fractals, vol. 22, no. 5, pp. 1031-1038, 2004. http://dx.doi.org/10.1016/j.chaos.2004.02.060 F. Yu, and C. Wang, "A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term", Engineering, Technology & Applied Science Research, vol. 2, no. 2, pp. 209-215, 2012. http://dx.doi.org/10.48084/etasr.86

Advances in Special Functions of Fractional Calculus, 2023, 39-51

39

CHAPTER 2

Improved Generalized Differential Transform Method for a Class of Linear Nonhomogeneous Ordinary Fractional Differential Equations İ. Onur KIYMAZ1,*and Ayşegül ÇETİNKAYA1 1

Deptartment of Mathematics, Ahi Evran University, Kırşehir, Turkey Abstract: In this paper, by using the generalized Taylor's formula we improved the generalized differential transform method, which is a useful tool for getting the approximate analytic solutions of fractional differential equations. With this improvement, solutions of a class of linear nonhomogeneous ordinary fractional differential equations, which could not be solved with generalized differential transform method before, will be achieved and the solutions obtained will contain more integers and fractional exponents

Keywords: Fractional Differential Equations, Generalized Taylor's Formula, Generalized Differential Transform Method. 2000 MSC: 65L05, 26A33. 1. INTRODUCTION In 1986, Zhou [1] presented the concept of differential transformation and used it for obtaining the solutions of linear and non-linear initial value problems in electric circuit analysis. The concept is derived from the Taylor series expansion. In 1999, Chen and Ho [2] proposed a new transformation for solving partial differential equations, which called two-dimensional differential transform. In 2008, Momani and Odibat [3] developed this method for finding the solution of linear fractional partial differential equations. This method is based on the generalized Taylor's formula which given by Odibat and Shawagfeh [4] in 2007 and called as generalized differential transform method (GDTM). Many other

author Onur KIYMAZ: Deptartment of Mathematics, Ahi Evran University, Kırşehir, Turkey; E-mail: [email protected] *Corresponding

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

40

Advances in Special Functions of Fractional Calculus

KIYMAZ and ÇETİNKAYA

authors used this method for solving fractional partial differential equations (see for example [5 - 9]). Recently, El-Ajou et al. [10] (see also [11]), introduced a general form of fractional power series. ∞ 𝑚−1

∑ ∑ 𝑎𝑗𝑛 (𝑥 − 𝑥0 )𝑗+𝑛α ,

(1.1)

𝑛=0 𝑗=0

where 𝑎𝑗𝑛 's are constants, 𝑚 ∈ ℕ, 𝑥 ≥ 𝑥0 and 0 ≤ 𝑚 − 1 < α ≤ 𝑚. They also obtained a general form of the generalized Taylor's formula. ∞ 𝑚−1

𝑓(𝑥) = ∑ ∑ 𝑛=0 𝑗=0

𝐷 𝑗 𝑫𝑛α 𝑓(𝑥0 ) (𝑥 − 𝑥0 )𝑗+𝑛α , Γ(𝑗 + 𝑛α + 1)

(1.2)

𝑑𝑗

where 𝐷 𝑗 = 𝑗, 𝑫𝑛α = 𝑫α 𝑫α ⋯ 𝑫α (n-times), and 𝑫α is the usual Caputo 𝑑𝑥 fractional derivative [12] which given for 𝑚 − 1 < α < 𝑚. 𝑫α 𝑓(𝑥) =

𝑥 𝑚 1 𝑑 ∫ 𝑓(𝑡)(𝑥 − 𝑡)𝑚−α−1 𝑑𝑡, Γ(𝑚 − α) 0 𝑑𝑡 𝑚

and for α = 𝑚 ∈ ℕ, 𝑫α 𝑓(𝑥) = 𝐷𝑚 𝑓(𝑥). Our motivation in this work is to improve the GDTM by moving from the abovementioned Taylor's formula (2). With the proposed method, more comprehensive solutions, which contains both integer and fractional orders of the unknown function, can be obtained for the following linear non homogenous fractional boundary value problem. 𝑫α 𝑢(𝑥) ± 𝜆𝑢(𝑥) = 𝑓(𝑥) 𝑢(𝑗) (0) = 𝑐𝑗

(1.3)

where 𝑥 ≥ 0, λ > 0, 𝑐𝑗 ∈ ℝ, 𝑗 = 0,1,2 … , 𝑚 − 1 and 𝑓(𝑥) is an analytic function in its domain.

Differential Equations

2. IMPROVED METHOD

Advances in Special Functions of Fractional Calculus

GENERALIZED

DIFFERENTIAL

41

TRANSFORM

Before the definition of the improved generalized differential transform method (IGDTM), we gave the following theorem and a basic identity for Caputo fractional derivative in case the reader does not have enough information about fractional derivatives. Theorem 2.1. Suppose that 𝑓(𝑥) = 𝑥 λ 𝑔(𝑥), where λ > 1 and 𝑔(𝑥) has the α𝑛 generalized power series expansion 𝑔(𝑥) = ∑∞ with radius of 𝑛=0 𝑎𝑛 𝑥 convergence 𝑅 > 0, 0 < α ≤ 1. If a) β < λ + 1 and α arbitrary or b) β ≥ λ + 1, α arbitrary and 𝑎𝑛 = 0 for 𝑛 = 0,1, … , 𝑚 − 1, where 𝑚 − 1 < β ≤ 𝑚, then we have 𝑫γ 𝑫β 𝑓(𝑥) = 𝑫γ+β 𝑓(𝑥)

for all 𝑥 ∈ (0, 𝑅) [13]. Lemma 2.1. Let 𝑚 − 1 < α < 𝑚 and λ > 𝑚 − 1 then 𝑫α 𝑥 λ =

Γ(λ+1) Γ(λ−α+1)

𝑥 λ−α .

For more details about fractional calculus, we refer the books [12, 14, 15] to the reader. Now, suppose that the function 𝑢(𝑥) can be represented as ∞ 𝑚−1

∞ 𝑚−1

𝑢(𝑥) = ∑ ∑ 𝑎𝑛𝑗 (𝑥 − 𝑥0 )𝑛α+𝑗 = ∑ ∑ 𝑈α (𝑛, 𝑗)(𝑥 − 𝑥0 )𝑛α+𝑗 , 𝑛=0 𝑗=0

(2.1)

𝑛=0 𝑗=0

where 𝑚 − 1 < α ≤ 𝑚. Then the one-dimensional improved generalized differential transform (IGDT) of the function 𝑢(𝑥) in (4) is given with.

42

KIYMAZ and ÇETİNKAYA

Advances in Special Functions of Fractional Calculus

𝑈α (𝑛, 𝑗) =

[𝐷 𝑗 𝑫𝑛α 𝑢(𝑥 − 𝑥0 )](𝑥 Γ(𝑛α + 𝑗 + 1)

0)

.

(2.2)

Besides, the function 𝑢(𝑥) in (4) is called as the inverse one-dimensional IGDT of 𝑈α (𝑛, 𝑗). Conveniently, throughout the paper we assumed that the center of the fractional power series (1) is zero since this can always be done via the linear change of variable (𝑥 − 𝑥0 ) ↦ 𝑥. Let 𝑈α , 𝑉α and 𝑊α are the IGDT of the functions u, v and w, respectively. For the class of problems given with (3) we need the following theorems: Theorem 2.2. If 𝑢(𝑥) = 𝑣(𝑥) ± 𝑤(𝑥) then 𝑈α (𝑛, 𝑗) = 𝑉α (𝑛, 𝑗) ± 𝑊α (𝑛, 𝑗),

and if 𝑢(𝑥) = λ𝑣(𝑥), λ ∈ ℝ then 𝑈α (𝑛, 𝑗) = λ𝑉α (𝑛, 𝑗).

Proof. The proof is trivial from the definition of transform in (4). Theorem 2.3. If 𝑢(𝑥) = 𝑥 𝑘α+𝑙 , 𝑘, 𝑙 ∈ ℤ+ , 0 ≤ 𝑙 < 𝑚 − 1 then 𝑈α (𝑛, 𝑗) = δ(𝑛 − 𝑘)δ(𝑗 − 𝑙).

Proof. Let 𝑢(𝑥) = 𝑥 𝑘α+𝑙 . Then ∞ 𝑚−1

𝑢(𝑥) = 𝑥

𝑘α+𝑙

= ∑ ∑ δ(𝑛 − 𝑘)δ(𝑗 − 𝑙)𝑥 𝑛α+𝑗 𝑛=0 𝑗=0

can be written in terms of Dirac-delta function. Theorem 2.4. If 𝑢(𝑥) = 𝑫α 𝑣(𝑥) where 𝑣(𝑥) satisfies the conditions in Theorem 2.1, then 𝑈α (𝑛, 𝑗) =

Γ((𝑛 + 1)α + 𝑗 + 1) 𝑉α (𝑛 + 1, 𝑗). Γ(𝑛α + 𝑗 + 1)

Differential Equations

Advances in Special Functions of Fractional Calculus

43

Proof. From the equation (5) we have, 𝑈α (𝑛, 𝑗)

=

[𝐷 𝑗 𝑫𝑛α 𝑫α 𝑣(𝑥)]𝑥=0 Γ(𝑛α + 𝑗 + 1) [𝐷 𝑫(𝑛+1)α 𝑣(𝑥)]𝑥=0 𝑗

= =

Γ(𝑛α + 𝑗 + 1) 𝑗 (𝑛+1)α 𝑣(𝑥)]𝑥=0 Γ((𝑛 + 1)α + 𝑗 + 1) [𝐷 𝑫 , Γ(𝑛α + 𝑗 + 1) Γ((𝑛 + 1)α + 𝑗 + 1)

which completes the proof. 3. APPLICATION OF THE METHOD We selected two examples to show the simplicity and effectiveness of the proposed method. But first we must mention that the Mittag-Leffler function with two variables is defined as: ∞

𝐸α,β (𝑥) = ∑ 𝑛=0

𝑥𝑛 , Γ(α𝑛 + β)

where 𝑅𝑒(α) > 0 [16]. Example 3.1. Assume that the IVP 𝑫𝛼 𝑢(𝑥) − 𝜆𝑢(𝑥) =

𝑥 𝛼+2 , Γ(𝛼 + 3)

𝒖(𝒋) (𝟎) = 𝒄𝒋 ,

(𝑥 ≥ 0; 𝜆 > 0; 𝑚 − 1 < 𝛼 ≤ 𝑚)

(𝒄𝒋 ∈ ℝ, 𝒋 = 𝟎, 𝟏, 𝟐 … , 𝒎 − 𝟏).

is given. The IGDT of the fractional differential equation (6) is Γ((𝑛 + 1)𝛼 + 𝑗 + 1) 𝛿(𝑛 − 1)𝛿(𝑗 − 2) 𝑈𝛼 (𝑛 + 1, 𝑗) = 𝜆𝑈𝛼 (𝑛, 𝑗) + , Γ(𝑛𝛼 + 𝑗 + 1) Γ(𝛼 + 3)

and the IGDT of the initial values (7) are 𝑈α (0, 𝑗) =

𝑐𝑗 , Γ(𝑗 + 1)

(𝑗 = 0,1,2 … , 𝑚 − 1).

(3.1) (3.2)

44

Advances in Special Functions of Fractional Calculus

KIYMAZ and ÇETİNKAYA

Then we have for 𝑛 = 0 𝑗=0

⇒

𝑈𝛼 (1,0) =

Γ(1) 𝜆𝑐0 [𝜆𝑈𝛼 (0,0)] = , Γ(𝛼 + 1) Γ(𝛼 + 1)

𝑗=1

⇒

𝑈𝛼 (1,1) =

𝜆𝑐1 Γ(2) [𝜆𝑈𝛼 (0,1)] = , Γ(𝛼 + 2) Γ(𝛼 + 2)

𝑗=2

⇒

𝑈𝛼 (1,2) =

Γ(3) 𝜆𝑐2 [𝜆𝑈𝛼 (0,2)] = , Γ(𝛼 + 3) Γ(𝛼 + 3) ⋮

𝑈𝛼 (1, 𝑚 − 1) =

𝑗 =𝑚−1 ⇒

Γ(𝑚) 𝜆𝑐𝑚−1 [𝜆𝑈𝛼 (0, 𝑚 − 1)] = . Γ(𝛼 + 𝑚) Γ(𝛼 + 𝑚)

For 𝑛 = 1 we have

𝑗=2

⇒

𝑗=0 ⇒

𝑈𝛼 (2,0) =

Γ(𝛼 + 1) 𝜆2 𝑐0 [𝜆𝑈𝛼 (1,0)] = , Γ(2𝛼 + 1) Γ(2𝛼 + 1)

𝑗=1 ⇒

𝑈𝛼 (2,1) =

Γ(𝛼 + 2) 𝜆2 𝑐1 [𝜆𝑈𝛼 (1,1)] = , Γ(2𝛼 + 2) Γ(2𝛼 + 2)

𝑈𝛼 (2,2) =

Γ(𝛼 + 3) 1 𝜆2 𝑐2 1 [𝜆𝑈𝛼 (1,2) + ]= + , Γ(2𝛼 + 3) Γ(2𝛼 + 3) Γ(2𝛼 + 3) Γ(𝛼 + 3) ⋮

𝑗 =𝑚−1

⇒

𝑈𝛼 (2, 𝑚 − 1) =

Γ(𝛼 + 𝑚) 𝜆2 𝑐𝑚−1 [𝜆𝑈𝛼 (1, 𝑚 − 1)] = . Γ(2𝛼 + 𝑚) Γ(2𝛼 + 𝑚)

And for 𝑛 = 2 we have 𝑗=0 ⇒

𝑈𝛼 (3,0) =

Γ(2𝛼 + 1) 𝜆3 𝑐0 [𝜆𝑈𝛼 (2,0)] = , Γ(3𝛼 + 1) Γ(3𝛼 + 1)

𝑗=1 ⇒

𝑈𝛼 (3,1) =

Γ(2𝛼 + 2) 𝜆3 𝑐1 [𝜆𝑈𝛼 (2,1)] = , Γ(3𝛼 + 2) Γ(3𝛼 + 2)

Differential Equations

𝑗=2 ⇒

Advances in Special Functions of Fractional Calculus

𝑈𝛼 (3,2) =

45

Γ(2𝛼 + 3) 𝜆3 𝑐2 𝜆 [𝜆𝑈𝛼 (2,2)] = + , Γ(3𝛼 + 3) Γ(3𝛼 + 3) Γ(3𝛼 + 3) ⋮

𝑗 =𝑚−1

⇒

𝑈𝛼 (3, 𝑚 − 1) =

Γ(2𝛼 + 𝑚) 𝜆3 𝑐𝑚−1 [𝜆𝑈𝛼 (2, 𝑚 − 1)] = . Γ(3𝛼 + 𝑚) Γ(3𝛼 + 𝑚)

Continuing this way, for 𝑗 = 0,1,2, … , 𝑚 − 1 we have 𝜆𝑛 𝑐𝑗 , Γ(𝑛𝛼 + 𝑗 + 1) (𝑛, 𝑈𝛼 𝑗) = 𝜆𝑛 𝑐𝑗 𝛿(𝑗 − 2)𝜆𝑛−2 + , {Γ(𝑛𝛼 + 𝑗 + 1) Γ(𝑛𝛼 + 𝑗 + 1)

𝑛 = 0, 1 𝑛 = 2,3,4, . . .

So, after some calculations, we get the solution from (4) as: 𝑚−1

𝑢(𝑥) = ∑ 𝑐𝑗 𝑥 𝑗 𝐸𝛼,𝑗+1 (𝜆𝑥 𝛼 ) + 𝑥 2𝛼+2 𝐸𝛼,2𝛼+3 (𝜆𝑥 𝛼 ).

(3.3)

𝑗=0

Example 3.2. Assume that the IVP 𝑫𝛼 𝑢(𝑥) + 𝑢(𝑥) = 𝑥 𝛼+3 + 𝑢(𝑗) (0) = 𝑐𝑗 ,

Γ(𝛼 + 4) 3 𝑥 , 6

(𝑥 ≥ 0; 𝑚 − 1 < 𝛼 ≤ 𝑚),

(𝑐𝑗 ∈ ℝ, 𝑗 = 0,1,2 … , 𝑚 − 1).

(3.5)

is given. The IGDT of the fractional differential equation (9) is Γ((𝑛 + 1)𝛼 + 𝑗 + 1) 𝑈𝛼 (𝑛 + 1, 𝑗) Γ(𝑛𝛼 + 𝑗 + 1) = −𝑈𝛼 (𝑛, 𝑗) + 𝛿(𝑛 − 1)𝛿(𝑗 − 3) + 𝛿(𝑛)𝛿(𝑗 − 3)

and the IGDT of the initial values (10) are 𝑈α (0, 𝑗) =

Then we have for 𝑛 = 0

𝑐𝑗 , Γ(𝑗 + 1)

(𝑗 = 0,1,2 … , 𝑚 − 1).

(3.4)

Γ(𝛼 + 4) 6

46

Advances in Special Functions of Fractional Calculus

KIYMAZ and ÇETİNKAYA

𝑗=0 ⇒

𝑈𝛼 (1,0) =

−𝑐0 Γ(1) [−𝑈𝛼 (0,0)] = , Γ(𝛼 + 1) Γ(𝛼 + 1)

𝑗=1 ⇒

𝑈𝛼 (1,1) =

−𝑐1 Γ(2) [−𝑈𝛼 (0,1)] = , Γ(𝛼 + 2) Γ(𝛼 + 2)

𝑗=2 ⇒

𝑈𝛼 (1,2) =

−𝑐2 Γ(3) [−𝑈𝛼 (0,2)] = , Γ(𝛼 + 3) Γ(𝛼 + 3)

𝑗=3 ⇒

𝑈𝛼 (1,3) =

Γ(𝛼 + 4) −𝑐3 Γ(4) [−𝑈𝛼 (0,3) + ]= + 1, 6 Γ(𝛼 + 4) Γ(𝛼 + 4) ⋮

𝑗 =𝑚−1

𝑈𝛼 (1, 𝑚 − 1) =

⇒

Γ(𝑚) −𝑐𝑚−1 [−𝑈𝛼 (0, 𝑚 − 1)] = . Γ(𝛼 + 𝑚) Γ(𝛼 + 𝑚)

For 𝑛 = 1 we have 𝑗=0

⇒

𝑈𝛼 (2,0) =

Γ(𝛼 + 1) 𝑐0 [−𝑈𝛼 (1,0)] = , Γ(2𝛼 + 1) Γ(2𝛼 + 1)

𝑗=1

⇒

𝑈𝛼 (2,1) =

Γ(𝛼 + 2) 𝑐1 [−𝑈𝛼 (1,1)] = , Γ(2𝛼 + 2) Γ(2𝛼 + 2)

𝑗=2

⇒

𝑈α (2,2) =

Γ(α + 3) 𝑐2 [−𝑈α (1,2)] = , Γ(2α + 3) Γ(2α + 3)

𝑗=3

𝑈𝛼 (2,3) =

⇒

Γ(𝛼 + 4) 𝑐3 [−𝑈𝛼 (1,3) + 1] = , Γ(2𝛼 + 4) Γ(2𝛼 + 4) ⋮

𝑗 =𝑚−1 ⇒

𝑈𝛼 (2, 𝑚 − 1) =

Γ(𝛼 + 𝑚) 𝑐𝑚−1 [−𝑈𝛼 (1, 𝑚 − 1)] = . Γ(2𝛼 + 𝑚) Γ(2𝛼 + 𝑚)

And for 𝑛 = 2 we have 𝑗=0

⇒

𝑈𝛼 (3,0) =

Γ(2𝛼 + 1) −𝑐0 [−𝑈𝛼 (2,0)] = , Γ(3𝛼 + 1) Γ(3𝛼 + 1)

Differential Equations

Advances in Special Functions of Fractional Calculus

𝑗=1

⇒

𝑈𝛼 (3,1) =

−𝑐1 Γ(2𝛼 + 2) [−𝑈𝛼 (2,1)] = , Γ(3𝛼 + 2) Γ(3𝛼 + 2)

𝑗=2

⇒

𝑈𝛼 (3,2) =

−𝑐2 Γ(2𝛼 + 3) [−𝑈𝛼 (2,2)] = , Γ(3𝛼 + 3) Γ(3𝛼 + 3)

𝑗=3

⇒

𝑈𝛼 (3,3) =

47

−𝑐3 Γ(2𝛼 + 4) [−&𝑈𝛼 (2,3)] = , Γ(3𝛼 + 4) Γ(3𝛼 + 4) ⋮

𝑗 =𝑚−1 ⇒

𝑈𝛼 (3, 𝑚 − 1) =

Γ(2𝛼 + 𝑚) −𝑐𝑚−1 [−𝑈𝛼 (2, 𝑚 − 1)] = . Γ(3𝛼 + 𝑚) Γ(3𝛼 + 𝑚)

Continuing this way, for 𝑗 = 0,1,2, … , 𝑚 − 1 we have 𝑈α (𝑛, 𝑗) =

(−1)𝑛 𝑐𝑗 Γ(𝑛α + 𝑗 + 1)

+ δ(𝑛 − 1)δ(𝑗 − 2).

So, after some calculations, we get the solution from (4) as: 𝑚−1

𝑢(𝑥) = ∑ 𝑐𝑗 𝑥 𝑗 𝐸α,𝑗+1 (−𝑥 α ) + 𝑥 α+3 . 𝑗=0

CONCLUDING REMARKS AND OBSERVATIONS In Example 3.1, the first term on the right-hand side of equation (8) 𝑚−1

∑ 𝑐𝑗 𝑥 𝑗 𝐸α,𝑗+1 (λ𝑥 α ) 𝑗=0

is the complementary solution and the second term 𝑥 2α+2 𝐸α,2α+3 (λ𝑥 α )

is a particular solution of the fractional differential equation (6).

The same problem also solved by Odibat and Shawagfeh [4] (for 𝑓(𝑥 ) = 0 and 0 < 𝛼 ≤ 1) and by El-Ajou et al. [10] (for 𝑓(𝑥 ) = 0 and 𝑚 < 𝛼 ≤ 𝑚 −

48

KIYMAZ and ÇETİNKAYA

Advances in Special Functions of Fractional Calculus

1) before. Our result (8) is in complete agreement with the results obtained in the above-mentioned papers for 𝑓 (𝑥 ) = 0. In fact, this example is a special case of the fractional differential equation. 𝑫α 𝑢(𝑥) = λ𝑢(𝑥) + 𝑓(𝑥),

(𝑥 > 0; α > 0; λ > 0)

with the initial conditions. 𝑢(𝑗) (0) = 𝑐0 ,

(𝑗 = 0,1,2 … , 𝑚 − 1; − 1 < α ≤ 𝑚).

Luchko and Gorenflo [17] proved this IVP has a unique solution of the form. 𝑚−1

𝑥

𝑢(𝑥) = ∑ 𝑐𝑗 𝑥 𝑗 𝐸𝛼,𝑗+1 (𝜆𝑥 𝛼 ) + ∫ (𝑥 − 𝑡)𝛼−1 𝐸𝛼,𝛼 (𝜆(𝑥 − 𝑡)𝛼 )𝑓(𝑡)𝑑𝑡.

(4.1)

0

𝑗=0

For 𝑓(𝑥) = 0, both results (8) and (11) are equal. Also, if we consider the two results, we observe that for 𝑓(𝑥) =

𝑥 α+2 Γ(α+3)

we have

𝑥

∫ 𝑡 α+2 (𝑥 − 𝑡)α−1 𝐸α,α (λ(𝑥 − 𝑡)α )𝑑𝑡 = Γ(α + 3)𝑥 2α+2 𝐸α,2α+3 (λ𝑥 α ). 0

The result can be easily checked for α = λ = 1: ∞

𝑥 𝑥

3 −𝑡

𝑒 ∫ 𝑡 𝑒 0

𝑥

𝑑𝑡 = 6𝑒 −

(𝑥 3

2

+ 3𝑥 + 6𝑥 + 6) = 6 ∑ 𝑛=0

𝑥 𝑛+4 = Γ(4)𝑥 4 𝐸1,5 (𝑥). (𝑛 + 4)!

Besides, Example 3.2 also solved in [18, 19] numerically, by taking the initial values 𝑢(𝑗) (0) = 0 for 𝑗 = 0,1,2, … , 𝑚 − 1. The analytic solution of this problem is 𝑥 α+3 , which is the same as our result. The two examples given above cannot be solved by GDTM because of the nonhomogeneous terms. So, our proposed method IGDTM can be used easily for the class of problems (3). Further studies can be done by changing the Caputo fractional derivative with its various extensions like [20, 21, 22]. We are working on both a new improvement that will appeal to much wider classes of problems, including non-linear ones and

Differential Equations

Advances in Special Functions of Fractional Calculus

49

the improved two-dimensional differential transform for the problems which includes fractional partial differential equations. CONSENT FOR PUBLICATON Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest. ACKNOWLEDGEMENT This work is supported by the Ahi Evran University Scientific Research Projects Coordination Unit with project number FEF.A4.18.017. REFERENCES [1]

J.K. Zhou, Differential Transformation and Its Applications for Elec- trical Circuits., Huazhong University Press: Wuhan, China, 1986. (in Chinese)

[2]

C. K. Chen, and S. H. Ho, "Solving partial differential equations by two- dimensional differential transform method", Appl. Math. Comput., vol. 106, no. 2-3, pp. 171-179, 1999.

[3]

S. Momani, and Z. Odibat, "A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula", J. Comput. Appl. Math., vol. 220, no. 1-2, pp. 85-95, 2008. http://dx.doi.org/10.1016/j.cam.2007.07.033

[4]

Z.M. Odibat, and N.T. Shawagfeh, "Generalized Taylor’s formula", Appl. Math. Comput., vol. 186, no. 1, pp. 286-293, 2007. http://dx.doi.org/10.1016/j.amc.2006.07.102

[5]

C.A., and Kıymaz I.O., "The solution of the time-fractional diffusion equation by the generalized differential transform method", Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2349-2354, 2013.

[6]

C.A., Kıymaz I.O., and amlı J.C¸ "Solutions of nonlinear PDE’s of fractional order with generalized differential transform method", In International Mathematical Forum, vol. 6, no. 1, pp. 39-47, 2011.

[7]

V.S. Erturk, S. Momani, and Z. Odibat, "Application of generalized differential transform method to multi-order fractional differential equations", Commun. Nonlinear Sci. Numer. Simul., vol. 13, no. 8, pp. 1642-1654, 2008. http://dx.doi.org/10.1016/j.cnsns.2007.02.006

50

Advances in Special Functions of Fractional Calculus

KIYMAZ and ÇETİNKAYA

[8]

L. Jin-Cun, and H. Guo-Lin, "New approximate solution for time- fractional coupled KdV equations by generalised differential transform method", Chin. Phys. B, vol. 19, no. 11, pp. 110-203, 2010.

[9]

Z. Odibat, and S. Momani, "A generalized differential transform method for linear partial differential equations of fractional order", Appl. Math. Lett., vol. 21, no. 2, pp. 194-199, 2008. http://dx.doi.org/10.1016/j.aml.2007.02.022

[10]

A. El-Ajou, O. Abu Arqub, and M. Al-Smadi, "A general form of the generalized Taylor’s formula with some applications", Appl. Math. Comput., vol. 256, pp. 851-859, 2015. http://dx.doi.org/10.1016/j.amc.2015.01.034

[11]

A. El-Ajou, O. Arqub, Z. Zhour, and S. Momani, "New results on fractional power series: theories and applications", Entropy (Basel), vol. 15, no. 12, pp. 5305-5323, 2013. http://dx.doi.org/10.3390/e15125305

[12]

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, "Theory and applications of fractional differential equations" Elsevier, Amsterdam etc., 2006.

[13]

S. Momani, Z. Odibat, and V.S. Ertu¨rk, "Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation", Phys. Lett. A, vol. 370, no. 56, pp. 379-387, 2007. http://dx.doi.org/10.1016/j.physleta.2007.05.083

[14]

I. Podlubny, Fractional differential equations., Academic Press: New York, 1999.

[15]

S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications., Gordon and Breach Science Publishers: Singapore, 1993.

[16]

R. Gorenflo, A.A. Kilbas, F. Mainardi, and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications., Springer-Verlag: Berlin, Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-43930-2

[17]

Y. Luchko, and R. Gorenflo, "An operational method for solving frac- tional differential equations with the Caputo derivatives", Acta Math. Vietnam, vol. 24, no. 2, pp. 207-233, 1999.

[18]

H.M. Baskonus, and H. Bulut, "On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method", Open Math., vol. 13, no. 1, 2015.000010151520150052. http://dx.doi.org/10.1515/math-2015-0052

[19]

J. Cao, and C. Xu, "A high order schema for the numerical solution of the fractional ordinary differential equations", J. Comput. Phys., vol. 238, pp. 154-168, 2013. http://dx.doi.org/10.1016/j.jcp.2012.12.013

[20]

A. C¸ etinkaya, I˙. O. Kıymaz, P. Agarwal and R. P. Agarwal, “A com- parative study on generating function relations for generalized hyper- geometric functions via generalized fractional operators”, Adv. Differ. Equ., vol. 2018, no. 156, 2018.

[21]

İ.O. Kıymaz, A. Çetinkaya, and P. Agarwa, "An extension of Caputo fractional derivative operator and its applications", J. Nonlinear Sci. Appl., vol. 9, no. 6, pp. 3611-3621, 2016. http://dx.doi.org/10.22436/jnsa.009.06.14

Differential Equations

[22]

Advances in Special Functions of Fractional Calculus

51

İ.O. Kıymaz, A. Çetinkaya, P. Agarwa, and S. Jain, "On a new extension of Caputo fractional derivative operator", Advances in real and complex analysis with applications, Birkhauser: Basel, 2017.

52

Advances in Special Functions of Fractional Calculus, 2023, 52-61

CHAPTER 3

Incomplete 𝑲𝟐 -function Dharmendra Kumar Singh1 and Vijay Laxmi Verma1, * 1

Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India Abstract: This chapter aims to introduce the incomplete 𝐾2 -Function. Incomplete hypergeometric function, incomplete confluent hypergeometric function, and incomplete Mittag-Leffler function can be deduced as special cases of our findings. Some fractional integral formulae illustrate various avenues of their applications.

Keywords: Incomplete pochhammer symbol, 𝐾2 function, Incomplete 𝐾2 function, Incomplete hypergeometric function, Incomplete Mittag-Leffler function. 1. INTRODUCTION In 1993, Miller and Ross [1] introduced a function. 𝐸𝑥 [𝑣, 𝑎] =

𝑑 −𝑣 𝑑𝑥 −𝑣

𝑒 𝑎𝑥 = 𝑥 𝑣 𝑒 𝑎𝑥 𝛾 ∗ (𝑣, 𝑎𝑥) = ∑∞ 𝑛=0

𝑎𝑛 𝑥 𝑛+𝑣 Γ(𝑛+𝑣+1)

,𝑣 ∈ 𝐶

(1.1)

based on the solution of the functional order initial value problem, where 𝛾 ∗ (𝑣, 𝑎𝑥) is the incomplete gamma function and divergent for |𝑥| = 1 if 1 ≤ 𝑅(𝛾). An extension of this function was introduced by Sharma and Dhakar [2] in the following form (𝑝;𝑞)

𝐾2 (𝑎1 , … , 𝑎𝑝 ;

(𝑣;𝑎)

(𝑝;𝑞)

𝑏1 , … , 𝑏𝑞 ; 𝑥) = 𝐾2 (𝑥) (𝑣;𝑎)

*Corresponding

author Vijay Laxmi Verma: Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

(𝑎1 )𝑛 …(𝑎𝑝 )

= ∑∞ 𝑛=0 (𝑏

𝑛

1 )𝑛 …(𝑏𝑞 )𝑛

𝑎𝑛 𝑥 𝑛+𝑣

(1.2)

Γ(𝑛+𝑣+1)

𝑛 𝑛 = 1 + ∑∞ 𝑛=1 ∏𝑖=1 (𝑎𝑖 )𝑛 / ∏𝑗=1 (𝑏𝑗 )𝑛

53

𝑎𝑛 𝑥 𝑛+𝑣

(1.3)

Γ(𝑛+𝑣+1)

where𝑣 ∈ 𝐶 and in which no denominator parameter 𝑏𝑗 is allowed to be zero or a negative integer. If any parameter 𝑎𝑖 in (1.3) is zero or a negative integer, the serves terminate. An application of the elementary ratio test to the power series on the right side of (1.3) shows at once that. (i) If 𝑝 > 𝑞 + 1 the series is convergent for all 𝑥. (ii) If 𝑝 = 𝑞 + 1and |𝑥| = 1, the series can converge in some cases. Let 𝛾 = 𝑝 𝑞 ∑𝑖=1 𝑎𝑖 − ∑𝑗=1 𝑏𝑗 it can be show that when 𝑝 = 𝑞 + 1 the series is absolutely convergent for |𝑥| = 1if 𝑅(𝛾) < 0, conditionally convergent for 𝑥 = −1 if 0 ≤ 𝑅(𝛾) < 1 and divergent for |𝑥| = 1 is 1 ≤ 𝑅(𝛾) Recently, Singh and Porwal [3] introduced the incomplete Mittag-Leffler function with the help of an incomplete Pochhammer Symbol in the following way, [𝛿,𝑘]

𝐸(𝛼,𝛽) (𝑥)

[𝛿,𝑘]

= ∑∞ 𝑛=0

𝐸(𝛼,𝛽) (𝑥) = ∑∞ 𝑛=0

[𝛿;𝑘]𝑛 𝑥 𝑛 Γ(𝛼𝑛+𝛽) 𝑛!

(𝛿;𝑘)𝑛 𝑥 𝑛 Γ(𝛼𝑛+𝛽) 𝑛!

,

where, 𝛼, 𝛽, 𝛿 ∈ 𝐶; Re(𝛼) > 0, 𝑅𝑒(𝛽) > 0, 𝑅𝑒(𝛿) > 0.

For details of the Mittag-Leffler function, see [4 - 8]. Motivated with work [3], now we introduce incomplete K 2 -Function in the following form (𝑝; 𝑞) 𝐾2 [(𝑎1 ; 𝑘) … , 𝑎𝑝 ; 𝑏1 , … , 𝑏𝑞 ; 𝑥] (𝑣, 𝑎)

(1.4)

(1.5)

54

Advances in Special Functions of Fractional Calculus

= ∑∞ 𝑛=0

Singh and Verma.

(𝑎1 ;𝑘)𝑛 …(𝑎𝑝 )

𝑎𝑛 𝑥 𝑛+𝑣

(𝑏1 )𝑛 …(𝑏𝑞 )

Γ(𝑛+𝑣+1)

𝑛

𝑛

,𝑣 ∈ 𝐶

(1.6)

and (𝑝; 𝑞) 𝐾2 [(𝑎1 ; 𝑘) … , 𝑎𝑝 ; 𝑏1 , … , 𝑏𝑞 ; 𝑥] (𝑣, 𝑎)

= ∑∞ 𝑛=0

[𝑎1 ;𝑘]𝑛 …(𝑎𝑝 ) (𝑏1 )𝑛 …(𝑏𝑞 )

𝑛

𝑛

𝑎𝑛 𝑥 𝑛+𝑣 Γ(𝑛+𝑣+1)

,𝑣 ∈ 𝐶

(1.7)

where𝑣 ∈ 𝐶 and the domain of convergents will be the same as of equations (1.2) and (1.3). Where [𝜆; 𝑘]𝑣 and (𝜆; 𝑘)𝑣 represent incomplete Pochhammer Symbol which is introduced by Srivastava et al. [9] and defined as follows: (𝜆; 𝑘)𝑣 =

[𝜆; 𝑘]𝑣 =

𝛾(𝜆+𝑣,𝑘) Γ(𝜆)

Γ(𝜆+𝑣,𝑘) Γ(𝜆)

, (𝜆, 𝑣 ∈ 𝐶; 𝑘 ≥ 0)

, (𝜆, 𝑣 ∈ 𝐶; 𝑘 ≥ 0)

(1.8) (1.9)

and these incomplete Pochhammer symbols satisfy the following decomposition relation: (𝜆; 𝑘)𝑣 + [𝜆; 𝑘]𝑣 = (𝜆)𝑣 ; (𝜆, 𝑣 ∈ 𝐶; 𝑘 ≥ 0)

(1.10)

where the Pochhammer Symbol (𝜆)𝑣 (𝜆, 𝑣 ∈ 𝐶) is given, in general, by (𝜆)𝑣 =

Γ(𝜆 + 𝑣) 1 (𝑣 = 0; 𝜆 ∈ 𝐶) ={ 𝜆(𝜆 + 1) … (𝜆 + 𝑛 − 1)(𝑣 ∈ 𝑁; 𝜆 ∈ 𝐶) Γ(𝜆)

If 𝐴𝑝 the array of 𝑝 parameters like 𝑎1 , 𝑎2 , … , 𝑎𝑝 . Then the Pochhammer Symbol (𝐴𝑝 )𝑛 , and the incomplete Pochhammer Symbols (𝐴𝑝 ; 𝑘)𝑛 and [𝐴𝑝 ; 𝑘]𝑛 are defined by:

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

55

(𝐴𝑝 )𝑛 = (𝑎1 )𝑛 (𝑎2 )𝑛 , … , (𝑎𝑝 )𝑛 (𝐴𝑝 ; 𝑘)𝑛 = (𝑎1 ; 𝑘)𝑛 (𝑎2 )𝑛 … (𝑎𝑝 )𝑛

(1.11)

[𝐴𝑝 ; 𝑘]𝑛 = [𝑎1 ; 𝑘]𝑛 (𝑎2 )𝑛 … (𝑎𝑝 )𝑛 } and its decomposition formula is defined as: (𝐴𝑝 ; 𝑘)𝑛 + [𝐴𝑝 ; 𝑘]𝑛 = [(𝑎1 ; 𝑘)𝑛 + [𝑎1 ; 𝑘]𝑛 ](𝑎2 )𝑛 … (𝑎𝑝 )𝑛 = (𝑎1 )𝑛 (𝑎2 )𝑛 … (𝑎𝑝 )𝑛 = (𝐴𝑝 )𝑛

(1.12)

Throughout this paper, we need the following well-known facts and rules. 1. The Riemann-Liouville fractional integral [10,11]. (𝐼+𝛼 𝑓)(𝑥) =

𝑥 1 ∫ (𝑥 Γ(𝛼) 0

− 𝑡)𝛼−1 𝑓(𝑡)𝑑𝑡,

(𝛼 ∈ 𝐶, Re(𝛼) > 0).

(1.13)

2. The Riemann-Liouville fractional derivative [10,11]. 𝑑

𝑛

(𝐷+𝛼 𝑓)(𝑥) = ( ) 𝑑𝑥

𝑥 1 ∫ (𝑥 Γ(𝑛−𝛼) 0

− 𝑡)𝑛−𝛼−1 𝑓(𝑡)𝑑𝑡,

(1.14)

The last definition can be written in the form. 𝑑

𝑛

(𝐷+𝛼 𝑓)(𝑥) = ( ) (𝐼+𝑛−𝛼 𝑓)(𝑥), 𝑛 = [Re(𝛼)] + 1 𝑑𝑥

(1.15)

3. The Beta function is given by 1

𝐵(𝛼, 𝛽) = ∫0 𝑡 𝛼−1 (1 − 𝑡)𝛽−1 𝑑𝑡,

min{𝑅(𝛼); 𝑅(𝛽)} > 0

Also, 𝐵(𝛼, 𝛽) =

Γ(𝛼)Γ(𝛽) , Γ(𝛼 + 𝛽)

{𝛼, 𝛽 ∈ 𝐶}

(1.16)

56

Advances in Special Functions of Fractional Calculus

Singh and Verma.

2. SPECIAL CASES (i) If we put 𝑣 = 0 in (1.6) and use (1.11), then ∞ (𝑝; 𝑞) (𝑝; 𝑞) (𝐴𝑝 ; 𝑘)𝑛 𝑎𝑛 𝑥 𝑛+0 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝐾2 (𝑥) = ∑ (𝐵𝑞 )𝑛 Γ(𝑛 + 0 + 1) (𝑣, 𝑎) (𝑣, 𝑎) 𝑛=0 ∞ (𝑝; 𝑞) (𝑝; 𝑞) (𝐴𝑝 ; 𝑘)𝑛 𝑎𝑛 𝑥 𝑛 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝐾2 (𝑥) = ∑ (𝐵𝑞 )𝑛 Γ(𝑛 + 1) (𝑣, 𝑎) (𝑣, 𝑎) 𝑛=0

(𝑝; 𝑞) (𝑝; 𝑞) (𝑎𝑝 ;𝑘) 𝑎𝑛 𝑥 𝑛 𝑛 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝐾2 (𝑥) = ∑∞ 𝑛=0 (𝑏 ) Γ(𝑛+1), 𝑞 𝑛 (𝑣, 𝑎) (𝑣, 𝑎)

(2.1)

(𝑝; 𝑞) where 𝐾2 (𝑥) is known as a generalized incomplete hypergeometric function [9]. (𝑣, 𝑎) (ii) If 𝑝 = 1, 𝑞 = 1, 𝑣 = 0 and 𝑎 = 1, then (1.6) reduces to incomplete confluent hypergeometric function [9]. ∞ (1; 1) (𝑎1 ; 𝑘)𝑛 (1)𝑛 𝑥 𝑛+0 𝐾2 [(𝑎1 ; 𝑘); 𝑏1 ; 𝑥] = ∑ (𝑏1 )𝑛 Γ(𝑛 + 0 + 1) (0,1) 𝑛=0

(1; 1) 𝐾2 [(𝑎1 ; 𝑘); 𝑏1 ; 𝑥] = (0; 1)

1 𝐹1 [(𝑎1 ; 𝑘); 𝑏1 ; 𝑥]

= ∑∞ 𝑛=0

(𝑎1 ;𝑘)𝑛

𝑥𝑛

(𝑏1 )𝑛 Γ(𝑛+1)

,

where 𝑅(𝑥) > 0 when 𝑥 = 0 𝑅(𝑎1 ) > 0. (iii) If we put 𝑝 = 2, 𝑞 = 1, 𝑣 = 0and 𝑎 = 1, then we get ∞ (2; 1) (𝑎 ; 𝑘) (𝑎 ) (1)𝑛 𝑥 𝑛+0 𝐾2 [(𝑎1 ; 𝑘), 𝑎2 ; 𝑏1 ; 𝑥] = ∑ 1 𝑛 2 𝑛 (𝑏1 )𝑛 Γ(𝑛 + 0 + 1) (0,1) 𝑛=0

(2.2)

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

57

∞ (2; 1) (𝑎1 ; 𝑘)𝑛 (𝑎2 )𝑛 𝑥𝑛 𝐾2 [(𝑎1 ; 𝑘), 𝑎2 ; 𝑏1 ; 𝑥] = ∑ (𝑏1 )𝑛 Γ(𝑛 + 1) (0,1) 𝑛=0

(2; 1) 𝐾2 [(𝑎1 ; 𝑘), 𝑎2 ; 𝑏1 ; 𝑥 ] = (0,1)

2 𝐹1 [(𝑎1 ; 𝑘 ); 𝑎2 ; 𝑏1 ; 𝑥 ],

(2.3)

Where 𝑅(𝑏1 ) > 0, 𝑅(𝑎2 ) > 0, 𝑅(𝑥) > 0 and 2 𝐹1 [(𝑎1 ; 𝑘), 𝑎2 ; 𝑏1 ; 𝑥] represent incomplete hypergeometric function introduced by Srivastava et al. [9]. (iv) If we take 𝑝 = 1, 𝑞 = 1, 𝑏1 = 1, 𝑎 = 1and 𝑣 = 0, then (1.6) we arrive at a form of equation (1.4) ∞ (1; 1) (𝑎 ; 𝑘) (1)𝑛 𝑥 𝑛+0 𝐾2 [(𝑎1 ; 𝑘); 𝑏1 ; 𝑥] = ∑ 1 𝑛 (0; 1) Γ(𝑏1 )𝑛 (0,1) 𝑛=0

∞ (2; 1) (𝑎 ; 𝑘) 𝑥𝑛 𝐾2 [(𝑎1 ; 𝑘); 1; 𝑥] = ∑ 1 𝑛 (1)𝑛 Γ(𝑛 + 1) (0,1) 𝑛=0

(2; 1) (𝑎1 ;𝑘)𝑛 𝑥 𝑛 (𝑎1 ;𝑘) 𝐾2 [(𝑎1 ; 𝑘); 1; 𝑥] = ∑∞ = 𝐸(1,1) (𝑥), 𝑛=0 Γ(𝑛+1) 𝑛! (0,1) Which represents incomplete Mettag-Leffler function [3]. (v) If 𝑝 = 1, 𝑞 = 0, 𝑣 = 0, 𝑎 = 1, then reduces to (1.6) ∞

(1;1)

𝐾2 [(𝑎1 ; 𝑘); 0; 𝑥] = ∑

(0;1)

∞

= ∑ 𝑛=0

𝑛=0 𝑛

(𝑎1 ; 𝑘)𝑛 𝑥 Γ(𝑛 + 1)

= (1 − 𝑥)−(𝑎1;𝑘) , where|𝑥| ≤ 1

(𝑎1 ; 𝑘)𝑛 (1)𝑛 𝑛+0 𝑥 Γ(𝑛 + 0 + 1)𝑛

(2.4)

58

Advances in Special Functions of Fractional Calculus

Singh and Verma.

3. RELATIONS WITH RIEMANN - LIOUVILLE FRACTIONAL CALCULUS OPERATORS In this section, we derive relations between incomplete 𝐾2 -Function and the operators of Riemann-Liouville Fractional Calculus. The relations are presented in the form of the two theorems as follows. Theorem 3.1. Let 𝛼 > 0, 𝑣 ∈ 𝐶 and 𝐼𝑥𝛼 be the operator Riemann-Liouville fractional integral; then there hold the relations. (𝑝; 𝑞) (𝑝; 𝑞) 𝐾2 𝐼𝑥𝛼 𝐾2 [(𝑎1 ; 𝑘), … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝑥 𝛼 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] (𝑣, 𝑎) (𝛼 + 𝑣; 𝑎)

(3.1)

Proof. By using (1.6) and (1.13) ∞

𝑥 (𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑡 𝑛+𝑣 𝑛 𝛼 𝛼−1 𝐼𝑥 𝑓(𝑥) = ∫ (𝑥 − 𝑡) ∑ 𝑑𝑡 Γ(𝛼) 0 (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑛=0

𝑛

The interchange of the order of integration and summation is permissible under the conditions stated along with the theorem. ∞

𝑥 (𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑛 = ∑ ∫ (𝑥 − 𝑡)𝛼−1 𝑡 𝑛+𝑣 𝑑𝑡 Γ(𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 0

𝐼𝑥𝛼 𝑓(𝑥)

𝑛=0 ∞

𝑛

𝑛=0

𝑛

𝑥 (𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑡 𝛼−1 𝑛+𝑣 𝑛 𝛼−1 = ∑ ∫ 𝑥 𝑡 𝑑𝑡 (1 − ) Γ(𝛼) 𝑥 (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 0

𝐼𝑥𝛼 𝑓(𝑥) 𝑡

Let = 𝑢 𝑥

𝐼𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑥 𝑛 ∞ = ∑𝑛=0 𝑥 𝛼+𝑛+𝑣 ∫0 (1 Γ(𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) − 𝑢)𝛼−1 𝑢𝑛+𝑣−1+1 𝑑𝑢

With the help of (1.16), we arrive at

𝑛

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

59

∞

𝐼𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 Γ(𝛼)Γ(𝑛 + 𝑣 + 1) 𝑛 = ∑ 𝑥 𝛼+𝑛+𝑣 Γ(𝛼) Γ(𝛼 + 𝑛 + 𝑣 + 1) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑛=0

𝐼𝑥𝛼 𝑓(𝑥)

∞

𝑛

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 𝑎𝑛 𝑥 𝑛+𝑣 1 𝑛 =𝑥 ∑ Γ(𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝛼 + 𝑛 + 𝑣 + 1) 𝛼

𝑛

𝑛=0

(𝑝; 𝑞) (𝑝; 𝑞) 𝑎 𝛼 𝐾2 𝐼𝑥 𝐾2 [(𝑎1 ; 𝑘), … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝑥 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] (𝛼 + 𝑣; 𝑎) (𝑣, 𝑎) This is the required result. Theorem 3.2. Let 𝛼 > 0, 𝑣 ∈ 𝐶 and 𝐷𝑥𝛼 be the operator of the Rieman-Liouville fractional derivative, then there holds the relation. 𝐷𝑥𝛼

(𝑝; 𝑞) (𝑝; 𝑞) 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝑥 −𝛼 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] (3.2) (𝑣; 𝑎) (𝛼 + 𝑣; 𝑎)

Proof. Using (1.15) on the left side of (3.2) 𝐷𝑥𝛼 𝑓(𝑥) =

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑑𝑛 𝑥 𝑎𝑛 𝑡 𝑛+𝑣 𝑛 𝑛−𝛼−1 ∞ ∫ ( 𝑥 − 𝑡) ∑ 𝑑𝑡 𝑛=0 Γ(𝑛 − 𝛼) 𝑑𝑥 𝑛 0 (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑛

The interchange of the order of integration and summation is permissible under the conditions stated along with the theorem. ∞

𝐷𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 𝑎𝑛 𝑑𝑛 𝑥 1 𝑛 ∑ ∫ (𝑥 − 𝑡)𝑛−𝛼−1 𝑡 𝑛+𝑣 𝑑𝑡 = Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑑𝑥 𝑛 0 𝑛=0 ∞

𝑛

𝑛=0

𝑛

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑑𝑛 𝑛 𝐷𝑥𝛼 𝑓(𝑥) = ∑ Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑑𝑥 𝑛 𝑥

𝑡 𝑛−𝛼−1 𝑛+𝑣 × ∫ 𝑥 𝑛−𝛼−1 (1 − ) 𝑡 𝑑𝑡 𝑥 0 𝑡

Let = 𝑢 𝑥

60

Advances in Special Functions of Fractional Calculus

Singh and Verma.

∞

𝐷𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑑𝑛 𝑛 = ∑ Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑑𝑥 𝑛 𝑛

𝑛=0

1

× ∫ 𝑥 𝑛−𝛼−1+𝑛+𝑣+1 (1 − 𝑢)𝑛−𝛼−1 𝑢𝑛+𝑣 𝑑𝑢 0

∞

𝐷𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 𝑑 𝑛 2𝑛−𝛼+𝑣 𝑛 ∑ 𝑥 = Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) 𝑑𝑥 𝑛 𝑛

𝑛=0

1

× ∫ (1 − 𝑢)𝑛−𝛼−1 𝑢𝑛+𝑣 𝑑𝑢 0

∞

𝐷𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 Γ(2𝑛 − 𝛼 + 𝑣 + 1) 𝛼+𝑣+𝑛 𝑛 = ∑ 𝑥 Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) Γ(𝑛 − 𝛼 + 𝑣 + 1) 𝑛

𝑛=0

Γ(𝑛 − 𝛼)Γ(𝑛 + 𝑣 + 1) × Γ(𝑛 + 𝑣 + 1 − 𝑛 − 𝛼) ∞

𝐷𝑥𝛼 𝑓(𝑥)

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 1 𝑎𝑛 Γ(2𝑛 − 𝛼 + 𝑣 + 1) 𝑛+𝑣+𝛼 𝑛 = ∑ 𝑥 Γ(𝑛 − 𝛼) (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 + 𝑣 + 1) Γ(𝑛 − 𝛼 + 𝑣 + 1) 𝑛

𝑛=0

Γ(𝑛 − 𝛼)Γ(𝑛 + 𝑣 + 1) × Γ(2𝑛 − 𝛼 + 𝑣 + 1) ∞ 𝑛 −𝛼 (𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) ⋅ 𝑥 𝑛+𝑣 𝑛 𝑎 𝑥 𝛼 𝐷𝑥 𝑓(𝑥) = ∑ (𝑏1 )𝑛 … (𝑏𝑞 ) Γ(𝑛 − 𝛼 + 𝑣 + 1) 𝑛

𝑛=0

∞

𝐷𝑥𝛼 𝑓(𝑥)

=𝑥

−𝛼

∑ 𝑛=0

𝐷𝑥𝛼

(𝑎1 ; 𝑘)𝑛 … (𝑎𝑝 ) 𝑛 (𝑏1 )𝑛 … (𝑏𝑞 ) 𝑛

𝑎𝑛 𝑥 𝑛+𝑣 Γ(𝑛 − 𝛼 + 𝑣 + 1)

(𝑝; 𝑞 ) (𝑝; 𝑞 ) −𝛼 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] 𝐾2 [(𝑎1 ; 𝑘) … 𝑎𝑝 ; 𝑏1 … 𝑏𝑞 ; 𝑥] = 𝑥 (𝑣; 𝑎) (𝑣; 𝑎)

This is required result. CONCLUSION The purpose of this chapter is to present a 𝐾2 function in a new way in the world of special functions, which can be useful to the students of Science and Engineering.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

61

CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8]

[9]

[10] [11]

K.S. Miller, and B. Ross, An introduction to the fractional Calculus and fractional Differential equations., John Wiley and sons, 1993. K. Sharma and V.S. Dhakar, "On fractional Calculus and Certain Result involving K2 Function", Global Journal Of Science Frontier Research, vol. 11, no. 5, pp. 17-22, 2011. D.K. Singh, and S. Porwal, "Incomplete Mittag-Leffler functions", Acta Universitatis Apulensis, vol. 34, 2013.151162. A. Wiman, "Über die Nallstelian der functionen", Acta Math., vol. 29, no. 0, pp. 217-234, 1905. http://dx.doi.org/10.1007/BF02403204 G.M. Mittag-Leffler, Sur La nuovelle function Ea(x), vol. 137, C.R.: Acad Sci Paris, 1903, pp. 554-558.Series II. G.M. Mittag-Leffler, "Sur La representation analytiquede’unebrancheuniformeune function monogene, Acta", Malh., vol. 29, pp. 101-181, 1905. T.R. Prabhakar, "A singular integral equation with a generalized Mittag-Leffler function in the kernel", Yokohama Math. J., vol. 19, no. 1, pp. 7-15, 1971. T.R. Prabhakar, "Two singular integral equations involving confluent hypergeometric functions", Math. Proc. Camb. Philos. Soc., vol. 66, no. 1, pp. 71-89, 1969. http://dx.doi.org/10.1017/S0305004100044728 H.M. Srivastava M. AslamChaudhry and Ravi P. Agarwal, "The incomplete Pochhammer Symbols and their applications to hypergeometric and related function, Integral trans", Spec. Function, vol. 23, no. 9, pp. 659-683, 2012. R.P. Agarwal, "A propos d’une Note M. Pierre Humbert", C.R. Acad. Sc. Paris, vol. 236, pp. 2031-2032, 1953. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional integrals and derivatives.Theory and application., Gordon and Breach Science Publisher, 1993.

62

Advances in Special Functions of Fractional Calculus, 2023, 62-72

CHAPTER 4

Some Results on Incomplete Hypergeometric Functions Dharmendra Kumar Singh1,* and Geeta Yadav1 1

Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India Abstract: Hypergeometric functions are extensions and generalizations of the geometric series, and the process of generalization of hypergeometric series started in the 19th century itself. Thus, the subject of hypergeometrics has a rich history and led to renewed interest. Many mathematicians have presented the hypergeometric function in different ways and explained its properties. Recently, Srivastava et al. [9] represented hypergeometric functions in different forms with the help of incomplete pochhammer symbols. This paper is an attempt to present some new results for the incomplete hypergeometric function.

Keywords: Generalized incomplete hypergeometric function, incomplete gamma function, incomplete pochhammer symbols, and decomposition formula. 1. INTRODUCTION 1.1. Incomplete Hypergeometric Function

Incomplete hypergeometric function was introduced and studied by H.M. Srivastava and Agarwal [1], p.675, equations (4.1) and (4.2)], and defined as:

𝑝 𝛾𝑞

(𝑎1 , 𝑥), 𝑎2 , . . . , 𝑎𝑝 (𝑎1 ,𝑥)𝑛 (𝑎2 )𝑛 ...(𝑎)𝑛 𝑧 𝑛 [ |𝑧] = ∑∞ 𝑛=0 (𝑏1 )𝑛 ...(𝑏𝑞 )𝑛 𝑛! 𝑏1 , . . . , 𝑏𝑞

(1.1)

(𝑎1 , 𝑥), 𝑎2 . . . , 𝑎𝑝 [𝑎1 ,𝑥]𝑛 (𝑎2 )𝑛 ...(𝑎𝑝 )𝑛 𝑧 𝑛 [ |𝑧] = ∑∞ , 𝑛=0 (𝑏1 )𝑛 ...(𝑏𝑞 )𝑛 𝑛! 𝑏1 , . . . , 𝑏𝑞

(1.2)

and 𝑝 𝛤𝑞

*Corresponding

author Dharmendra Kumar Singh: Department of Mathematics, University Institute of Engineering and Technology, CSJM University, Kanpur, India; E-mail: [email protected] Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

63

where [𝑎1 ; 𝑥]𝑣 and (𝑎1 ; 𝑥)𝑣 represent incomplete pochhammer symbols which are defined as follows, (𝜆; 𝑥)𝑣 =

𝛾(𝜆+𝑣,𝑥) 𝛤(𝜆)

(𝜆, 𝑣 ∈ 𝐶; 𝑥 ≥ 0)

(1.3)

(𝜆, 𝑣 ∈ 𝐶; 𝑥 ≥ 0),

(1.4)

and [𝜆; 𝑥]𝑣 =

𝛤(𝜆+𝑣,𝑥) 𝛤(𝜆)

and these incomplete pochhammer symbols (𝜆; 𝑥)𝑣 and [𝜆; 𝑥]𝑣 satisfy the following decomposition formula (𝜆; 𝑥)𝑣 + [𝜆; 𝑥]𝑣 = (𝜆)𝑣

(𝜆, 𝑣 ∈ 𝐶; 𝑥 ≥ 0),

Here, the incomplete Gamma functions, 𝛾(𝑠, 𝑥) and (𝑠, 𝑥) , are defined as 𝑥

𝛾(𝑠, 𝑥) = ∫0 𝑡 𝑠−1 𝑒 −𝑡 𝑑𝑡

(𝑅(𝑠) > 0; 𝑥 ≥ 0)

(1.5)

(𝑅(𝑠) > 0; 𝑥 ≥ 0 when 𝑥 = 0),

(1.6)

and ∞

𝛤(𝑠, 𝑥) = ∫𝑥 𝑡 𝑠−1 𝑒 −𝑡 𝑑𝑡 and satisfy the following formula

𝛾(𝑠, 𝑥) + 𝛤(𝑠, 𝑥) = 𝛤(𝑠).

(𝑅(𝑠) > 0)

(1.7)

1.2. Incomplete Wright Function Incomplete Wright function,

𝑝 𝛹𝑞

and 𝑝 𝛹𝑞 , was introduced by Singh and Porwal ̲

[2] and defined as: 𝑝 𝛹𝑞

[

= ∑∞ 𝑘=0 𝑝 𝛹𝑞 ̲

[

[𝛼1 , 𝐴1 , 𝑥]. . . . (𝛼𝑝 , 𝐴𝑝 ) |𝑧] (𝛽1 , 𝐵1 ). . . (𝛽𝑞 , 𝐵𝑞 )

𝛤(𝛼1 +𝐴1 𝑘,𝑥)𝛤(𝛼2 +𝐴2 𝑘)...𝛤(𝛼𝑝 +𝐴𝑝 𝑘) 𝑧 𝑘 𝛤(𝛽1 +𝐵1 𝑘)𝛤(𝛽2 +𝐵2 𝑘)...𝛤(𝛽𝑞 +𝐵𝑞 𝑘) 𝑘!

(1.8)

(𝛼1 , 𝐴1 , 𝑥). . . . (𝛼𝑝 , 𝐴𝑝 ) |𝑧] (𝛽1 , 𝐵1 ). . . (𝛽𝑞 , 𝐵𝑞 )

= ∑∞ 𝑘=0

𝛾(𝛼1 +𝐴1 𝑘,𝑥)𝛤(𝛼2 +𝐴2 𝑘)...𝛤(𝛼𝑝 +𝐴𝑝 𝑘) 𝑧 𝑘 𝛤(𝛽1 +𝐵1 𝑘)𝛤(𝛽2 +𝐵2 𝑘)...𝛤(𝛽𝑞 +𝐵𝑞 𝑘) 𝑘!

,

(1.9)

64

Advances in Special Functions of Fractional Calculus

Singh and Yadav

Where, |𝜆; 𝑥)𝑛 | ≤ |(𝜆)𝑛 | and |[𝜆; 𝑥]𝑛 | ≤ |(𝜆)𝑛 |. Decomposition of (1.8) and (1.9) gives a well-known Wright function 𝑝 𝛹𝑞 [3 - 7] who presented its asymptotic expansion for a large value of the argument z under the condition. ∑𝑞𝑗=1 𝛽𝑗 − ∑𝑝𝑖=1 𝛼𝑖 > −1.

(1.10a)

If these conditions are satisfied, the series is in (1.8) and (1.9) is convergent for any z ∈ C. 1.3. Hypergeometric Function The hypergeometric function [8], 2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧) is defined as: ∞ 2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧) = ∑𝑛=0

(𝑎)𝑛 (𝑏)𝑛 𝑧 𝑛 (𝑐)𝑛

𝑛!

, |𝑧| < 1

(1.10b)

Where 𝑎, 𝑏, 𝑐 are complex numbers and 𝑐 ≠ 0, −1, −2, … and the generalized hypergeometric function, in a classical sense, has been defined [5] as:

a1 ,..., a p F | z p Fq a1 ,..., a p ; b1 ,..., bq ; z p q b1 ,..., bq

= ∑∞ 𝑘=0 Where (𝑎𝑖 )𝑛 =

𝛤(𝑎𝑖 +𝑛) 𝛤(𝑎𝑖 )

(𝑎1 )𝑘 ...(𝑎𝑝 )𝑘 𝑧 𝑘 (𝑏1 )𝑘 ...(𝑏𝑞 )𝑘 𝑘!

,

(1.11)

and no denominator parameter equal to zero or negative

integer. 2. THEOREMS Theorem 2.1. If 𝑎, 𝑏, 𝑐 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0; 𝜏 ∈ 𝑁, then 𝑐

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑧)

=𝑐

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 1; 𝑧) + 𝑧

𝑑 𝑑𝑧

Proof. From the left side of equation (2.1) 𝑧

𝑑 𝑑𝑧

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 1; 𝑧)

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 1; 𝑧)

(2.1)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

65

∞

𝑑 [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 =𝑧 ∑ 𝑑𝑧 (𝑐 + 1)𝑛 𝑛! 𝑛=0

∞

=∑ 𝑛=0

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑛𝑧 𝑛 𝛤(𝑐 + 1 + 𝑛)⁄𝛤(𝑐 + 1) 𝑛!

∞

= ∑[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑛=0

𝑐𝛤(𝑐)(𝑛 + 𝑐 − 𝑐) 𝑧 𝑛 (𝑐 + 𝑛)𝛤(𝑐 + 𝑛) 𝑛!

∞

= ∑[𝑎; 𝑥]𝑛 (𝑏)𝑛 { 𝑛=0 ∞

= ∑[ 𝑎; 𝑥]𝑛 (𝑏)𝑛 { 𝑛=0

𝑐𝛤(𝑐)(𝑛 + 𝑐) 𝑐 2 𝛤(𝑐) 𝑧𝑛 − } (𝑛 + 𝑐)𝛤(𝑐 + 𝑛) 𝛤(𝑐 + 𝑛 + 1) 𝑛! 𝑐 𝑧𝑛 𝑐 − } (𝑐)𝑛 (𝑐 + 1)𝑛 𝑛!

∞

𝑐[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑐[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 = ∑[ − ] (𝑐)𝑛 (𝑐 + 1)𝑛 𝑛! 𝑛=0 ∞

∞

𝑐[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 𝑐[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 =∑ −∑ (𝑐)𝑛 𝑛! (𝑐 + 1)𝑛 𝑛! 𝑛=0

𝑛=0

∞

∞

𝑛=0

𝑛=0

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 −𝑐∑ =𝑐∑ 𝑛! (𝑐 + 1)𝑛 𝑛! (𝑐)𝑛 =𝑐

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑧)

−𝑐

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 1; 𝑧).

Theorem 2.2. If a, b, c 𝛿 ∈C; Re(a) > 0, Re(b) > 0, Re(c) > 0, Re(𝛿) > 0 then 𝛤(𝑐+𝛿) 𝛤(𝛿)

1

∫0 𝑢𝑐−1 (1 − 𝑢)𝛿−1

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑧𝑢)𝑑𝑢

= 𝛤(𝑐) 2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐 + 𝛿; 𝑧) (2.2)

Proof. From the left side of the equation 𝛤(𝑐 + 𝛿) 1 𝑐−1 ∫ 𝑢 (1 − 𝑢)𝛿−1 𝛤(𝛿) 0 =

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑧𝑢)𝑑𝑢 ∞

𝛤(𝑐 + 𝛿) 1 𝑐−1 [𝑎; 𝑥]𝑛 (𝑏)𝑛 (𝑧𝑢)𝑛 ∫ 𝑢 (1 − 𝑢)𝛿−1 ∑ 𝑑𝑢 𝑛! 𝛤(𝛿) 0 (𝑐)𝑛 ∞

𝑛=0

𝛤(𝑐 + 𝛿) [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 1 𝑐+𝑛−1 ∑ ∫ 𝑢 (1 − 𝑢)𝛿−1 𝑑𝑢. = 𝛤(𝑐 + 𝑛) 𝑛! 0 𝛤(𝛿) 𝑛=0 𝛤(𝑐)

66

Advances in Special Functions of Fractional Calculus

Singh and Yadav

∞

𝛤(𝑐 + 𝛿) [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 𝛤(𝑐 + 𝑛)𝛤(𝛿) = ∑ . 𝛤(𝑐 + 𝑛) 𝑛! 𝛤(𝑐 + 𝑛 + 𝛿) 𝛤(𝛿) 𝑛=0 𝛤(𝑐) ∞

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 . 𝛤(𝑐 + 𝑛 + 𝛿) 𝑛! 𝑛=0 𝛤(𝑐 + 𝛿)

= 𝛤(𝑐) ∑ ∞

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑧 𝑛 = 𝛤(𝑐) ∑ . (𝑐 + 𝛿)𝑛 𝑛! 𝑛=0

= 𝛤(𝑐)2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐 + 𝛿; 𝑧). Theorem 2.3. If 𝑎, 𝑏, 𝑐, 𝛿, 𝜆 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0, 𝑅𝑒(𝛿) > 0 then, 𝛤(𝑐+𝛿) 𝛤(𝛿)

𝑥

∫𝑡 ( 𝑥 − 𝑠)𝛿−1 (𝑠 − 𝑡)𝑐−1

= (𝑥 − 𝑡)𝛿+𝑐−1 𝛤(𝑐)

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝜆(𝑠

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

− 𝑡))𝑑𝑠

(2.3)

+ 𝛿; 𝜆(𝑥 − 𝑡))

Proof. From equation (2.3)

=

𝑥 𝛤(𝑐 + 𝛿) ∫ ( 𝑥 − 𝑠)𝛿−1 (𝑠 − 𝑡)𝑐−1 2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝜆(𝑠 − 𝑡))𝑑𝑠 (𝑥 − 𝑡)𝛿−1 𝛤(𝛿) 𝑡 𝛤(𝑐 + 𝛿) 𝑥 (𝑥 − 𝑡) − (𝑠 − 𝑡) 𝛿−1 ∫ { } (𝑠 − 𝑡)𝑐−1 2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝜆(𝑠 − 𝑡))𝑑𝑠. (𝑥 − 𝑡) 𝛤(𝛿) 𝑡

Put 𝑢 =

𝑠−𝑡 𝑥−𝑡

=

, we arrive at

𝛤(𝑐 + 𝛿) 1 ∫ ( 1 − 𝑢)𝛿−1 𝑢𝑐−1 (𝑥 − 𝑡)𝑐−1 𝛤(𝛿) 0

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝜆(𝑥

− 𝑡)𝑢)(𝑥 − 𝑡)𝑑𝑢

∞

𝛤(𝑐 + 𝛿) 1 [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑢𝑛 𝜆𝑛 (𝑥 − 𝑡)𝑛 = ∫ ( 1 − 𝑢)𝛿−1 𝑢𝑐−1 (𝑥 − 𝑡)𝑐 ∑ 𝑑𝑢 𝛤(𝛿) 0 (𝑐)𝑛 𝑛! ∞

= (𝑥 − 𝑡)𝑐

𝑛=0 1

𝛤(𝑐 + 𝛿) [𝑎; 𝑥]𝑛 (𝑏)𝑛 ∑ (∫ 𝛤(𝑐 + 𝑛) 𝛤(𝛿) 𝑛=0 𝛤(𝑐)

(1 − 𝑢)𝛿−1 𝑢𝑐+𝑛−1 𝑑𝑢) 0

𝜆𝑛 (𝑥 − 𝑡)𝑛 𝑛!

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

67

∞

= (𝑥 − 𝑡)𝑐

𝛤(𝑐 + 𝛿) [𝑎; 𝑥]𝑛 (𝑏)𝑛 𝛤(𝛿)𝛤(𝑐 + 𝑛) 𝜆𝑛 (𝑥 − 𝑡)𝑛 ∑ . . 𝛤(𝑐 + 𝑛) 𝛤(𝑐 + 𝑛 + 𝛿) 𝑛! 𝛤(𝛿) 𝑛=0 𝛤(𝑐) ∞

𝑐

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝜆𝑛 (𝑥 − 𝑡)𝑛 . 𝛤(𝑐 + 𝑛 + 𝛿) 𝑛! 𝑛=0 𝛤(𝑐 + 𝛿)

= 𝛤(𝑐). (𝑥 − 𝑡) ∑ ∞ 𝑐

= 𝛤(𝑐). (𝑥 − 𝑡) ∑ 𝑛=0

= (𝑥 − 𝑡)𝑐 𝛤(𝑐)

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝜆𝑛 (𝑥 − 𝑡)𝑛 . (𝑐 + 𝛿)𝑛 𝑛!

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 𝛿; 𝜆(𝑥 − 𝑡)).

Theorem 2.4. If 𝑎, 𝑏, 𝑐 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0 then 𝑧

∫0 𝑡 𝑐−1

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑤𝑡)𝑑𝑡 =

𝑧𝑐 𝑐

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

Proof. Using equation (2.4) 𝑧

∫ 𝑡 𝑐−1 0 𝑧

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑤𝑡)𝑑𝑡 ∞

= ∫ 𝑡 𝑐−1 ∑ 0 ∞

=

𝑛=0

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑤 𝑛 𝑡 𝑛 . 𝑑𝑡 (𝑐)𝑛 𝑛! 𝑧

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑤 𝑛 𝑡 𝑛+𝑐 . ∑ ( ) 𝑛! 𝑛 + 𝑐 0 (𝑐)𝑛

𝑛=0 ∞

=

∑ 𝑛=0 𝑐

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑤 𝑛 𝑧 𝑛+𝑐 . . (𝑐)𝑛 𝑛! 𝑛 + 𝑐 ∞

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑤𝑛 𝑛 . .𝑧 𝛤(𝑐 + 𝑛) 𝑛! 𝑛=0 (𝑐 + 𝑛) 𝛤(𝑐)

= 𝑧 ∑ ∞

𝑧𝑐 [𝑎; 𝑥]𝑛 (𝑏)𝑛 (𝑤𝑧)𝑛 = ∑ . 𝑐 𝛤(𝑐 + 𝑛 + 1)/𝑐𝛤𝑐 𝑛! 𝑛=0

+ 1; 𝑤𝑧).

(2.4)

68

Advances in Special Functions of Fractional Calculus

Singh and Yadav

∞

𝑧𝑐 [𝑎; 𝑥]𝑛 (𝑏)𝑛 (𝑤𝑧)𝑛 = ∑ . (𝑐 + 1)𝑛 𝑛! 𝑐 𝑧𝑐 = 𝑐

𝑛=0

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐

+ 1; 𝑤𝑧)

Theorem 2.5. (Laplace Transform) ∞

∫0 𝑒 −𝑠𝑧 𝑧 𝛼−1 =

𝑠 −𝛼 𝛤(𝑐)

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑥𝑧

𝜎

)𝑑𝑧

𝑥 3 𝛹1 [([𝑎; 𝑥],1), (𝑏, 1), (𝛼, 𝜎); (𝑐, 1); 𝑠 𝜎 ]

𝛤(𝑎)𝛤(𝑏)

(2.5)

Where 𝑎, 𝑏, 𝑐, 𝛼, 𝜎 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0, 𝑅𝑒(𝛼) > 0, 𝑅𝑒(𝜎) > 𝑥 0, 𝑅𝑒(𝑠) > 0 and | 𝜎 | < 1. 𝑠

Proof. From equation (2.5) ∞

∫ 𝑒 −𝑠𝑧 𝑧 𝛼−1 0

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑥𝑧

∞

𝜎

)𝑑𝑧

[𝑎; 𝑥]𝑛 (𝑏)𝑛 (𝑥𝑧 𝜎 )𝑛 . ) 𝑑𝑧 (𝑐)𝑛 𝑛! 𝑛=0 0 ∞ ∞ 𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛)𝛤(𝑐) (𝑥𝑧 𝜎 )𝑛 = ∫ 𝑒 −𝑠𝑧 𝑧 𝛼−1 ( ∑ . ) 𝑑𝑧 𝑛! 𝛤(𝑎)𝛤(𝑐 + 𝑛)𝛤(𝑏) 𝑛=0 0 ∞

= ∫ 𝑒 −𝑠𝑧 𝑧 𝛼−1 ( ∑

∞

𝛤(𝑐) 𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛) ∞ −𝑠𝑧 𝛼+𝜎𝑛−1 𝑥 𝑛 = ∑ ∫ 𝑒 𝑧 𝑑𝑧. 𝛤(𝑎)𝛤(𝑏) 𝛤(𝑐 + 𝑛) 𝑛! 0 𝑛=0 ∞

𝛤(𝑐) 𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛) 𝑥 𝑛 𝑠 −𝛼 = ∑ . 𝛤(𝛼 + 𝜎𝑛) 𝛤(𝑎)𝛤(𝑏) 𝛤(𝑐 + 𝑛) 𝑛! 𝑠 𝜎𝑛 −𝛼

=

𝛤(𝑐)𝑠 𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛)𝛤(𝛼 + 𝜎𝑛) 𝑥 𝑛 ( 𝜎) ∑ 𝛤(𝑎)𝛤(𝑏) 𝛤(𝑐 + 𝑛)𝑛! 𝑠 −𝛼

=

𝑛=0 ∞

𝑠 𝛤(𝑐) 𝛤(𝑎)𝛤(𝑏)

𝑛=0

𝑥

3 𝛹1 [([𝑎; 𝑥],1), (𝑏, 1), (𝛼, 𝜎); (𝑐, 1); 𝜎 ] . 𝑠

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

69

Theorem 2.6 (Whittaker Transform) ∞

∫ 𝑡𝜌−1 𝑒 −1⁄2𝑝𝑡 𝑊𝜆,𝜇 (𝑝𝑡)

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝑤𝑡

𝛿

)𝑑𝑡

0

=

𝑝−𝜌 𝛤(𝑐) 𝛤(𝑎)𝛤(𝑏)

1 4 𝛹2 [([𝑎; 𝑥],1), (𝑏, 1), (2

± 𝜇 + 𝜌, 𝛿); (𝑐, 1), (1 − 𝜆 + 𝜌, 𝛿);

𝜔 𝑝𝛿

].

(2.6)

Where 𝑎, 𝑏, 𝑐, 𝜌, 𝛿, 𝑝 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0𝑅𝑒(𝜌) > 0, 𝑅𝑒(𝛿) > 0, 𝑅𝑒(𝑝) > 0. Proof. To obtain Whittaker transform [9], we use the following integral. ∞

∫ 𝑒 −𝑡/2 𝑡 𝑣−1 𝑊𝜆,𝜇 (𝑡)𝑑𝑡 = 0

𝛤(1⁄2 + 𝜇 + 𝑣)𝛤(1⁄2 − 𝜇 + 𝑣) 𝛤(1 − 𝜆 + 𝑣)

Where Re (𝑣 ± 𝜇) > -1/2. Substituting 𝑝𝑡 = 𝑣 on the L.H.S. of equation (2.6), it reduces to

∞ 𝑣 𝜌−1 ∫ 𝑒 −1⁄2𝑣 ( ) 𝑊𝜆,𝜇 (𝑣) 𝑝 0 ∞

=∑ 𝑛=0

=

2 𝐹1 ([𝑎; 𝑥], 𝑏; 𝑐; 𝜔 (

𝑣 𝛿 1 ) ) 𝑑𝑣 𝑝 𝑝

[𝑎; 𝑥]𝑛 (𝑏)𝑛 𝑛 1 𝛿𝑛 ∞ 𝛿𝑛 𝑣 𝜌−1 −1⁄2𝑣 1 𝜔 ( ) ∫ 𝑣 ( ) 𝑒 𝑊𝜆,𝜇 (𝑣) 𝑑𝑣 𝑝 (𝑐)𝑛 𝑝 𝑝 0 ∞

𝛤(𝑐) 𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛) 1 𝜔 𝑛 ∞ −1⁄2𝑣 𝛿𝑛+𝜌−1 ∑ ( ) ∫ 𝑒 𝑣 𝑊𝜆,𝜇 (𝑣)𝑑𝑣 𝑝𝜌 𝑝 𝛿 𝛤(𝑏)𝛤(𝑎) 𝛤(𝑐 + 𝑛)𝑛! 0 𝑛=0

∞

𝛤(𝑎 + 𝑛, 𝑥)𝛤(𝑏 + 𝑛) 𝛤(1⁄2 + 𝜇 + 𝜌 + 𝛿𝑛)𝛤(1⁄2 − 𝜇 + 𝜌 + 𝛿𝑛) 1 𝜔 𝑛 𝛤(𝑐) = 𝑝−𝜌 ∑ { } ( 𝛿) 𝛤(𝑏)𝛤(𝑎) 𝛤(𝑐 + 𝑛) 𝛤(1 − 𝜆 + 𝜌 + 𝛿𝑛) 𝑛! 𝑝 𝑛=0

𝛤(𝑐) = 𝑝−𝜌 𝛤(𝑏)𝛤(𝑎)

4 𝛹2

1 1 𝜔 [([𝑎; 𝑥],1), (𝑏, 1), ( + 𝜇 + 𝜌, 𝛿), ( − 𝜇 + 𝜌, 𝛿); (𝑐, 1), (1 − 𝜆 + 𝜌, 𝛿); 𝛿 ] . 2 2 𝑝

70

Advances in Special Functions of Fractional Calculus

Theorem 2.7 then .

If

Singh and Yadav

𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0, 𝑅𝑒(𝑑) > 0,

1

𝛤(𝑐)

∫ 𝑡 𝑑−1 (1 − 𝑡)𝑐−𝑑−1 𝐹([𝑎; 𝑧], 𝑏; 𝑑; 𝑥𝑡)𝑑𝑡 𝛤(𝑑)𝛤(𝑐−𝑑) 0

𝐹([𝑎; 𝑧], 𝑏; 𝑐; 𝑥) =

(2.7)

Proof. With (2.7) ∞

1 1 𝛤(𝑐) [𝑎; 𝑧]𝑛 (𝑏)𝑛 (𝑥𝑡)𝑛 𝛤(𝑐) ∫ 𝑡 𝑑−1 (1 − 𝑡)𝑐−𝑑−1 𝐹([𝑎; 𝑧], 𝑏; 𝑑; 𝑥𝑡)𝑑𝑡 = ∫ 𝑡 𝑑−1 (1 − 𝑡)𝑐−𝑑−1 ∑ 𝑑𝑡 𝑛! (𝑑)𝑛 𝛤(𝑑)𝛤(𝑐 − 𝑑) 0 𝛤(𝑑)𝛤(𝑐 − 𝑑) 0 𝑛=0

∞

=

𝛤(𝑐) [𝑎; 𝑧]𝑛 (𝑏)𝑛 (𝑥)𝑛 1 𝑛+𝑑−1 (1 − 𝑡)𝑐−𝑑−1 𝑑𝑡 ∑ ∫ 𝑡 𝛤(𝑑)𝛤(𝑐 − 𝑑) (𝑑)𝑛 𝑛! 0 𝑛=0

∞

𝛤(𝑐) [𝑎; 𝑧]𝑛 (𝑏)𝑛 (𝑥)𝑛 = ∑ 𝐵(𝑛 + 𝑑, 𝑐 − 𝑑) 𝛤(𝑑)𝛤(𝑐 − 𝑑) (𝑑)𝑛 𝑛! 𝑛=0 ∞

𝛤(𝑐) [𝑎; 𝑧]𝑛 (𝑏)𝑛 (𝑥)𝑛 𝛤(𝑛 + 𝑑)𝛤(𝑐 − 𝑑) = ∑ 𝛤(𝑑)𝛤(𝑐 − 𝑑) 𝛤(𝑛 + 𝑐) 𝑛! (𝑑)𝑛 𝑛=0

∞

=∑ 𝑛=0

[𝑎; 𝑧]𝑛 (𝑏)𝑛 (𝑑)𝑛 𝑥 𝑛 (𝑐)𝑛 (𝑑)𝑛 𝑛!

∞

=∑ 𝑛=0

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 𝑛! (𝑐)𝑛

= 𝐹([𝑎; 𝑧], 𝑏; 𝑐; 𝑥).

Theorem 2.8. If 𝑎, 𝑏, 𝑐 ∈ 𝐶; 𝑅𝑒(𝑎) > 0, 𝑅𝑒(𝑏) > 0, 𝑅𝑒(𝑐) > 0 then 1

𝐹([𝑎; 𝑧], 𝑏; 𝑐 + 1; 𝑥) = 𝑐 ∫0 𝐹 ([𝑎; 𝑧], 𝑏; 𝑐; 𝑥𝑡)𝑡 𝑐−1 𝑑𝑡

(2.8)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

71

Proof. Using equation (2.8) 1

𝑐 ∫ 𝐹 ([𝑎; 𝑧], 𝑏; 𝑐; 𝑥𝑡)𝑡 𝑐−1 𝑑𝑡 0

1 ∞

= 𝑐∫ ∑ 0 𝑛=0 ∞

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 𝑡 𝑛 𝑐−1 𝑡 𝑑𝑡 (𝑐)𝑛 𝑛!

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 1 𝑛+𝑐−1 =𝑐∑ ∫ 𝑡 𝑑𝑡 𝑛! 0 (𝑐)𝑛 𝑛=0 ∞

=𝑐∑ 𝑛=0 ∞

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 1 (𝑐)𝑛 𝑛! (𝑛 + 𝑐)

𝑥𝑛 [𝑎; 𝑧]𝑛 (𝑏)𝑛 𝛤(𝑐 + 𝑛) 𝑛! )(𝑛 + 𝑐) 𝑛=0 ( 𝛤(𝑐)

=𝑐∑ ∞

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 =∑ (𝑛 + 𝑐)𝛤(𝑛 + 𝑐) 𝑛! 𝑛=0 𝑐𝛤(𝑐) ∞

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 =∑ 𝛤(𝑛 + 1 + 𝑐) 𝑛! 𝑛=0 𝛤(𝑐 + 1) ∞

=∑ 𝑛=0

[𝑎; 𝑧]𝑛 (𝑏)𝑛 𝑥 𝑛 (𝑐 + 1)𝑛 𝑛!

= 𝐹([𝑎; 𝑧], 𝑏; 𝑐 + 1; 𝑥). CONCLUSION This chapter aims to present a wide variety of interesting formulas of incomplete hypergeometric functions in a flexible and accessible format. Using the same in engineering and science can achieve even more important results. CONSENT FOR PUBLICATION Not applicable.

72

Advances in Special Functions of Fractional Calculus

Singh and Yadav

CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1]

[2] [3] [4] [5]

[6]

[7] [8]

[9]

H.M. Srivastava, M.A. Chaudhary, and R.P. Agarwal, "The incomplete pochhammer symbols and their applications to hypergeometric and related functions", Integral Transforms Spec. function, vol. 23, no. 9, pp. 659-683, 2012. D.K. Singh, and S. Porwal, "Incomplete Mittag-Leffler function", Acta UniversitatisApulensis, vol. 34, pp. 151-162, 2013. A.A. Kilbas, "Fractional calculus of the generalized Wright function", Fract. Calc. Appl. Anal., vol. 8, no. 2, pp. 113-126, 2005. A.A. Kilbas, M. Saigo, and J.J. Trujillo, "On the generalized Wright Function", Fract. Calc. Appl. Anal., vol. 5, no. 4, pp. 437-460, 2002. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations., vol. 204, Elsevier, 2006. http://dx.doi.org/10.1016/S0304-0208(06)80001-0 F. Al- Musallam and V. K. Tuan, A finite and infinite Whittaker integral transform, “An international journal computers and mathematics with applications”, vol. 46, no. 12, pp. 1847-1859, 2003. E.M. Wright, "The asymptotic expansion of the generalized hypergeometric function", J. Lond. Math. Soc., vol. s1-10, no. 4, pp. 286-293, 1935. E.D. Rainville, Special Functions, Macmillan, New York., 1st edThe Mcmillan Company: N.Y., 1960. http://dx.doi.org/10.1112/jlms/s1-10.40.286 Erdelyi, Higher Transcendental Functions., McGraw-Hill: New York, 1953.

Advances in Special Functions of Fractional Calculus, 2023, 73-93

73

CHAPTER 5

Transcendental Bernstein Series: Interpolation and Approximation Z. Avazzadeh1, H. Hassani2*, J.A. Tenreiro Machado3, P. Agarwal2 and E. Naraghirad4 1

Department of Mathematical Sciences, University of South Africa, Florida, South Africa

2

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India

3

Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, Porto 431, 4249-015, Portugal 4

Department of Mathematics, Yasouj University, Yasouj, Iran

Abstract: This paper adopts the transcendental Bernstein series (TBS), a set of basis functions based on the Bernstein polynomials (BP), for approximating analytical functions. The TBS is more accurate than the BP method, particularly in approximating functions including one or more transcendental terms. The numerical results reveal also the applicability and higher computational efficiency of the new approach.

Keywords: Transcendental functions; Bernstein polynomials; Transcendental Bernstein series. 1. INTRODUCTION Generalized polynomials have been proven to be valuable tools in several areas of mathematics [1–6]. Several applied sciences adopted the Bernstein polynomials (BP) as a powerful practical tool [7–12]. Moreover, BP has an important role in approximation theory. Draganov [13] proved that several forms of the BP with integer coefficients reveal the property of simultaneous approximation, since the BP approximates the functions and its derivatives. Qian et al. [14] provided a uniform approximation of polynomials and BP. Javadi et al. [15] introduced the *Corresponding

author H. Hassani: Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India; E-mail: [email protected] and [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

74

Advances in Special Functions of Fractional Calculus

Singh and Yadav

shifted orthonormal BP and derived the operational matrices of integration and delays to solve generalized pantograph equations. Chen et al. [16] proposed a method for the numerical solution of a class of variable order fractional linear cable equations and obtained two kinds of operational matrices of BP. Acu and Muraru [17] introduced a bivariate generalization of the Bernstein-Schurer Kantorovich operators based on q-integers and discussed a Bohmann-Korovkin-type approximation theorem for these operators. Bataineh [18] used the BP method and its operational matrices to obtain analytical solutions to variational problems. Mirzaee and Hoseini [19] considered a combination of BP and block-pulse functions to approximate the solution of the optimal control problem for systems governed by a class of nonlinear Volterra integral equations. In the area of computer graphics, in the follow-up of Bézier curves and surfaces, the BP can be employed with a high degree of accuracy [20–25]. Sorokina [26] developed Bernstein-Bézier techniques for analyzing polynomial spline fields in n variables. Lewanowicz et al. [27] derived a set of recurrence relations satisfied by the Bezier coefficients of dual bivariate BP and proposed an efficient algorithm for evaluating these coefficients. Winkel [28] studied the ā-BP and ā-Béziercurves based on an interpretation of the ā-BPby means of the convolution of parameters. Winkler and Yang [29] described the application of a structure-preserving matrix method to the deconvolution of two BP basis. Aside from computer applications, BP has been adopted in the solution of elliptic and hyperbolic differential equations based on the Galerkin and collocation methods [30–35]. Hereafter, stemming from the BP formalism, the transcendental Bernstein series (TBS) and their properties are discussed. Indeed, as the set of basis functions, the TBS can approximate analytical functions as discussed in the follow-up. In Section 2, we review the definition and some important properties of the BP, which will be used in the next sections. In Section 3, we introduce the TBS and investigate some fundamental properties of the TBS. In addition, we prove two convergence theorems and apply the TBS to approximate analytical functions. In Section 4, we expand four test functions in terms of the TBS, and we investigate the practical efficiency of the method. Section 5 is dedicated to a brief conclusion. 2. BERNSTEIN POLYNOMIALS To improve the readability of the follow-up, we review the BP definition and some fundamental properties. Definition 2.1. The BP of degree 𝑚 are defined by [36, 37].

Transcendental Bernstein

Advances in Special Functions of Fractional Calculus

𝑚 𝐵𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖 , 𝑖

0 ≤ 𝑖 ≤ 𝑚,

75

(2.1)

𝑚 𝑚! Where ( ) = . By using the binomial expansion 𝑖!(𝑚−𝑖)! 𝑖 𝑚−𝑖

(1 − 𝑡)

𝑚−𝑖

= ∑(−1)𝑘 ( 𝑘=0

𝑚−𝑖 𝑘 )𝑡 , 𝑘

(2.2)

we have the following formula 𝑚 𝑚 𝑚 − 𝑖 𝑖+𝑘 𝑚−𝑖 ) 𝑡 . (2.3) 𝐵𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖 = ∑𝑘=0 (−1)𝑘 ( ) ( 𝑖 𝑖 𝑘

In general, a given function u(t) can be approximated by means of the first 𝑚 + 1BP as T u(t)= ∑m i=0 di Bi,m (t)= D Φm(t) ,

(2.4)

where 𝐷𝑇 = [𝑑0 𝑑1 . .. 𝑑𝑚 ]. Furthermore, we have 𝑇

Φ𝑚 (𝑡) = [𝐵0,𝑚 (𝑡)𝐵1,𝑚 (𝑡) … 𝐵𝑚,𝑚 (𝑡)] = 𝐴 𝑇𝑚 (𝑡),

𝐴=

𝑎01 … 𝑎0𝑚 𝑎00 𝑎10 𝑎11 … 𝑎1𝑚 , 𝑇𝑚 (𝑡) = [1 𝑡 𝑡 2 . .. 𝑡 𝑚 ]𝑇 , ⋮ ⋮ ⋮ ⋮ (𝑎𝑚0 𝑎𝑚1 … 𝑎𝑚𝑚 )

(2.5)

(2.6)

and 𝑚 𝑚−𝑖 (−1)𝑗−𝑖 ( ) ( ) , 𝑖 ≤ 𝑗, 𝑗−𝑖 𝑖 𝑎𝑖𝑗 = { 0, 𝑖 > 𝑗,

which represents the matrix form of approximation based on BP. 2.1. Properties of BP Some relevant properties of BP are listed as follows:

(2.7)

76

Advances in Special Functions of Fractional Calculus

Singh and Yadav

1. 2.

𝑚 𝐵𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖 ≥ 0 (𝑡 ∈ [0,1]). 𝑖 (1 − t)𝐵𝑖,𝑚−1 (𝑡) + 𝑡 𝐵𝑖−1,𝑚−1 (𝑡) = 𝐵𝑖,𝑚 (𝑡).

3.

𝐵𝑖,𝑚−1 (𝑡) = (

4.

The set {𝐵0,𝑚 (𝑡), 𝐵1,𝑚 (𝑡), … , 𝐵𝑚,𝑚 (𝑡)}, is linearly independent.

5.

𝑑𝐵𝑖,𝑚 dt

𝑚 − 1 𝐵𝑖,𝑚 (𝑡) )[ 𝑚 + (𝑖) 𝑖

𝐵𝑖+1,𝑚 (𝑡) ] 𝑚 (𝑖+1)

(𝑡) = 𝑚 (𝐵𝑖−1,𝑚−1 (𝑡) − 𝐵𝑖,𝑚−1 (𝑡)).

Proof: The readers are referred to see [37]. 3. TRANSCENDENTAL BERNSTEIN SERIES Definition 3.1. The TBS, denoted by ℬ𝑖,𝑚 (𝑡), are defined from the following formula: 𝑚 ℬ𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖+𝛾𝑖 (1 − 𝑡)𝑚−𝑖 , 𝑖

0 ≤ 𝑖 ≤ 𝑚, 𝛾𝑖 ≥ 0.

Using the binomial expansion in (2.2), we obtain: 𝑚 𝑚 𝑚 − 𝑖 𝑖+𝑘+𝛾𝑖 𝑚−𝑖 )𝑡 ℬ𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖+𝛾𝑖 (1 − 𝑡)𝑚−𝑖 = ∑𝑘=0 (−1)𝑘 ( ) ( . 𝑖 𝑖 𝑘

(3.1)

Similarly, the given function 𝑢(𝑡) can be approximated with the first 𝑚 + 1 terms of TBS as 𝑢(𝑡) = 𝐶 𝑇 ɸ𝑚 (𝑡) = 𝐶 𝑇 𝐇Ѱ𝑚 (𝑡),

(3.2)

where 𝐶 𝑇 = [𝑐0 𝑐1 . .. 𝑐𝑚 ], 𝑇

ɸ𝑚 (𝑡) = [ℬ0,𝑚 (𝑡)ℬ1,𝑚 (𝑡)ℬ2,𝑚 (𝑡) . .. ℬ𝑚,𝑚 (𝑡)] = 𝐇Ѱ𝑚 (𝑡),

and

(3.3)

Transcendental Bernstein

Advances in Special Functions of Fractional Calculus

𝑚 𝑚−𝑖 (−1)𝑗−𝑖 ( ) ( ) , 𝑖 ≤ 𝑗, ℎ00 ℎ01 ℎ02 … ℎ0𝑚 𝑗−𝑖 𝑖 𝐇 = ℎ10 ℎ11 ℎ12 … ℎ1𝑚 , ℎ𝑖𝑗 = { ⋮ ⋮ ⋮ ⋮ ⋮ 𝑖 > 𝑗. 0, ℎ ℎ ℎ ℎ ( 𝑚0 𝑚1 𝑚2 … 𝑚𝑚 )

77

(3.4)

Also, Ѱ𝑚 (𝑡) is a (𝑚 + 1)-vector as follows Ѱ𝑚 (𝑡) ≜ [𝜓0 (𝑡)𝜓1 (𝑡) . .. 𝜓𝑚 (𝑡)]𝑇 ,

(3.5)

𝜓𝑖 (𝑡) = 𝑡 𝑖+𝛾𝑖 , 𝑖 = 0, 1, . . . , 𝑚.

(3.6)

where

We note that for approximating the function 𝑢(𝑡) in equation (3.2), the coefficients 𝑐𝑖 and the control parameters 𝛾𝑖 are unknown and should be determined. 3.1. Properties of TBS Here, we review some properties of the TBS following the properties of the BP, presented in the previous section, as follows 𝑚 1. ℬ𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖+𝛾𝑖 (1 − 𝑡)𝑚−𝑖 𝑖

≥ 0 (𝑡 ∊ [0,1]).

2. 𝑡 𝛾𝑖−1 (1 − 𝑡)ℬ𝑖,𝑚−1 (𝑡) + 𝑡 1+𝛾𝑖 ℬ𝑖−1,𝑚−1 (𝑡) = 𝑡 𝛾𝑖−1 ℬ𝑖,𝑚 (𝑡). 3. ℬ𝑖,𝑚−1 (𝑡) =

(𝑚−1) 𝑡 𝛾𝑖+1 𝑖 [ 𝑡 𝛾𝑖+1 (𝑚) 𝑖

ℬ𝑖,𝑚 (𝑡) +

t𝛾𝑖 𝑚 ℬ𝑖+1,𝑚 (𝑡)] . (𝑖+1)

4. The set {ℬ0,𝑚 (𝑡), ℬ1,𝑚 (𝑡), . . . , ℬ𝑚,𝑚 (𝑡)}, where𝑖 + 𝛾𝑖 ≠ 𝑗 + 𝛾𝑗 , is linearly independent. 5.

𝑑ℬ𝑖,𝑚 𝑑𝑡

(𝑡) =

𝑚(𝑖+𝛾𝑖 )𝑡 𝛾𝑖 𝑖𝑡 𝛾𝑖−1

ℬ𝑖−1,𝑚−1 (𝑡) − 𝑚ℬ𝑖,𝑚−1 (𝑡).

Proof: 1. The proof is trivial. 2. Using the definition of ℬ𝑖,𝑚−1 and ℬ𝑖−1,𝑚−1

78

Advances in Special Functions of Fractional Calculus

Singh and Yadav

𝑚 − 1 𝑖+𝛾𝑖 +𝛾𝑖−1 (1 )𝑡 − 𝑡)𝑚−𝑖 𝑖 𝑚 − 1 𝑖+𝛾𝑖 +𝛾𝑖−1 𝑡 1+𝛾𝑖 ℬ𝑖−1,𝑚−1 (𝑡) = ( )𝑡 (1 − 𝑡)𝑚−𝑖 𝑖−1 𝑡 𝛾𝑖−1 (1 − 𝑡)ℬ𝑖,𝑚−1 (𝑡) = (

and substitution in the formula gives 𝑡

𝛾𝑖−1

(1 − 𝑡)ℬ𝑖,𝑚−1 (𝑡) + 𝑡 1+𝛾𝑖 ℬ𝑖−1,𝑚−1 (𝑡) = [(𝑚 − 1) + (𝑚 − 1)] 𝑡 𝑖+𝛾𝑖 +𝛾𝑖−1 (1 − 𝑡)𝑚−𝑖 𝑖 𝑖−1

𝑚 = ( ) 𝑡 𝑖+𝛾𝑖 +𝛾𝑖−1 (1 − 𝑡)𝑚−𝑖 𝑖 𝛾𝑖−1 𝑚 ( ) 𝑡 𝑖+𝛾𝑖 (1 − 𝑡)𝑚−𝑖 =𝑡 𝑖 = 𝑡 𝛾𝑖−1 ℬ𝑖,𝑚 (𝑡). 3. We have 𝑡 𝛾𝑖+1 𝑚 ℬ𝑖,𝑚 (𝑡) (𝑖)

= 𝑡 𝑖+𝛾𝑖 +𝛾𝑖+1 (1 − 𝑡)𝑚−𝑖 .

𝑡 𝛾𝑖 𝑚 ℬ𝑖+1,𝑚 (𝑡) (𝑖+1)

= 𝑡 𝑖+1+𝛾𝑖 +𝛾𝑖+1 (1 − 𝑡)𝑚−(𝑖+1) .

Summing the above equalities leads to t 𝛾𝑖+1 t 𝛾𝑖 𝑖+𝛾𝑖 +𝛾𝑖+1 (1 (𝑡) + − 𝑡)𝑚−(𝑖+1) [1 − 𝑡 + 𝑡] 𝑚 ℬ𝑖,𝑚 𝑚 ℬ𝑖+1,𝑚 (𝑡) = 𝑡 ( ) ( ) 𝑖 𝑖+1 t 𝛾𝑖+1 (𝑡). = 𝑡 𝑖+𝛾𝑖 +𝛾𝑖+1 (1 − 𝑡)𝑚−(𝑖+1) = ℬ 𝑚 − 1 𝑖,𝑚−1 ) ( 𝑖 4. Suppose that the set {ℬ0,𝑚 (𝑡), ℬ1,𝑚 (𝑡), . . . , ℬ𝑚,𝑚 (𝑡)}, where 𝑖 + 𝛾𝑖 ≠ 𝑗 + 𝛾𝑗 , is not linearly independent. Then, we have𝑐0 , 𝑐1 , . . . , 𝑐𝑚 ≠ 0, 𝑖 + 𝛾𝑖 ≥ 0, such that the linear combinatory of them is zero 𝑐0 ℬ0,𝑚 + 𝑐1 ℬ1,𝑚 + ⋯ + 𝑐𝑚 ℬ𝑚,𝑚 = 0, Or 𝑚 𝑚 𝑚 𝑐0 ( ) 𝑡 𝛾0 (1 − 𝑡)𝑚 + 𝑐1 ( ) t1+𝛾1 (1 − 𝑡)𝑚−1 + ⋯ + 𝑐𝑚 ( ) 𝑡 𝑚+𝛾𝑚 (1 − 𝑡)𝑚−𝑚 = 0. 0 𝑚 1

Transcendental Bernstein

Advances in Special Functions of Fractional Calculus

79

Without loss of the generality, we may assume that 𝛾0 < 1 + 𝛾1 < 2 + 𝛾2 < ⋯ < 𝑚 + 𝛾𝑚 , which implies that 𝑚 𝑚 𝑐0 ( ) t 𝛾0 −𝑚− 𝛾𝑚 (1 − 𝑡)𝑚 + 𝐶1 ( ) t1+𝛾1 −𝑚− 𝛾𝑚 (1 − 𝑡)𝑚−1 + ⋯ + 𝐶𝑚 = 0. 0 1 Taking the limit 𝑡 → 1, we deduce that 𝑐𝑚 = 0. Continuing this process, we conclude that 𝑐𝑖 = 0, 𝑖 = 0,1,2, … , 𝑚. This completes the proof of (4). 5. Straight forwardly, using the definition of ℬ𝑖,𝑚 , we have d d 𝑚 𝑖+𝛾 ℬ𝑖,𝑚 (𝑡) = ( ) 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖 𝑑𝑡 𝑑𝑡 𝑖 𝑚 𝑚 = (𝑖 + 𝛾𝑖 ) ( ) 𝑡 𝑖+𝛾𝑖 −1 (1 − 𝑡)𝑚−𝑖 − (𝑚 − 𝑖) ( ) 𝑡 𝑖+𝛾𝑖 (1 − 𝑡)𝑚−𝑖−1 𝑖 𝑖 (𝑖 + 𝛾𝑖 ) 𝑚! 𝑖+𝛾 −1 (𝑚 − 𝑖) 𝑚! 𝑚 𝑖+𝛾 = 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖 − ( ) 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖−1 𝑖! (𝑚 − 𝑖)! 𝑖! (𝑚 − 𝑖)! 𝑖 =

𝑖 𝑚! 𝛾𝑖 𝑚! 𝑡 𝑖+𝛾𝑖 −1 (1 − 𝑡)𝑚−𝑖 + 𝑡 𝑖+𝛾𝑖 −1 (1 − 𝑡)𝑚−𝑖 𝑖! (𝑚 − 𝑖)! 𝑖! (𝑚 − 𝑖)!

+

𝑚 (𝑚 − 1)! 𝑖+𝛾 𝑡 𝑖 (1 − 𝑡)𝑚−𝑖−1 𝑖! (𝑚 − 𝑖 − 1)!

𝑖 𝛾𝑖 ℬ𝑖,𝑚 (𝑡) + ℬ𝑖,𝑚 (𝑡) + 𝑚ℬ𝑖,𝑚−1 (𝑡) 𝑡 𝑡 𝑖 + 𝛾𝑖 = ℬ𝑖,𝑚 (𝑡) + 𝑚ℬ𝑖,𝑚−1 (𝑡) 𝑡 𝑚 (𝑖 + 𝛾𝑖 )𝑡 𝛾𝑖 = ℬ𝑖−1,𝑚−1 (𝑡) − 𝑚ℬ𝑖,𝑚−1 (𝑡). 𝑖𝑡 𝛾𝑖−1 =

3.2. Convergence Analysis In this subsection, we provide a theorem addressing the convergence analysis of the proposed method. Theorem 1. Let g ∶ [0,1] → ℝ be a continuous function. Then, for every𝑡 ∈ (0,1) and Ԑ > 0, there exists a TBS ℬ𝑚 (𝑡) such that

80

Advances in Special Functions of Fractional Calculus

Singh and Yadav

|g(𝑡) − ℬ𝑚 (𝑡)| < Ԑ. Proof. Let Ԑ > 0 be arbitrarily chosen. In view of the Weierstrass theorem [38], there exists a polynomial 𝑃𝑚 (𝑡) = 𝑎0 + 𝑎1 𝑡 + ⋯ + 𝑎𝑚 𝑡 𝑚 , 𝑡 ∈ (0,1) such that sup |g(t) − 𝑃𝑚 (𝑡)|

0, there exist 𝑁0 , 𝑁1 , … , 𝑁𝑚 in ℕ, such that |1 − 𝑐𝑚𝑘 𝑡 𝛾𝑘,𝑛 |

0 there exists a positive integer 𝑁0 such that ∀ n ≥ 𝑁0 , sup |ℬ𝑛 (𝑓)(t) − ℬ𝑛 (𝑡)𝑓(𝑡)| < Ԑ . t ∈ [0,1] Proof. Let 𝑥 ∈ [0,1] be fixed. Then, we have 𝑛

𝑛 𝑖 ℬ𝑛 (𝑓) − ℬ𝑛 (𝑡)𝑓(𝑡) = ∑ ( ) 𝑥 𝑖+ 𝛾𝑖 (1 − 𝑡)𝑛−𝑖 𝑓 ( ) 𝑖 𝑛 𝑛

𝑖=0

𝑛 −𝑓(𝑡) ∑ ( ) 𝑡 𝑖+ 𝛾𝑖 (1 − 𝑡)𝑛−𝑖 𝑖 𝑖=0

𝑛

𝑛 𝑖 = ∑ ( ) 𝑡 𝑖+ 𝛾𝑖 (1 − 𝑡)𝑛−𝑖 [ 𝑓 ( ) − 𝑓(𝑡)]. 𝑛 𝑖 𝑖=0

This implies that

𝑛

𝑛 𝑖 |ℬ𝑛 (𝑓) − ℬ𝑛 (𝑡)𝑓(𝑡)| ≤ ∑ ( ) 𝑡 𝑖+ 𝛾𝑖 (1 − 𝑡)𝑛−𝑖 |𝑓 ( ) − 𝑓(𝑡)| 𝑖 𝑛 𝑖=0

𝑛

𝑛 𝑖 ≤ ∑ ( ) 𝑡 𝑖 (1 − 𝑡)𝑛−𝑖 |𝑓 ( ) − 𝑓(𝑡)|. 𝑖 𝑛 𝑘=0

82

Advances in Special Functions of Fractional Calculus

Singh and Yadav

Now, for any δ > 0, we separate the last sum into two sums ∑1 and ∑2 , the first 𝑖 for those terms where | − 𝑡| < δ, and the second for the remaining terms, so that 𝑛

𝑘

| − 𝑡| ≥ δ . If f is continuous at 𝑥 ∈ [0,1], then 𝑛

∀ Ԑ > 0, ∃ δ > 0, 𝑦 ∈ [0,1], |y − 𝑡| < δ ⟹ |𝑓(𝑦) − 𝑓(𝑡)|

𝛼, 𝑓(𝑧)

𝑧 ∈ 𝔻.

A set Ω is said to be convex if it is starlike with respect to each of its points, that is if the line segment joining any two points of Ω lies entirely in Ω. A function f ∈𝒜 is called convex, denoted by f ∈𝒦 if f is univalent in 𝔻 and f(𝔻) is a convex domain. For a given 0 ≤ α < 1, a function f ∈𝒜 is called convex function of order α, denoted by 𝒦 (α), if 𝑧𝑓 ′′ (𝑧) ℜ𝔢 (1 + ′ ) > 𝛼, 𝑓 (𝑧)

𝑧 ∈ 𝔻.

It is well known that 𝒮 ∗ (0) = 𝒮 ∗ and 𝒦 (0) = 𝒦. We recall [1] that the function zg′(z) is starlike if and only if the function g(z) is convex. In 1916, L. Bieberbach’s 𝑛 conjectured that: The coefficient of each function𝑓(𝑧) = 𝑧 + ∑∞ 𝑛=2 𝑎𝑛 𝑧 , satisfy |𝑎𝑛 | ≤ 𝑛,

(𝟏. 𝟐)

and proved the bound for the case n = 2. In 1985 L. de Branges proved the Biberbach Conjecture with the help of the theory of Special Functions. The equality in (1.2) can be obtained for the Koebe function and its rotations, defined by: 𝐾(𝑧) =

𝑧 = 𝑧 + 𝑧 2 + 𝑧 3 +. . . . . . . . . . . . . .. (1 − 𝑧)2

(𝟏. 𝟑)

Using mapping properties, one can easily show that the Koebe function maps the unit disk 𝔻 (Fig. (a)) onto ℂ− (−∞, ¼) (Fig. (b)). It is easy to see that K(z) is a starlike but not convex function. The Koebe function plays a role of extremal function for many problems related to the class 𝒮 ∗ . Further, the function defined by 𝑓0 (𝑧) =

𝑧 = 𝑧 + 𝑧 2 + 𝑧 3 +. . . . . . . . . . . . . .. (1 − 𝑧)

(𝟏. 𝟒)

is convex as well as starlike function. This function maps the unit disk 𝔻 (Fig. (c)) onto domain such that ℜ(𝑓0 (𝑧)) > −1/2 (Fig. (d)). 𝑓0 (𝑧) plays a role of extremal function for many problems related to class 𝒦.

96

Advances in Special Functions of Fractional Calculus

Bansal and Jain

(b) Image of 𝔻 under z/(1 − 𝑧)2

(a) Unit Disc 𝔻

(c) Unit Disc 𝔻

(d) Image of 𝔻 under z/(1 − 𝑧)

A function f ∈𝒜 is said to be convex in the direction of the imaginary axis if f(𝔻) intersects every line parallel to the imaginary axis either in an interval or not at all. Given a convex function g ∈𝒦 with 𝑔(𝑧) ≠ 0and α < 1, a function f ∈𝒜, is called close-to-convex of order α with respect to convex function g, denoted by 𝐶𝑔 (𝛼),if ℜ𝔢 (

𝑓 ′ (𝑧) ) > 𝛼, 𝑔′ (𝑧)

𝑧 ∈ 𝔻.

(𝟏. 𝟓)

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

97

The class 𝐶𝑔 (0), is the class of functions close-to-convex with respect to g. Geometrically, a function f ∈𝒜 belongs to 𝒞 if the complement E of the imageregion F = {f(z) : |z| < 1} is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). One can easily see that the following function is close-to-convex by looking at the image of the unit disk (Fig. (e))under the map (Fig. (f)) 𝑓1 (𝑧) =

𝑧(1 − 𝑧/2) 1−𝑧

(𝟏. 𝟔)

The Noshiro-Warschawski theorem implies that close-to-convex functions are univalent in 𝔻, but not necessarily the converse. It is easy to verify that 𝒦⊂𝒮 ∗ ⊂𝒞. For more details, see [8].

(e) Unit Disc 𝔻

98

Advances in Special Functions of Fractional Calculus

(f) Image of 𝔻 under

Bansal and Jain

z−𝑧 2 /2 (1−𝑧)

It is well known that special functions play an important role in the function theory, especially after the solution of the famous Bieberbach conjecture by De-Branges see [2]. A number of authors have investigated geometric properties of the shifted hypergeometric functions. For instance, sufficient conditions for those functions to be starlike or convex were found by Merkes and Scott [3], Lewis [4], Ruscheweyh and Singh [5], Miller and Mocanu [6], Silverman [7], Ponnusamy and Vuorinen [8], K𝑢̈ stner [9, 10], Ponnusamy [11], H𝑎̈ sto, Ponnusamy and Vuorinen [12] and Sugawa [13]. Most of known results in this line, deals with z2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧) for real parameters a, b, c. Recently, several researchers have studied the geometric properties of Bessel functions [14-17], Struve functions [18, 19], and Lommel functions [20]. In the present chapter we aim to find some sufficient conditions under which z1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧) belongs to UCV (α, β) and 𝑆𝑝 (α, β). Here 1 F2 a, b, c; z is a special case of generalized hypergeometric function for p = 1 and q = 2, defined by: ∞

z1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧) = ∑ 𝑛=0

(𝑎)𝑛 𝑧 𝑛+1 . (𝑏)𝑛 (𝑐)𝑛 𝑛!

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

99

Here both b and c are not allowed to be zero or negative integer. It can be easily seen that w(z) = 1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧) is a solution of third order homogeneous differential equation: 𝑧 2 𝑤′′′(𝑧) + (𝑏 + 𝑐 + 1)𝑧𝑤′′(𝑧) + (𝑏𝑐 − 𝑧)𝑤′(𝑧) − 𝑎𝑤(𝑧) = 0.

(𝟏. 𝟖)

The well-known Struve function of order 𝜐 is defined by: ∞

(−1)𝑛 𝑧 2𝑛+𝜐−1 ( ) 𝐻𝜐 (𝑧) = ∑ Γ(𝑛 + 𝜐 + 3/2)Γ(𝑛 + 3/2) 2

(𝟏. 𝟗)

𝑛=0

which is a particular solution of the following non-homogeneous Bessel differential equation: 2

2

4(𝑧/2)𝜐+1

2

𝑧 𝑤′′(𝑧) + 𝑧𝑤′(𝑧) + (𝑧 − 𝜐 )𝑤 =

√𝜋Γ(𝜐 + 1/2)

(𝟏. 𝟏𝟎)

The modified Struve function 𝐿𝜐 (𝑧) is defined by (see [21], p.353]) 𝐿𝜐 (𝑧) = −𝒾𝑒 −𝒾𝜐 π/2 𝐻𝜐 (𝒾𝑧) ∞

=∑ 𝑛=0

=

𝑧 2𝑛+𝜐−1 1 ( ) Γ(𝑛 + 𝜐 + 3/2)Γ(𝑛 + 3/2) 2 𝑧 𝜐−1

√𝜋Γ(𝜐+3/2)2𝜐−2

1 𝐹2 (1, 𝜐

3 3 𝑧2

+ , ; ) 2 2

4

(1.11)

𝐿𝜐 (𝑧) does not belong to the class 𝒜, so we use the following normalization 𝕃𝜐 (𝑧) = √𝜋Γ(𝜐 + 3/2)2𝜐−2 𝑧 2−𝜐 𝐿𝜐 (𝑧).

(𝟏. 𝟏𝟐)

It is easy to see that 𝕃𝜐 (𝑧) = 𝑧

3 3 𝑧2 ) 1 𝐹2 (1, 𝜐 + , ; 2 2 4

(𝟏. 𝟏𝟑)

Lommel function of the first kind 𝑠𝜇,𝜐 (𝑧) is a particular solution of the inhomogeneous Bessel differential equation (see, for details ):

100

Advances in Special Functions of Fractional Calculus

Bansal and Jain

𝑧 2 𝑤′′(𝑧) + 𝑧𝑤′(𝑧) + (𝑧 2 − 𝜐 2 )𝑤(𝑧) = 𝑧 𝜇+1

(𝟏. 𝟏𝟒)

and it can be expressed in terms of a hypergeometric series S𝜇,𝜐 (z) =

z 𝜇+1 (𝜇 − 𝜈 + 1)(𝜇 + 𝜈 + 1)

1 𝐹2 (1,

𝜇 − 𝜈 + 3 𝜇 + 𝜈 + 3 −𝑧 2 , ; ), (𝟏. 𝟏𝟓) 2 2 4

where μ ± ν is non-negative odd integer. The modified Lommel function is defined by: T𝜇,𝜐 (z) = −𝒾 1−𝜇 S𝜇,𝜐 (𝒾z) z 𝜇+1 = (𝜇 − 𝜈 + 1)(𝜇 + 𝜈 + 1)

𝜇 − 𝜈 + 3 𝜇 + 𝜈 + 3 𝑧2 , ; ). 1 𝐹2 (1, 2 2 4

(𝟏. 𝟏𝟔)

For further information on modified Lommel functions, the reader is referred to Rollinger [22] and Shafer [23]. T𝜇,𝜐 (𝑧) does not belong to class A, so we use the following normalization 𝕋𝜇,𝜐 (𝑧) = (𝜇 − 𝜈 + 1)(𝜇 + 𝜈 + 1)𝑧 −𝜇 T𝜇,𝜐 (𝑧).

(𝟏. 𝟏𝟕)

In view of this, it is easy that 𝕋𝜇,𝜐 (𝑧) = z

1 𝐹2 (1,

𝜇 − 𝜈 + 3 𝜇 + 𝜈 + 3 𝑧2 , ; ). 2 2 4

(𝟏. 𝟏𝟖)

In a special case, when a = b, (1.7) reduces to ∞

1 𝑧𝑛 , 0 𝐹1 (−, 𝑐; 𝑧) = ∑ (𝑐)𝑛 𝑛!

(𝟏. 𝟏𝟗)

𝑛=0

which is a solution of second order homogeneous differential equation: 𝑧 2 𝑤′′(𝑧) + 𝑐𝑤′(𝑧) − 𝑤(𝑧) = 0.

(𝟏. 𝟐𝟎)

One can easily see the following relationship between 0 F1 function and modified Bessel function:

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

z0 F1 , 1; z 2 / 4 2 Γ 1 z 1I z

z,

101

(𝟏. 𝟐𝟏)

where the modified Bessel function is defined by: ∞

𝐼𝜐 (𝑧) = ∑ 𝑛=0

1 𝑧 2𝑛+𝜐−1 ( ) , Γ(𝑛 + 1)Γ(𝑛 + 𝜐 + 1) 2

(𝟏. 𝟐𝟐)

and 𝕀𝜐 (𝑧) is normalized modified Bessel function. 2. SUFFICIENCY CONDITIONS FOR UNIFORMLY CONVEXITY A function f is said to be uniformly convex (UCV), if the image of every circular arc γ contained in 𝔻, with center also in 𝔻, is convex. Analytically, UCV family is characterized as follows: 𝑈𝐶𝑉 = {𝑓 ∈ 𝒜:

𝑧𝑓 ′′ (𝑧) 𝑧𝑓 ′′ (𝑧) ℜ {1 + ′ }>| ′ |, 𝑧 ∈ 𝔻}, 𝑓 (𝑧) 𝑓 (𝑧)

(𝟐. 𝟏)

for more details, see [24]. Using the Alexander transformation, Ronning [24] defined a new class called 𝑆𝑝 as follows: 𝑧𝑓 ′ (𝑧) 𝑧𝑓 ′ (𝑧) 𝑆𝑝 = {𝑓 ∈ 𝒜: ℜ { }>| − 1|, 𝑧 ∈ 𝔻}. 𝑓(𝑧) 𝑓(𝑧)

(𝟐. 𝟐)

Kanas and Wísniowska [25] defined the class k − UCV as k − UCV = {𝑓 ∈ 𝒜:

ℜ {1 +

𝑧𝑓′′ (𝑧) 𝑓′ (𝑧)

} > 𝑘|

𝑧𝑓′′ (𝑧) 𝑓′ (𝑧)

|, (0 ≤ 𝑘 < ∞), 𝑧 ∈ 𝔻} .

(𝟐. 𝟑)

In another paper, Kanas and Wísniowska [26] extended the class 𝑆𝑝 as k − ST = {𝑓 ∈ 𝒜: ℜ {

𝑧𝑓 ′ (𝑧) 𝑧𝑓 ′ (𝑧) } > 𝑘| − 1| , 𝑓(𝑧) 𝑓(𝑧)

(0 ≤ 𝑘 < ∞), 𝑧 ∈ 𝔻} . (𝟐. 𝟒)

Various properties of the classes k−UCV and k−ST were extensively studied by Kanas and Srivastava [27]. Further, Bharathi et al. [28] extended the class k − UCV to UCV (α, β) with k = α ≥ 0 and0 ≤ β < 1 as:

102

Advances in Special Functions of Fractional Calculus

UCV(α, β) = {𝑓 ∈ 𝒜:

ℜ {1 +

𝑧𝑓′′ (𝑧) 𝑓′ (𝑧)

} > α|

𝑧𝑓′′ (𝑧) 𝑓′ (𝑧)

Bansal and Jain

| + β, 𝑧 ∈ 𝔻}

(2.5)

They give a sufficient condition for a function f ∈𝒜 to be in UCV (α, β) in terms of the coefficient of the function. It can be easily seen that UCV(0, β) = 𝒦(β)

(𝟐. 𝟔)

The class 𝑆𝑝 (α, β) was also discussed in [28], as 𝑆𝑝 (α, β) = {𝑓 ∈ 𝒜: ℜ {

𝑧𝑓 ′ (𝑧) 𝑧𝑓 ′ (𝑧) } > α| − 1| + β, 𝑧 ∈ 𝔻}. 𝑓(𝑧) 𝑓(𝑧)

(𝟐. 𝟕)

It can be easily verified that 𝑆𝑝 (0, β) = 𝑆 ∗ (β)

(𝟐. 𝟖)

In this section, we find conditions on a, b, and c such that z 1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧)belongs to the classes UCV (α, β)and 𝑆𝑝 (α, β). For this, we require the following Lemmas: Lemma 2.1 ([28], Theorem 2.3, p.21). If ∞

∑ [𝑛(𝑛(1 + α) − (α + β))]|𝑎𝑛 | ≤ 1 − β,

(𝟐. 𝟗)

𝑛=2

then the function of the form (1.1) is in UCV (α, β). Lemma 2.2 ([28], Theorem 2.6, p.23). If ∞

∑ [𝑛(𝑛(1 + α) − (α + β))]|𝑎𝑛 | ≤ 1 − 𝛽,

(𝟐. 𝟏𝟎)

𝑛=2

then the function of the form (1.1) is in 𝑆𝑝 (α, β). Lemma 2.3 ([29], Lemma 4, p.767). Let the function f(z) be of the form (1.1), then a sufficient condition for f satisfy ℜ𝑒 𝑖𝜂 (𝑓(𝑧)/𝑧 − 𝛽) > 0 in 𝔻 𝑖𝑠

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

103

∞

∑ |𝑎𝑛 | ≤ (1 − 𝛽) cos 𝜂(|𝜂| < 𝜋/2, 𝛽 < 1),

(𝟐. 𝟏𝟏)

𝑛=2

The above condition is also necessary if η = 0 and 𝑎𝑛 < 0 for all n ≥ 2. To prove our main result, we need to prove the following lemma: Lemma 2.4. For b, c≠0,-1,-2,…… ∞

∑ 𝑛=0

(n + 1)(𝑎)𝑛 𝑎 = (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 𝑏𝑐

1 𝐹2 (𝑎

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

+ 1, 𝑏 + 1, 𝑐 + 1; 1) +

(𝟐. 𝟏𝟐)

and ∞

∑ 𝑛=0

(𝑛 + 1)2 (𝑎)𝑛 (𝑎)2 = (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)2 (𝑐)2

1 𝐹2 (𝑎

+ 2, 𝑏 + 2, 𝑐 + 2; 1) +

3𝑎 𝑏𝑐

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

(𝟐. 𝟏𝟑)

Proof: A simple calculation gives ∞

∞

∞

𝑛=0

𝑛=0

𝑛=0

n(𝑎)𝑛 (𝑎)𝑛 (n + 1)(𝑎)𝑛 =∑ +∑ ∑ (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 ∞

=∑ 𝑛=1 ∞

=∑ 𝑛=0

=

𝑎 𝑏𝑐

(𝑎)𝑛 + (𝑏)𝑛 (𝑐)𝑛 (n − 1)!

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

(𝑎)𝑛+1 + (𝑏)𝑛+1 (𝑐)𝑛+1 (n)!

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

1 𝐹2 (𝑎

+ 1, 𝑏 + 1, 𝑐 + 1; 1) +

1 𝐹2 (𝑎, 𝑏, 𝑐; 1).

Writing (𝑛 + 1)2 = 𝑛(𝑛 − 1) + 3𝑛 + 1, the LHS of (2.13) can be written as: ∞

∞

∞

∞

𝑛=0

𝑛=0

𝑛=0

𝑛=0

(𝑛 + 1)2 (𝑎)𝑛 𝑛(𝑛 − 1)(𝑎)𝑛 n(𝑎)𝑛 (𝑎)𝑛 ∑ =∑ +3∑ +∑ (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

104

Advances in Special Functions of Fractional Calculus ∞

=∑ 𝑛=2 ∞

=∑ 𝑛=0

=

Bansal and Jain

(𝑎)𝑛 3𝑎 + (𝑏)𝑛 (𝑐)𝑛 (𝑛 − 2)! 𝑏𝑐

1 𝐹2 (𝑎

+ 1, 𝑏 + 1, 𝑐 + 1; 1) +

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

(𝑎)𝑛+2 3𝑎 + (𝑏)𝑛+2 (𝑐)𝑛+2 (𝑛)! 𝑏𝑐

1 𝐹2 (𝑎

+ 1, 𝑏 + 1, 𝑐 + 1; 1) +

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

(𝑎)2 (𝑏)2 (𝑐)2

1 𝐹2 (𝑎

+ 2, 𝑏 + 2, 𝑐 + 2; 1) +

3𝑎 𝑏𝑐

1 𝐹2 (𝑎

+ 1, 𝑏 + 1, 𝑐 + 1; 1) +

1 𝐹2 (𝑎, 𝑏, 𝑐; 1)

This completes the proof of Lemma 2.4. Theorem 2.1. Let b, 𝑐 > 0, 𝛼 ≥ 0,0 ≤ 𝛽 < 1 and (1 + 𝛼)

(|𝑎|)2

1 𝐹2 (|𝑎|

(𝑏)2 (𝑐)2

+ 2, 𝑏 + 2, 𝑐 + 2; 1)+(3 + 2𝛼 − 𝛽)

1, 𝑏 + 1, 𝑐 + 1; 1)+(1 − 𝛽)

1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1)

1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧)∈

then the function z

1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧).

Proof: Set f(z)=z that

|𝑎| 𝑏𝑐

1 𝐹2 (|𝑎|

+

≤ 2(1 − 𝛽),

UCV (α, β).

Then by using Lemma 2.1, it suffices to show

∞

𝑆: = ∑ (𝑛 + 1)[(𝑛 + 1)(1 + 𝛼) − (𝛼 + 𝛽)]| 𝑛=1

(𝑎)𝑛 | ≤ 1 − 𝛽. (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

(𝟐. 𝟏𝟒)

From the fact that|(𝑎)𝑛 | ≤ (|𝑎|)𝑛 , we observe that ∞

𝑆 ≤ ∑ (𝑛 + 1)[(𝑛 + 1) − (𝛼 + 𝛽)](1 + 𝛼) 𝑛=1 ∞

∞

= (1 + 𝛼) ∑ (𝑛 + 1)2 𝑛=0

= (1 + 𝛼)

(|𝑎|)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

(|𝑎|)𝑛 (|𝑎|)𝑛 − (𝛼 + 𝛽) ∑ (𝑛 + 1) +𝛽−1 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 𝑛=0

(|𝑎|)2 (𝑏)2 (𝑐)2

1 𝐹2 (|𝑎|

+ 2, 𝑏 + 2, 𝑐 + 2; 1)

|𝑎| 𝐹 (|𝑎| + 1, 𝑏 + 1, 𝑐 + 1; 1) 𝑏𝑐 1 2 + (1 − 𝛽) 1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1) + 𝛽 − 1 + (3 + 2𝛼 − 𝛽)

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

105

using Lemma 2.1 and the hypothesis of Theorem 2.1, S is less than 1 − β. This completes the proof. Theorem 2.2. Let b, 𝑐 > 0, 𝛼 ≥ 0,0 ≤ 𝛽 < 1 and (1 + 𝛼)

|𝑎| 𝑏𝑐

1 𝐹2 (|𝑎|

then the function z Proof: Set f(z)=z

1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1)

+ 1, 𝑏 + 1, 𝑐 + 1; 1)+(1 − 𝛽)

1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧)∈𝑆𝑝 1 𝐹2 (𝑎, 𝑏, 𝑐; 𝑧)Then

≤ 2(1 − 𝛽),

(α, β).

by using Lemma 2.2, it suffices to show that

∞

𝑇: = ∑ [(𝑛 + 1)(1 + 𝛼) − (𝛼 + 𝛽)]| 𝑛=1

(|𝑎|)𝑛 | ≤ 1 − 𝛽. (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

(𝟐. 𝟏𝟓)

From the fact that|(𝑎)𝑛 | ≤ (|𝑎|)𝑛 , we observe that ∞

𝑇 ≤ ∑ [(𝑛 + 1) − (𝛼 + 𝛽)](1 + 𝛼) 𝑛=1

(|𝑎|)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

∞

∞

𝑛=0

𝑛=0

(|𝑎|)𝑛 (|𝑎|)𝑛 = (1 + 𝛼) ∑ (𝑛 + 1) − (𝛼 + 𝛽) ∑ +𝛽−1 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛 =(1 + 𝛼)

(|𝑎|) (𝑏)(𝑐)

1 𝐹2 (|𝑎|

+ 1, 𝑏 + 1, 𝑐 + 1; 1) + (1 − 𝛽)

1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1)

+𝛽−1

using Lemma 2.2 and the hypothesis of Theorem 2.2, T is less than 1 − β. This completes the proof. Theorem 2.3. Let b, 𝑐 > 0,|𝜂| < 𝜋/2,0 ≤ 𝛽 < 1 and 1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1)

− 1 ≤ (1 − 𝛽) cos 𝜂,

then we have ℜ𝑒 𝑖𝜂 1 F2 a, b, c;1 > 0 for 𝑧 ∈ 𝔻. Proof: Set f(z) = z 1 F2 a, b, c; z Then by using Lemma 2.3, it suffices to show that

106

Advances in Special Functions of Fractional Calculus ∞

𝑅: = ∑ | 𝑛=1

Bansal and Jain

(|𝑎|)𝑛 | ≤ 1 − 𝛽 cos 𝜂 . (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

(𝟐. 𝟏𝟔)

From the fact that|(𝑎)𝑛 | ≤ (|𝑎|)𝑛 , we observe that ∞

𝑅≤∑ 𝑛=1 ∞

=∑ 𝑛=0

=

(|𝑎|)𝑛 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

(|𝑎|)𝑛 −1 (𝑏)𝑛 (𝑐)𝑛 (1)𝑛

1 𝐹2 (|𝑎|, 𝑏, 𝑐; 1)

− 1,

using Lemma 2.3 and the hypothesis of Theorem 2.2, R is less than 1 − β. This completes the proof. Setting a = b in Theorem 2.1, Theorem 2.2 and Theorem 2.3, we have Corollary 2.1. Let c > 0, then a sufficient condition for the function z1𝐹2 (𝑎, 𝑏, 𝑐; 𝑧) belong to UCV (α, β) is that (1 + 𝛼)

1 (𝑐)2

0 𝐹1 (−, 𝑐

+ 2; 1)+(3 + 2𝛼 − 𝛽)

1 𝑐

0 𝐹1 (−, 𝑐

+ 1; 1)+(1 − 𝛽)

0 𝐹1 (−, 𝑐; 1)

≤

2(1 − 𝛽).

Corollary 2.2. Let c > 0, then a sufficient condition for the function z 0𝐹1 (−; c; z) belong to 𝑆𝑝 (α, β). is that (1 + 𝛼)

1 𝑏𝑐

0 𝐹1 (−, 𝑐

+ 1; 1)+(1 − 𝛽)

0 𝐹1 (−, 𝑐; 1)

≤ 2(1 − 𝛽).

Corollary 2.3. Let b,𝑐 > 0, |𝜂| < 𝜋/2,0 ≤ 𝛽 < 1 and 0 F1 , c;1 −1 ≤ (1 − 𝛽) cos 𝜂, then we have Rei 0 F1 , c;1 > 0 for 𝑧 ∈ 𝔻.

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

107

3. SUFFICIENCY CONDITIONS FOR Re1 F2 a, b, c; z 1/ 2 In order to establish the results for this section, we need the following Lemma: Lemma 3.1. [30] Let {an}n be a sequence of nonnegative real numbers such that 𝑎1 = 1, and that for n ≥ 2, the sequence {𝑎𝑛 } is a convex decreasing, i.e. 0 ≥ 𝑎𝑛+2 − 𝑎𝑛+1 ≥ 𝑎𝑛+1 − 𝑎𝑛 , for all n ∈ N. Then ∞

ℜ𝔢 (∑ 𝑎𝑛 𝑧 𝑛−1 ) > 1/2(𝑧 ∈ 𝔻).

(𝟑. 𝟏)

𝑛=1

Theorem 3.1. Let a, b, c > 0, a ≤ bc, and 2bc(b + 1)(c + 1) ≥ a[4(b + 1)(c + 1) − (a + 1)],

(3.2)

then

Re1 F2 a, b, c; z 1/ 2. for 𝑧 ∈ 𝔻. Proof: In view of Lemma 3.1, it is sufficient to show that the sequence {𝐴𝑛 }∞ 𝑛=1 = {

∞ (𝑎)𝑛−1 (𝐴 = 1) } (𝑏)𝑛−1 (𝑐)𝑛−1 (𝑛 − 1)! 𝑛=1 1

is a convex decreasing sequence. We first prove that the above sequence is decreasing sequence, i.e., 𝐴𝑛 − 𝐴𝑛−1 = 𝐴𝑛 [ =

(𝑎 + 𝑛 − 1) ] (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛

𝐴𝑛 𝑀(𝑛), (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛

Where, 𝑀(𝑛) = (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛 − (𝑎 + 𝑛 − 1)

108

Advances in Special Functions of Fractional Calculus

Bansal and Jain

= 𝑛3 + 𝑛2 (𝑏 + 𝑛 − 2) + 𝑛(𝑏𝑐 − 𝑏 − 𝑐) − 𝑎 + 1 = 𝑛2 (𝑛 − 2) + 1 + 𝑛(𝑏 + 𝑐)(𝑛 − 1) + (𝑛𝑏𝑐 − 𝑎) ≥0

(𝐼𝑛 𝑣𝑖𝑒𝑤 𝑜𝑓 𝑏𝑐 ≥ 𝑎).

Thus, we have M(n) ≥ 0 for all n ≥ 1, provided that a, b, c > 0, and a ≤ bc and so the sequence {𝐴𝑛 } is non-increasing. Next we show that the sequence {𝐴𝑛 }∞ 𝑛=1 is convex decreasing, and for that we need to show that 𝐴𝑛 − 2𝐴𝑛+1 + 𝐴𝑛+2 ≥ 0, for all n ≥ 1. As 𝐴𝑛 =

(𝑎)𝑛−1 (𝑛 ≥ 2, 𝐴1 = 1), (𝑏)𝑛−1 (𝑐)𝑛−1 (𝑛 − 1)!

Therefore, That 𝐴𝑛 − 2𝐴𝑛+1 + 𝐴𝑛+2 ≥ 0 ⇔ 𝐴𝑛 [1 − 2

(𝑎 + 𝑛 − 1) (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛 (𝑎 + 𝑛)(𝑎 + 𝑛 − 1) + ]≥0 (𝑏 + 𝑛)(𝑏 + 𝑛 − 1)(𝑐 + 𝑛)(𝑐 + 𝑛 − 1)𝑛(𝑛 + 1)

In view of (3.2), the above is true for n = 1. For n ≥ 2, the difference of the first two-term is 1−2

(𝑎 + 𝑛 − 1) 𝑁(𝑛) = (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛 (𝑏 + 𝑛 − 1)(𝑐 + 𝑛 − 1)𝑛

Where, 𝑁(𝑛)=𝑛3 + 𝑐𝑛2 + 𝑏𝑛2 − 2𝑛2 + 𝑏𝑐𝑛 − 𝑏𝑛 − 𝑐𝑛 − 𝑛 − 2𝑎 + 2 = (𝑛3 − 2𝑛2 − 𝑛 + 2) + 𝑛[𝑛(𝑏 + 𝑐) − 𝑏 − 𝑐] + 𝑛𝑏𝑐 − 2𝑎 In view of bc ≥ a, it is easy to see that N(n) ≥ 0 for all n ≥ 2, and hence applying the Lemma 3.1, we get the desired result.

Sufficient Conditions

Advances in Special Functions of Fractional Calculus

109

Substituting a = b in Theorem 3.1, we have Corollary 3.1. Let 𝑐 ≥

1+√7 2

then Re 0 F1 , c; z 1/ 2,

for 𝑧 ∈ 𝔻 Remark 3.1. Applying above Corollary 3.1 for positive half-integers greater than equal to 2, we obtain the following inequality for c = 5/2:

sinh 2 z 3 1/ 2, z Re 0 F1 ,5 / 2; z Re 2 cosh 2 z 8 z z

(𝟑. 𝟑)

Similarly, for c = 7/2, Corollary 4.7 gives the inequality: ℜ𝔢 (

sinh(2√𝑧) 3 1 (2 cosh(2√𝑧) − )) > , (𝑧 ∈ 𝔻). 8𝑧 2 √𝑧

(𝟑. 𝟒)

CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENTS Declared none. REFERENCES [1] [2]

P.L. Duren, Univalent Functions, Springer-Verlag, 1983. L. Branges, "A proof of the Bieberbach conjecture", Acta Math., vol. 154, no. 1-2, pp. 137152, 1985. http://dx.doi.org/10.1007/BF02392821

110

[3]

[4]

[5]

[6]

[7]

[8]

[9] [10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

Advances in Special Functions of Fractional Calculus

Bansal and Jain

E.P. Merkes, and W.T. Scott, "Starlike hypergeometric functions", Proc. Am. Math. Soc., vol. 12, no. 6, pp. 885-888, 1961. http://dx.doi.org/10.1090/S0002-9939-1961-0143950-1 J.L. Lewis, "Applications of a convolution theorem to Jacobi polynomails", SIAM J. Math. Anal., vol. 10, no. 6, pp. 1110-1120, 1979. http://dx.doi.org/10.1137/0510102 S. Ruscheweyh, and V. Singh, "On the order of starlikeness of hypergeometric functions", J. Math. Anal. Appl., vol. 113, no. 1, pp. 1-11, 1986. http://dx.doi.org/10.1016/0022-247X(86)90329-X S.S. Miller, and P.T. Mocanu, "Univalence of Gaussian and confluent hypergeometric functions", Proc. Am. Math. Soc., vol. 110, no. 2, pp. 333-342, 1990. http://dx.doi.org/10.1090/S0002-9939-1990-1017006-8 H. Silverman, "Starlike and convexity properties for hypergeometric functions", J. Math. Anal. Appl., vol. 172, no. 2, pp. 574-581, 1993. http://dx.doi.org/10.1006/jmaa.1993.1044 S. Ponnusamy, and M. Vuorinen, "Univalence and convexity properties for gaussian hypergeometric functions", Rocky Mt. J. Math., vol. 31, no. 1, pp. 327-353, 2001. http://dx.doi.org/10.1216/rmjm/1008959684 R. K𝑢̈stner, "Mapping properties of hypergeometric functions and convolutions of starlike or convex functions oforder α", Comput. Methods Funct. Theory, vol. 2, pp. 597-610, 2002. R. Küstner, and R. stner, "On the order of starlikeness of the shifted Gauss hypergeometric function", J. Math. Anal. Appl., vol. 334, no. 2, pp. 1363-1385, 2007. http://dx.doi.org/10.1016/j.jmaa.2007.01.011 S. Ponnusamy, "Close-to-convexity properties of Gaussian hypergeometric functions", J. Comput. Appl. Math., vol. 88, no. 2, pp. 327-337, 1998. http://dx.doi.org/10.1016/S0377-0427(97)00221-5 P. Hästö, S. Ponnusamy, M. Vuorinen, and M. Vuorinen, "Starlikeness of the Gaussian hypergeometric functions", Complex Var. Elliptic Equ., vol. 55, no. 1-3, pp. 173-184, 2010. http://dx.doi.org/10.1080/17476930903276134 T. Sugawa, "A self-duality of strong starlikeness", Kodai Math. J., vol. 28, no. 2, pp. 382389, 2005. http://dx.doi.org/10.2996/kmj/1123767018 Á. Baricz, "Functional inequalities involving special functions II", J. Math. Anal. Appl., vol. 327, no. 2, pp. 1202-1213, 2007. http://dx.doi.org/10.1016/j.jmaa.2006.05.006 Á. Baricz, "Geometric properties of generalized Bessel functions", Publ. Math. (Debrecen), vol. 73, no. 1-2, pp. 155-178, 2008. http://dx.doi.org/10.5486/PMD.2008.4126 Á. Baricz, and S. Ponnusamy, "Starlikeness and convexity of generalized Bessel functions", Integr. Transforms Spec. Funct., vol. 21, no. 9, pp. 641-653, 2010. http://dx.doi.org/10.1080/10652460903516736 A. Baricz, P. A. Kup𝑎n, R. Sz𝑎sz, "The radius of starlikeness of normalized Bessel function of first kind", Proc. Am. Math. Soc., vol. 142, no. 6, pp. 2019-2025, 2014.

Sufficient Conditions

[18]

[19]

[20]

[21] [22]

[23] [24] [25]

[26] [27]

[28]

[29]

[30]

Advances in Special Functions of Fractional Calculus

111

http://dx.doi.org/10.1090/S0002-9939-2014-11902-2 H. Orhan, and N. Yagmur, "Geometric properties of generalized Struve functions, An. Stiint", Univ. Al. I. Cuza Iasi.Mat. (N.S.). http://dx.doi.org/10.2478/aicu-2014-0007 N. Yagmur, and H. Orhan, "Partial sums of generalized Struve functions", Miskolc Math. Notes, vol. 17, no. 1, p. 657, 2016. http://dx.doi.org/10.18514/MMN.2016.1419 M. Çağlar, and E. Deniz, "Partial sums of the normalized Lommel functions", Math. Inequal. Appl., vol. 18, no. 3, pp. 1189-1199, 2015. http://dx.doi.org/10.7153/mia-18-92 S. Zhang, and J. Jin, Computation of Special Functions., Wiley Interscience Publication: New York, 1996. C.N. Rollinger, "Lommel functions with imaginary argument", Q. Appl. Math., vol. 21, no. 4, pp. 343-349, 1964. http://dx.doi.org/10.1090/qam/153883 R.E. Shafer, Lommel functions of imaginary argument, Technical report., Lawrence Radiation Laboratory: Livermore, 1963. F. Ronning, "Uniformly convex functions and a corresponding class of starlike functions", Proc. Amer. Math. Soc., vol. 118, no. 1, pp. 189-196, 1993. S. Kanas, A. Wisniowska, and A. niowska, "Conic regions and k-uniform convexity", J. Comput. Appl. Math., vol. 105, no. 1-2, pp. 327-336, 1999. http://dx.doi.org/10.1016/S0377-0427(99)00018-7 S. Kanas, and Wi, A. niowska, "Conic domains and starlike functions", Rev. Roumaine Math. Pures Appl., vol. 45, pp. 647-657, 2000. S. Kanas, and H.M. Srivastava, "Linear operators associated with k -uniformly convex functions", Integr. Transforms Spec. Funct., vol. 9, no. 2, pp. 121-132, 2000. http://dx.doi.org/10.1080/10652460008819249 R. Bharati, R. Parvatham, and A. Swaminathan, "On subclasses of uniformly convex functions and corresponding classof starlike functions", Tamkang Journal of Mathematics, vol. 28, no. 1, pp. 17-32, 1997. http://dx.doi.org/10.5556/j.tkjm.28.1997.4330 Y.C. Kim, and S. Ponnusamy, "Sufficiency for Gaussian hypergeometric functions to be uniformly convex", Int. J. Math. Math. Sci., vol. 22, no. 4, pp. 765-773, 1999. http://dx.doi.org/10.1155/S0161171299227652 L. F𝑒jer, "Untersuchungen, 𝑢̈ber Potenzreihen mit mehrfach monotoner Koeffizientenfolge", Acta Litterarum Sci., vol. 8, pp. 89-115, 1936.

112

Advances in Special Functions of Fractional Calculus, 2023, 112-120

CHAPTER 7

From Abel Continuity Theorem to Paley-Wiener Theorem S. Yu1, P. Agarwal2 and S. Kanemitsu3,* 1 2

Taishan College, Shandong University, Jinan, 250110, P.R.China Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India

3

KSCSTE-Kerala School of Mathematics, Kunnamangalam, Kozhikode, Kerala, 673571 India Abstract: In this note we reveal that the missing link among a few crucial results in analysis, Abel continuity theorem, convergence theorem on (generalized) Dirichlet series, Paley-Wiener theorem is the Laplace transform with Stieltjes integration. By this discovery, the reason why the domains of Stoltz path and of convergence look similar is made clear. Also as a natural intrinsic property of Stieltjes integral, the use of partial summation in existing proofs is elucidated. Secondly, we shall reveal that a basic part of the proof of Paley-Wiener theorem is a version of the Laplace transform.

Keywords: Laplace transform, Stieltjes integral, Abel continuity theorem, PaleyWiener theorem, conformal mapping, 2010 MSC: 130E99, 44A10, 40A05. 1. INTRODUCTION Let {n } be an increasing sequence of real numbers for which we may suppose

1 0 . For complex coefficients an , the series

f ( s ) an e n s

(1.1)

n 1

convergent in some half-plane, is called a generalized Dirichlet series. a 1. If n log n with log denoting the principal value, f ( s ) ns is (an ordinary) n 1 n Dirichlet series. *Corresponding

author S. Kanemitsu: KSCSTE-Kerala School of Mathematics, Kunnamangalam, Kozhikode, Kerala, 673571 India; E-mail: [email protected] Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Paley-Wiener Theorem

2. If

Advances in Special Functions of Fractional Calculus

113

n n and e s w , f ( w) f ( log w) a wn is the power series. n n 1

In all literature [1], [2], [3], etc. the convergence theorem for generalized Dirichlet series, Theorem 1 and the Abel continuity theorem, Corollary 1 are regarded as independent and proofs are given separately. Cf. also [4] (cf. [5]). In [6] it is shown that Theorem 1 entails Corollary 1 via a counterpart, Corollary 2 together with conformality of the analytic mapping e s w , thus revealing the reason why the convergence domains are angular domains of a similar shape. The proof uses a general form of the partial summation [6, Lemma 2] for a generalized sequence {n } , thus unifying all existing proofs. In this note we employ a general treatment by (Lebesgue-) Stieltjes integrals to attain two objects at a stretch. I.e. we follow [7] to introduce Corollary 3 whose discrete version leads to Theorem 1. In the proof, integration by parts is used which is a more general version of the partial summation. Then on one hand we cover Abel continuity theorem by the convergence theorem, Corollary 3, for Laplace transforms and conformality, revealing the reason why convergence domains being similar. On the other hand, we shall show that the basic part of the Paley-Wiener theorem (cf. e.g. [8]) is laid by the Laplace transform method. Then we appeal to two fundamental results, the Plancherel formula and the Fourier inversion formula to conclude the theorem. By finding this hidden link of Laplace transform, we are able to treat these two remote-looking objects of Paley-Wiener theorem and Abel continuity theorem in a unified way, up to some auxiliary fundamental results. The Paley-Wiener theorem has recently been highlighted in view of its essential application to signal restoration. In both well-known approaches by sampling [7], [10], [11] and by Bernstein polynomials [12] the Paley-Wiener theorem plays a fundamental role. This is a typical example of ideas indoctrinated previously with the established methods and attitudes of the discipline, can sometimes point to unorthodox, though remarkably simple, solutions to those problems. Theorem 1. If the series (1.1) is convergent for s s0 0 it0 , then f (s) is uniformly convergent in the right half-plane 0 in the wide sense and represents an analytic function there. More precisely, let D be an angular domain

0 0, arg( s s0 )

(1.2)

114

Advances in Special Functions of Fractional Calculus

with 0

2

0

2

Yu et al.

. Then f ( s ) is uniformly convergent on D in the wide

sense. Corollary 1. (Abel continuity theorem)

Suppose a power series f ( z ) an z n converges at the point z0 on its circle of n 1

z convergence. Draw two chords (inside the circle) that start from 0 and form an z angle with the tangent at 0 of the circle 0 . Let be the (closure of) 2 intersection of this angular subdomain and the disc of convergence. Then f ( z ) approaches f ( z0 ) as z z0 in the angular domain inside . This is often said as z approaches to z0 along Stoltz path. Corollary 2. (Counterpart of Abel continuity theorem) f(s) approaches to f(s0) as s s0 in the angular domain (1.2) Lemma 1. (i) The Stieltjes integral

b

a

f dg exists if f is continuous and g is of bounded

variation and linear in f and g. The role can be changed in view of Item (ii). It holds that

b

a

dg ( x) g (b) g (a).

(1.3)

(ii) The formula for integration by parts holds true:

b

a

b

f ( x)dg ( x) [ f ( x) g ( x)]ba g ( x)df ( x), a

(1.4)

provided that f is continuous and g is of bounded variation or g is continuous and f is of bounded variation.

Paley-Wiener Theorem

Advances in Special Functions of Fractional Calculus

(iii) If g is a step function with jumps sum:

b

a

an

at

xn

f ( x)dg ( x)

a xn b

(iv) If f(x) is continuous, ( x) L L1

115

, the Stieltjes integral reduces to the

f ( xn )an .

(1.5)

n! ( x) L L1 on and r ! n r !

x

g ( x) (u)du x, c [a, b] c

,

Then

b

a

b

b

a

a

f ( x)dg ( x) f ( x) ( x)dx f ( x) g ( x)dx

(1.6)

where the last integral exists as a Lebesgue integral. Lemma 2. Let

f (s) e sx dg ( x) 0

(1.7)

and u

h(u) e s0 x dg ( x) u 0. 0

(1.8)

If lim sup | h(u ) | M

(1.9)

0u

with s0 0 it0 s0 0 it0 , then (1.7) is convergent at s0 and

0

e sx dg ( x) (s s0 ) e( s s0 ) x dh( x) , 0

(1.10)

116

Advances in Special Functions of Fractional Calculus

Yu et al.

the integral on the right being absolutely convergent. Remark 1. From Lemma 2 it follows that the domain of convergence is a half-plane. For our purpose we need a stronger results that follow. In most of existing literature on (generalized) Dirichlet series, the above proof is used with integration by parts as partial summation. Theorem 2. ({[13], Theorem 4.3, p. 54}) If the integral (1.7) is convergent at s s0 and H 0, K 1 are constants, then the integral is uniformly convergent in the domain D :| s s0 | K ( 0 )e H ( 0 ) , 0

(1.11)

Corollary 3. If the integral

f (s) e sx dg ( x) 0

(1.12)

1 1 where has the same meaning cos in Theorem 1, then the integral is uniformly convergent in the angular domain

is convergent at s s0 0 it0 and K

Ds0 :| s s0 | K ( 0 ), 0 .

(1.13)

Suppose f ( z ) is an integral function satisfying the conditions

| f ( z ) | Ce A|z| ,

0 A

(1.14)

and

f ( x) o(1) as z x . For a complex number α, let ( r ) denote a ray starting from the originα

( 1.15)

Paley-Wiener Theorem

Advances in Special Functions of Fractional Calculus

117

(r ) : z rei , 0 r .

(1.16)

s P {s | Re( se ) A},

(1.17)

Let

so that Pa is the rotated right half-plane obtained from the right half-plane {s | Re s A} by rotation in the positive direction. Let ( w) be defined by ( s) ( f , s) e sz f ( z )dz, s P .

(1.18)

In view of (1.16) and (1.17) the Weierstrass M-test applies and therefore the integral (1.18) converges absolutely and represents an analytic function in P for every

. 0 is the Laplace transform

0 ( f , s) [ f ](s) e sx f ( x)dx, : Re s a. 0

(1.19)

The integral (1.19) is convergent at s=0 in view of (1.17). Hence Corollary 3 applies and it is analytic in the angular domain (1.13). Lemma 3. Suppose 0 . Then in the intersection P P , we have ( f , s ) ( f , s ).

(1.20)

Theorem 3. Every ( f , s) in (1.18) is an integral representation of the Laplace transform (1.19) for all α Proof follows from Corollary 3 and Lemma 3.

118

Advances in Special Functions of Fractional Calculus

Yu et al.

2. PALEY-WIENER THEOREM In this section we elucidate the proof of the Paley-Wiener theorem given in Rudin [14]. There are some books and papers related to this theorem (cf. e.g. [8], pp.38-40). Let

f ( z)

1 2

A

A

F ( )ei z d, z

(2.1)

where A 0 and F L2 ( A, A) . One can show that f is an integral function. It satisfies the growth condition

| f ( z ) |

1 2

A

A

|F ( ) | e y d

1 A| y| A e |F ( ) |d, A 2

(2.2)

Where we write z=x+iy. Hence denoting the last integral by C, we deduce (1.16). An integral function f that satisfies condition (1.14) is said to be of exponential type (or of order 1 à la Hadamard). Thus we have Proposition 1. Every f of the form (2.1) is an integral function which satisfies (1.14) and whose 2 restriction is in L (by the Plancherel theorem). The remarkable theorem of Paley-Wiener asserts that the converse also holds. Theorem 4. (Paley-Wiener theorem) Suppose A and C are positive constants, that f is an integral function of exponential type, i.e. satisfying (1.14) for all values of z, and the boundary condition

| f ( x) |2 dx .

(2.3)

Paley-Wiener Theorem

Advances in Special Functions of Fractional Calculus

119

Then there exists a boundary function F L2 (, ) such that

f ( z)

1 2

A

A

F ( )ei z d

(2.4)

for all} values of z. Proof follows from the results stated above including Lemma 3 etc. More details including proofs of results can be found in [15]. Conclusion. By applying the Laplace transform with Stieltjes integration, we have established the following theorem. Theorem 5. Both the Abel continuity theorem (Corollary 1) and the Paley-Wiener theorem (Theorem 4) are based on similar grounds and may be treated in a unified way up to some auxiliary fundamental results, where in the former we use the discrete form as partial summation and in the latter, a generalized Laplace transform along a ray. CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1] [2] [3] [4]

G.H. Hardy, and M. Riesz, "The general theory of Dirichlet’s series", CUP, Cambridge, 1915reprint: Hafner, New York, 1972. J-P. Serre, A course in arithmetic., Springer Verl: New York, Heidelberg, Berlin, 1973. http://dx.doi.org/10.1007/978-1-4684-9884-4 T. Tatuzawa, Theory of functions., Kyoritsu-shuppan: Tokyo, 1989. A. Walfisz, "Uber Summabilitatssatze von Marcel Riesz", Math. Ann., vol. 30, pp. 130-148, 1930.

120

[5] [6]

[7] [8] [9] [10] [11] [12]

[13] [14] [15]

Advances in Special Functions of Fractional Calculus

Yu et al.

S. Kanemitsu, and H. Tsukada, "Contributions to the theory of zeta-functions modular relation supremacy", World Sci., Singapore, 2014. F-H. Li, and S. Kanemitsu, "Around boundary functions of the right half-plane and the unit disc", Advances in special functions and analysis of differential equations., CRC Press: Chapman & Hall, 2019, pp. 179-196. D.W. Widder, The Laplace transform., Princeton UP: New Jersey, 1946. T. Kakita, Introduction to the theory of Schwarz distribution., Nihon-hyoron-sha: Tokyo, 1985. P.M. Woodward, Probability and information theory, with applications to radar., Pergamon Press: London, 1953. A. Papoulis, The Fourier integral and its applications., McGraw-Hill, 1962. H. J. Weaver, Applications of discrete and continuous Fourier transforms., Wiley: New York, 1983. C.R. Giardina, "Band-limited signal extrapolation by truncated Bernstein polynomials", J. Math. Anal. Appl., vol. 104, no. 1, pp. 264-273, 1984. http://dx.doi.org/10.1016/0022-247X(84)90047-7 D.W. Widder, "A generalization of Dirichlet’s series and of Laplace’s integrals by means of a Stieltjes integral", Trans. Am. Math. Soc., vol. 31, pp. 694-743, 1929. W. Rudin, Real and complex analysis., 3rd edMacGraw-Hill: New York, 1987. H-Y. Li, W-B. Li, and S. Kanemitsu, From complex analysis to meta-science: A stroll around the boundary and beyond behavior, similarity and duality., World Sci, 2023. http://dx.doi.org/10.1142/13302

Advances in Special Functions of Fractional Calculus, 2023, 121-135

121

CHAPTER 8

A New Class of Truncated Exponential-GouldHopper-based Genocchi Polynomials Ghazala Yasmin1 and Hibah Islahi2,* 1

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India 2

Institute of Applied Sciences, Mangalayatan University, Aligarh-202145, India

Abstract: The present paper introduces a hybrid family of truncated exponentialGould-Hopper-based Genocchi polynomials by means of generating function and series definition. Some significant properties of these polynomials are established. In addition, graphs of truncated exponential-Gould-Hopper-based Genocchi polynomials are drawn using Matlab. Thereafter, the distribution of zeros of these polynomials is shown.

Keywords: Truncated exponential-Gould-Hopper polynomials, Genocchi polynomials, Monomiality principle, Operational techniques. 1. INTRODUCTION AND PRELIMINARIES In its various forms, multivariable and generalized forms of the special functions have been an object of speculation and application in recent years. Most of the special functions and their generalizations are suggested by physical problems. We recall the 3-variable truncated exponential-based Gould-Hopper polynomials (𝑠) (3VTEGHP), denoted by 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧), defined by means of the following generating function [1]:

exp xt zt s 1 yt r

� ,er H n s x y, z n 0

tn n!

*Corresponding

(1.1)

author Hibah Islahi: Institute of Applied Sciences, Mangalayatan University, Aligarh202145, India; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

122

Advances in Special Functions of Fractional Calculus

Yasmin and Islahi

and posses the following equivalent forms of series representation in terms of 2 (𝑟) variable truncated exponential polynomials (2VTEP) [2], denoted by 𝑒𝑛 (𝑥, 𝑦); (𝑠) Gould-Hopper polynomials (GHP) [3], denoted by 𝐻𝑛 (𝑥, 𝑧); and in terms of 𝑥, 𝑦, 𝑧: (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

(𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

𝑛

[ ] 𝑠

= 𝑛! ∑𝑘=0

(𝑟)

𝑧 𝑘 𝑒𝑛−𝑠𝑘 (𝑥,𝑦)

𝑛

[𝑟]

= 𝑛! ∑𝑚=0

𝑘!(𝑛−𝑠𝑘)!

,

(1.2)

(𝑠)

𝑦 𝑚 𝐻𝑛−𝑟𝑚 (𝑥,𝑧)

(1.3)

(𝑛−𝑟𝑚)!

and (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

= 𝑛! ∑𝑠𝑘+𝑟𝑚≤𝑛 𝑘,𝑚=0

𝑥 𝑛−𝑠𝑘−𝑟𝑚 𝑦𝑚 𝑧 𝑘 𝑘!(𝑛−𝑠𝑘−𝑟𝑚)!

,

(1.4)

respectively. (𝑠)

It is shown in [1] that the 3VTEGHP 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) are quasimonomial [4, 5], and their multiplicative and derivative operators are given by: ̂ 𝑒 (𝑟) 𝐻 (𝑠) = 𝑥 + 𝑟𝑦𝜕𝑦 𝑦𝜕𝑥𝑟−1 + 𝑠𝑧𝜕𝑥𝑠−1 𝑀

(1.5)

𝑃̂𝑒 (𝑟) 𝐻 (𝑠) = 𝜕𝑥 ,

(1.6)

and

respectively. (𝑠)

Now since 𝑒 (𝑟) 𝐻0 (𝑥, 𝑦, 𝑧) = 1, so monomiality principle implies that the (𝑠) 3VTEGHP 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) can be constructed as: (𝑠) (𝑥, 𝑦, 𝑧) 𝑒 (𝑟) 𝐻𝑛

𝑛

𝑛

̂ (𝑟) (𝑠) {1} = (𝑥 + 𝑟𝑦𝜕𝑦 𝑦𝜕𝑥𝑟−1 + 𝑠𝑧𝜕𝑥𝑠−1 ) {1}, =𝑀 𝑒 𝐻

(1.7)

which yields the series definition (1.4). (𝑠)

In view of identity (1.7), the exponential generating function of the GHP 𝐻𝑛 (𝑥, 𝑦) can be given by:

Genocchi Polynomials

Advances in Special Functions of Fractional Calculus

123

𝑛

𝑡 (𝑠) ̂ 𝑒 (𝑟)𝐻 (𝑠) 𝑡){1} = ∑∞ exp(𝑀 𝑛=0 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) , 𝑛!

(1.8)

which gives generating function (1.1). The operational representation of 3VTEGHP (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

(𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

is:

= exp(𝑧𝜕𝑥𝑠 + 𝑦𝜕𝑦 𝑦𝜕𝑥𝑟 ) {𝑥 𝑛 }.

The operational representation which links the 3VTEGHP (𝑟) (𝑠) 2VTEP 𝑒𝑛 (𝑥, 𝑦) and GHP 𝐻𝑛 (𝑥, 𝑦) is: (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

(1.9)

(𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) with the

(𝑟)

= exp(𝑧𝜕𝑥𝑠 ) {𝑒𝑛 (𝑥, 𝑦)}

(1.10)

and (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

(𝑠)

= exp(𝑦𝜕𝑦 𝑦𝜕𝑥𝑟 ) {𝐻𝑛 (𝑥, 𝑧)},

(1.11)

respectively. (𝑠)

The integral representation for the 3VTEGHP 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) in terms of 2iterated Gould-Hopper polynomials (2IGHP) [6] is: (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

∞

(𝑠)

= ∫0 𝑒 −𝑢 𝐻 (𝑟) 𝐻𝑛 (𝑥, 𝑦𝑢, 𝑧)𝑑𝑢.

(1.12)

The research on Genocchi numbers and Genocchi polynomials can be traced back to Angelo Genocchi (1817-1889). During these very recent years, Genocchi numbers and Genocchi polynomials are extensively studied in many different contexts in mathematics and physics, such as, elementary number theory, analytic number theory, theory of modular forms, p-adic analytic number theory, different topology, and quantum physics. The generating function of Genocchi polynomials 𝐺𝑛 (𝑥) are given by 2𝑡

(

) exp(𝑥𝑡) = ∑∞ 𝑛=0 𝐺𝑛 (𝑥)

𝑒 𝑡 +1

The Genocchi numbers 𝐺𝑛 are given as

𝑡𝑛 𝑛!

, |𝑡| < 𝜋.

(1.13)

124

Advances in Special Functions of Fractional Calculus

Yasmin and Islahi

𝐺𝑛 : = 𝐺𝑛 (0).

(1.14)

Series representation of Genocchi polynomials is given by: 𝑛

𝐺𝑛 (𝑥) = ∑𝑛𝑘=0 𝐶𝑘 𝐺𝑘 𝑥 𝑛−𝑘 .

(1.15)

The Genocchi polynomials 𝐺𝑛 (𝑥) are quasi-monomial. Their multiplicative and derivative operators are given by ̂𝐺 = 𝑥 + 𝑀

exp(𝐷𝑥 )(1−𝐷𝑥 )+1

(1.16)

𝐷𝑥 (exp(𝐷𝑥 )+1)

and 𝑃̂𝐺 = 𝐷𝑥 ,

(1.17)

respectively. In the present paper, a hybrid class of 3-variable truncated exponential-GouldHopper-based Genocchi polynomials is introduced, and many significant properties of these polynomials are established. In addition, numbers related to these polynomials are explored. Further, the shapes of this hybrid class of polynomials are demonstrated graphically. 2. VARIABLE TRUNCATED EXPONENTIAL-GOULD-HOPPER BASED GENOCCHI POLYNOMIALS To introduce the 3-variable truncated exponential-Gould-Hopper-based Genocchi polynomials (3VTEGHGP) denoted by 𝑒 (𝑟)𝐻 (𝑠) 𝐺𝑛 (𝑥, 𝑦, 𝑧), the following theorem is proved: Theorem 2.1. The generating function for the 3VTEGHGP given as: 2t

(

)

et +1

exp(xt+zts ) 1−ytr

tn

= ∑∞ n=0 e(r) H(s) Gn (x, y, z) . n!

e(r) H(s) Gn (x, y, z)

is

(2.1)

̂ 𝑒 (𝑟)𝐻 (𝑠) of the 3VTEGHP Proof. In place of 𝑥, putting the multiplicative operator 𝑀 (𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧) in the generating function (1.13) of Genocchi polynomials 𝐺𝑛 (𝑥), we get

Genocchi Polynomials

(

2𝑡

Advances in Special Functions of Fractional Calculus 𝑡𝑛

̂ 𝑒 (𝑟) 𝐻 (𝑠) )𝑡) = ∑∞ ̂ ) exp((𝑀 𝑛=0 𝐺𝑛 (𝑀𝑒 (𝑟) 𝐻 (𝑠) ) .

𝑒 𝑡 +1

125

(2.2)

𝑛!

Now using equation (1.5) on the right-hand side (r.h.s.) and equation (1.8) on the left-hand side (l.h.s.) of equation (2.2), we find (

2𝑡

(𝑠)

) ∑∞ 𝑛=0 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

𝑒 𝑡 +1

𝑡𝑛 𝑛!

𝑡𝑛

𝑟−1 = ∑∞ + 𝑠𝑧𝜕𝑥𝑠−1 ) . (2.3) 𝑛=0 𝐺𝑛 (𝑥 + 𝑟𝑦𝜕𝑦 𝑦𝜕𝑥 𝑛!

Finally, in the l.h.s. using generating function (1.1) of 3VTEGHP and in the r.h.s denoting the resultant by 𝑒 (𝑟) 𝐻 (𝑠) 𝐺𝑛 (𝑥, 𝑦, 𝑧), that is

(𝑠) 𝑒 (𝑟) 𝐻𝑛 (𝑥, 𝑦, 𝑧)

̂ 𝑒 (𝑟) 𝐻 (𝑠) ) = 𝐺𝑛 (𝑥 + 𝑟𝑦𝜕𝑦 𝑦𝜕𝑥𝑟−1 + 𝑠𝑧𝜕𝑥𝑠−1 ) =𝑒 (𝑟) 𝐻 (𝑠) 𝐺𝑛 (𝑥, 𝑦, 𝑧), 𝐺𝑛 (𝑀

(2.4)

yields assertion (2.1). Remark 2.1. We remark that, Equation (2.4) gives the operational correspondence (s) between the 3VTEGHP e(r) Hn (x, y, z) and 3VTEGHGP e(r) H(s) Gn (x, y, z). The next result gives the series definition of the 3VTEGHGP Theorem 2.2. The series representation of 3VTEGHGP by: e(r) H(s) Gn (x, y, z)

n

e(r) H(s) Gn (x, y, z).

e(r) H(s) Gn (x, y, z)

(s)

= n! ∑nk=0 Ck Gk e(r) Hn−k (x, y, z).

is given

(2.5)

̂ e(r) H(s) in the series representation (1.15) of Genocchi Proof. Replacing x by M polynomials Gn (x), we have ̂ e(r) H(s) ) = ∑nk=0 n Ck Gk (M ̂ e(r) H(s) )n−k . Gn (M

(2.6)

Using relation (2.4) in the l.h.s. and equation (1.7) in the r.h.s. of equation (2.6), we get assertion (2.5). Similarly, we can find the following equivalent forms of the series representation (s) of 3VTEGHGP e(r) H(s) Gn (x, y, z) in terms of 2VTEP ern (x, y), GHP Hn (x, z), Genocchi polynomials Gn (x) and in terms of x, y, z, respectively : e(r) H(s)

Gn (x, y, z) = n! ∑k+sm≤n k,m=0

Gk zm ern−k−sm (x,y) k!m!(n−k−sm)!

,

(2.7)

126

Advances in Special Functions of Fractional Calculus

e(r) H(s)

Yasmin and Islahi

Gn (x, y, z) = n! ∑k+rm≤n k,m=0

e(r) H(s) Gn (x, y, z)

(s)

Gk ym Hn−k−rm (x,z) k!(n−k−rm)!

m p rm+sp≤n y z Gn−rm−sp (x) (n−rm−sp)!

= n! ∑m,p=0

,

(2.8)

(2.9)

and e(r) H(s) Gn (x, y, z)

k+rm+sp≤n Gk xn−k−rm−sp ym zp . k!p!(n−k−rm−sp)!

= n! ∑k,m,p=0

(2.10)

To deal 3VTEGHGP e(r) H(s) Gn (x, y, z) in the context of monomiality principle, we first determine multiplicative and derivative operators: Theorem 2.3. The multiplicative and derivative operators of 3VTEGHGP e(r) H(s) Gn (x, y, z) are: exp(∂x )(1−∂x )+1 ̂ e(r) H(s)G = x + ry ∂y y ∂r−1 M + sz ∂s−1 + x x

(2.11)

∂x (exp(∂x )+1)

and ̂e(r) H(s)G = ∂x , P

(2.12)

respectively. Proof. Using equation (2.1), consider the identity tn

tn

n!

n!

∞ ∂x (∑∞ n=0 e(r) H(s) Gn (x, y, z) ) = t (∑n=0 e(r) H(s) Gn (x, y, z) ).

(2.13)

Differentiating equation (2.2) partially with respect to t, we get ̂ e(r)H(s) + exp(∂x )(1−∂x )+1) ( (M ∂x (exp(∂x )+1)

̂ = ∑∞ n=0 Gn (Me(r) H(s) )

2t

̂ e(r) H(s) t) ) exp(M

et +1

tn−1 (n−1)!

.

(2.14)

Using equation (2.2) on l.h.s. and then using relation (2.4) on both sides of equation (2.14), we obtain

Genocchi Polynomials

Advances in Special Functions of Fractional Calculus t ̂ e(r) H(s) + exp(t)(1−t)+1) ∑∞ (M n=0 e(r) H(s) Gn (x, y, z) t(exp(t)+1)

127

n

n!

= ∑∞ n=0 e(r) H(s) Gn (x, y, z)

tn−1 (n−1)!

.

(2.15) (s)

Now, putting the value of the multiplicative operator of 3VTEGHP e(r) Hn (x, y, z) from (1.5) and using identity (2.13) in the l.h.s. gives (x + ry ∂y y ∂r−1 + sz ∂s−1 + x x

exp(∂x )(1−∂x )+1

tn

∂x (exp(∂x )+1)

(n)!

= ∑∞ n=0 e(r) H(s) Gn (x, y, z)

) ∑∞ n=0 e(r) H(s) Gn (x, y, z)

tn−1 (n−1)!

.

(2.16)

On comparing powers of t, we get (x + ry ∂y y ∂r−1 + sz ∂s−1 + x x

exp(∂x )(1−∂x )+1 ∂x (exp(∂x )+1)

)

e(r) H(s)

Gn (x, y, z)

=e(r) H(s) Gn+1 (x, y, z),

(2.17)

which on using the monomiality principle gives assertion (2.11). Further, equating coefficients of the same powers of t on both sides of the identity (2.13), we find ∂x { e(r) H(s) Gn (x, y, z)} = n e(r) H(s) Gn−1 (x, y, z),

(2.18)

which in view of the monomiality principle, yields assertion (2.12). Remark 2.2. We remark that equations (2.17) and (2.18) are the differential recurrence relations satisfied by the 3VTEGHGP e(r) H(s) Gn (x, y, z). Theorem 2.4 The 3VTEGHGP equation:

e(r) H(s) Gn (x, y, z)

(x ∂x + ry ∂y y ∂rx + sz ∂sx + ∂x (

satisfy the following differential

exp(∂x )(1−∂x )+1 ∂x (exp(∂x )+1)

) − n)

e(r) H(s)

Gn (x, y, z) = 0.

(2.19)

Proof. Using expressions (2.11) and (2.12) and in view of monomiality principle, we get assertion (2.19).

128

Advances in Special Functions of Fractional Calculus

Yasmin and Islahi

3. OPERATIONAL AND INTEGRAL REPRESENTATIONS Now, we derive some operational representations for 3VTEGHGP Gn (x, y, z).

e(r) H(s)

Theorem 3.1. The operational representation connecting 3VTEGHGP Gn (x, y, z) and Genocchi polynomials Gn (x) is given by:

e(r) H(s)

e(r) H(s) Gn (x, y, z)

= exp(z ∂sx + y ∂y y ∂rx ){Gn (x)}.

(3.1)

(s)

Proof. Using operational representation (1.9) of 3VTEGHP e(r) Hn (x, y, z) in the r.h.s. of the series definition (2.5) of 3VTEGHGP e(r) H(s) Gn (x, y, z), we get e(r) H(s) Gn (x, y, z)

n

= n! ∑nk=0 Ck Gk exp(z ∂sx + y ∂y y ∂rx ){x n−k },

(3.2)

which on using the series representation (1.15) of Genocchi polynomials Gn (x) on the r.h.s., gives assertion (3.1). Theorem 3.2. The following operational representation connecting 3VTEGH GP e(r) H(s) Gn (x, y, z) and 2-variable truncated exponential-Genocchi polynomials (2VTEGP) [7] denoted by e(r) Gn (x, y) holds true: e(r) H(s) Gn (x, y, z)

= exp(z ∂sx )e(r) {Gn (x, y)}.

(3.3) (s)

Proof. Using operational representation (1.10) of 3VTEGHP e(r) Hn (x, y, z) in the r.h.s. of the series definition (2.5) of 3VTEGHGP e(r) H(s) Gn (x, y, z), we get e(r) H(s) Gn (x, y, z)

n

(r)

= n! ∑nk=0 Ck Gk exp(z ∂sx ){en (x, y)}.

(3.4)

(r)

As 2VTEP en (x, y) is quasi-monomial, so by using monomiality principle and series representation (1.15) of Genocchi polynomials Gn (x) on the r.h.s., gives assertion (3.3). Theorem 3.3. The operational representation connecting 3VTEGHGP e(r) H(s) Gn (x, y, z) and Gould-Hopper-Genocchi polynomials (GHGP) [8], denoted by H(s) Gn (x, y) is given by: e(r) H(s) Gn (x, y, z)

= exp(y ∂y y ∂rx )

G (x, z). H(s) n

(3.5)

Genocchi Polynomials

Advances in Special Functions of Fractional Calculus

129

(s)

Proof. Using operational representation (1.11) of 3VTEGHP e(r) Hn (x, y, z) in the r.h.s. of the equation (2.5) of 3VTEGHGP e(r) H(s) Gn (x, y, z), we get e(r) H(s) Gn (x, y, z)

n

(s)

= n! ∑nk=0 Ck Gk exp(y ∂y y ∂rx ) Hn (x, z).

(3.6)

(s)

As GHP Hn (x, z) is quasi-monomial, so by using monomiality principle and series representation (1.15) of Genocchi polynomials Gn (x) on the r.h.s., gives assertion (3.5). Recall that the generating function definition of 2-iterated Gould-Hopper (s) polynomials (2IGHP) H(r) Hn (x, y, z) [6] is tn

(s)

exp(xt + yt r + zt s ) = ∑∞ n=0 H(r) Hn (x, y, z) . n!

(3.7)

(s)

It is shown in [6], that 2IGHP H(r) Hn (x, y, z) are quasimonomial with respect to the multiplicative and derivative operators given by ̂ H(r) H(s) = x + ry ∂x + sz ∂s−1 M x

(3.8)

̂H(r)H(s) = ∂x , P

(3.9)

and

respectively, From monomiality principle, the generating function of the 2IGHP can be given by

(s) H(r) Hn (x, y, z)

n

t (s) ̂ H(r)H(s) t){1} = ∑∞ exp(M n=0 H(r) Hn (x, y, z) . n!

(3.10)

̂ H(r) H(s) of the 2IGHP H(r) Hn(s) (x, y, z) in place of Putting multiplicative operator M x in the generating function (1.13) of Genocchi polynomials Gn (x), we get 2t

(

tn

̂ H(r) H(s) t) = ∑∞ ̂ ) exp(M n=0 Gn (MH(r) H(s) ) .

et +1

n!

(3.11)

Now using equation (3.10) in l.h.s. and denoting the resultant in the r.h.s. by H(r) H(s) Gn (x, y, z), we find

130

Advances in Special Functions of Fractional Calculus 2t

(

(s)

) ∑∞ n=0 H(r) Hn (x, y, z)

tn

et +1

n!

Yasmin and Islahi tn

= ∑∞ n=0 H(r) H(s) Gn (x, y, z) ,

(3.12)

n!

which on using equation (3.7) in the l.h.s. yields the generating function for the new family of polynomials called 2-iterated Gould-Hopper-Genocchi polynomials (2IGHGP), denoted by H(r) H(s) Gn (x, y, z): (

2t

tn

) exp(xt + yt r + zt s ) = ∑∞ n=0 H(r) H(s) Gn (x, y, z) .

et +1

(3.13)

n!

Now, we will establish an integral representation of the 3VTEGHGP e(r) H(s) Gn (x, y, z) in terms of 2IGHGP H(r) H(s) Gn (x, y, z): Theorem 3.4. The following integral representation for the 3VTEGHGP e(r) H(s) Gn (x, y, z) in terms of 2IGHGP holds true: e(r) H(s) Gn (x, y, z)

∞

= ∫0 e−u H(r)H(s) Gn (x, yu, z)du.

Proof. Recall that the integral representation of 3VTEGHP given by (s) e(r) Hn (x, y, z)

∞

(3.14)

(s) e(r) Hn (x, y, z)

(s)

= ∫0 e−u H(r) Hn (x, yu, z)du.

[8] is

(3.15)

(s)

Using generating function (1.1) of 3VTEGHP e(r) Hn (x, y, z) in the generating function (2.1) of 3VTEGHGP e(r) H(s) Gn (x, y, z) , we obtain ∑∞ n=0 e(r) H(s) Gn (x, y, z)

tn n!

2t

=(

et +1

tn n!

=(

∞

2t et +1

tn n!

∞

tn

(s)

n!

2t

= ∫0 (

(s) e(r) Hn (x, y, z)

−u ) ∑∞ n=0 (∫0 e H(r) Hn (x, yu, z)du) .

Again using generating function (3.7) of 2IGHP ∑∞ n=0 e(r) H(s) Gn (x, y, z)

(3.16)

n!

which on using integral representation (3.15) of 3VTEGHP ∑∞ n=0 e(r) H(s) Gn (x, y, z)

tn

(s)

) ∑∞ n=0 e(r) Hn (x, y, z) ,

(s) H(r) Hn (x, y, z),

(3.17)

we get

) exp(−u + xt + yut r + zt s )du.

et +1

Finally, using generating function (3.13) of 2IGHGP

gives

H(r) H(s) Gn (x, y, z)

(3.18)

gives

Genocchi Polynomials

Advances in Special Functions of Fractional Calculus

∑∞ n=0 e(r) H(s) Gn (x, y, z)

tn n!

∞

(s)

tn

−u = ∑∞ n=o (∫0 e H(r) Hn (x, y, z)du) ,

131

(3.19)

n!

which on comparing powers of t gives assertion (3.14). 4. GRAPHICAL REPRESENTATION AND ROOTS Over the years, interest in solving mathematical and physical problems with the aid of computers has been increasing. We can explore concepts much more easily than in the past by using computers. The ability to manipulate and create figures on the computer screen enables us to produce and visualize many problems, look for patterns and examine the properties of figures. In this section, we display the shapes and find numbers of some members of the new family of polynomials discussed in section 3, by making it one variable. Next, we investigate the zeros of these polynomials using Matlab. Taking r = y = 1, z = −1 s = 2 and x → 2x, in both sides of equation (2.9) and denoting the resultant 1-variable truncated exponential Hermite Genocchi polynomials in the l.h.s. by eH Gn (x), we find. eH

p m+2p≤n (−1) Gn−m−2p (2x) , (n−m−2p)!

Gn (x) = n! ∑m,p=0

(4.1)

which on putting values of Genocchi polynomials Gn (x) in equation (4.1), we obtain expressions for 1-variable truncated exponential Hermite Genocchi polynomials eH Gn (x) for different values of n. The expressions of the first five G (x) are mentioned in Table 1. eH n Table 1. Expressions of the first five

𝐞𝐇

𝐆𝐧 (𝐱)

𝐞𝐇

𝐆𝐧 (𝐱).

0

1

2

3

4

0

1

4x + 1

12x 2 + 6x − 3

32x 3 + 24x 2 − 24x + 1

In view of equation (1.14), taking appropriate values in equation (4.1) and denoting the 1-variable truncated exponential Hermite Genocchi numbers on the left-hand side by eH Gn we get their values. The expressions of the first five eH Gn are given in Table 2.

132

Advances in Special Functions of Fractional Calculus

Table 2. Expressions of first five n 𝐞𝐇

𝐆𝐧

𝐞𝐇

Yasmin and Islahi

𝐆𝐧 .

0

1

2

3

4

0

1

1

−3

1

With the help of Matlab and by using the expressions for n = 1,2,3,4, the graph is drawn in Fig. (1) .

Fig. (1). Shapes of

eH

eH

Gn (x) from Table 1 for

Gn for n=1,2,3 and 4.

The zeros of eH Gn (x) can be computed using Matlab and are presented in Table 3 for n = 1,2,3 and 4. Table 3. Zeros of

𝐞𝐇

𝐆𝐧 (𝐱) for 𝐧 = 𝟏, 𝟐, 𝟑 and 𝟒.

Degree n

1

2

𝐆𝐧 (𝐱)

–

−0.2500

𝐞𝐇

3

4

−0.8090; 0.3090 −1.3311; 0.0436; 0.5374

To get an idea of the behaviour of zeros of the polynomials zeros for n = 1,2,3,4 in Fig. (2).

eH

Gn (x) we plot their

Genocchi Polynomials

Fig. (2). Zeros of .

eH

Advances in Special Functions of Fractional Calculus

133

Gn (x) for n=2, 3 and 4.

In order to make the above discussion more clear, the combined graphs of shape and zeros of eH Gn (x) are drawn for n = 1,2,3,4 in Fig. (3). It is to be noted that these figures give scientists an unconstrained ability to generate a visual mathematical analysis of the behaviour of eH Gn (x). In this article, the approach presented is general and opens new prospects to deal with other hybrid classes of special polynomials. The result established in this paper may find applications in solving the existing as well as new emerging problems of certain branches of mathematics, physics and engineering.

134

Advances in Special Functions of Fractional Calculus

Fig. (3). Shape and zeros of

eH

Yasmin and Islahi

G4 (x).

CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1]

[2]

G. Yasmin, and H. Islahi, "On amalgamation of truncated exponential and Gould-Hopper polynomials", Tbi. Math. J., vol. 14, no. 1, pp. 55-70, 2021. http://dx.doi.org/10.32513/tmj/1932200815 G. Dattoli, M. Migliorati, and H.M. Srivastava, "A class of Bessel summation formulas and associated operational methods", Fract. Calc. Appl. Anal., vol. 7, no. 2, pp. 169-176, 2004.

Genocchi Polynomials

[3]

[4]

[5]

[6] [7]

[8]

Advances in Special Functions of Fractional Calculus

135

H.W. Gould, and A.T. Hopper, "Operational formulas connected with two generalizations of Hermite polynomials", Duke Math. J., vol. 29, no. 1, pp. 51-63, 1962. http://dx.doi.org/10.1215/S0012-7094-62-02907-1 G. Dattoli, "Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle", Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics, 2000pp. 147-164. J.F. Steffensen, "The poweroid, an extension of the mathematical notion of power", Acta Math., vol. 73, no. 0, pp. 333-366, 1941. http://dx.doi.org/10.1007/BF02392231 G. Yasmin, and H. Islahi, "Finding mixed families of special polynomials associated with Gould-Hopper matrix polynomials", J. Ine. Spec. Fun., vol. 11, no. 1, pp. 43-63, 2020. S. Khan, G. Yasmin, and N. Ahmad, "A note on truncated exponential-based appell polynomials", Bull. Malays. Math. Sci. Soc., vol. 40, no. 1, pp. 373-388, 2017. http://dx.doi.org/10.1007/s40840-016-0343-1 S. Khan, and N. Raza, "General-Appell polynomials within the context of monomiality principle", Int. J. Anal., vol. 2013, pp. 1-11, 2013. http://dx.doi.org/10.1155/2013/328032

136

Advances in Special Functions of Fractional Calculus, 2023, 136-156

CHAPTER 9

Computational Preconditioned Gauss-Seidel via Half-Sweep Approximation to Caputo's TimeFractional Differential Equations Andang Sunarto1,*, Jumat Sulaiman2, and Jackel Vui Lung Chew3 1

Tadris Matematika, Universitas Islam Negeri (UIN) Fatmawati Sukarno, Bengkulu, 38211, Indonesia 2

Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu, 88400, Malaysia 3

Faculty of Computing and Informatics, Universiti Malaysia Sabah Labuan International Campus, Labuan, 87000, Malaysia Abstract: In this paper, we derived a finite difference approximation equation from the discretization of the one-dimensional linear time-fractional diffusion equations with Caputo's time-fractional derivative. A linear system is generated by implementing Caputo's finite difference approximation equation on the specified solution domain. Then, the linear system is solved using the proposed half-sweep preconditioned Gauss-Seidel iterative method. The effectiveness of the method is studied, and the efficiency is analyzed compared to the existing preconditioned Gauss-Seidel, also known as the fullsweep preconditioned Gauss-Seidel and the classic Gauss-Seidel iterative method. A few examples of the mathematical problem are delivered to compare the performance of the proposed and existing methods. The finding of this paper showed that the proposed method is more efficient and effective than the full-sweep preconditioned Gauss-Seidel and Gauss-Seidel methods.

Keywords: Caputo's fractional derivative, Implicit scheme, Half-sweep, Preconditioned, Gauss-Seidel, Iterative method.

*Corresponding

author Andang Sunarto: Tadris Matematika, Universitas Islam Negeri (UIN) Fatmawati Sukarno, Bengkulu, 38211, Indonesia; Tel: +62 0736 51276; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

137

1. INTRODUCTION Fractional partial differential equations (FPDEs) have been actively studied by many researchers nowadays. Several studies illustrated the importance of FPDEs to model complex phenomena [1-4]. The capability of FPDEs to capture anomalous phenomena, which some of the past PDEs have failed, attracted many researchers to apply in natural science fields other than pure mathematics [5-8]. By considering pure mathematics alone, various methods have been proposed to obtain accurate and efficient numerical solutions for the FPDEs [9-12]. The present paper focuses on developing an efficient numerical method to solve one-dimensional linear time-fractional diffusion equations via Caputo's timefractional derivative. A numerical method is proposed to improve the effectiveness of the preconditioned Gauss-Seidel (GS) iteration studied by [13]. The contribution of the paper is to present the effectiveness of the half-sweep computation approach [14] in deriving a better version of the preconditioned Gauss-Seidel iterative method, which can be named the half-sweep preconditioned Gauss-Seidel (HSPGS) method. The half-sweep computation approach is a computational complexity reduction approach that has been extended from the finite difference method. This approach is capable of reducing the complexity of solving a large system of equations generated from linear and nonlinear PDEs [15-18]. The reported results from [15-18] motivated this study to investigate the efficacy of half-sweep computation in solving an FPDE. The theory and concept of iterative methods have contributed to numerical analysis and computation since the early 20th century. Several researchers introduced and explained different families of iterative methods [19-21]. Among the families of efficient iterative methods, the preconditioned iterative methods are widely accepted as one of the efficient methods for solving equations [22-25]. Thus, the paper presents the formulation of a preconditioned iterative method using the halfsweep computation approach and Gauss-Seidel iteration. The effectiveness of the proposed HSPGS method is investigated and compared against the existing methods, such as the full-sweep preconditioned Gauss-Seidel [13] and the classic Gauss-Seidel methods.

138

Advances in Special Functions of Fractional Calculus

Sunarto et al.

2. IMPLICIT APPROXIMATION WITH CAPUTO'S TIME-FRACTIONAL Let us consider a time-FPDE to be defined as 𝜕 2 𝑈(𝑥, 𝑡) 𝜕𝑈(𝑥, 𝑡) 𝜕 𝛼 𝑈(𝑥, 𝑡) = 𝑎(𝑥) + 𝑏(𝑥) + 𝑐(𝑥)𝑈(𝑥, 𝑡), 2 𝛼 𝜕𝑥 𝜕𝑥 𝜕𝑡

(𝟏)

where 𝑎(𝑥), 𝑏(𝑥) and 𝑐(𝑥) are known functions or constants, whereas 𝛼 is a parameter that refers to the fractional-order of time derivative. A discrete approximation equation to the mathematical problem shown by equation (1) can be formulated using finite differences and Caputo's time-fractional derivative. Some basic definitions of the fractional derivative theory are used to approximate the fractional derivative in equation (1), which can be stated as [26]: Definition 1. The Riemann-Liouville fractional integral operator, 𝐽𝛼 of order-𝛼 is defined as 𝑥 1 𝐽 𝑓(𝑥) = ∫ (𝑥 − 𝑡)𝛼 𝑓(𝑡) 𝑑𝑡, 𝛼 > 0, 𝑥 > 0. 𝛤(𝛼) 0 𝛼

(𝟐)

Definition 2. Caputo's fractional partial derivative operator, 𝐷𝛼 of order -𝛼 is defined as 𝐷𝛼 𝑓(𝑥) =

𝑥 1 𝑓 (𝑚) (𝑡) ∫ 𝑑𝑡, 𝛼 > 0, 𝛤(𝑚 − 𝛼) 0 (𝑥 − 𝑡)𝛼−𝑚+1

(𝟑)

with 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ 𝑁, 𝑥 > 0. This paper aims to obtain the numerical solution of equation (1) using the implicit approximation based on Caputo's definition. The solution domain is subjected to Dirichlet boundary conditions, and we consider a nonlocal fractional derivative operator. The paper considers the following general notation of boundary and initial conditions: 𝑈(0, 𝑡) = 𝑔0 (𝑡), 𝑈(ℓ, 𝑡) = 𝑔1 (𝑡), 𝑈(𝑥, 0) = 𝑓(𝑥),

(𝟒)

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

139

where 𝑔0 (𝑡), 𝑔1 (𝑡), and 𝑓(𝑥), are given functions based on the exact or analytical solutions. Using Caputo's definition, the time-fractional derivative in equation (1) can be approximated by 𝑛

𝐷𝑡𝛼 𝑈𝑖,𝑛

(𝛼)

≅ 𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝑈𝑖,𝑛−𝑗+1 − 𝑈𝑖,𝑛−𝑗 ) ,

(𝟓)

𝑗=1

which is formulated after the following simplifications: 𝜎𝛼,𝑘 =

1 , 𝛤(1 − 𝛼)(1 − 𝛼)𝑘 𝛼

(𝟔)

and (𝛼)

𝜔𝑗

= 𝑗1−𝛼 − (𝑗 − 1)1−𝛼 .

(𝟕)

For the full discretization of equation (1), the paper uses the uniform partitioning of the solution domain. Defining some positive integers m and n to represent the grid points in space and time directions for the finite differences. The distance between any two grid points is defined as ℎ = 𝛥𝑥 = 𝛾⁄𝑚 and 𝑘 = 𝛥𝑡 = 𝑇/𝑛, respectively. The resultant network of points is individually denoted as 𝑈𝑖,𝑗 = 𝑈(𝑥𝑖 , 𝑡𝑗 ) where the grid points in the space interval [0, 𝛾] are labelled by 𝑥𝑖 = 𝑖ℎ, 𝑖 = 0,1,2, . . . , 𝑚 and the grid points in the time interval [0, 𝑇] are labelled 𝑡𝑗 = 𝑗𝑘, 𝑗 = 0,1,2, . . . , 𝑛. By using equation (5) and the implicit half-sweep finite difference scheme, Caputo's time-fractional implicit approximation to equation (1) is formulated into 𝑛 (𝛼)

𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝑈𝑖,𝑛−𝑗+1 − 𝑈𝑖,𝑛−𝑗 ) = 𝑎𝑖 𝑗=1

+𝑏𝑖 where 𝑖 = 1,2. . . , 𝑚 − 1.

1 (𝑈 − 2𝑈𝑖,𝑛 + 𝑈𝑖+2,𝑛 ) 4ℎ2 𝑖−2,𝑛

1 (𝑈 − 𝑈𝑖−2,𝑛 ) + 𝑐𝑖 𝑈𝑖,𝑛 , 4ℎ 𝑖+2,𝑛

(𝟖)

140

Advances in Special Functions of Fractional Calculus

Sunarto et al.

The implicit approximation equation shown by equation (8) is similar to the fully implicit finite difference approximation equation, which is consistent with firstorder accuracy in time and second-order in space. Equation (6) can be rewritten according to the specified time level. For 𝑛 ≥ 2, we have 𝑛 (𝛼)

𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝑈𝑖,𝑛−𝑗+1 − 𝑈𝑖,𝑛−𝑗 ) = ( 𝑗=1

+ (𝑐𝑖 −

𝑎𝑖 𝑏𝑖 − ) 𝑈𝑖−2,𝑛 2 4ℎ 4ℎ

𝑎𝑖 𝑎𝑖 𝑏𝑖 ) 𝑈𝑖,𝑛 + ( 2 + ) 𝑈𝑖+2,𝑛 , 2 2ℎ 4ℎ 4ℎ

(𝟗)

and further manipulations give the simplest form as follows: 𝑛 (𝛼)

𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝑈𝑖,𝑛−𝑗+1 − 𝑈𝑖,𝑛−𝑗 ) = 𝑝𝑖 𝑈𝑖−2,𝑛 + 𝑞𝑖 𝑈𝑖,𝑛 + 𝑟𝑖 𝑈𝑖+2,𝑛 ,

(𝟏𝟎)

𝑗=1

where the coefficients of the three grid points 𝑈𝑖−2,𝑛 , 𝑈𝑖,𝑛 and 𝑈𝑖+2,𝑛 are calculated using the equations: 𝑝𝑖 =

𝑎𝑖 𝑏𝑖 𝑎𝑖 𝑎𝑖 𝑏𝑖 − , 𝑞 = 𝑐 − , 𝑟 = + . 𝑖 𝑖 𝑖 4ℎ2 4ℎ 2ℎ2 4ℎ2 4ℎ

(𝟏𝟏)

Also, for 𝑛 = 1, a linear equation corresponding to Caputo's time-fractional implicit approximation to equation (1) can be expressed as −𝑝𝑖 𝑈𝑖−2,1 + 𝑞𝑖∗ 𝑈𝑖,1 − 𝑟𝑖 𝑈𝑖+2,1 = 𝑓𝑖,1 , 𝑖 = 1,2, … , 𝑚 − 1, (𝛼)

where 𝜔𝑗

(𝟏𝟐)

= 1, 𝑞𝑖∗ = 𝜎𝛼,𝑘 − 𝑞𝑖 , and 𝑓𝑖,1 = 𝜎𝛼,𝑘 𝑈𝑖,1 .

A tridiagonal linear system can be constructed for equation (12). By considering all grid points, a matrix form of a linear system can be expressed as 𝐴𝑈 = 𝑓 , ~

where

~

(𝟏𝟑)

Computational Preconditioned

𝑞1∗ −𝑝2 𝐴=

−𝑟1 𝑞2∗ −𝑝3

Advances in Special Functions of Fractional Calculus

−𝑟2 𝑞3∗ ⋱

−𝑟3 ⋱ −𝑝𝑚−2

[ 𝑈 = [𝑈11 ~

𝑈21

𝑈31

,

⋱ ∗ 𝑞𝑚−2 −𝑝𝑚−1

⋯

141

(𝟏𝟒)

−𝑟𝑚−2 ∗ 𝑞𝑚−1 ](𝑚−1)×(𝑚−1)

𝑈𝑚−2,1

𝑈𝑚−1,1 ]𝑇 ,

(𝟏𝟓)

and 𝑓 = [𝑈11 + 𝑝1 𝑈01

𝑈21

𝑈31

⋯ 𝑈𝑚−2,1

𝑈𝑚−1,1 + 𝑝𝑚−1 𝑈𝑚,1 ]𝑇 .

(𝟏𝟔)

~

3. STABILITY ANALYSIS Before the implementation of Caputo's time-fractional implicit approximation to a time-FPDE, we studied the stability properties of the implicit approximation equation shown in equation (8). The stability analysis is done using the method of Von-Neumann and Lax equivalence theorem. The implicit approximation in equation (8) is stable if it follows that the numerical solution obtained by the approximation equation converges to the exact solution as both space and time step sizes approach zero. Based on our theoretical work, the following theorem is established. Theorem 1 Caputo's time-fractional implicit approximation (equation (8)) with 0 < 𝛼 < 1 on the finite domain 0 ≤ 𝑥 ≤ 1 for all 𝑡 ≥ 0 is consistent and unconditionally stable. Proof Suppose that the solution of equation (1) has the form of 𝑈𝑗𝑛 = 𝜉𝑛 𝑒 𝑖𝜔𝑗ℎ , 𝑖 = √−1, 𝜔 ∈ ℝ.

(𝟏𝟕)

Substituting equation (17) to equation (8) and doing some manipulations yields

142

Advances in Special Functions of Fractional Calculus

Sunarto et al.

𝑛

𝜎𝛼,𝑘 𝜉𝑛−1 𝑒

𝑖𝜔𝑗ℎ

(𝛼)

− 𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝜉𝑛−𝑗+1 𝑒 𝑖𝜔𝑗ℎ − 𝜉𝑛−𝑗 𝑒 𝑖𝜔𝑗ℎ ) 𝑗=2

= −𝑝𝑖 𝜉𝑛 𝑒 𝑖𝜔(𝑗−2)ℎ + (𝜎𝛼,𝑘 − 𝑞𝑖 )𝜉𝑛 𝑒 𝑖𝜔𝑗ℎ − 𝑟𝑖 𝜉𝑛 𝑒 𝑖𝜔(𝑗+2)ℎ .

(𝟏𝟖)

Simplifying and reordering over equation (18) produces 𝑛 (𝛼)

𝜎𝛼,𝑘 𝜉𝑛−1 − 𝜎𝛼,𝑘 ∑ 𝜔𝑗 (𝜉𝑛−𝑗+1 − 𝜉𝑛−𝑗 ) 𝑗=2

= 𝜉𝑛 (((−𝑝𝑖 − 𝑟𝑖 ) 𝑐𝑜𝑠(𝜔ℎ)) + (𝜎𝛼,𝑘 − 𝑞𝑖 )) ,

(𝟏𝟗)

and the equation can be reduced to the following amplification factor, (𝛼)

𝜉𝑛−1 + ∑𝑛𝑗=2 𝜔𝑗 (𝜉𝑛−𝑗 − 𝜉𝑛−𝑗+1 ) 𝜉𝑛 = . (𝑝𝑖 + 𝑟𝑖 ) 𝑞𝑖 𝑐𝑜𝑠(𝜔ℎ) + (1 + 𝜎 𝜎𝛼,𝑘 ) 𝛼,𝑘

(𝟐𝟎)

Based on equation (20), it can be observed that (1 +

(−𝑝𝑖 − 𝑟𝑖 ) 𝑞𝑖 𝑐𝑜𝑠(𝜔ℎ) − ) ≥ 1, 𝜎𝛼,𝑘 𝜎𝛼,𝑘

(𝟐𝟏)

and for any values of 𝛼, 𝑛, 𝜔, ℎ and 𝑘, we have 𝜉1 ≤ 𝜉0 ,

(𝟐𝟐)

(𝛼)

(𝟐𝟑)

and also, we have 𝑛

𝜉𝑛 ≤ 𝜉𝑛−1 + ∑ 𝜔𝑗 (𝜉𝑛−𝑗 − 𝜉𝑛−𝑗+1 ), 𝑛 ≥ 2. 𝑗=2

Thus, letting 𝑛 = 2, the inequality (equation (23)) implies (𝛼) 𝜉2 ≤ 𝜉1 + 𝜔2 (𝜉0 − 𝜉1 ).

(𝟐𝟒)

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

143

Repeating the above process, which is similar to equation (24), gives 𝜉𝑗 ≤ 𝜉𝑗−1 , 𝑗 = 1,2, … 𝑛 − 1.

(𝟐𝟓)

Finally, we have 𝑛 (𝛼)

𝜉𝑛 ≤ 𝜉𝑛−1 + ∑ 𝜔𝑗 (𝜉𝑛−𝑗 − 𝜉𝑛−𝑗+1 ) ≤ 𝜉𝑛−𝑗 .

(𝟐𝟔)

𝑗=2

Since each term in the summation, as in equation (26), is negative, then the resultant inequality becomes 𝜉𝑛 ≤ 𝜉𝑛−1 ≤ 𝜉𝑛−2 ≤ ⋯ ≤ 𝜉1 ≤ 𝜉0 .

(𝟐𝟕)

𝜉𝑛 = |𝑈𝑗𝑛 | ≤ 𝜉0 = |𝑈𝑗0 | = |𝑓𝑗 |,

(𝟐𝟖)

Thus,

which entails ‖𝑈𝑗𝑛 ‖ ≤ ‖𝑓𝑗 ‖𝑊e have proven the stability properties of Caputo's time-fractional implicit approximation shown by equation (8). 4. HALF-SWEEP PRECONDITIONED GAUSS-SEIDEL This section shows the formulation of the proposed HSPGS method to solve the linear system as in equation (13). The formulation begins with a transformation of the linear system (equation (13)) into a preconditioned linear system that has the form of: 𝐴∗ 𝑥 = 𝑓 ∗ , ~

(𝟐𝟗)

~

where 𝐴∗ = 𝑃𝐴𝑃𝑇 , 𝑓 ∗ = 𝑃𝑓 , and 𝑈 = 𝑃𝑇 𝑥 . ~

~

~

~

We use matrix 𝑃, which is a preconditioned matrix, that is defined as: 𝑃 = 𝐼 + 𝑆, where

(𝟑𝟎)

144

Advances in Special Functions of Fractional Calculus

0 −𝑟1 0 0 0 0 𝑆= 0 0 0 0 [0 0

0 −𝑟2 0 ⋱ 0 0

0 0 −𝑟3 ⋱ 0 0

0 0 0 ⋱ 0 0

Sunarto et al.

0 0 0 , 0 −𝑟𝑚−1 0 ](𝑚−1)×(𝑚−1)

(𝟑𝟏)

and the matrix 𝐼 is an identity matrix. Next, let the coefficient matrix 𝐴∗ as in equation (29), be expressed in a summation of the three different type matrices as follows. 𝐴∗ = 𝐷 − 𝐿 − 𝑉.

(𝟑𝟐)

Each matrix, 𝐷, 𝐿 and 𝑉, represent the coefficients at the diagonal, lower triangular and upper triangular part of the matrix 𝐴∗ , respectively. Hence, we formulate the HSPGS iterative method using equation (32) and express the iterative formula as 𝑥 (𝑘+1) = (𝐷 − 𝐿)−1 𝑉𝑥 (𝑘) + (𝐷 − 𝐿)−1 𝑓 ∗ , ~

~

(𝟑𝟑)

~

where 𝑥 (𝑘+1) represents an unknown vector at (𝑘 + 1)-th iteration. ~

To facilitate the programming code of the HSPGS method, we designed the following algorithm. 4.1. Algorithm 1: HSPGS Iterative Method 1) Initialize 𝑈 = 0 and set the convergence criterion 𝜀 = 10−10 . ~

2) For 𝑗 = 1,2, … , 𝑛. For 𝑖 = 1,2, … , 𝑚 − 1, run iteratively 𝑥 (𝑘+1) = (𝐷 − 𝐿)−1 𝑉𝑥 (𝑘) + (𝐷 − 𝐿)−1 𝑓 ∗ , ~ ~ ~ 𝑈 (𝑘+1) = 𝑃𝑇 𝑥 (𝑘+1) . ~ ~ 3) Check convergence ‖𝑈 (𝑘+1) − 𝑈 (𝑘) ‖ ≤ 𝜀. ~

4) Display outputs.

~

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

145

5. NUMERICAL EXPERIMENT This study conducted a numerical experiment using two examples of the timeFPDE. The effectiveness and efficiency of the proposed HSPGS method are measured and compared side-by-side to both FSPGS and GS iterative methods. The effectiveness of the methods to obtain the approximate solutions is evaluated based on the maximum absolute errors (𝐸𝑚𝑎𝑥 ) while the efficiency of the methods is measured using the degree of reduction of iterations (𝑘) and programming execution time. The execution time is taken per unit seconds (𝑠). The effectiveness and efficiency study is based on the solution of the two examples under three different values of time-fractional derivative orders, 𝛼 = 0.25, 0.50 and 0.75. Besides that, the numerical experiment uses different mesh sizes, m = 128, 256, 512, 1024, and 2048. For the convergence of the numerical solutions, the experiment uses the stopping criterion at 𝜀 =10−10 for all examples with different values of 𝛼. Below are the following examples used in the numerical experiment. Example 1: Let us consider an initial-boundary value problem to be given as 𝜕 𝛼 𝑈(𝑥, 𝑡) 𝜕 2 𝑈(𝑥, 𝑡) = , 0 < 𝛼 ≤ 1,0 ≤ 𝑥 ≤ 𝛾, 𝑡 > 0, 𝜕𝑡 𝛼 𝜕𝑥 2

(𝟑𝟒)

and the desired approximate solutions are computed by using the following fractional boundary conditions 2𝑘𝑡 𝛼 2𝑘𝑡 𝛼 2 𝑈(0, 𝑡) = , 𝑈(ℓ, 𝑡) = ℓ + , 𝛤(𝛼 + 1) 𝛤(𝛼 + 1)

(𝟑𝟓)

and the initial condition is 𝑈(𝑥, 0) = 𝑥 2 .

(𝟑𝟔)

Noticed that by substituting 𝛼 = 1 into equation (34), one can get a onedimensional linear diffusion equation or expressed in the form of

146

Advances in Special Functions of Fractional Calculus

𝜕𝑈(𝑥, 𝑡) 𝜕 2 𝑈(𝑥, 𝑡) = , 0 ≤ 𝑥 ≤ 𝛾, 𝑡 > 0. 𝜕𝑡 𝜕𝑥 2

Sunarto et al.

(𝟑𝟕)

Consequently, the fractional boundary conditions, as in equation (35), become 𝑈(0, 𝑡) = 2𝑘𝑡, 𝑈(ℓ, 𝑡) = ℓ2 + 2𝑘𝑡.

(𝟑𝟖)

The analytical solution of equation (34) for 0 < 𝛼 ≤ 1 is given by 𝑈(𝑥, 𝑡) = 𝑥 2 + 2𝑘

𝑡𝛼 . 𝛤(𝛼 + 1)

(𝟑𝟗)

Example 2: Let us consider the following time-FPDE: 𝜕 𝛼 𝑈(𝑥, 𝑡) 1 2 𝜕 2 𝑈(𝑥, 𝑡) = 𝑥 , 0 < 𝛼 ≤ 1,0 ≤ 𝑥 ≤ 𝛾, 𝑡 > 0, 𝜕𝑡 𝛼 𝜕𝑥 2 2

(𝟒𝟎)

where the boundary conditions are given by 𝑈(0, 𝑡) = 0, 𝑈(1, 𝑡) = 𝑒 𝑡 ,

(𝟒𝟏)

𝑈(𝑥, 0) = 𝑥 2 .

(𝟒𝟐)

and the initial condition is

Based on equation (40), taking 𝛼 = 1 will reduce the equation to a diffusion equation with the form of 𝜕𝑈(𝑥, 𝑡) 1 2 𝜕 2 𝑈(𝑥, 𝑡) = 𝑥 , 0 ≤ 𝑥 ≤ 𝛾, 𝑡 > 0, 𝜕𝑡 2 𝜕𝑥 2

(𝟒𝟑)

and the analytical solution of equation (43) is 𝑈(𝑥, 𝑡) = 𝑥 2 𝑒 𝑡 . Meanwhile, the analytical solution of equation (40) is

(𝟒𝟒)

Computational Preconditioned

𝑈(𝑥, 𝑡) = 𝑥 2 [1 +

Advances in Special Functions of Fractional Calculus

𝑡 2𝛼 𝑡 3𝛼 𝑡𝛼 + + + ⋯]. 𝛤(𝛼 + 1) 𝛤(2𝛼 + 1) 𝛤(3𝛼 + 1)

147

(𝟒𝟓)

All results of numerical experiments, for examples 1 and 2, obtained from the implementation of GS, FSPGS and HSPGS iterative methods, have been recorded in Tables 1 and 2. In addition, Figs. (1 and 2) show the methods' comparison in the number of iterations and programming execution time for solving Example 1. Similarly, Figs. (3 and 4) show the numerical comparison results in the number of iterations and programming execution time for solving Example 2. Table 1. Numerical results for Example 1 at different values of 𝜶 and 𝒎. 𝜶 = 𝟎. 𝟐𝟓 𝒎

GS

12

FS

8

PG

𝒌

𝒔

21017

37.01

7292

35.86

1966

5.64

77231

332.11

26884

261.56

S HS PG S

GS 25

𝜶 = 𝟎. 𝟓𝟎 𝑬𝒎𝒂𝒙 9.97 e-5

9.96 e-5

9.96 e-5

1.00 e-4

𝒌

𝒔

13601

23.92

4715

2.23

1270

1.59

50095

213.28

17417

16.68

𝜶 = 𝟎. 𝟕𝟓 𝑬𝒎𝒂𝒙 9.85 e-5

9.84 e-5

9.84 e-5

9.90 e-5

𝒌

𝒔

6695

12.1

2319

1.93

625

1.03

24732

104.04

8585

12.37

𝑬𝒎𝒂𝒙 1.30 e-4

1.30 e-4

1.30 e-4

1.30 e-4

6 FS PG S

9.98 e-5

9.87 e-5

1.30 e-4

148

Advances in Special Functions of Fractional Calculus

Sunarto et al.

(Table 1) cont.....

HS PG

37.36

281598

2522.20

98422

1916.28

26884

272.45

101714

18485.4

1.09

0

3

e-4

14064.4

1.04

4

e-4

S

GS

9.96

7292

e-5

1.02 e-4

4715

10.28

183181

162.08

63298

123.01

17417

95.09

663971

2454.53

232784

1007.47

63928

893.24

9.87 e-5

1.01 e-5

2319

8.23

90783

831.58

31619

62.78

8585

46.08

330622

5870.9

115617

820.93

31619

636.78

1.30 e-4

1.32 e-4

51 2

FS PG S HS PG S

GS

1024

FS PG

357258

S HS PG

2048

FS PG S

e-4

9.98 e-5

1.04

9.96 e-5

9.87 e-5

1.08 e-4

1.03 e-5

9.95

1.31 e-4

1.31 e-4

1.40 e-4

1.35 e-4

1.35

98422

2025.13

363163

58914.3

1.38

238094

17795.2

1.38

119252

8794.2

1.71

8

0

e-4

6

5

e-4

8

6

e-4

1.36

115015

e-4

3

362784

1305.5

S

GS

1.00

118329 3

4104.17

e-4

3239.84

e-5

1.34 e-5

e-4

1.35 e-4

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

149

(Table 1) cont.....

HS PG

339197

3121.13

S

1.36 e-4

232784

2511.66

1.34 e-5

115627

807.13

1.35 e-4

Table 2. Numerical results, for example 2 at different values of 𝜶 and 𝒎. 𝜶 = 𝟎. 𝟐𝟓

𝜶 = 𝟎. 𝟓𝟎

𝜶 = 𝟎. 𝟕𝟓

𝒎 𝒌

𝒔

230579

12.46

𝑬𝒎𝒂𝒙

𝒌

𝒔

182947

10.99

1.95 GS FS 8.48

PGS

7.00

5.05

PGS

e-2 817596

110.24

FS 96.54

PGS

PGS

FS PGS

PGS

FS PGS

PGS

1.40 9884872

664.92 e-4

492,97

1.37 32602

420.11

e-2

e-1

8.30 389.57

e-2

e-1

8.29 69108

582.43

30.08

e-2

1.95 39068

1.37 2420

964.92

e-2

HS

e-1

8.29 11884877

791.55

184.75

e-2

1.95 142635

1.37 8911

57.19

e-2 1024

e-1

8.29 5162

1487.01

397.32

e-2

e-2 9767783

1.37 1482921

277.23

1.09 GS

e-1

8.29 18957

108.69

8.50

e-2

1.95 10624

1.37 655

797.32

e-2

HS

e-1

8.29 2282930

648.25

15.95

e-2

1.95 39608

1.37 2420

12.12

e-2 512

e-4

8.29 1398

1071.25

35.98

e-2

e-2 2853149

1.30 880921

35.69

1.95 GS

e-1

8.29 5162

19.09

2.59

e-2

1.95 2873

1.37 178

53.98

e-2

HS

e-1

8.29 100946

1.95 10624

1.37 4.44

e-2

e-2 256

e-1 655

5.11

390.80

1.37 8911

e-2

𝑬𝒎𝒂𝒙 1.37

8.28 378

1.95 GS

9.98

e-2

1.95 774

112911 8.28

1398 e-2

HS

𝒔

e-2

1.95 2873

𝒌

8.28

e-2 128

𝑬𝒎𝒂𝒙

189.50 e-1

150

Advances in Special Functions of Fractional Calculus

Sunarto et al.

(Table 2) cont.....

1.38 GS

32773526

3266.51

8.29 29754285

2106.87

e-2 FS 2543.23

PGS 1326.21

1781.32

e-1 1.37

116801

951,53 e-1

8.30 67817

e-2

3

e-2

1.95

128676

1.37

8.29 240051

e-2

HS PGS

e-2

1.95 487355

2048

1585,2 17752282

920,14

1.37 33318

e-2

511,32 e-1

Fig. (1). Methods' comparison in the number of iterations at mesh sizes 𝑀 = 128, 256, 512, 1024 and 2048 and time-fractional orders 𝛼 = 0.25, 0.50 and 0.75, for solving Example 1.

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

151

Fig. (2). Methods' comparison in the programming execution time at mesh sizes 𝑀 = 128, 256, 512, 1024 and 2048 and time-fractional orders 𝛼 = 0.25, 0.50 and 0.75, for example 1.

152

Advances in Special Functions of Fractional Calculus

Sunarto et al.

Fig. (3). Methods' comparison in the number of iterations at mesh sizes 𝑀 = 128, 256, 512, 1024 and 2048 and time-fractional orders 𝛼 = 0.25, 0.50 and 0.75 , for solving Example 2.

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

153

Fig. (4). Methods' comparison in the programming execution time at mesh sizes 𝑀 = 128, 256, 512, 1024 and 2048 and time-fractional orders 𝛼 = 0.25, 0.50 and 0.75, for solving Example 2.

154

Advances in Special Functions of Fractional Calculus

Sunarto et al.

CONCLUDING REMARKS The paper presented several numerical solutions to the time-fractional diffusion problems using the formulated Caputo's implicit finite difference approximation equation. The study observed that the implicit approximation equation produced a sparse and large tridiagonal linear system. A high computational complexity occurs when many points are computed at a time. The paper proposed HSPGS iterative method as an efficient solution to the problem. From the experimental study, the result was significant at 𝛼 = 0.25 in which the number of iterations declined drastically by 71.99-91.07% after the HSPGS iterative method was used in the numerical computation. Moreover, in terms of the programming execution time, the HSPGS method runs much faster by 69.78-95.82% compared to the existing FSPGS and GS methods. Besides that, the accuracy of the three iterative methods is comparable, and it can be concluded that their numerical solutions are in good agreement with the tested analytical solutions. CONSENT FOR PUBLICATON Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest. ACKNOWLEDGEMENT The authors are very grateful to the anonymous referee for valuable comments and suggestions. REFERENCES [1]

M. Farman, A. Akgül, T. Abdeljawad, P.A. Naik, N. Bukhari, and A. Ahmad, "Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel", Alex. Eng. J., vol. 61, no. 3, pp. 2062-2073, 2022. http://dx.doi.org/10.1016/j.aej.2021.07.040

[2]

D.L. Suthar, H. Habenom, and M. Aychluh, "Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia", Results Phys., vol. 32, 2022.105022. http://dx.doi.org/10.1016/j.rinp.2021.105022 PMID: 34868831

Computational Preconditioned

Advances in Special Functions of Fractional Calculus

155

[3]

X. Liu, S. Ullah, A. Alshehri, and M. Altanji, "Mathematical assessment of the dynamics of novel coronavirus infection with treatment: A fractional study", Chaos Solitons Fractals, vol. 153, 2021.111534. http://dx.doi.org/10.1016/j.chaos.2021.111534 PMID: 34751202

[4]

F. Muhammad Khan, A. Ali, Z. Ullah Khan, M.R. Alharthi, and A.H. Abdel-Aty, "Qualitative and Quantitative study of Zika virus epidemic model under Caputo’s fractional differential operator", Phys. Scr., vol. 96, no. 12, 2021.124030 http://dx.doi.org/10.1088/1402-4896/ac259e

[5]

L.I. Moroz, and A.G. Maslovskaya, "Numerical simulation of an anomalous diffusion process based on a scheme of a higher order of accuracy", Math. Models Comput. Simul., vol. 13, no. 3, pp. 492-501, 2021. http://dx.doi.org/10.1134/S207004822103011X

[6]

V. Bohaienko, and V. Bulavatsky, "Mathematical modeling of solutes migration under the conditions of groundwater filtration by the model with the k-caputo fractional derivative", Fractal and Fractional, vol. 2, no. 4, p. 28, 2018. http://dx.doi.org/10.3390/fractalfract2040028

[7]

J.E. Escalante-Martínez, J.F. Gómez-Aguilar, C. Calderón-Ramón, A. Aguilar-Meléndez, and P. Padilla-Longoria, "A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators", Int. J. Biomath., vol. 11, no. 1, 2018.1850014. http://dx.doi.org/10.1142/S1793524518500146

[8]

X.J. Yang, "Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat transfer problems", Therm. Sci., vol. 21, no. 3, pp. 1161-1171, 2017. http://dx.doi.org/10.2298/TSCI161216326Y

[9]

G. Sayyar, S.M. Hosseini, and F. Mostajeran, "A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives", Comput. Math. Appl., vol. 104, pp. 3443, 2021. http://dx.doi.org/10.1016/j.camwa.2021.11.002

[10]

A. Kamran, A. Ahmadian, S. Salahshour, and M. Salimi, "A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative", Phys. Scr., vol. 96, no. 12, 2021.124015. http://dx.doi.org/10.1088/1402-4896/ac1ccf

[11]

A. Torres-Hernandez, F. Brambila-Paz, and C. Torres-Martínez, "Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black–Scholes", Comput. Appl. Math., vol. 40, no. 7, p. 245, 2021. http://dx.doi.org/10.1007/s40314-021-01634-z

[12]

J. Shen, M. Stynes, and Z.Z. Sun, "Two finite difference schemes for multi-dimensional fractional wave equations with weakly singular solutions", Comput. Methods Appl. Math., vol. 21, no. 4, pp. 913-928, 2021.

156

Advances in Special Functions of Fractional Calculus

Sunarto et al.

http://dx.doi.org/10.1515/cmam-2020-0101 [13]

A. Sunarto, and J. Sulaiman, "Preconditioned Gauss-Seidel Method for the Solution of Timefractional Diffusion Equations", Proceedings of the 2nd International Conference on Applied Science, Engineering and Social Sciences, 2019. http://dx.doi.org/10.5220/0009882502680273

[14]

A.R. Abdullah, "The four point explicit decoupled group (EDG) Method: a fast poisson solver", Int. J. Comput. Math., vol. 38, no. 1-2, pp. 61-70, 1991. http://dx.doi.org/10.1080/00207169108803958 M.M. Xu, J. Sulaiman, and L.H. Ali, "Linear rational finite difference approximation for second-order linear fredholm integro-differential equations using the half-sweep SOR iterative method", International Journal of Engineering Trends and Technology, vol. 69, no. 6, pp. 136-143, 2021. http://dx.doi.org/10.14445/22315381/IJETT-V69I6P221

[15]

Advances in Special Functions of Fractional Calculus, 2023, 157-184

157

CHAPTER 10

Krasnoselskii-type Theorems for Monotone Operators in Ordered Banach Algebra with Applications in Fractional Differential Equations and Inclusion Nayyar Mehmood1,* and Niaz Ahmad1 1

Department of Mathematics & Statistics International Islamic University H-10 Islamabad, Pakistan Abstract: This chapter discusses Krasnoselskii-type fixed point results for monotone operators. It is well known that the monotone operators are not continuous on the whole domain, so we will find the solutions of discontinuous operator equations and inclusions. The presented fixed point results may be considered as variants of the Krasnoselskii fixed point theorem in a more general setting. The results of Darbo, Schauder and Bohnentblust-Karlin are also generalized. We prove these results for the case of singlevalued and set-valued monotone operators. We use our main result for single-valued operators to obtain the existence of solutions of anti-periodic ABC fractional BVP. The fixed point result for set-valued monotone operators is used to discuss the existence of solutions of a given fractional integral inclusion in ordered Banach spaces.

Keywords: Krasnoselskii's fixed point theorem, Set valued mappings, Convex, Compact, Closed sets, Banach spaces, Fractional differential equations, Atangana and Baleanu derivatives. 1. INTRODUCTION Fractional calculus is one of the most emerging fields of mathematics. Many physical models have been studied more accurately by virtue of fractional calculus. Some classical fractional derivatives are Riemann Liouville, Hadamard, Caputo and Grunwald-Letnikov; all of these have many applications [1]. Many of them have a singular Kernel, which creates some flaws when applied to some physical *Corresponding

author Nayyar Mehmood: Department of Mathematics & Statistics. International Islamic University, H-10, Islamabad, Pakistan; Tel: +92-519019948; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

158

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

problems. To overcome this problem, Caputo and Febrizio [2] introduced a new fractional derivative with a non-singular kernel. It also involves some ambiguities that were later removed by Atangana and Baleanu [3] by introducing a more general form of the fractional derivative with a non-singular kernel, using the MittageLeffler function. Many applications can be seen in the literature. Topological fixed point theory is one of the most emerging fields in nonlinear analysis. It was initiated about a century ago. Utilizing the topological methods in the theory of differential equations, Poincare initiated the idea of fixed point theory [4]. After that, the most celebrated fixed point theorem of Brouwer was presented in 1910 [5]. Schauder generalized this result for the case of infinite dimensional Banach spaces [6]. The following is the theorem of Schauder. Theorem 1 Let be a convex and compact subset of a Banach space H . Suppose is a continuous mapping of into H . Then possesses at least one fixed point. The famous Banach contraction principle was proved in 1922 [7], which gave the theory more strength and a new direction to the stability and existence theory of nonlinear operators. This principle is given as follows. Theorem 2 Let , d be a complete metric space and suppose is a self-mapping of such that d , hd , , for all , ,

For some, h 0,1 . Then possesses only one fixed point. While studying the theory of perturbed differential equations and the article of Schauder, Krasnoselskii [8] came to know that the inversion of a perturbed differential equation might have formed a sum of contractive and compact operators. This thought ended with a famous Krasnoselskii's fixed point theorem for the sum of two operators. The result of Krasnoselskii's fixed point theorem is given below. Theorem 3 Let be a nonempty convex and closed subset of a Banach space H. Suppose , are mappings of into H such that: i , , implies ,

ii

is a contraction mapping,

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

159

iii

is continuous and compact. Then possesses at least one fixed point. Remark 4 If be a zero operator and the condition of convexity is relaxed, then it is Banach contraction theorem. If is a zero operator with the condition of compactness, then it is the fixed point theorem of Schauder. Many generalizations and extensions of Krasnoselskii's fixed point theorems are presented in the literature [9-13]. Different directions have been considered to generalize this result, for example, generalizing the space, weakening compactness or continuity, or taking set-valued maps instead of single-valued operators, etc. A novel way of generalization is to use partial ordering on space with some other conditions. This kind of generalization was considered by Ran and Reuring [14], in 2004. Their main result is stated as follows. Theorem 5 Let the lower or upper bound of a partially ordered set exists. Suppose d is a complete metric on , and is a monotone and continuous self-mapping on ,

i

h 0,1 and for all such that d , hd , ;

ii

0 such that either 0 Q0 or 0 Q0 . Then possesses only one fixed point. This theorem was applied to matrix equations to develop solvability results for these equations. A modified variant of the above result by weakening the continuity of the mappings was given by Nieto and Rodriguez-Lopez [15] as follows. Theorem 6 Let the lower or upper bound of a partially ordered set exists. Suppose d is a complete metric on and is a monotone self-mapping on ,

i h 0,1 ii 0

and for all such that d , hd , ; such that either 0 Q0 or 0 Q0 ;

160

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

iii There is a monotone sequence

m satisfying m or m . Then Then possesses only one fixed point. We mention here that the above theorem was separately proved in [15], for increasing and decreasing operators, however, we have combined both cases just to avoid repetition, and used the term monotone. The above theorem was used to study interesting applications of ordinary differential equations with periodic boundary conditions for the existence of the solution. For further exploration in this direction, we need the following definitions. Suppose K is a convex and nonempty closed subset of a Banach space H . K is said to be cone, if K where K and for all 0. This cone K defines a partial order ≼ in the Banach space H . We say 𝜆 ≼ 𝜇 iff 𝜇 − 𝜆 ∈ 𝐾 while 𝜆 ≪ 𝜇 means 𝜇 − 𝜆 ∈ 𝑖𝑛𝑡(𝐾), where 𝑖𝑛𝑡 represents interior of K. This order relation enjoys the following properties as well: (i) 𝜆 ≼ 𝜇 implies 𝛼𝜆 ≼ 𝛼𝜇 for all 0 and (ii) 𝜆 + 𝑧 ≼ 𝜇 + 𝑧 for all z H , while 𝜆 ≼ 𝜇 and 𝜇 ≼ 𝜆 implies 𝜆 = 𝜇 It is important to note that if 𝐻 is an ordered Banach space with partial order ≼ then the set 𝐾 = {𝜆 ∈ 𝐻: 𝜆 ≽ 0} is a cone [16]. This chapter contains three more sections. In Section 2, Krasnoselskii-type results for two operators are discussed, in which the first result is about the fixed points of the sum of two monotone operators, while the second is about the fixed points of the product of two monotone operators. In Section 3, two more subsections are given, in which the first subsection is devoted to generalization of Darbo's fixed point theorem for the case of multivalued monotone operators, and the second subsection is about a generalization of Krasnoselskii's fixed point theorem for the case of two multivalued monotone operators. In the Section 4, applications of our results are presented. The first application concerns solutions for the ABC fractional anti-periodic boundary value problem (BVP). We use our main result to obtain the solution of fractional BVP. At last, we consider a general form of integral inclusion associated with the above-said BVP and find the solution using a well-known theorem. To understand the remaining results, some crucial definitions and results are given. Definition 7 [17] Let K be a cone in an ordered Banach space H . K is said to be reproducing if for all λ∈ 𝐻 there exist 𝑢, 𝑣 ∈ 𝐾 such that 𝜆 = 𝑢 − 𝑣. The following lemma describes the properties of reproducing cones.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

161

Lemma 8 [17] Let K be cone in an ordered Banach space 𝐻. Then the following conditions are equivalent:

i For all 𝜆 ∈ 𝐻, there exists 𝜇 ≼ 𝜃 (𝜇 ≽ 𝜃) such that 𝜆 ≼ 𝜇 or (𝜆 ≽ 𝜇);

ii

K is reproducing;

(iii) For any set { 𝜆, 𝜇} ⊆ 𝐻 has an upper or lower bound. Definition 9 [17] Let : K K be a monotone mapping with the following conditions;

i

K I K ;

ii iii

m m , where means composition of to itself m times;

m 1

0 when 0.

Denote the class 𝝍 of all functions satisfying the above Definition 9. Theorem 10 [17] Let H be a partially ordered Banach space and K be a reproducing normal cone. If Q is monotone self-mapping on H such that 𝑄𝜆 − 𝑄𝜇 ≼ 𝜓(𝜆 − 𝜇)

(A)

for all 𝜇 ≼ 𝜆 and some 𝜓 ∈ 𝝍 Then Q possesses only one fixed point z . Furthermore, for any 𝜆 ∈ 𝐻,

lim Q m z.

m

We call a self-monotone mapping Q satisfying A a -contraction. Theorem 11 Let K be a reproducing normal cone of a partially ordered Banach algebra H . If 𝑄 is monotone self mapping on 𝐻 such that 𝑄𝜆 − 𝑄𝜇 ≼ 𝑤𝜓(𝜆 − 𝜇)

(B)

162

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

for all 𝜇 ≼ 𝜆 and some w , 𝜓 ∈ 𝜓. Then Q possesses only one fixed point 𝑧. Furthermore, for any 𝜆 ∈ 𝐻,

lim Q m z.

m

We call a self-monotone mapping Q satisfying B a w -contraction. 2. KRASNOSELSKII-TYPE RESULTS FOR TWO MONOTONE SELF OPERATORS This section presents two fixed-point theorems. First, the existences of fixed points for the sum of two monotone operators in ordered Banach space are discussed. After that, the fixed point result in ordered Banach space H for the product of two monotone operators with reproducing cones K has been presented. Theorem 12. Let C be a convex and closed chain in an ordered Banach space H with reproducing normal cone 𝐾. Suppose P, Q are self monotone operators such that

i Q is a -contraction ; ii P is continuous and P C is precompact; iii Q P C for all , C ; 1 iv I Q is continuous. Then the operator P Q possesses a fixed point in C. Proof For all, C, observe that Q C for all P C , because C is closed.

For

each,

P C ,

define

an

operator

Q : C C

by

Q Q . As Q is monotone, therefore for 𝑤 ≼ 𝑧 implies 𝑄(𝑤) ≼ 𝑄(𝑧) which further implies. 𝑄(𝑤) + 𝜇 ≼ 𝑄(𝑧) + 𝜇. Therefore Q is also a monotone operator. Now we show that Q is a contraction. Let 𝑤 ≼ 𝑧 and Q is a contraction. Then 𝑄𝜇 (𝑤) − 𝑄𝜇 (𝑧) = 𝑄(𝑤) + 𝜇 − 𝑄(𝑧) − 𝜇

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

163

= 𝑄(𝑤) − 𝑄(𝑧) ≼ 𝜓(𝑤 − 𝑧), which implies Q is also a -contraction. We claim that there exists z0 C such that z0 Q z0 . For this fixed P C , either Q K or Q K. If

Q K , then there exists z0 K such that (using i in Definition 9),

I z0 Q or 𝑧0 = 𝑄𝜇 (𝜃) + 𝜓(𝑧0 ). As 𝑧0 ≽ 𝜃 and Q is -contraction, therefore 𝑄𝜇 (𝑧0 ) − 𝑄𝜇 (𝜃) ≼ 𝜓(𝑧0 ) which implies 𝑄𝜇 (𝑧0 ) ≼ 𝜓(𝑧0 ) + 𝑄𝜇 (𝜃) = 𝑧0 . Now, if Q K then using Lemma 8 there exists w K such that 𝑄𝜇 (𝜃) ≼ 𝑤.

Let z0 be the solution of I z w, then 𝑄𝜇 (𝑧0 ) − 𝑄𝜇 (𝜃) ≼ 𝜓(𝑧0 ), which implies 𝑄𝜇 (𝑧0 ) ≼ 𝜓(𝑧0 ) + 𝑄𝜇 (𝜃) ≼ 𝜓(𝑧0 ) + 𝑤 = 𝑧0 .

Therefore Q satisfies all conditions of Theorem 10 and hence possesses a unique fixed point 𝑧 i.e.,

z Q z Q z . or

z Qz .

164

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

The operator I Q : P C C exists, since for each P C , there exists 1

unique C such that Q . The composition of two continuous functions

I Q

1

P : C C is continuous. Also, the set

Therefore I Q

1

I Q

1

P C is compact.

P is a compact and continuous operator. By Tychonoff Theo-

rem [18], I Q P possesses a fixed point say w. This implies w Qw Pw. This proves the theorem. 1

Corollary 13 Let C be a convex and closed chain in an ordered Banach space H . Suppose P, Q are self-monotone operators such that

i Q is a -contraction; ii P is continuous and P C is precompact; iii Q P C for all , C ; 1 iv I Q is continuous. If for each P C , there exists z0 such that 𝑧0 ≼ 𝑄(𝑧0 ) + 𝜇. Then the operator P Q possesses a fixed point in C. Corollary 14 Let H be an ordered complete space with norm ‖⋅‖ and suppose P, Q are self-monotone operators such that

i

P is continuous and compact;

ii

There exists a positive continuous linear operator M with spectral radius

𝑟(𝑀) < 1 such that for 𝜆 ≤ 𝜇, Q Q M

iii Q P C for all , C 1 iv I Q is continuous. If for each P C , there exist 𝑧0 such that 𝑧0 ≼ 𝑄(𝑧0 ) + 𝜇. Then the operator

P Q possesses a fixed point in C.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

165

Corollary 15 [8] Let be a convex and closed nonempty subset of a Banach space 𝐻. Suppose 𝑃, 𝑄 are self mappings such that (i) 𝑃 is a contraction mapping; (ii) 𝑄 is compact and continuous; (iii) 𝜇, 𝜆 ∈ 𝛿 implies 𝑃(𝜆) + 𝑄(𝜇) ∈ 𝛿. Then 𝑃 + 𝑄 possesses a fixed point in 𝛿. Now we state and prove the second theorem related to Dhage's fixed point theorem, but in this case, in Banach Algebra for the product of two monotone operators. Theorem 16 Let C K be a convex, closed and bounded chain in ordered Banach algebra H with reproducing normal cone K . Suppose P, Q are self-monotone operators such that

i Q ii

is a -contraction ;

P is continuous and P C is precompact;

iii Q P C

iv

I Q

for all , C;

1

is continuous.

Then the operator PwQw w has a solution in C. Proof For C, observe that 𝑄(𝜆)𝐼(𝜇) ∈ 𝐶 for all P C , as C is closed. For each P C , defines an operator Q : C C by Q Q I . Since Q is monotone, therefore for 𝑤 ≼ 𝑧 implies 𝜃 ≼ 𝑄(𝑤) − 𝑄(𝑧) which further implies 𝜃 ≼ 𝑄(𝑤)𝜇 − 𝑄(𝑧)𝜇. Therefore Q is also a monotone operator. We show that Q is a contraction. For this consider

166

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

Q w Q z Q w Q z w z ,

Since C is bounded so , and therefore, from Theorem 11, Q possesses a unique fixed point z Q z Q z , or

I Q

1

1

I Q

I Q

1

P is continuous. Also

I Q

1

P C is compact, which implies

P is completely continuous and has a fixed point, say

P( z )

Also, the operator

( z) . As a composition of two continuous functions is continuous,

therefore, I Q

I Q

: P C C exists, since for each P C , there exists a unique z C

such that

z .

z Q( z)

I Q

1

P( z ) z, that is

or Q( z) P( z) z. This completes the proof.

3. FIXED POINT RESULTS FOR SET VALUED MONOTONE MAPPINGS This section presents some fixed point results for set-valued monotone operators. First, some definitions and results will be recalled. From onwards, we associate the subscripts cl , b, cp and cv for the class of closed, bounded, compact and convex subsets of the given normed/Banach space H , that is cv , cp H means the class of all convex and compact subsets of H . Definition 17 [19] Suppose H and H 1 are two Banach spaces. A multivalued mapping G from H into H1 is called upper semi-continuous (u.s.c) (resp, lower semi-continuous (l.s.c)) if r H : G(r) M (resp, r H : G(r) M ) is an open set in H for every open subset M of H1. In the theory of multivalued operators, the role of u.s.c and l.s.c operators are important and also used in many results to find existence of fixed points. To find fixed points of u.s.c multivalued operator, the famous result of Kakutani-Fan [20] is stated as follows. Theorem 18 Let H be Banach space and suppose is compact subset of H . Let G from into cv , cp H be an u.s.c multivalued operator. Then G possesses a fixed point.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

Definition 19 [19] A multivalued operator G from into

cp

167

H is said to be

compact, if G H is compact subset of H 1 . G is said to be totally bounded, if for any bounded subset N of H , the set G N is totally bounded subset of H 1 . Remark 20 [19] Every multivalued operator is totally bounded if it is compact and the converse may not hold. Definition 21 [19] If a multivalued operator G from H into totally bounded, then G is completely continuous.

cp

H1 is u.s.c and

Definition 22 [19] The graph of a multivalued operator Q from H into H1 is defined by Gr Q t , s H H1 : s Q(t ) ,

and if every sequence rn , sn in Gr Q , such that rn , sn r, s Gr Q , then graph is closed . If the graph is closed, then Q is also closed. Lemma 23 [19] A multivalued operator G from H into if it is compact and closed.

cl

H1 is u.s.c if and only

The generalization of the Kakutani theorem is given by Bohnentblust-Karlin, in which the domain of the operator is relaxed. The statement is presented as follows. Theorem 24 [21] Suppose H is a Banach space and is convex and closed subset of H . Let G from into cp , cv H be an u.s.c compact operator. Then 𝐺 possesses a fixed point. The next theorem is more general than the above theorem and the statement of the theorem is presented as follows. Theorem 25 [22] Suppose H is a Banach space and is convex, bounded and closed subset of H . Let G from into cp H be closed and compact multivalued operator. Then G possesses a fixed point.

168

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

Using the notion of condensing operators, the compactness of the operator can be weakened. In the existence theory, the role of condensing operators are very important and useful. A ball measure of noncompactness , was presented by Kauratowski [23], whereas Sadovskii [24] defined the Hausdorff measure of noncompactness . For given Banach space H , consider a bounded subset N , then

n

N inf 0 : N Si , D Si , i 0

where D represents diameter of the set and

n

N 0 : N N ri , for some ri H , i 0

where N ri , represents open balls with centered at ri and radius . Recently, in [25, 26], the authors defined a generalized measure of noncompactness, which is presented as follows. Definition 26 A function from b H into [0, ) is said to be a measure of noncompactness, if M1 , M 2 b H , the following conditions are satisfied;

A1 M1 0, implies M1 is precompact, A2 M 1 M 1 , where M 1 represents closure of M1 , A3 is nondecreasing, i.e., if M 1 M 2 then M1 M 2 ,

A4 Cons M1 M1 , where Cons M1 represents convex hull of M 1 , M n 0, which implies that A5 If M n is a sequence in b H such that lim n

M M n is nonempty. n 1

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

169

Moreover is said to be sub-linear if A6 kM1 k M1 for F , and

A7 M1 M 2 M1 M 2 for

M 1 , M 2 cl , b H .

Note that, the Hausdorff measure of noncompactness and the ball measure of noncompactness , both are special cases of . Definition 27 [23] A multivalued operator G from H into

cl , b

H is said to be

- k -set contraction, if for any N b H , a set G N b H satisfies

G N k N , for some k 0,1 . 3.1. A Generalization of Darbo's Fixed Point Theorem for Monotone Operators Definition 28. [27] Suppose H is an ordered Banach space with a partial order ≼ induced by the cone K . Let from H into H be a set-valued mapping such that i u 1 v means that for each r u s v such that 𝑟 ≼ 𝑠 , while u 2 v means that for each s v r u such that 𝑟 ≼ 𝑠. By u mean u

ii

1

v and u

2

v, we

v.

A set-valued mapping is said to be increasing if for all u, v H with

𝑢 ≼ 𝑣 implies that u

v.

iii

A set-valued mapping is said to be decreasing if for all u, v H with 𝑢 ≼ 𝑣 implies that v u.

iv

A set-valued mapping is a monotone if for 𝑤 ≼ 𝑧 implies that 𝑟 ≼ 𝑠 for each r w and s z . The following definitions and theorems will be useful to prove our main results.

170

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

Definition 29 [28] Suppose from H into H is set-valued map, where H be an ordered Banach space, then is said to be an L -contraction if there exists L from H into H a linear operator with spectral radius r L 1 and L K K such that for u, v H with 𝑢 ≼ 𝑣

i for each

w u there exists z v such that

0 ≼ 𝑧 − 𝑤 ≼ 𝐿(𝑣 − 𝑢);

ii

for each z v there exists w u such that

0 ≼ 𝑧 − 𝑤 ≼ 𝐿(𝑣 − 𝑢). Lemma 30 For every L -contraction from H into H is set-valued map , there exists z0 H such that 𝑧0 ≼ 𝛤𝑧0 𝑜𝑟 𝛤𝑧0 ≼ 𝑧0 Proof The proof can be seen in [28]. Theorem 31 Suppose H is a Banach space , then every L contraction on H possesses a fixed point. Definition 32 Suppose K is the reproducing normal cone of a Banach space H . A multivalued monotone mapping G on H is said to be monotone - k -set contraction if M , N cl , b H , G satisfies 𝜇(𝐺𝑀) ≤ 𝑘𝜇(𝑀) for all 𝑀 ≼ 𝑁, for some k [0,1).

Next, we present a generalized version of the Bohnentblust-Karlin theorem, which weakens the compactness condition on the operator. Theorem 33 [25] Suppose H a Banach space is a convex, bounded and closed subset of H . Let G from into cl , cv H be -condensing and u.s.c operator. Then G possesses a fixed point.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

171

The generalization of the well-known results of Darbo [23] and [29] are presented as follows. Theorem 34 Every monotone - k -set contraction which is bounded, closed and convex valued has a fixed point. Proof From Lemma 30, there exists a 𝑢0 ∈ 𝐾 such that 𝑢0 ≼ 𝐺𝑢0 , so that there exists u1 Gu0 such that u0 ? u1 , which further implies 𝐺𝑢0 ≼ 𝐺𝑢1 . Hence for

u1 Gu0 0 there exists u2 Gu1 such that 𝑢1 ≼ 𝑢2 and so on. Define n 1 Cons G un , n 0,1, 2,3,.... Then clearly, we get

.... n1 n ... 2 1 0 . Which implies that

n

is a nonincreasing sequence of bounded, closed and

convex subsets of a Banch space H . Now, we consider

n 1 Cons G un

G un k Gun 1

k Cons G un 1

k 2 Gun 2 : k n u0 0

as n , using A5 , in definition of , we conclude that

lim n1 0, n

172

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

which implies that n is nonempty. As n for all 𝑛 ∈ 𝑁 therefore n 1

F lim F n1 0. Hence is convex and compact and from Lemma n

23, the operator is u.s.c. By virtue of the Kakutani theorem G possesses a fixed point. Corollary 35 Every continuous - k -set ( - k -set, - k -set), contraction on a bounded, convex and closed subset of a Banach space H possesses a fixed point. 3.2. Krasnoselskii-type Results For The Sum Of Two Monotone Multivalued Operators We present the Krasnoselskii-type fixed point theorem for monotone multivalued operators with the help of Theorem 34. Definition 36 A multivalued operator G from H into

cl , cv

H is called a

multivalued k -contraction if there exists k 0,1 such that

H D Gr, Gs k.d r, s for all r, s H , where H D represents the Hausdorff Pompeiu distance [30]. In the next result, we investigate fixed points for the sum of two monotone operators, which is stated as follows. Theorem 37 Let H be a Banach space and be convex, bounded and closed chain in H . Suppose be a measure of noncompactness and S ,G be monotone multivalued operators from into cl , cv such that,

a G is closed and compact; b S is closed and monotone - k -set contraction, for some c S (r ) G(r ) , for 𝑟 ∈ 𝛿.

k 0,1 ;

Then a solution of the operator inclusion r S (r ) G(r ) exists. Proof Define from into

cl , cv

by

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

173

r Sr Gr for every r . As S and G are convex and closed, so their sum is also closed and convex. Therefore : cl , cv is well defined. Now for any 𝑢 ≼ 𝑣 we have 𝛤𝑢 = 𝑆𝑢 + 𝐺𝑢 ≼ 𝑆𝑣 + 𝐺𝑣 = (𝑆 + 𝐺) = 𝛤𝑣 . Therefore is also a monotone operator. Further, since N is any nonempty subset of , we have N which implies that N is bounded and by definition

N S N G N . Using the properties of , consider

N S N G N S N G N , since G is compact therefore S N S N k N , using the fact that S is -k -set contraction. Therefore satisfies all conditions of Theorem 34, Hence there exists z z , which is the required solution of operator inclusion z Sz Gz. 4. APPLICATIONS In this section, applications of the above results are presented. First, we consider an ABC fractional anti-periodic BVP, and using Theorem 12, we discuss the existence of a solution to this problem. It is known that the Mittage-Leffler (M-L) function is the solution of the following fractional differential equation,

D a , for 0 1. D Now we recall recently proposed Atangana--Baleanu (AB) fractional derivative of Caputo type as defined in [3].

174

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

Definition 38 Let H1 a, b and (0,1), then ABC fractional derivative of order is defined by ABC b

N

t Dt t E 1 b 1

t

d .

N is normalization function and satisfy The associated integral is defined by Where

AB a t

I

t

N 0 N 1 1.

1 1 t t d . N N a t

The following proposition will be useful for the next lemma. Proposition 39 [31] For (t ) defined on [a, b] and (m, m 1] for some 𝑚 ∈ ℕ0 we have 1) (aABR D 2) ( aAB I

3) ( aAB I

AB a

I )(t ) (t ). m 1

ABR a

D )(t ) (t )

ABC a

D )(t ) (t )

s 0

m

s 0

zs (a) s!

(t a ) s .

zs (a) s!

(t a ) s .

We consider the following ABC fractional BVP

ABC

Db t t , t , 1 2, 0 t T ,

(1)

with anti-periodic boundary conditions

0 T and 0 T . The following lemma is very crucial for the existence theorem of this section.

(2)

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

175

Lemma 40 Suppose : 0,T is a continuous and t H 1 0, T is a function, which is the solution of the following linear anti-periodic BVP

ABC

Db t t , 1 2, 0 t T ,

with boundary conditions 2 , given by T

t h t G t , d . 0

Where ( 1) (T 2 t ) ) (T ) 2 2 (2 4 N 1 ( ) N 1 N (11) ( ) (t ) 1 2 N (11) ( ) (T ) 1 , 0 t T G t , 2 ) 2 (2 4(N1) 1(T (2t )) (T ) 2 N 1 2 N (11) ( ) (T ) 1 , 0 t T 2

(3)

and

h t

T 2t 2 T . 4 N 1

(4)

Proof We have

ABC

Db t t ,

1 2, 0 t T .

Applying above Proposition 39, we have

t c1 c2t

(2 ) ( 1) d (t ) 1 d N 1 0 N 1 ( ) 0 t

Differentiating on both sides we have

t

(5)

176

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

(2 ) ( 1)2 t c2 t (t ) 2 d . N 1 N 1 ( ) 0 t

(6)

Using (2) with the assumption 0 0, we get T 1 (2 ) ( 1) 2 c1 T (T ) 2 d T 4 N 1 N 1 ( ) 0

(2 ) ( 1) d (T ) 1 d 2 N 1 0 2 N 1 ( ) 0 T

T

and T 1 (2 ) ( 1) 2 c2 T (T ) 2 d . 2 N 1 N 1 ( ) 0

Putting values of c1 and c2 in equation (5), we get

t

T 2t 2 T ( 1)2 (T 2t ) T (T ) 2 d 4 N 1 4 N 1 ( ) 0 (2 ) ( 1) d (T ) 1 d 2 N 1 0 2 N 1 ( ) 0 T

T

(2 ) ( 1) d (t ) 1 d . N 1 0 N 1 ( ) 0 t

t

After simplification, we have T

t h t G t , d . 0

Where G t , and h t are given in (3) and (4), respectively. This proves the lemma. The following lemma will be useful in the next results.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

177

Lemma 41 In the above theorem, we have

i G t, is decreasing with respect to t , and ii G t , NT 3T1 (()) : . Finally, we have the associated nonlinear problem given by T

t h t G t , , d .

(7)

0

In the next theorem, the existence of the solution (using Theorem 12) of the above problem 7 , which is equivalent to the problem 1 - (2) is discussed . Theorem 42 Suppose 𝜉: [0. 𝑇] × ℝ → ℝ is continuous and H 1 0, T . If

i

ii

T 2 N 1

1,

M sup t , t . t 0, T

Then problem (7) possesses a solution. Proof We consider the Banach space H H 1 0, T with ordered defined by ξ ≼ ζ

if and only if t t , for all t 0, T . We define operator Q : H H by T

Q t h t G t , , d 0

and let T

k t G t , , d . 0

178

Advances in Special Functions of Fractional Calculus T

T

0

0

Mehmood and Ahmad

k t1 G t1 , , d G t2 , , d k t2 .

Since G is decreasing in its first argument. Also, note that

I h

1

k 4 N 1 T 2 T 4 N 1 2 2 T

is continuous. Now we prove h is a -contraction, for this, consider

T 2t1 2 T T 2t2 2 T 4 N 1 4 N 1 2 T t t 2 1 2 N 1 T t2 t1 for all t1 t2 . 2 N 1

h t1 h t2

Therefore h is a -contraction with t

T 2 N 1

1, t 0, T .

Next, we will show that k is continuous and compact. The continuity of G and implies the continuity of k . Consider T

k t G t , , d 0 T

, d 0

T : r.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

179

Consider r x H : x r is a convex, bounded and closed subset, then for all r , we have k r , therefore k is uniformly bounded on r . Now for 0 t1 t2 T , consider k t1 k t2 t2 ( 1) 2 (T 2t1 ) t1 ( 1) 2 (T 2t2 ) 2 2 ( ) , ( ) ( ) , ( ) T d T d 0 4 N 1 ( ) 4 N 1 ( ) 0

t t2 (2 ) 1 , d , d 2 N 1 0 0

t2 t1 ( 1) 1 1 (t1 ) , d (t2 ) , d N 1 ( ) 0 0

t2 t1 ( 1) 1 1 (T ) , d (T ) , d 2 N 1 ( ) 0 0

T T (2 ) , d , d 2 N 1 t1 t2

T ( 1) 2 (T 2t ) T ( 1) 2 (T 2t2 ) 2 2 1 ( T ) , d ( T ) , d 4 N 1 ( ) t2 4 N 1 ( ) t1

T T ( 1) 1 1 (T ) , d (T ) , d . 2 N 1 ( ) t1 t2

Also as k t1 k t2 t2 T ( 1) 2 t1 2 2 M (T ) d (T ) d 0 0 4 N 1 ( )

180

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

t t2 t2 t1 (2 ) 1 ( 1) 1 1 d d (t1 ) d (t2 ) d 2 N 1 0 N 1 ( ) 0 0 0 t2 t1 ( 1) 1 1 (T ) d (T ) d 2 N 1 ( ) 0 0

T T T T ( 1) 2 T (2 ) 2 2 d d ( T ) d ( T ) d t 4 N 1 ( ) t t 2 N 1 t1 2 2 1

T T ( 1) 1 1 (T ) d (T ) d . 2 N 1 ( ) t1 t2

After simplification, we get k t1 k t2 T ( 1) M (T t1 ) 1 (T t2 ) 1 4 N 1 ( ) (2 ) ( 1) t1 t2 t1 t2 2 N 1 N 1 ( )

( 1) (T t1 ) (T t2 ) 2 N 1 ( )

(2 ) T ( 1) (T t1 ) 1 (T t2 ) 1 t1 t2 2 N 1 4 N 1 ( )

( 1) (T t1 ) (T t2 ) , 2 N 1 ( )

as t1 t2 , then right-hand side of the above inequality tends to zero and is independent of r , therefore, k is relatively compact on r . Thus from Arzela Ascoli's theorem, it follows that k is compact on r , hence all conditions of Theorem 12 are satisfied. So given ABC fractional BVP 7 has a solution.

Krasnoselskii-type

Advances in Special Functions of Fractional Calculus

181

Now we consider the multivalued case of the above problem (7), which is given by T

t h t G t , , d

(8)

0

where T G t , d G t , u d : u S , . 0 0

T

Where 𝜉: [0, 𝑇] × ℝ → 2ℝ \{𝜑} is a set valued mapping. For each H define the set of selections of by

S , x ABCI : x , a.e 0, T .

Where ABCI denotes the collection of all ABC fractional integrable functions and assumes that S , is nonempty for each H . Theorem 43. Assume that all the following conditions hold.

i 𝜉: [0, 𝑇] × ℝ → 2ℝ \{𝜑} be a compact valued mapping, ii ; for all X , , is measurable, iii for all 0,T and H , sup x L1 , x ,

iv

T 1, where is defined in Lemma 41,

iv for each

v G t , , d H , there exists T

0

u G t , , d H T

0

such that T

0 u t v t d . 0

182

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

Then the ABC fractional inclusion (8) has a solution. T

Proof Here we have L d , where 0

T 3 T ( ) N 1 ( )

, then L is a linear and as

T 1 , so r L 1 and all conditions of Theorem 31 are satisfied to obtain the

solution of fractional integral inclusion 8 . CONCLUSION Monotone operators are very important since they are not continuous on the whole domain. This property attracted us to find the fixed point results for the sum and product of monotone single/mutivalued operators. We conclude our investigation by remarking that the results presented in this chapter may constitute a base for a class of discontinuous operator equations and inclusions to find the criterion for the existence of solutions. CONSENT FOR PUBLICATON Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest. ACKNOWLEDGEMENT The authors are very grateful to the anonymous referee for valuable comments and suggestions. REFERENCES [1]

[2] [3]

I. Podlubny, "Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications", Elsevier, 1998. M. Caputo, and M. Fabrizio, "A new definition of fractional derivative without singular kernel", Progr. Fract. Differ. Appl., vol. 1, no. 2, pp. 1-13, 2015. A. Atangana, and D. Baleanu, "New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model", Therm. Sci., vol. 20, no. 2, pp. 763769, 2016. http://dx.doi.org/10.2298/TSCI160111018A

Krasnoselskii-type

[4] [5]

[6]

[7] [8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

Advances in Special Functions of Fractional Calculus

183

H. Poincar e, “Sur les courbes définies par les équations différentielles,” Journal de Mathématiques Pures et Appliquées, vol. 1, pp. 167-244, 1885. L.E.J. Brouwer, "Uber abbildung von mannigfaltigkeiten", Math. Ann., vol. 71, no. 1, pp. 97115, 1911. http://dx.doi.org/10.1007/BF01456931 J. Schauder, "Der fixpunktsatz in funktionalraümen", Stud. Math., vol. 2, no. 1, pp. 171-180, 1930. http://dx.doi.org/10.4064/sm-2-1-171-180 S. Banach, "Sur les opérations dans les ensembles abstraits et leur application aux équations integrals", Fund. math, vol. 3, no. 1, pp. 133-181, 1922. M. Krasnoselskii, "American mathematical society", Translation, vol. 10, pp. 345-409, 1958. T.A. Burton, and C. Kirk, "A fixed point theorem of krasnoselskii-schaefer type", Math. Nachr., vol. 189, no. 1, pp. 23-31, 1998. http://dx.doi.org/10.1002/mana.19981890103 T.A. Burton, "A fixed-point theorem of Krasnoselskii", Appl. Math. Lett., vol. 11, no. 1, pp. 85-88, 1998. http://dx.doi.org/10.1016/S0893-9659(97)00138-9 J. Garcia-Falset, K. Latrach, E. Moreno-Gálvez, and M.A. Taoudi, "Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness", J. Differ. Equ., vol. 252, no. 5, pp. 3436-3452, 2012. http://dx.doi.org/10.1016/j.jde.2011.11.012 S.K. Kashyap, B.K. Sharma, A. Banerjee, and S.C. Shrivastava, "On Krasnoselskii fixed point theorem and fractal", Chaos Solitons Fractals, vol. 61, pp. 44-45, 2014. http://dx.doi.org/10.1016/j.chaos.2014.02.003 E.A. Ok, "Fixed set theorems of Krasnoselskiĭ type", Proc. Am. Math. Soc., vol. 137, no. 2, pp. 511-518, 2008. http://dx.doi.org/10.1090/S0002-9939-08-09332-5 A.C.M. Ran, and M.C.B. Reurings, "A fixed point theorem in partially ordered sets and some applications to matrix equations", Proc. Am. Math. Soc., vol. 132, no. 5, pp. 1435-1443, 2003. http://dx.doi.org/10.1090/S0002-9939-03-07220-4 J.J. Nieto, and R. Rodríguez-López, "Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations", Order, vol. 22, no. 3, pp. 223-239, 2005. http://dx.doi.org/10.1007/s11083-005-9018-5 A. Al-Rawashdeh, W. Shatanawi, and M. Khandaqji, "Normed Ordered and -Metric Spaces", Int. J. Math. Math. Sci., vol. 2012, pp. 1-11, 2012. http://dx.doi.org/10.1155/2012/272137 Y. Feng, and H. Wang, "Characterizations of reproducing cones and uniqueness of fixed points", Nonlinear Anal., vol. 74, no. 16, pp. 5759-5765, 2011. http://dx.doi.org/10.1016/j.na.2011.05.067 A. Tychonoff, "Ein Fixpunktsatz", Math. Ann., vol. 111, no. 1, pp. 767-776, 1935. http://dx.doi.org/10.1007/BF01472256

184

[19] [20]

[21] [22]

[23] [24]

[25] [26]

[27]

[28]

[29] [30]

[31]

Advances in Special Functions of Fractional Calculus

Mehmood and Ahmad

K. Deimling, "Nonlinear functional analysis", Courier Corporation, 2010. S. Kakutani, "A generalization of Brouwer’s fixed point theorem", Duke Math. J., vol. 8, no. 3, pp. 457-459, 1941. http://dx.doi.org/10.1215/S0012-7094-41-00838-4 HF. Bohnenblust, and S. Karlin, "On a theorem of Ville", Contributions to the Theory of Games, Princeton: University Press, 2016, pp. 155-160. R.P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, vol. 141, Cambridge University Press, 2001. http://dx.doi.org/10.1017/CBO9780511543005 RR, Akhmerov, MI, Kamenskii, AS, Potapov, AE. Rodkina, BN. Sadovskii, “Measures of Noncompactness and Condensing Operators,” Operator theory, vol. 55, pp. 1-244, 1992. B.N. Sadovskii, "A fixed-point principle", Funct. Anal. Appl., vol. 1, no. 2, pp. 151-153, 1968. http://dx.doi.org/10.1007/BF01076087 B.C. Dhage, “Multi-valued mappings and fixed points nonlinear", Funct. Anal. Appl., vol. 10, no. 3, pp. 359-378, 2005. B.C. Dhage, "Multi-valued mappings and fixed points II", Tamkang Journal of Mathematics, vol. 37, no. 1, pp. 27-46, 2006. http://dx.doi.org/10.5556/j.tkjm.37.2006.177 A. Azam, M. Rashid, and N. Mehmood, "Set-valued ordered contractions with applications in differential inclusions", J. Anal., vol. 27, no. 3, pp. 673-695, 2019. http://dx.doi.org/10.1007/s41478-018-0107-4 Y. Feng, and Y. Wang, "Fixed points of multi-valued monotone operators and the solvability of a fractional integral inclusion", Fixed Point Theory Appl., vol. 2016, no. 1, p. 64, 2016. http://dx.doi.org/10.1186/s13663-016-0554-z G. Darbo, "Puntiuniti in trasformazioni a codominio non compatto", Rendiconti del SeminariomatematicodellaUniversità di Padova, vol. 24, pp. 84-92, 1955. S. Nadler, "Multi-valued contraction mappings", Pac. J. Math., vol. 30, no. 2, pp. 475-488, 1969. http://dx.doi.org/10.2140/pjm.1969.30.475 T. Abdeljawad, "A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel", J. Inequal. Appl., vol. 2017, no. 1, p. 130, 2017. http://dx.doi.org/10.1186/s13660-017-1400-5 PMID: 28680233

Advances in Special Functions of Fractional Calculus, 2023, 185-218

185

CHAPTER 11

General Fractional Order Quadratic Functional Integral Equations: Existence, Properties of Solutions, and Some of their Applications Ahmed M.A. El-Sayed1* and Hind H.G. Hashem2 1

Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, 21526 Alexandria 21526, Egypt 2

College of Science, Qassim University, P.O. Box 6644 Buraidah 51452, Saudi Arabia Abstract: In this chapter, we are interested in a certain class of integral equations, namely the quadratic integral equation. In this case, the unknown function is treated by some operators, then a pointwise multiplication of such operators is applied. The study of such a kind of problem was begun in the early 60’s due to the mathematical modeling of radiative transfer. The main objective was to present a special method or technique and results concerning various existence for a certain quadratic integral equation.

Keywords: Quadratic integral equation, Carathéodory Theorem, Continuous solution, Iterative scheme, Maximal and minimal solutions, Comparison Theorem, 𝛿 −Approximate solutions, Hybrid functional 𝜙 −differential equation, Pantograph functional 𝜙 − differential equation. 1. INTRODUCTION Quadratic integral equations have many useful applications and problems in the real world. For example, quadratic integral equations are often applicable in the theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory. Many authors studied the existence of solutions for several classes of nonlinear quadratic integral equations (see, e.g. [1-13] and [12-28]). Corresponding author Ahmed M.A. El-Sayed: Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, 21526 Alexandria 21526, Egypt ; Tel: 002035458664; E-mail: [email protected] *

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

186

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Let 𝐽 = [0, 𝑇], 𝜙: 𝐽 → 𝑅 be increasing and absolutely continuous and 𝜓𝑖 : 𝐽 → 𝐽, 𝑖 = 1,2 be continuous. Let 𝛽 ∈ (0,1) and 𝑡 ∈ 𝐽. Here, we study the existence of continuous solutions 𝑥 ∈ 𝐶(𝐽) of the 𝜙 − fractional-orders quadratic functional integral equation 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) . ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝜙 ′ (𝑠)𝑑𝑠, 𝑡 ∈ 𝐽, 𝛽 ∈ (0,1]. (1)

We discuss some properties of the solutions, prove the existence of maximal and minimal solutions of the quadratic integral equation (1) and introduce some particular cases and applications. Brownian motion is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober. For this reason, it is proposed to call such a family of diffusive processes Erdélyi- Kober fractional diffusion [29]. An Erdélyi-Kober operator is a fractional integration operator introduced by Arthur Erdélyi (1940) and Hermann Kober (1940). The Erdélyi-Kober fractional integral is given by [23-25] 𝑡 (𝑡 𝑚 − 𝑠 𝑚 )𝛼−1

𝛼 𝑓(𝑡) = ∫0 𝐼𝑚

Γ(𝛼)

𝑚 𝑠 𝑚−1 𝑓(𝑠) 𝑑𝑠, 𝑚 > 0, 𝛼 > 0,

which generalizes the Riemann fractional integral (when 𝑚 = 1) and its generalized fractional derivative of order 𝛼, like: 𝛼 1−𝛼 , 𝑚 > 0, 𝛼 ∈ (0,1). 𝐷𝑚 𝑓(𝑡) = 𝐷𝑚 𝐼𝑚

For the properties of Erdélyi-Kober operators, see [30] and [31] for examples. As a particular case of equation (1), we can consider the Erdélyi-Kober functional quadratic integral equation 𝑡 (𝑡 𝑚 − 𝑠 𝑚 )𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓(𝑡))) ∫0

when 𝜙(𝑡) = 𝑡 𝑚 , 𝑚 > 0.

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓(𝑠)))𝑚 𝑠 𝑚−1 𝑑𝑠, 𝑡 ∈ 𝐽

(2)

General Fractional Order

Advances in Special Functions of Fractional Calculus

187

As applications, we consider the nonlinear hybrid functional 𝜙 −differential equation of fractional order 𝑥(𝑡)−𝑎(𝑡)

𝛽

𝐷𝜙 (

𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))

) = 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝛽 ∈ (0,1).

The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. We also consider the 𝜙 −differential equation of pantograph-type delay of fractional order 𝛽

𝑥(𝑡)−𝑎(𝑡)

𝐷𝜙 (

𝑓1 (𝑡,𝑥(𝜎1 𝑡))

) = 𝑓2 (𝑡, 𝑥(𝜎2 𝑡)), 𝑡 ∈ 𝐽, 𝛽 ∈ (0,1).

where 𝜎1 , 𝜎2 ∈ (0,1). 2. PRELIMINARIES Let 𝐿1 = 𝐿1 (𝐽) be the class of Lebesgue integrable functions on 𝐽 = [0, 𝑇] with the standard norm and let 𝐶 = 𝐶(𝐽) be the space of all real-valued functions defined and continuous on 𝐽 with the standard supremum norm. This section collects some definitions and results needed in our further investigations. Assume that the function 𝑓(𝑡, 𝑥) = 𝑓: (0,1) × 𝑅 → 𝑅 satisfies Carathèodory conditions, i.e., measurable in 𝑡 for any 𝑥 ∈ 𝑅 and continuous in 𝑥 for almost all 𝑡 ∈ 𝐽 . Then to every function 𝑥(𝑡) being measurable on the interval 𝐽 we may assign the function (𝐹𝑥)(𝑡) = 𝑓(𝑡, 𝑥(𝑡)), 𝑡 ∈ (0,1), the operator 𝐹 defined in such a way, is called the superposition (or Nemytskii) operator with the generating function 𝑓. This operator is one of the simplest and most important operators investigated in the nonlinear functional analysis and in the theories of differential, integral and functional equations (see [32], [3-7] and [33]). Furthermore, for every 𝑓 ∈ 𝐿1 and every 𝜓: 𝐽 → 𝐽 , we define the superposition operator generated by the functions 𝑓 and 𝜓

188

Advances in Special Functions of Fractional Calculus

(𝐹𝑥(𝜓))(𝑡) = 𝑓(𝑡, 𝑥(𝜓(𝑡))),

El-Sayed and Hashem

𝑡 ∈ 𝐽.

In the sequel, we need the following theorem (see [3] and [7]). Theorem 1 The superposition operator 𝐹 maps 𝐿1 into itself if and only if |𝑓(𝑡, 𝑥)| ≤ 𝑐(𝑡) + 𝑘|𝑥| 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ 𝐽 and 𝑥 ∈ 𝑅, where 𝑐(𝑡) is a function from 𝐿1 and 𝑘 is a nonnegative constant. This theorem was proved by Kranosel’skii [33] in the case of the interval [𝑎, 𝑏]. The investigation for the case of an unbounded domain is due to Apple and Zabrejko [32]. In what follows, we will need the following property of the superposition operator in the space 𝐶[𝑎, 𝑏] of all continuous functions acting from 𝐼 = [𝑎, 𝑏] into 𝑅 with the standard maximum norm (see [34]). Lemma 1 Assume that 𝐹 is the superposition operator generated by the function 𝑓: [𝑎, 𝑏] × 𝑅 → 𝑅. Then 𝐹 transforms the space 𝐶(𝐼) into itself and is continuous if and only if the function 𝑓 is continuous on the set 𝐼 × 𝑅. Definition 1 The fractional-order integral of order 𝛽 of the function 𝑓 is defined on [𝑎, 𝑏] by (see [30], [35], [36] and [31]). 𝑡 (𝑡−𝑠)𝛽−1

𝛽

𝐼𝑎 𝑓(𝑡) = ∫𝑎

Γ(𝛽)

𝑓(𝑠)𝑑𝑠, 𝑡 > 𝑎

(3)

𝛽

and when 𝑎 = 0, we have 𝐼𝛽 𝑓(𝑡) = 𝐼0 𝑓(𝑡), 𝑡 > 0. Definition 2 The Riemann-Liouville fractional-order derivative of order 𝛽 ∈ (0,1) of the function 𝑓 is given by (see [30, 31, 35 and 36]) 𝑅𝐷

𝛽

𝑓(𝑡) =

𝑑 1−𝛽 𝐼 𝑓(𝑡). 𝑑𝑡

An Erdélyi-Kober operator is a fractional integration operation introduced by Arthur Erdélyi (1940) and Hermann Kober (1940).

General Fractional Order

Advances in Special Functions of Fractional Calculus 𝑡 (𝑡 𝑚 − 𝑠 𝑚 )𝛼−1

𝛼 𝐼𝑚,𝑎 𝑓(𝑡) = ∫𝑎

Γ(𝛼)

189

𝑚 𝑠 𝑚−1 𝑓(𝑠) 𝑑𝑠.

The Erdélyi-Kober fractional integral is defined in many literature [33] and [2325]. The aim of the short note [29] is to highlight that the generalized grey Brownian motion(ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason, here it is proposed to call the such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides fast and slow anomalous diffusion models. For the properties of Erdélyi-Kober operators, see [30, 31 and 33] for examples. Now, we shall denote by 𝐿1𝜙 = 𝐿1𝜙 [𝑎, 𝑏] the space of all real functions defined 𝑏

on [𝑎, 𝑏]. Such that 𝜙′(𝑡) 𝑓(𝑡) ∈ 𝐿1 and ∫𝑎 | 𝜙′(𝑡) 𝑓(𝑡) | 𝑑𝑡 ≤ ∞. where 𝜙 is an increasing function and absolutely continuous on [𝑎, 𝑏], and we introduce the norm [38] 𝑏

|| 𝑓(𝑡) ||𝐿1 = ∫𝑎 | 𝜙′(𝑡) 𝑓(𝑡) | 𝑑𝑡 𝜙

𝑡 ∈ [𝑎, 𝑏].

Definition 3 [38] The 𝜙 − fractional integral of order 𝛼 ≥ 0 of the function 𝑓( 𝑡 ) ∈ 𝐿1𝜙 is defined as: 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛼−1

𝛼 𝐼𝑎,𝜙 𝑓(𝑡) = ∫𝑎

Γ(𝛼)

𝜙′(𝑠) 𝑓(𝑠) 𝑑𝑠.

𝛼 𝐼𝑎,𝜙 may be known as the fractional integral of the function 𝑓(𝑡) with respect to 𝜙(𝑡), which is defined for any monotonic increasing function 𝜙(𝑡) ≥ 0, having a continuous derivative.

Definition 4 The 𝜙 − fractional derivative of order 𝛼 ∈ (0,1) of the function 𝑓(𝑡) ∈ 𝐿1𝜙 is defined as 𝛼 𝐷𝑎,𝜙 𝑓(𝑡) = (

1

𝑑

𝜙′(𝑡) 𝑑𝑡

𝑡 (𝜙(𝑡) − 𝜙(𝑠))−𝛼−1

) ∫𝑎

Γ(−𝛼)

𝜙 ′ (𝑠) 𝑓(𝑠) 𝑑𝑠, 𝜙 ′ (𝑡) = 0, 𝑎 < 𝑡 < 𝑏 (4)

defined for any monotonic increasing function 𝜙(𝑡) ≥ 0, having a continuous derivative.

190

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

If 𝜙 ′ (𝑡) ≠ 0, 𝑎 < 𝑡 < 𝑏 then the operator 𝐼𝜙𝛼 is easily expressed via the usual Riemann-Liouville fractional integration (see [35]). So many properties of the operator 𝐼𝜙𝛼 , in particular, the semigroup property 𝛽

𝛼+𝛽

𝐼 𝛼𝜙 𝐼 𝜙 𝑓(𝑡) = 𝐼 𝜙

𝑓(𝑡),

follows directly from the corresponding properties of the Riemann-Liouville fractional integral. When 𝜙(𝑡) = 𝑡 , we obtain the Riemann-Liouville fractional integral 𝐼𝑎𝛼 𝑡 (𝑡 − 𝑠)𝛼−1

𝐼𝑎𝛼 𝑓(𝑡) = ∫𝑎

Γ(𝛼)

𝑓(𝑠) 𝑑𝑠.

When 𝜙(𝑡) = 𝑡 𝑚 , 𝑚 > 0, we obtain the Erdelyi-Kober (see [30, 31 and 33]) fractional order operator 𝐼𝑡𝛼𝑚 ,𝑎 . Now, for the continuation in 𝐿1𝜙 of the fractional integral to the usual ones, we have the following lemmas. Lemma 2 If 𝑓(𝑡) ∈ 𝐿1𝜙 , then 𝛼 𝑛 𝑓(𝑡) = 𝐼𝑎,𝜙 𝑓(𝑡) 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦, 𝑛 = 1,2,3, . .. lim 𝐼𝑎,𝜙

𝛼→𝑛

𝑡

1 where 𝐼𝑎,𝜙 𝑓(𝑡) = ∫𝑎 𝜙′(𝑠) 𝑓(𝑠) 𝑑𝑠. 𝛼 maps 𝐿1𝜙 into itself continuously. Lemma 3 𝐼𝑎,𝜙

Lemma 4 Let 𝑓(𝑡) ∈ 𝐿1 . If 𝑓(𝑡) is bounded and measurable on [𝑎, 𝑏], then 𝛽

𝐼𝑎,𝜙 𝑓(𝑡) |𝑡 = 𝑎 = 0. For more properties of this integral operator, see [30, 31 and 38].

General Fractional Order

Advances in Special Functions of Fractional Calculus

191

3. EXISTENCE OF SOLUTIONS 3.1. Carathéodory Theorem for the Quadratic Integral Equation (1) Now we shall generalize these results and obtain similar ones for the functional quadratic 𝜙 −integral of fractional order (1) under the following assumption: i. 𝑎 ∶ 𝐽 → 𝑅+ is continuous and sup|𝑎(𝑡)| = 𝑘. 𝑡∈𝐽

ii. 𝑓2 ∶ 𝐽 × 𝑅 → 𝑅+ satisfies Carathéodory condition (i.e., measurable in 𝑡 for all 𝑥 ∈ 𝑅 and continuous in 𝑥 for all 𝑡 ∈ 𝐽 ). iii. There exist a functions 𝑚2 ∈ 𝐿1 (𝐽) and a nonnegative constant 𝑏 such that iv. |𝑓2 (𝑡, 𝑥(𝑡))| ≤ |𝑚2 (𝑡)| + 𝑏|𝑥|. v. 𝜙: 𝐽 → 𝑅+ , is increasing and absolutely continuous. vi. 𝜓𝑖 : 𝐽 → 𝐽, 𝑖 = 1,2 are continuous. 𝛾 vii. 𝐼𝜙 𝑚2 ≤ 𝑀, ∀𝛾 ≤ 𝛽. 𝑟 is a positive solution of the inequality: 𝜆1 𝑏 𝑇 𝛽 Γ(𝛽+1)

𝑟2 + [

𝑀 𝜆1 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

+

𝑏 𝑀1 𝑇 𝛽 Γ(𝛽+1)

]𝑟 + 𝑘 +

𝑀1 𝑀 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

≤ 𝑟.

viii. 𝑓1 ∶ 𝐽 × 𝑅 → 𝑅+ is continuous and satisfies Lipchitz condition with constant 𝜆1 REMARK 1 |𝑓1 (𝑡, 𝑥) − 𝑓1 (𝑡, 0)| ≤ 𝜆1 |𝑥 − 0| |𝑓1 (𝑡, 𝑥)| ≤ |𝑓1 (𝑡, 0)| + 𝜆1 |𝑥| ≤ 𝑚1 (𝑡) + 𝜆1 |𝑥|, 𝑚1 (𝑡) = |𝑓1 (𝑡, 0)|, ∀ 𝑥 ∈ 𝑅, 𝑡 ∈ 𝐽, with 𝑀1 = max|𝑚1 (𝑡)| 𝑡∈𝐽

To prove the existence of at least one solution for equation (1), firstly, we construct an iterative scheme (as done in the original Carathéodory theorem). Theorem 2 Let assumptions (i)-(viii) be satisfied, then the functional quadratic integral equation of fractional order (1) has at least one positive solution 𝑥 ∈ 𝐶(𝐽).

192

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Proof. Consider the ball 𝑆𝑟 in the space 𝐶(𝐽) defined as 𝑆𝑟 = {𝑥 ∈ 𝐶(𝐽): |𝑥(𝑡)| ≤ 𝑟 𝑓𝑜𝑟 𝑡 ∈ 𝐽}. 1

Define the sequence {𝑥𝑛 (𝑡)}, 𝑡 ∈ [0, 𝑇 − ] 𝑛

𝑥 𝑛 (𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥𝑛 (𝜓1 (𝑡))) 1

1 𝛽−1 𝑡+ (𝜙(𝑡+𝑛)−𝜙(𝑠)) . ∫0 𝑛 𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠. Γ(𝛽)

1

The sequence {𝑥𝑛 (𝑡)}, 𝑡 ∈ [0, 𝑇 − ] is uniformly bounded 𝑛

|𝑥𝑛 (𝑡)| ≤ |𝑎(𝑡)| 1

1 𝛽−1 𝑡+𝑛 (𝜙(𝑡+𝑛)−𝜙(𝑠))

+(𝜆1 |𝑥𝑛 (𝜓1 (𝑡))| + 𝑀1 ) ∫0

Γ(𝛽)

|𝑚2 (𝑠) +

𝑏|𝑥𝑛 (𝜓2 (𝑠))|𝜙′(𝑠)𝑑𝑠, ≤ |𝑎(𝑡)| + (𝜆1 |𝑥𝑛 (𝜓1 (𝑡))| + 𝑀1 ) 1

1 𝛽−1 𝑡+𝑛 (𝜙(𝑡+𝑛)−𝜙(𝑠))

. ∫0

Γ(𝛽)

[|𝑚2 (𝑠)| + 𝑏|𝑥𝑛 (𝜓1 (𝑠))|]𝜙′(𝑠)𝑑𝑠 1

1 𝛽−𝛾−1 𝑡+𝑛 (𝜙(𝑡+𝑛)−𝜙(𝑠))

≤ 𝑘 + (𝜆1 𝑟 + 𝑀1 ) 𝑀 ∫0

Γ(𝛽−𝛾)

𝜙′(𝑠)𝑑𝑠

1

1 𝛽−1 𝑡+𝑛 (𝜙(𝑡+𝑛)−𝜙(𝑠))

+𝑏 𝑟 (𝜆1 𝑟 + 𝑀1 ) ∫0

Γ(𝛽)

𝜙′(𝑠)𝑑𝑠

1

≤ 𝑘 + (𝜆1 𝑟 + 𝑀1 )𝑀

(𝜙(𝑡+𝑛))𝛽−𝛾

≤ 𝑘 + 𝑀 (𝜆1 𝑟 + 𝑀1 ) ≤

𝜆1 𝑏 𝑇 𝛽 Γ(𝛽+1)

𝑟2 + [

Γ(𝛽−𝛾+1) 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

𝑀 𝜆1 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

+

1

+ 𝑏 𝑟 (𝜆1 𝑟 + 𝑀1 )

+ 𝑏 𝑟 (𝜆1 𝑟 + 𝑀1 )

𝑏 𝑀1 𝑇 𝛽 Γ(𝛽+1)

]𝑟 +𝑘+

(𝜙(𝑡+𝑛))𝛽 Γ(𝛽+1) 𝑇𝛽

Γ(𝛽+1)

𝑀1 𝑀 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

≤ 𝑟.

General Fractional Order

Advances in Special Functions of Fractional Calculus

193

Also, we shall show that the sequence is equi-continuous. 1

For 𝑡1 , 𝑡2 ∈ [0, 𝑇 − ] such that 𝑡1 < 𝑡2 , then 𝑛

|𝑥𝑛 (𝑡2 ) − 𝑥𝑛 (𝑡1 )| = |𝑎(𝑡2 ) − 𝑎(𝑡1 ) 1

1 𝛽−1 𝑡 + (𝜙(𝑡2 +𝑛)−𝜙(𝑠)) +𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) ∫0 2 𝑛 𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠 Γ(𝛽)

1

1 𝛽−1 𝑡 +𝑛 (𝜙(𝑡1 +𝑛)−𝜙(𝑠))

−𝑓1 (𝑡1 , 𝑥𝑛 (𝜓1 (𝑡1 ))) ∫0 1

𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠| ≤

Γ(𝛽)

|𝑎(𝑡2 ) − 𝑎(𝑡1 )| 1

1 𝛽−1 𝑡 +𝑛 (𝜙(𝑡2 +𝑛)−𝜙(𝑠))

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) ∫0 2

Γ(𝛽)

|𝑓2 (𝑠, 𝑥𝑛 (𝜓(𝑠)))|𝜙′(𝑠)𝑑𝑠

1

1 𝛽−1 𝑡 +𝑛 (𝜙(𝑡1 +𝑛)−𝜙(𝑠))

−𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) ∫0 1

Γ(𝛽)

𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠|

1

1 𝛽−1 𝑡 + (𝜙(𝑡1 +𝑛)−𝜙(𝑠)) +|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) ∫0 1 𝑛 𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠 Γ(𝛽)

−𝑓1 (𝑡1 , 𝑥𝑛 (𝜓1 (𝑡1 ))) ∫

1 𝑡1 + (𝜙(𝑡 𝑛 1

0

1 𝑛

𝑡 +

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 )))| ∫0 1

1 + ) − 𝜙(𝑠))𝛽−1 𝑛 𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠| ≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| Γ(𝛽)

1 𝑛

1 𝑛

(𝜙(𝑡2 + )−𝜙(𝑠))𝛽−1 −(𝜙(𝑡1 + )−𝜙(𝑠))𝛽−1 Γ(𝛽)

1

1

𝑡2 +𝑛 (𝜙(𝑡2 +𝑛)−𝜙(𝑠))𝛽−1

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 )))| ∫

1

𝑡1 +𝑛

|𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))|𝜙′(𝑠)𝑑𝑠

Γ(𝛽) 𝑡 +

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) − 𝑓1 (𝑡1 , 𝑥𝑛 (𝜓1 (𝑡1 )))| ∫0 1

1 𝑛

|𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))|𝜙′(𝑠)𝑑𝑠

1 𝑛

(𝜙(𝑡2 + )−𝜙(𝑠))𝛽−1 Γ(𝛽)

|𝑓2 (𝑠, 𝑥𝑛 (𝜓2 (𝑠)))|𝜙′(𝑠)𝑑𝑠.

194

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Then we get, | 𝑥𝑛 (𝑡2 ) − 𝑥𝑛 (𝑡1 ) | ≤ | 𝑎(𝑡2 ) − 𝑎(𝑡1 ) | + (𝜆1 |𝑥(𝜓1 (𝑡2 ))| + 𝑀1 ) .∫

1 𝑡1 + (𝜙(𝑡 𝑛 2

0

1 1 + ) − 𝜙(𝑠))𝛽−1 − (𝜙(𝑡1 + ) − 𝜙(𝑠))𝛽−1 𝑛 𝑛 [𝑚2 (𝑠) + 𝑏|𝑥𝑛 (𝜓(𝑠))|]𝜙′(𝑠)𝑑𝑠 Γ(𝛽)

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 )))| ∫

1 𝑡2 + (𝜙(𝑡 𝑛 2

1 𝑡1 + 𝑛

1 + ) − 𝜙(𝑠))𝛽−1 𝑛 [𝑚2 (𝑠) + 𝑏|𝑥𝑛 (𝜓2 (𝑠))|]𝜙′(𝑠)𝑑𝑠 Γ(𝛽) 𝑡 +

+|𝑓1 (𝑡2 , 𝑥𝑛 (𝜓1 (𝑡2 ))) − 𝑓1 (𝑠, 𝑥𝑛 (𝜓1 (𝑡1 )))| ∫0 1

1 𝑛

1 𝑛

(𝜙(𝑡2 + )−𝜙(𝑠))𝛽−1

[𝑚2 (𝑠) + 𝑏|𝑥𝑛 (𝜓2 (𝑠))|]𝜙′(𝑠)𝑑𝑠

Γ(𝛽)

≤ | 𝑎(𝑡2 ) − 𝑎(𝑡1 ) | + (𝜆1 𝑟 + 𝑀1 ) 1

.[

1

𝑀 (𝜙(𝑡2 +𝑛)−𝜙2 (𝑡1 +𝑛))𝛽−𝛾 Γ(𝛽−𝛾+1) 1

+(𝜆1 𝑟 + 𝑀1 ) [

1

+

Γ(𝛽+1)

1

𝑀 (𝜙(𝑡2 +𝑛)−𝜙(𝑡1 +𝑛))𝛽−𝛾 Γ(𝛽−𝛾+1) 1 𝑛

]

1

+ 1 𝑛

Γ(𝛽−𝛾+1)

1

𝑏 𝑟 (𝜙(𝑡2 +𝑛)−𝜙(𝑡1 +𝑛))𝛽

𝑀 (𝜙(𝑡2 + )−𝜙(𝑡1 + ))𝛽−𝛾

+|𝑥𝑛 (𝜓1 (𝑡2 )) − 𝑥𝑛 (𝜓1 (𝑡1 ))| [

1

𝑏 𝑟 (𝜙(𝑡2 +𝑛)−𝜙(𝑡1 +𝑛))𝛽

Γ(𝛽+1)

+

1 𝑛

1 𝑛

]

𝑏 𝑟 (𝜙(𝑡2 + )−𝜙(𝑡1 + ))𝛽 Γ(𝛽+1)

]

Which implies that, |𝑡2 − 𝑡1 | → 0 ⇒ |𝑥𝑛 (𝑡2 ) − 𝑥𝑛 (𝑡1 )| → 0 and this proves the equi-continuity of the sequence {𝑥𝑛 (𝑡)}. Then, {𝑥𝑛 (𝑡)} is a sequence of equi-continuous and uniformly bounded functions. Hence, by ArzelaAscoli Theorem [39], then there exists a subsequence {𝑥𝑛𝑘 (𝑡)} of continuous functions, which converge uniformly to a continuous function 𝑥 as 𝑘 → ∞ . Now we show that this limit function is the required solution. From assumptions (ii), (iii) and (viii), we have |𝑓1 (𝑡, 𝑥𝑛𝑘 (𝜓1 (𝑡)))| ≤ 𝜆1 𝑟 + 𝑀1 ∈ 𝐿1 ,

General Fractional Order

Advances in Special Functions of Fractional Calculus

195

and |𝑓2 (𝑡, 𝑥𝑛𝑘 (𝜓2 (𝑡)))| ≤ |𝑚2 (𝑡)| + 𝑏 𝑟 ∈ 𝐿1 , and the functions 𝑓𝑖 (𝑠, 𝑥𝑛𝑘 (𝜓𝑖 (𝑠))), 𝑖 = 1,2 are continuous in the second argument, 𝑖. 𝑒., . 𝑓𝑖 (𝑠, 𝑥𝑛𝑘 (𝜓𝑖 (𝑠))) → 𝑓𝑖 (𝑠, 𝑥(𝜓𝑖 (𝑠))) 𝑎𝑠 𝑘 → ∞. For 𝑠 ∈ (0, 𝑡) and 𝑡 ∈ 𝐽 (𝜙(𝑡 +

1 1 ) − 𝜙(𝑠)) > (𝜙(𝑡) − 𝜙(𝑠)) ⇒ (𝜙(𝑡 + ) − 𝜙(𝑠))𝛽−1 < (𝜙(𝑡) − 𝜙(𝑠))𝛽−1 , 𝑛𝑘 𝑛𝑘

therefore the sequence {(𝜙(𝑡 +

1 𝑛𝑘

) − 𝜙(𝑠))𝛽−1 𝑓2 (𝑠, 𝑥𝑛𝑘 (𝜓(𝑠)))}, 𝛽 ∈ (0,1]

satisfies Lebesgue dominated convergence theorem [39]. 1 𝑛𝑘

𝑡+

𝑓1 (𝑡, 𝑥𝑛𝑘 (𝜓1 (𝑡))) ∫0

1

(𝜙(𝑡+𝑛 )−𝜙(𝑠))𝛽−1 𝑘

Γ(𝛽)

𝑓2 (𝑠, 𝑥𝑛𝑘 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

⇒ 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠,

Also 𝑥𝑛𝑘 (𝑡) may be written as 𝑡+

𝑥𝑛𝑘 (𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥𝑛𝑘 (𝜓1 (𝑡))) ∫0

1 𝑛𝑘

(𝜙(𝑡+

1 )−𝜙(𝑠))𝛽−1 𝑛𝑘

Γ(𝛽)

𝑓2 (𝑠, 𝑥𝑛𝑘 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠

1

(𝜙(𝑡+𝑛 )−𝜙(𝑠))𝛽−1 𝑡 𝑘 +𝑓1 (𝑡, 𝑥𝑛𝑘 (𝜓1 (𝑡))) ∫𝑡+ 1 𝑓2 (𝑠, 𝑥𝑛𝑘 (𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠 Γ(𝛽) 𝑛 𝑘

and 1

1

𝑡+𝑛 (𝜙(𝑡+𝑛 )−𝜙(𝑠))𝛽−1

∫𝑡

𝑘

𝑘

Γ(𝛽) 1

𝑡+𝑛

≤ ∫𝑡

𝑘

|𝑓2 (𝑠, 𝑥(𝜓2 (𝑠)))|𝜙′(𝑠)𝑑𝑠

1

(𝜙(𝑡+𝑛 )−𝜙(𝑠))𝛽−1 𝑘

Γ(𝛽)

(|𝑚2 (𝑠)| + 𝑏𝑟)𝜙′(𝑠) 𝑑𝑠

196

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

→ 0 𝑎𝑠 𝑘 → ∞. Then we have 𝑥(𝑡) = lim 𝑥𝑛𝑘 (𝑡 ) 𝑘→∞

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

= 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛼)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠)))𝜙′(𝑠)𝑑𝑠

which proves the existence of positive solution 𝑥 ∈ 𝐶(𝐽) of the quadratic integral equation (1). 4. SPECIAL CASES AND REMARKS In Section 3, we have proved an existing result for the functional quadratic 𝜙 −integral equation of fractional order (1) under Carathèodory and growth conditions which in turn gives the existence as well as the existence of many key integral and functional equations that arise in nonlinear analysis and its applications. Corollary 1 Let the assumptions (i)-(viii) be satisfied with 𝜓1 (𝑡) = 𝜓2 (𝑡) = 𝜓(𝑡), then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓(𝑡))) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓(𝑠))) 𝜙′(𝑠) 𝑑𝑠, 𝑡 ∈ 𝐽.

Corollary 2 Let the assumptions (i)-(viii) be satisfied with 𝜓1 (𝑡) = 𝑡, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓(𝑠))) 𝜙′(𝑠) 𝑑𝑠, 𝑡 ∈ 𝐽,

which is the same results in [40]. Corollary 3 Let the assumptions (i)-(viii) be satisfied with 𝜓1 (𝑡) = 𝜓2 (𝑡) = 𝑡, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝑠)) 𝜙′(𝑠) 𝑑𝑠, 𝑡 ∈ 𝐽.

General Fractional Order

Advances in Special Functions of Fractional Calculus

197

Corollary 4 Let the assumptions (i)-(viii) be satisfied with 𝜙(𝑡) = 𝑡 𝑚 , 𝑚 > 0, then there exists at least one solution for the Erdélyi-Kober functional quadratic equation of fractional order 𝑡 (𝑡 𝑚 −𝑠 𝑚 )𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝑚𝑠 𝑚−1 𝑑𝑠, 𝑡 ∈ 𝐽.

Corollary 5 Let the assumptions (i)-(viii) be satisfied with 𝜙 = 𝑡 𝑚 , 𝑚 > 0 𝜓1 (𝑡) = 𝑡, 𝜓2 (𝑡) = 𝜓(𝑡), then there exists at least one solution for the ErdélyiKober functional quadratic equation of fractional order 𝑡 (𝑡 𝑚 −𝑠 𝑚 )𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓(𝑠))) 𝑚𝑠 𝑚−1 𝑑𝑠, 𝑡 ∈ 𝐽.

which is the same result in [27] Corollary 6 Let the assumptions (i)-(vii) be satisfied with 𝜙 = 𝑡 𝑚 , 𝑚 > 0 𝜓1 (𝑡) = 𝜓(𝑡), 𝜓2 (𝑡) = 𝑡, then there exists at least one solution for the ErdélyiKober functional quadratic equation of fractional order 𝑡 (𝑡 𝑚 −𝑠 𝑚 )𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓(𝑡))) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝑠)) 𝑚𝑠 𝑚−1 𝑑𝑠, 𝑡 ∈ 𝐽.

Corollary 7 Let the assumptions (i)-(vii) be satisfied with 𝜙 = 𝑡 𝑚 , 𝑚 > 0, 𝜓1 (𝑡) = 𝜓2 (𝑡) = 𝑡, then there exists at least one solution for the Erdélyi-Kober functional quadratic equation of fractional order 𝑡 (𝑡 𝑚 −𝑠 𝑚 )𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝑠)) 𝑚𝑠 𝑚−1 𝑑𝑠, 𝑡 ∈ 𝐽.

Corollary 8 Let the assumptions (i)-(vii) be satisfied with 𝜙 = 𝑡, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡 (𝑡−𝑠)𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝑑𝑠, 𝑡 ∈ 𝐽.

which is the same results in [12]. Corollary 9 Let the assumptions (i)-(vii) be satisfied with 𝜙(𝑡) = 𝑡, 𝜓1 (𝑡) = 𝜓2 (𝑡) = 𝑡, then there exists at least one solution for the functional quadratic integral equation of fractional order

198

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

𝑡 (𝑡−𝑠)𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝑠)) 𝑑𝑠, 𝑡 ∈ 𝐽.

The same result is obtained in [11]. Corollary 10 Let the assumptions (i)-(vii) be satisfied with 𝜙(𝑡) = 𝑡, 𝜓2 (𝑡) = 𝑡 and 𝑓1 = 1, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡 (𝑡−𝑠)𝛽−1

𝑥(𝑡) = 𝑎(𝑡) + ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝑠)) 𝑑𝑠, 𝑡 ∈ 𝐽.

The same result is obtained in [41]. Letting 𝛽 → 1, we get Corollary 11 Let the assumptions (i)-(vii) be satisfied with 𝛽 → 1, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0 𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝑑𝑠, 𝑡 ∈ 𝐽. Corollary 12 Let the assumptions (i)-(vii) be satisfied with 𝛽 → 1 and 𝜓1 = 𝑡 then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0 𝑓2 (𝑠, 𝑥(𝜓(𝑠))) 𝑑𝑠, 𝑡 ∈ 𝐽. Corollary 13 Let the assumptions (i)-(vii) be satisfied with 𝛽 → 1, and 𝜓1 (𝑡) = 𝜓2 (𝑡) = 𝑡, then there exists at least one solution for the functional quadratic integral equation of fractional order 𝑡

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝑡)) ∫0 𝑓2 (𝑠, 𝑥(𝑠)) 𝑑𝑠, 𝑡 ∈ 𝐽. This equation was studied in [18] by using Picard and Adomian methods. 5. PROPERTIES OF SOLUTIONS In this section, we give sufficient conditions for the uniqueness of the solution of the quadratic functional integral equation (1) and study some of its properties and features.

General Fractional Order

Advances in Special Functions of Fractional Calculus

199

5.1. Uniqueness of Solutions of QFIE (1) Let us assume the following assumptions i. 𝑎 ∶ 𝐽 → 𝑅+ is continuous and sup|𝑎(𝑡)| = 𝑘 𝑡∈𝐽

ii. 𝑓2 ∶ 𝐽 × 𝑅 → 𝑅+ satisfies Carathéodory condition (i.e., measurable in 𝑡 for all 𝑥 ∈ 𝑅 and continuous in 𝑥 for all 𝑡 ∈ 𝐽 ). iii. 𝑓1 ∶ 𝐽 × 𝑅 → 𝑅+ is continuous. iv. There exist two nonnegative constants 𝜆1 , 𝜆2 ∈ 𝑅 such that |𝑓𝑖 (𝑡, 𝑥) − 𝑓𝑖 (𝑡, 𝑦)| ≤ 𝜆𝑖 |𝑥 − 𝑦|, 𝑖 = 1,2. i. 𝜙: 𝐽 → 𝑅+ be increasing and absolutely continuous. ii. 𝜓𝑖 : 𝐽 → 𝐽, 𝑖 = 1,2 be continuous. 𝛾 iii. 𝐼𝜙 𝑚2 ≤ 𝑀, ∀𝛾 ≤ 𝛽. iv. 𝑟 is a positive solution of the inequality: 𝜆1 𝑏 𝑇 𝛽 Γ(𝛽+1)

𝑟2 + [

𝑀 𝜆1 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

+

𝑏 𝑀1 𝑇 𝛽 Γ(𝛽+1)

]𝑟 +𝑘+

𝑀1 𝑀 𝑇 𝛽−𝛾 Γ(𝛽−𝛾+1)

≤ 𝑟.

Theorem 3 Let the assumptions (I)-(𝑉𝐼𝐼𝐼) be satisfied. If [[𝜆1 𝑟 + 𝑀1 ]

𝜆2 𝑇 𝛽 Γ(𝛽+1)

𝑀 𝑇 𝛽−𝛾2

+ 𝜆1 [

Γ(𝛽−𝛾2 +1)

+

𝜆2 𝑇 𝛽 𝑟 Γ(𝛽+1)

]] < 1,

then the quadratic integral equation (1) has a unique positive solution 𝑥 ∈ 𝐶(𝐽). Proof |𝑓𝑖 (𝑡, 𝑥) − 𝑓𝑖 (𝑡, 0)| ≤ 𝜆𝑖 |𝑥 − 0| |𝑓𝑖 (𝑡, 𝑥)| ≤ |𝑓𝑖 (𝑡, 0)| + 𝜆𝑖 |𝑥| ≤ 𝑚𝑖 (𝑡) + 𝜆𝑖 |𝑥|, 𝑚𝑖 (𝑡) = |𝑓𝑖 (𝑡, 0)|, ∀ 𝑥 ∈ 𝑅, 𝑡 ∈ 𝐽. Equation (1) can be written as 𝛽

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) 𝛽−𝛾 𝛾 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑡

= 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))𝐼𝜙

∈ 𝐽.

(5)

200

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Define the operator Ϝ by: 𝑡

Ϝ𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓(𝑡))) ∫ 0

(𝜙(𝑡) − 𝜙(𝑠))𝛽−1 𝑓2 (𝑠, 𝑥(𝜓(𝑠))) 𝜙′(𝑠) 𝑑𝑠, Γ(𝛽)

𝑡 ∈ 𝐽. The operator Ϝ maps 𝐶(𝐽) into itself. For this, let 𝑡1 , 𝑡2 ∈ 𝐽, 𝑡1 < 𝑡2 such that |𝑡2 − 𝑡1 | ≤ 𝛿, then in a similar way as done before using the condition (𝑣𝑖 ⋆ ) and the relation (5) , we can prove that | (Ϝ𝑥)(𝑡2 ) − (Ϝ𝑥)(𝑡1 ) | → 0

𝑎𝑠 𝑡2 → 𝑡1

which proves that Ϝ: 𝐶(𝐽) → 𝐶(𝐽). Now, to show that Ϝ is a contraction. Let 𝑥, 𝑦 ∈ 𝐶(𝐽), then we have 𝑡

|Ϝ𝑥(𝑡) − Ϝ𝑦(𝑡)| = | 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫ 0

(𝜙(𝑡) − 𝜙(𝑠))𝛽−1 𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠 Γ(𝛽)

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

−𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

𝑓2 (𝑠, 𝑦(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠|

Γ(𝛽) 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

= | 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

− 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

+ 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0

Γ(𝛽)

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

−𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) ∫0 𝑡

≤ |𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))| ∫ 0

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠

𝑓2 (𝑠, 𝑦(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠 𝑓2 (𝑠, 𝑦(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠 𝑓2 (𝑠, 𝑦(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠|

(𝜙(𝑡) − 𝜙(𝑠))𝛽−1 |𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) − 𝑓2 (𝑠, 𝑦(𝜓2 (𝑠)))| 𝜙′(𝑠) 𝑑𝑠 Γ(𝛽) 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

+|𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) − 𝑓1 (𝑡, 𝑦(𝜓1 (𝑡)))| ∫0

Γ(𝛽)

𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

≤ 𝜆2 |𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))| ∫0

Γ(𝛽)

|𝑓2 (𝑠, 𝑦(𝜓2 (𝑠)))| 𝜙′(𝑠)𝑑𝑠

|𝑥(𝜓2 (𝑠)) − 𝑦(𝜓2 (𝑠))| 𝜙′(𝑠) 𝑑𝑠

General Fractional Order

Advances in Special Functions of Fractional Calculus 𝑡 (𝜙(𝑡)−𝜙(𝑠))𝛽−1

+𝜆1 |𝑥(𝜓1 (𝑡)) − 𝑦(𝜓1 (𝑡))| ∫0 ≤ [𝜆1 𝑟 + 𝑀1 ] ≤ [[𝜆1 𝑟 + 𝑀1 ]

𝜆2 𝑇 𝛽 ||𝑥−𝑦|| Γ(𝛽+1)

𝜆2 𝑇 𝛽 Γ(𝛽+1)

|𝑓2 (𝑠, 𝑦(𝜓2 (𝑠)))| 𝜙′(𝑠) 𝑑𝑠

Γ(𝛽)

+ 𝜆1 ||𝑥 − 𝑦||[

𝑀 𝑇 𝛽−𝛾2

Γ(𝛽−𝛾2 +1)

𝑀 𝑇 𝛽−𝛾2

+ 𝜆1 [

Γ(𝛽−𝛾2 +1)

+

201

𝜆2 𝑇 𝛽 𝑟 Γ(𝛽+1)

+

𝜆2 𝑇 𝛽 𝑟 Γ(𝛽+1)

]

]]||𝑥 − 𝑦||

Then |Ϝ𝑥(𝑡) − Ϝ𝑦(𝑡)| ≤ Λ||𝑥 − 𝑦||, Λ ∈ (0,1), where Λ = [[𝜆1 𝑟 + 𝑀1 ]

𝜆2 𝑇 𝛽 Γ(𝛽+1)

+ 𝜆1 [

𝑀 𝑇 𝛽−𝛾2 Γ(𝛽−𝛾2 +1)

+

𝜆2 𝑇 𝛽 𝑟 Γ(𝛽+1)

]].

Then Ϝ is a contraction. Therefore, by the Banach contraction fixed point Theorem [26], the operator Ϝ has a unique fixed point 𝑥 ∈ 𝐶(𝐽) (i.e., the quadratic integral equation (1) has a unique solution 𝑥 ∈ 𝐶(𝐽) ) which completes the proof. 5.2. Maximal and Minimal Solutions Definition 5 [37] Let 𝑞(𝑡) be a solution 𝑥(𝑡) of (1). Then 𝑞(𝑡) is said to be a maximal solution of (1) if every solution of (1) on 𝐽 satisfies the inequality 𝑥(𝑡) ≤ 𝑞(𝑡), 𝑡 ∈ 𝐽 . A minimal solution 𝑠(𝑡) can be defined in a similar way by reversing the above inequality i.e. 𝑥(𝑡) ≥ 𝑠(𝑡), 𝑡 ∈ 𝐽. We need the following lemma to prove the existence of maximal and minimal solutions of (1). Lemma 5 Let 𝑓𝑖 (𝑡, 𝑥), 𝑖 = 1,2 satisfy the assumptions in Theorem 2 and let 𝑥(𝑡), 𝑦(𝑡) be continuous functions on 𝐽 satisfying 𝛽

𝑥(𝑡) ≤ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡)))

202

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem 𝛽

𝑦(𝑡) ≥ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑦(𝜓1 (𝑡))) 𝐼𝜙 𝑓2 (𝑡, 𝑦(𝜓2 (𝑡))) where one of them is strict. Suppose 𝑓𝑖 (𝑡, 𝑥), 𝑖 = 1,2 are nondecreasing functions in 𝑥. Then 𝑥(𝑡) < 𝑦(𝑡), 𝑡 ∈ 𝐽.

(6)

Proof Let the conclusion (6) be false; then there exists 𝑡1 such that 𝑥(𝑡1 ) = 𝑦(𝑡1 ),

𝑡1 > 0

and 𝑥(𝑡) < 𝑦(𝑡),

0 < 𝑡 < 𝑡1 .

From the monotonicity of the functions 𝑓𝑖 in 𝑥, we get 𝛽

𝑥(𝑡1 ) ≤ 𝑎(𝑡1 ) + 𝑓1 (𝑡1 , 𝑥(𝜓1 (𝑡1 ))) 𝐼𝜙 𝑓2 (𝑡1 , 𝑥(𝜓2 (𝑡1 ))) 𝑡 (𝜙(𝑡1 ) − 𝜙(𝑠))𝛽 − 1 Γ(𝛽)

= 𝑎(𝑡1 ) + 𝑓1 (𝑡1 , 𝑥(𝜓1 (𝑡1 ))) ∫0 1

𝑡 (𝜙(𝑡1 ) − 𝜙(𝑠))𝛽 − 1

< 𝑎(𝑡1 ) + 𝑓1 (𝑡1 , 𝑦(𝜓1 (𝑡1 ))) ∫0 1

Γ(𝛽)

𝑓2 (𝑠, 𝑥(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠

𝑓2 (𝑠, 𝑦(𝜓2 (𝑠))) 𝜙′(𝑠) 𝑑𝑠

< 𝑦(𝑡1 ). This contradicts the fact that 𝑥(𝑡1 ) = 𝑦(𝑡1 ); then 𝑥(𝑡) < 𝑦(𝑡), 𝑡 ∈ 𝐽. As particular cases of Lemma 5, we remark on the following: • For 𝜙(𝑡) = 𝑡 𝑚 , 𝑚 > 0, all the assumptions of Lemma 5 are satisfied and 𝛽

𝑥(𝑡) ≤ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) 𝐼𝑚 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) 𝛽

𝑦(𝑡) ≥ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑦(𝜓1 (𝑡))) 𝐼𝑚 𝑓2 (𝑡, 𝑦(𝜓2 (𝑡)))

General Fractional Order

Advances in Special Functions of Fractional Calculus

203

where one of them is strict. Then 𝑥(𝑡) < 𝑦(𝑡), 𝑡 ∈ 𝐽. • For 𝜙(𝑡) = 𝑡, all the assumptions of Lemma 5 are satisfied and 𝑥(𝑡) ≤ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) 𝐼𝛽 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) 𝑦(𝑡) ≥ 𝑎(𝑡) + 𝑓1 (𝑡, 𝑦(𝜓1 (𝑡))) 𝐼𝛽 𝑓2 (𝑡, 𝑦(𝜓2 (𝑡))) where one of them is strict. Then 𝑥(𝑡) < 𝑦(𝑡)

𝑓𝑜𝑟 𝑡 ∈ 𝐽.

Theorem 4 Let the assumptions of Theorem 2 be satisfied. Furthermore, if 𝑓𝑖 (𝑡, 𝑥), 𝑖 = 1,2 are nondecreasing functions in 𝑥, then there exist maximal and minimal solutions of (1). Proof Firstly, we shall prove the existence of maximal solution of (1). Let 𝜖 > 0 be given. Now consider the fractional-order quadratic functional integral equation. 𝛽

𝑥𝜖 (𝑡) = 𝑎(𝑡) + 𝑓1,𝜖 (𝑡, 𝑥𝜖 (𝜓1 (𝑡))) 𝐼𝜙 𝑓2,𝜖 (𝑡, 𝑥𝜖 (𝜓2 (𝑡))),

(7)

where 𝑓1,𝜖 (𝑡, 𝑥𝜖 (𝜓1 (𝑡))) = 𝑓1 (𝑡, 𝑥𝜖 (𝜓1 (𝑡))) + 𝜖 and 𝑓2,𝜖 (𝑡, 𝑥𝜖 (𝜓2 (𝑡))) = 𝑓2 (𝑡, 𝑥𝜖 (𝜓2 (𝑡))) + 𝜖. Clearly the functions 𝑓𝑖𝜖 (𝑡, 𝑥𝜖 ) satisfy assumptions (ii), (iv) and | 𝑓1,𝜖 (𝑡, 𝑥𝜖 ) | ≤ 𝑀1 + 𝑏𝑟 + 𝜖 = 𝑀′. | 𝑓2𝜖 (𝑡, 𝑥𝜖 ) | ≤ 𝑚2 (𝑡) + 𝜖 + 𝑏 |𝑥| = 𝑚′(𝑡) + 𝑏 |𝑥|. Therefore, equation (7) has a continuous solution 𝑥𝜖 (𝑡) according to Theorem 2.

204

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Let 𝜖1 and 𝜖2 be such that 0 < 𝜖2 < 𝜖1 < 𝜖. Then 𝛽

𝑥𝜖1 (𝑡) = 𝑎(𝑡) + 𝑓1,𝜖1 (𝑡, 𝑥𝜖1 (𝜓1 (𝑡))) 𝐼𝜙 𝑓2,𝜖1 (𝑡, 𝑥𝜖1 (𝜓2 (𝑡))), 𝛽

𝑥𝜖1 (𝑡) = 𝑎(𝑡) + (𝑓1 (𝑡, 𝑥𝜖1 (𝜓1 (𝑡))) + 𝜖1 ) 𝐼𝜙 (𝑓2 (𝑡, 𝑥𝜖1 (𝜓2 (𝑡))) + 𝜖1 ), 𝛽

> 𝑎(𝑡) + (𝑓1 (𝑡, 𝑥𝜖1 (𝜓1 (𝑡))) + 𝜖2 ) 𝐼𝜙 (𝑓2 (𝑡, 𝑥𝜖1 (𝜓2 (𝑡))) + 𝜖2 ), 𝛽

𝑥𝜖2 (𝑡) = 𝑎(𝑡) + (𝑓1 (𝑡, 𝑥𝜖2 (𝜓1 (𝑡))) + 𝜖2 ) 𝐼𝜙 (𝑓2 (𝑡, 𝑥𝜖2 (𝜓2 (𝑡))) + 𝜖2 ).

(8) (9)

Applying Lemma 5, then (8) and (9) imply that 𝑥𝜖2 (𝑡) < 𝑥𝜖1 (𝑡)

𝑓𝑜𝑟 𝑡 ∈ 𝐽.

As shown before in the proof of Theorem 2, the family of functions 𝑥𝜖 (𝑡) defined by equation (7) is uniformly bounded and of equi-continuous functions. Hence to the Arzela-Ascoli Theorem, there exists a decreasing sequence 𝜖𝑛 such that 𝜖𝑛 → 0 as 𝑛 → ∞, and lim 𝑥𝜖𝑛 (𝑡) exists uniformly in 𝐽. We denote this limit by 𝑛→∞

𝑞(𝑡). From the continuity of the functions 𝑓𝑖,𝜖𝑛 , 𝑖 = 1,2 in the second argument, we get 𝛽

𝑞(𝑡) = lim 𝑥𝜖𝑛 (𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑞(𝜓1 (𝑡))) 𝐼𝜙 𝑓2 (𝑡, 𝑞(𝜓2 (𝑡))) 𝑛→∞

which proves that 𝑞(𝑡) is a solution of (1). Finally, we shall show that 𝑞(𝑡) is a maximal solution of (1). To do this, let 𝑥(𝑡) be any solution of (1). Then 𝛽

𝑥𝜖 (𝑡) = 𝑎(𝑡) + 𝑓1,𝜖 (𝑡, 𝑥𝜖 (𝜓1 (𝑡)))𝐼𝜙 𝑓2,𝜖 (𝑡, 𝑥𝜖 (𝜓2 (𝑡))) 𝛽

> 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥𝜖 (𝜓1 (𝑡))) 𝐼𝜙 𝑓2 (𝑡, 𝑥𝜖 (𝜓2 (𝑡))) and 𝛽

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))). Applying Lemma 5, we get

General Fractional Order

Advances in Special Functions of Fractional Calculus

𝑥𝜖 (𝑡) > 𝑥(𝑡)

205

𝑓𝑜𝑟 𝑡 ∈ 𝐽.

From the uniqueness of the maximal solution (see [46]), it is clear that 𝑥𝜖 (𝑡) tends to 𝑞(𝑡) uniformly in 𝑡 ∈ 𝐽 𝑎𝑠 𝜖 → 0. In a similar way, we can prove that there exists a minimal solution of (1). 5.3. Comparison Theorem An important technique is concerned with comparing a function satisfying a quadratic integral inequality of fractional order by the maximal and the minimal solutions of the corresponding fractional-order integral equation. Some of the results that are widely used are the following comparison theorems: Theorem 5 Let the assumptions of Theorem 4 be satisfied and 𝛽

𝑥(𝑡) ≤ 𝑎(𝑡) + 𝐼𝜙𝛼1 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡))) 𝐼𝜙2 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑡 ≥ 0,

(10)

where 𝑥(𝑡) is a continuous function on 𝐽. Suppose that 𝑞(𝑡) is the maximal solution of the fractional-order quadratic integral equation 𝛽

𝑢(𝑡) = 𝑎(𝑡) + 𝐼𝜙𝛼1 𝑓1 (𝑡, 𝑢(𝜓1 (𝑡))) 𝐼𝜙2 𝑓2 (𝑡, 𝑢(𝜓2 (𝑡))) existing on 𝐽. Then 𝑥(𝑡) ≤ 𝑞(𝑡), 𝑡 ∈ 𝐽. Proof Let 𝑢(𝑡, 𝜖) be any solution of 𝛽

𝑢(𝑡, 𝜖) = 𝑎(𝑡) + 𝐼𝜙𝛼1 𝑓1𝜖 (𝑡, 𝑢(𝜓1 (𝑡), 𝜖)). 𝐼𝜙2 𝑓2𝜖 (𝑡, 𝑢(𝜓2 (𝑡), 𝜖)) for 𝜖 > 0 sufficiently small. Since lim 𝑢(𝑡, 𝜖) = 𝑞(𝑡),

𝜖→0

then, it is enough to show that 𝑥(𝑡) < 𝑢(𝑡, 𝜖), 𝑡 ≥ 0.

(11)

206

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

Since 𝛽

𝑢(𝑡, 𝜖) = 𝑎(𝑡) + 𝐼𝜙𝛼1 (𝑓1 (𝑡, 𝑢(𝜓1 (𝑡))) + 𝜖). 𝐼𝜙2 (𝑓2 (𝑡, 𝑢(𝜓2 (𝑡))) + 𝜖) 𝛽

> 𝑎(𝑡) + 𝐼𝜙𝛼1 𝑓1 (𝑡, 𝑢(𝜓1 (𝑡))). 𝐼𝜙2 𝑓2 (𝑡, 𝑢(𝜓2 (𝑡))), 𝑡 ≥ 0. Hence, an application of Lemma 5 shows that 𝑥(𝑡) < 𝑢(𝑡, 𝜖). Taking the limit when 𝜖 → 0, we get 𝑥(𝑡) ≤ 𝑞(𝑡). Theorem 6 Let the assumptions of Theorem 4 be satisfied and reversing inequality (10). Then 𝑥(𝑡) ≥ 𝑠(𝑡), where 𝑠(𝑡) is the minimal solution of (11) on 𝐽. The 𝜙 −fractional order integral equation 𝛽

𝑥(𝑡) = 𝑎(𝑡) + 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑡 ∈ 𝐽,

(12)

is studied, and the same results are proved in [38] under the assumptions. (i*) 𝑎: 𝐽 → ℝ is continuous and bounded with 𝑘1 = sup |𝑎(𝑡)|. 𝑡∈𝐽

(ii*) 𝑓2 : 𝐽 × ℝ → ℝ satisfies the Carathèodory condition (i.e., measurable in 𝑡 for all 𝑥: 𝐽 → ℝ and continuous in 𝑥 for all 𝑡 ∈ 𝐽 ). (iii*)There exists a function 𝑚 ∈ 𝐿1 such that |𝑓2 (𝑡, 𝑥)| ≤ 𝑚(𝑡) (∀ (𝑡, 𝑥) ∈ 𝐽 × ℝ ) and 𝑘2 = sup𝐼𝛽 𝑚(𝑡) for any 𝛽 ≤ 𝛼. 𝑡∈𝐽

Moreover, 𝑓2 is monotonic non-decreasing in the second argument. (iv*) 𝜓: 𝐽 → 𝐽 is continuous.

General Fractional Order

Advances in Special Functions of Fractional Calculus

207

(iiv*) 𝜙: 𝐽 → 𝐽 be any monotonic increasing function having a continuous derivative. And the following results are proven. Theorem 7 [38] Let the assumptions (i*)-(iiv*) be satisfied. Then the fractional integral equation (12) has maximal and minimal solutions. Corollary 14 [38] Let the assumptions (i*)-(iiv*) be satisfied and 𝛽

𝑥(𝑡) ≤ 𝑎(𝑡) + 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑡 ≥ 0,

(13)

where 𝑥(𝑡) is a continuous function on 𝐽. Suppose that 𝑞(𝑡) is the maximal solution of the fractional-order integral equation 𝛽

𝑢(𝑡) = 𝑎(𝑡) + 𝐼𝜙 𝑓2 (𝑡, 𝑢(𝜓2 (𝑡)))

(14)

existing on 𝐽. Then 𝑥(𝑡) ≤ 𝑞(𝑡), 𝑡 ∈ 𝐽. Corollary 15 [38] Let the assumptions (i*)-(iiv*) be satisfied and reverse inequality (13). Then 𝑥(𝑡) ≥ 𝑠(𝑡), where 𝑠(𝑡) is the minimal solution of (14) on 𝐽 . 5.4. 𝜹 −Approximate Solutions Let us define an approximate solution of (12). Definition 6 [38] Let 𝑥(𝑡) be continuous on 𝐽 and satisfies 𝛽

| 𝑥(𝑡) − 𝑎(𝑡) − 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) | ≤ 𝛿(𝑡), where 𝛿 is continuous on 𝐽. Then 𝑥(𝑡) is said to be a 𝛿 −approximate solution of (12).

208

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

The following theorem shows the difference between an approximate solution and any other solution of (12). Theorem 8 [38] Let 𝑓2 (𝑡, 𝑥), 𝑔(𝑡, 𝑥) satisfy the assumptions of Theorem 4, 𝑔(𝑡, 𝑥) is monotonic nondecreasing in 𝑥 for each 𝑡, and | 𝑓2 (𝑡, 𝑥) − 𝑓2 (𝑡, 𝑦) | ≤ 𝑔(𝑡, | 𝑥 − 𝑦 |).

(15)

If 𝑥(𝑡, 𝛿) is a 𝛿 −approximate solution of (12) and 𝑦(𝑡) is any solution of (12). Then | 𝑥(𝑡, 𝛿) − 𝑦(𝑡)| ≤ 𝑞(𝑡), where 𝑞(𝑡) is the maximal solution of 𝛽

𝑢(𝑡) = 𝛿(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑢(𝜓2 (𝑡))). Proof Consider the function 𝑛(𝑡) = | 𝑥(𝑡, 𝛿) − 𝑦(𝑡) |, where 𝑥(𝑡, 𝛿) is 𝛿 −approximate solution of (12) and 𝑦(𝑡) is any solution of (12). Then, using the definition of 𝛿 −approximate solution and (15), we get 𝑛(𝑡) = | 𝑥(𝑡, 𝛿) − 𝑦(𝑡) | 𝛽

= |𝑥(𝑡, 𝛿) − 𝑎(𝑡) − 𝐼𝜙 𝑓2 (𝑡, 𝑦(𝜓2 (𝑡)))| 𝛽

≤ | 𝑥(𝑡, 𝛿) − 𝑎(𝑡) − 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡), 𝛿))| 𝛽

+ 𝐼𝜙 | 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡), 𝛿)) −

𝑓2 (𝑡, 𝑦(𝜓2 (𝑡))) |

𝛽

≤ 𝛿(𝑡) + 𝐼𝜙 𝑔(𝑡, | 𝑥(𝜓2 (𝑡), 𝛿) − 𝑦(𝑡) |) 𝛽

= 𝛿(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑛(𝑡)). By a direct application of Comparison Corollary 14, we get

General Fractional Order

Advances in Special Functions of Fractional Calculus

209

𝑛(𝑡) = | 𝑥(𝑡, 𝛿)– 𝑦(𝑡)| ≤ 𝑞(𝑡), 𝑡 ≥ 0. The next theorem offers an estimate of the growth of solutions of (12). Theorem 9 [38] Let 𝑓2 (𝑡, 𝑥) and 𝑔(𝑡, 𝑥) satisfy the assumptions in Theorem 4. If |𝑓2 (𝑡, 𝑥)| ≤ 𝑔(𝑡, |𝑥|);

(16)

then |𝑥(𝑡)| ≤ 𝑞(𝑡), 𝑡 ≥ 0, where 𝑥(𝑡) is any solution of (12) and 𝑞(𝑡) is the maximal solution of 𝛽

𝑢(𝑡) = ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑢(𝜓2 (𝑡))) such that | 𝑎(𝑡) | ≤ ℎ(𝑡),

(17)

𝑡 ∈ 𝐽.

Proof If 𝑛(𝑡) = | 𝑥(𝑡) | , we have by (16) the fractional-order integral inequality 𝛽

𝑛(𝑡) = | 𝑥(𝑡) | ≤ | 𝑎(𝑡) | + 𝐼𝜙 | 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) | 𝛽

≤ ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, | 𝑥(𝜓2 (𝑡)) |) 𝛽

= ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑛(𝑡)), and consequently, Comparison Corollary 14 gives 𝑛(𝑡) = | 𝑥(𝑡) | ≤ 𝑞(𝑡), 𝑡 ≥ 0, where 𝑞(𝑡) is the maximal solution of (17). Theorem 10 [38] Assume that: a) b) c)

𝑣1 , 𝑣2 , 𝑔 satisfy the assumptions of Theorem 4, 𝑔(𝑡, 𝑢) is monotonic nondecreasing in 𝑢 for each 𝑡, and |𝑣1 (𝑡, 𝑥) − 𝑣2 (𝑡, 𝑦)| ≤ 𝑔(𝑡, |𝑥 − 𝑦 |); (18) 𝑥(𝑡), 𝑦(𝑡) are any two solutions of 𝛽

𝑥(𝑡) = 𝑎1 (𝑡) + 𝐼𝜙 𝑣1 (𝑡, 𝑥(𝜓2 (𝑡))),

210

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

𝛽

𝑦(𝑡) = 𝑎2 (𝑡) + 𝐼𝜙 𝑣2 (𝑡, 𝑦(𝜓2 (𝑡))), respectively; d) 𝑞(𝑡) is the maximal solution of 𝛽

𝑢(𝑡) = ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑢(𝜓2 (𝑡))) such that | 𝑎1 (𝑡) − 𝑎2 (𝑡) | = ℎ(𝑡), 𝑡 ∈ 𝐽, where 𝑎1 , 𝑎2 , ℎ are continuous on 𝐽. Then | 𝑥(𝑡) − 𝑦(𝑡) | ≤ 𝑞(𝑡),

𝑡 ∈ 𝐽.

Proof The proof is an easy modification of the proof of Theorem 8. For, setting 𝑛(𝑡) = |𝑥(𝑡) − 𝑦(𝑡)| and using (18), we obtain 𝑛(𝑡) = | 𝑥(𝑡) − 𝑦(𝑡) | 𝛽

≤ |𝑎1 (𝑡) − 𝑎2 (𝑡) | + 𝐼𝜙 | 𝑣1 (𝑡, 𝑥(𝜓2 (𝑡))) − 𝑣2 (𝑡, 𝑦(𝜓2 (𝑡))) | 𝛽

≤ ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, | 𝑥(𝜓2 (𝑡)) − 𝑦(𝜓2 (𝑡)) |) 𝛽

= ℎ(𝑡) + 𝐼𝜙 𝑔(𝑡, 𝑛(𝑡)). The result follows from Corollary 14. Corollary 16 Let the assumptions in Theorem 10 be satisfied with 𝑎1 = 𝑎2 , then | 𝑥(𝑡) − 𝑦(𝑡) | ≤ 𝑞(𝑡), where 𝑞(𝑡) is the maximal solution of 𝛽

𝑢(𝑡) = 𝐼𝜙 𝑔(𝑡, 𝑢(𝜓2 (𝑡))).

(19)

General Fractional Order

Advances in Special Functions of Fractional Calculus

211

Theorem 11 [38] Let the assumptions in Theorem 10 be satisfied with 𝑎1 = 𝑎2 and 𝑣1 = 𝑣2 , then 𝑥(𝑡) = 𝑦(𝑡). If 𝑢(𝑡) = 0 is the only solution of the fractional-order integral equation (19). Proof Let 𝑥(𝑡), 𝑦(𝑡) be as in Theorem 10. Setting 𝑛(𝑡) = | 𝑥(𝑡) − 𝑦(𝑡) | and arguing as before, we get 𝑛(𝑡) ≤ 𝑞(𝑡),

𝑡 ∈ 𝐽,

where 𝑞(𝑡) is the maximal solution of (19). Since 𝑢(𝑡) = 0 is the only solution of (19), then 𝑛(𝑡) = | 𝑥(𝑡) − 𝑦(𝑡) | ≤ 0

⇒

𝑥(𝑡) = 𝑦(𝑡).

Remark 2 Clearly, when 𝑣1 = 𝑣2 condition (18) yields Perron’s condition, which implies a uniqueness theorem of Perron type. Remark 3 When 𝑔(𝑡, 𝑢) = 𝐾 𝑢 and 𝑣1 = 𝑣2 then condition (18) becomes Lipschitz condition. Now consider the following two fractional-order functional integral equations 𝛽

𝑥(𝑡) = 𝑎1 (𝑡) + 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))),

𝛽 ∈ (0,1),

𝑦(𝑡) = 𝑎2 (𝑡) + 𝐼𝜙𝛼 𝑓2 (𝑡, 𝑦(𝜓2 (𝑡))), 𝛽 ≥ 𝛼,

(20) (21)

where 𝑎1 (𝑡), 𝑎2 (𝑡) are continuous functions on 𝐽. The following is another comparison theorem more general than Theorem 10. Theorem 12 [38] Assume that: 1) 𝑓, 𝑔 satisfy the assumptions of Theorem 4, 𝑔(𝑡, 𝑢) is monotonic non-decreasing in 𝑢 for each 𝑡, and 𝛽−𝛼

| 𝐼𝜙

𝑓(𝑡, 𝑥) − 𝑓(𝑡, 𝑦) | ≤ 𝑔(𝑡, | 𝑥 − 𝑦 |);

2) 𝑥(𝑡), 𝑦(𝑡) are any two solutions of (20) and (21), respectively;

212

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

3) 𝑞(𝑡) is the maximal solution of 𝑢(𝑡) = ℎ(𝑡) + 𝐼𝜙𝛼 𝑔(𝑡, 𝑢(𝜓2 (𝑡))) such that | 𝑎1 (𝑡) − 𝑎2 (𝑡) | ≤ ℎ(𝑡), 𝑡 ∈ 𝐽, where ℎ(𝑡) is continuous on 𝐽. Then | 𝑥(𝑡) − 𝑦(𝑡) | ≤ 𝑞(𝑡),

𝑡 ∈ 𝐽.

Proof. Similarly, as in Theorem 10, putting 𝑛(𝑡) = | 𝑥(𝑡) − 𝑦(𝑡) | 𝛽−𝛼

≤ | 𝑎1 (𝑡) − 𝑎2 (𝑡) + 𝐼𝜙𝛼 |𝐼𝜙

𝑓(𝑡, 𝑥(𝜓2 (𝑡))) − 𝑓(𝑡, 𝑦(𝜓2 (𝑡)))|, 𝛽 ≥ 𝛼

≤ ℎ(𝑡) + 𝐼𝜙𝛼 𝑔(𝑡, | 𝑥(𝑡) − 𝑦(𝑡) |) = ℎ(𝑡) + 𝐼𝜙𝛼 𝑔(𝑡, 𝑛(𝑡)), It follows from Theorem 5 that | 𝑥(𝑡)– 𝑦(𝑡)| ≤ 𝑞(𝑡), 𝑡 ∈ 𝐽. 6. APPLICATIONS In this section, we consider some applications to our results. 6.1. Hybrid Functional 𝝓 −Differential Equation of Fractional Order Consider the 𝜙 − quadratic integral equation in the form 𝛽

𝑥(𝑡) = 𝑎(𝑡) + 𝑓1 (𝑡, 𝑥(𝜓1 (𝑡)))𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))) which can be reformulated as: 𝑥(𝑡)−𝑎(𝑡) 𝑓1 (𝑡,𝑥(𝜓1 (𝑡))) 1−𝛽

Operating both sides by 𝐼𝜙

𝛽

= 𝐼𝜙 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))).

, we have

(22)

General Fractional Order

Advances in Special Functions of Fractional Calculus 1−𝛽

𝐼𝜙

𝑓1 (𝑡,𝑥(𝜓1 (𝑡)))

1

Next, operating both sides by (

𝑥(𝑡)−𝑎(𝑡)

(

𝑑

𝜙′(𝑡) 𝑑𝑡

1

(

𝑑

𝜙′(𝑡) 𝑑𝑡

213

) = 𝐼𝜙1 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡)))

), we get

1−𝛽

𝑥(𝑡)−𝑎(𝑡)

)𝐼𝜙2 (

𝑓1 (𝑡,𝑥(𝜓1 (𝑡)))

) = 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))),

which can be written as 𝛽

𝑥(𝑡)−𝑎(𝑡)

𝐷𝜙 (

𝑓1 (𝑡,𝑥(𝜓1 (𝑡)))

) = 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑥(0) = 𝑎(0) 𝛽 ∈ (0,1).

(23)

And known as a nonlinear hybrid functional 𝜙 −differential equation of fractional order. The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. The consideration of hybrid differential equations is implicit in the works of Krasnoselskii [42] and extensively treated in several papers on hybrid differential equations with different perturbations. See [43-47]. Conversely, operating both 𝛽 sides of (23) by 𝐼𝜙 , we can prove the equivalence between (22) and (23), and hence we have the following results. Theorem 13 Let the assumptions of Theorem 2. Then the nonlinear hybrid functional 𝜙 −differential equation of fractional order (23) has at least one solution in 𝐶(𝐽). In particular cases of Theorem 13, we have Corollary 17 Let the assumptions of Theorem 2 with 𝑓1 = 1 Then the nonlinear hybrid functional 𝜙 −differential equation of fractional order 𝛽

𝐷𝜙 (𝑥(𝑡) − 𝑎(𝑡)) = 𝑓2 (𝑡, 𝑥(𝜓2 (𝑡))), 𝑥(0) = 𝑎(0) 𝛽 ∈ (0,1).

(24)

has at least one solution in 𝐶(𝐽). 6.2. Pantograph Functional 𝝓 −Differential Equation of Fractional Order The pantograph equation is a special type of functional differential equation with proportional delay. It arises in rather different fields of pure and applied mathematics, such as electrodynamics, control systems, number theory, probability,

214

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

and quantum mechanics. Many researchers have studied the pantograph-type delay differential equation using analytical and numerical techniques [48]. Pantograph equations are characterized by the presence of a linear functional argument. These equations arise in several applications, and often, the argument has a repelling fixed point at the origin. Marshall et al. [49] studied a related class of functional differential equations with nonlinear functional arguments and showed that, generically, solutions to such equations have a natural boundary. Marshall et al. [50] A natural boundary for solutions to the second order pantograph equation. G. Derfel, A. Iserles [51] studied a pantograph equation in the complex plane. As a particular case of Theorem 13, we obtain the existing result for 𝜙 −differential equation of pantograph-type delay of fractional order 𝛽

𝑥(𝑡)−𝑎(𝑡)

𝐷𝜙 (

𝑓1 (𝑡,𝑥(𝜎1 𝑡))

) = 𝑓2 (𝑡, 𝑥(𝜎2 𝑡)), 𝑥(0) = 𝑎(0) 𝑡 ∈ 𝐽, 𝛽 ∈ (0,1).

(25)

where 𝜎1 , 𝜎2 ∈ (0,1). CONCLUSION Quadratic integral equations(QIEs) have been investigated from different points of view and using different techniques (see [1–11] and [43-47, 52]). In this chapter, we have discussed a nonlinear quadratic integral equation of fractional order. Firstly, we have introduced fractional 𝜙 −integral and some of its properties. Then we derived some existence theorems for that equation. Moreover, we establish some particular cases and corollaries. Finally, we introduce some applications of the results presented in this work. CONSENT FOR PUBLICATON Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest.

General Fractional Order

Advances in Special Functions of Fractional Calculus

215

ACKNOWLEDGEMENT The authors are very grateful to the anonymous referee for valuable comments and suggestions. REFERENCES [1]

[2] [3]

[4]

[5] [6]

[7] [8]

[9]

[10]

[11]

[12]

I.K. Argyros, "Quadratic equations and applications to Chandrasekhar’s and related equations", Bull. Aust. Math. Soc., vol. 32, no. 2, pp. 275-292, 1985. http://dx.doi.org/10.1017/S0004972700009953 I.K. Argyros, "On a class of quadratic integral equations with perturbations", Funct. Approx., vol. 20, pp. 51-63, 1992. J. Banaś, "On the superposition operator and integrable solutions of some functional equation", Nonlinear Anal., vol. 12, no. 8, pp. 777-784, 1988. http://dx.doi.org/10.1016/0362-546X(88)90038-7 J. Banaś, "Integrable solutions of Hammerstein and Urysohn integral equations", J. Aust. Math. Soc. A Pure Math. Stat., vol. 46, no. 1, pp. 61-68, 1989. [Series A]. http://dx.doi.org/10.1017/S1446788700030378 J. Banas, and Z. Knap, "Integrable solutions of a functional integral equation", Revista Matehatica de la Universidad Compltense de Madrid, vol. 2, 1989no. 1, . J. Banaś, M. Lecko, and W.G. El-Sayed, "Existence theorems of some quadratic integral equation", J. Math. Anal. Appl., vol. 222, no. 1, pp. 276-285, 1998. http://dx.doi.org/10.1006/jmaa.1998.5941 J. Banas, and W.G. El-Sayed, Solvability of functional and integral equations in some classes of integrable functions., Politechnik. Rzeszowska: Rzeszow, 1993. J. Banaś, and A. Martinon, "Monotonic solutions of a quadratic integral equation of Volterra type", Comput. Math. Appl., vol. 47, no. 2-3, pp. 271-279, 2004. http://dx.doi.org/10.1016/S0898-1221(04)90024-7 J. Banaś, J. Caballero, J. Rocha, and K. Sadarangani, "Monotonic solutions of a class of quadratic integral equations of Volterra type", Comput. Math. Appl., vol. 49, no. 5-6, pp. 943952, 2005. http://dx.doi.org/10.1016/j.camwa.2003.11.001 J. Banaś, J. Rocha Martin, and K. Sadarangani, "On solutions of a quadratic integral equation of Hammerstein type", Math. Comput. Model., vol. 43, no. 1-2, pp. 97-104, 2006. http://dx.doi.org/10.1016/j.mcm.2005.04.017 J. Banaś, and B. Rzepka, "Monotonic solutions of a quadratic integral equation of fractional order", J. Math. Anal. Appl., vol. 332, no. 2, pp. 1371-1379, 2007. http://dx.doi.org/10.1016/j.jmaa.2006.11.008 A. M. A. El-Sayed, and H. H. G. Hashem, "Existence results for nonlinear quadratic functional integral equations of fractional order", Miskolc Mathematical Notes, vol. 14, no. 1, pp. 79-87, 2013.

216

[13] [14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24] [25]

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

http://dx.doi.org/10.18514/MMN.2013.578 A.M.A. El-Sayed, and H.H.G. Hashem, "Carathe'odory type theorem for a nonlinear quadratic integral equation", MATH. SCI. RES. J, vol. 12, no. 4, pp. 71-95, 2008. A. El-Sayed, and H.H.G. Hashem, "Integrable and continuous solutions of a nonlinear quadratic integral equation", Electron. J. Qual. Theory Differ. Equ., vol. 25, no. 25, pp. 1-10, 2008. http://dx.doi.org/10.14232/ejqtde.2008.1.25 A.M.A. El-Sayed, and H.H.G. Hashem, "Monotonic positive solution of nonlinear quadratic Hammerstein and Urysohn functional integral equations", Commentationes Mathematicae., vol. 48, no. 2, pp. 199-207, 2008. A.M.A. El-Sayed, and H.H.G. Hashem, "Weak maximal and minimal solutions for Hammerstein and Urysohn integral equations in reflexive Banach spaces", Differential Equation and Control Processes, no. 4, pp. 50-62, 2008. A. El-Sayed, and H.H.G. Hashem, "Monotonic solutions of functional integral and differential equations of fractional order", Electron. J. Qual. Theory Differ. Equ., no. 7, pp. 1-8, 2009. http://dx.doi.org/10.14232/ejqtde.2009.1.7 A.M.A. El-Sayed, H.H.G. Hashem, and E.A.A. Ziada, "Picard and Adomian methods for quadratic integral equation", Comput. Appl. Math., vol. 29, no. 3, pp. 447-463, 2010. http://dx.doi.org/10.1590/S1807-03022010000300007 A.M.A. El-Sayed, H.H.G. Hashem, and E.A.A. Ziada, "Picard and Adomian decomposition methods for a quadratic integral equation of fractional order", Mat. Apl. Comput., vol. 33, no. 1, pp. 95-109, 2014. http://dx.doi.org/10.1007/s40314-013-0045-3 W.G. El-Sayed, and B. Rzepka, "Nondecreasing solutions of a quadratic integral equation of Urysohn type", Comput. Math. Appl., vol. 51, no. 6-7, pp. 1065-1074, 2006. http://dx.doi.org/10.1016/j.camwa.2005.08.033 M. M. El-Borai, W. G. El-Sayed, and M. I. Abbas, "Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type", Int. J. Contemp. Math. Sci, vol. 2, no. 2, pp. 89-102, 2007. http://dx.doi.org/10.12988/ijcms.2007.07004 M. Borai, W. El-Sayed, and M. Abbas, "Monotonic solutions of some singular integral equations of Volterra type", Studia Sci. Math. Hungar., vol. 46, no. 3, pp. 297-316, 2009. http://dx.doi.org/10.1556/sscmath.2009.1094 A. Erdélyi, "On fractional integration and its application to the theory of Hankel transforms", Quart. J. Math., Oxford, Second Ser, vol. 11, pp. 293-303, 1940. http://dx.doi.org/10.1093/qmath/os-11.1.293 A. Erdélyi Arthur, ""On some functional transformations", Rendiconti del Seminario Matematico dell’Universitá e del Politecnico di Torino, vol. 10, pp. 217-234, 1950. A. Erdélyi, and H. Kober, "Some remarks on Hankel transforms", Quart. J. Math., Oxford, Second Ser, vol. 11, pp. 212-221, 1940. http://dx.doi.org/10.1093/qmath/os-11.1.212

General Fractional Order

[26]

[27]

[28]

[29]

[30] [31] [32] [33] [34]

[35] [36] [37] [38]

[39] [40]

[41]

[42]

Advances in Special Functions of Fractional Calculus

217

K. Goebel, and W.A. Kirk, Topics in Metric Fixed point theory., Cambridge University Press: Cambridge, 1990. http://dx.doi.org/10.1017/CBO9780511526152 H. H. G Hashem, and M. S. Zaki, "Caratheèodory theorem for quadratic integral equations of Erdyéli-Kober type", Journal of Fractional Calculus and its Applications, vol. 4, no. 5, pp. 1-8, 2013. H.H.G. Hashem, and A.R. Al-Rwaily, Asymptotic Stability of Solutions to a Nonlinear Urysohn Quadratic Integral Equation, 2013. http://dx.doi.org/10.1155/2013/259418 G. Pagnini, "Erdélyi-Kober fractional diffusion", Fract. Calc. Appl. Anal., vol. 15, no. 1, pp. 117-127, 2012. http://dx.doi.org/10.2478/s13540-012-0008-1 A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations., Elsevier, North-Holland, 2006. S.G. Samko, A.A. Kilbas, and O. Marichev, Integrals and Derivatives of Fractional Orders and Some of their Applications., Nauka i Teknika: Minsk, 1987. J. Appell and P.P. Zabrejko, "Continuity properties of the superposition operator", Preprint, University of Augsburg, No.131, (1986). M.A. Kransel’skii, "On the continuity of the operator 𝐹(𝑥)=𝑓(𝑥,𝑢(𝑥))", Dokl. Akad. Nauk, vol. 77, pp. 185-188, 1951. J. Appell, and P.P. Zabrejko, Nonlinear superposition operator., Cambridge University Press, 1990. http://dx.doi.org/10.1017/CBO9780511897450 I. Podlubny, Fractional Differential equations., San Diego, NewYork, London, 1999. B. Ross, and K.S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations., John Wiley: New York, 1993. V. Lakshmikantham, and S. Leela, Differential and Integral Inequalities., vol. Vol. 1, Academic press: New York, London, 1969. H.H.G. Hashem, "On the solution of a generalized fractional order integral equation and some applications", Journal of Fractional Calculus and Applications, vol. 6, no. 1, pp. 120130, 2015. R.F. Curtain, and A.J. Pritchard, Functional Analysis in Modern Applied Mathematics., Academic press, 1977. H. H. G. Hashem, and A. M. A. El-Sayed, "Existence results for a quadratic integral equation of fractional order by a certain function", Fixed Point Theory, vol. 21, no. 1, pp. 181-19, 2020. http://dx.doi.org/10.24193/fpt-ro.2020.1.13 A. M. A. El-Sayed, F. M. Gaafar, and H. H. G. Hashem, "On the maximal and minimal solutions of arbitrary-orders nonlinear functional integral and differenbtial equations", MATH. SCi, RES.J, vol. 8, no. 11, pp. 336-348, 2004. M.A. Kransel’skii, P.P. Zabrejko, J.I. Pustyl’nik, and P.J. Sobolevskii, Integral operators in spaces of summable functions., Noordhoff: Leyden, 1976.

218

[43]

[44] [45] [46] [47]

[48] [49]

[50]

[51]

[52]

Advances in Special Functions of Fractional Calculus

El-Sayed and Hashem

http://dx.doi.org/10.1007/978-94-010-1542-4 B.C. Dhage, and B.D. Karande, First order integro-differential equations in Banach algebras involving Caratheodory and discontinuous nonlinearities, 2005. http://dx.doi.org/10.14232/ejqtde.2005.1.21 B.C. Dhage, and D. O’Regan, "A fixed point theorem in Banach algebras with applications to functional integral equations", Funct. Differ. Equ., vol. 7, pp. 259-267, 2000. B.C. Dhage, "On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations", J. Math. Phys. Sci., vol. 22, no. 5, pp. 603-611, 1988. B.C. Dhage, "A fixed point theorem in Banach algebras involving three operators with applications", Kyungpook Math. J., vol. 44, no. 1, pp. 145-155, 2004. B.C. Dhage, "Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations", Differential Equations & Applications, vol. 2, no. 4, pp. 465486, 2010. http://dx.doi.org/10.7153/dea-02-28 M.M. Bahşi, and M. Çevik, Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms, 2015. J.C. Marshall, B. van Brunt, and G.C. Wake, "Natural boundaries for solutions to a certain class of functional differential equations", J. Math. Anal. Appl., vol. 268, no. 1, pp. 157-170, 2002. http://dx.doi.org/10.1006/jmaa.2001.7806 J.C. Marshall, B. van-Brunt, and G.C. Wake, "A natural boundary for solutions to the second order pantograph equation", J. Math. Anal. Appl., vol. 299, no. 2, pp. 314-321, 2004. http://dx.doi.org/10.1016/j.jmaa.2003.12.051 G. Derfel, and A. Iserles, "The pantograph equation in the complex plane", J. Math. Anal. Appl., vol. 213, no. 1, pp. 117-132, 1997. http://dx.doi.org/10.1006/jmaa.1997.5483 M.A. Darwish, "On solvability of some quadratic functional-integral equation in banach algebra", Commun. Appl. Anal., vol. 11, pp. 441-450, 2007.

Advances in Special Functions of Fractional Calculus, 2023, 219-243

219

CHAPTER 12

Non-linear Set-Valued Delay Functional Integral Equations of Volterra-Stieltjes Type: Existence of Solutions, Continuous Dependence and Applications A. M. A. El-Sayed1,2 , Sh. M Al-Issa2,3, *, and Y. M. Y. Omar4 1

Faculty of Science, Alexandria University, Alexandria, Egypt

2

Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon 3

Faculty of Arts and Sciences, Department of Mathematics, The International University of Beirut, Beirut, Lebanon 4

Faculty of Science, Omar Al-Mukhtar University, Libya Abstract: In this chapter, we established two existence theorems for the non-linear Volterra-Stieltjes integral inclusion. The continuous dependence of the solutions on the delay functions, 𝑔𝑖 (𝑖 = 1,2) and on the set of selections, will be proved. The nonlinear Chandrasekhar set-valued functional integral equation and a non-linear Chandrasekhar quadratic functional integral equation, also the set-valued fractional orders integral equation, are studied as an application. An initial value problem of fractional-orders set-valued integro-differential equation will be considered.

Keywords: Non-linear functional integral equation, Volterra-Stieltjes integral inclusion, Chandrasekhar quadratic integral equation, Function of bounded variation, Continuous dependence, Differential inclusion, Delay function. 1. INTRODUCTION The non-linear Volterra-Stieltjes type integral operator 𝑡

𝑇𝑥(𝑡) = ∫0 𝑓(𝑠, 𝑥(𝑠)) 𝑑𝑠 𝑔(𝑡, 𝑠), 𝑡 ∈ 𝐼 = [0, 𝑇] *Corresponding

author Sh. M Al-Issa: Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

220

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

has been studied, recently by J.Banas (see [1,2]), and has been studied by some authors(see [3, 4]), and references therein. Here we discuss with the non-linear set-valued delay functional integral equations of Volterra-Stieltjes type 𝜑(𝑡)

𝑥(𝑡) ∈ 𝑎(𝑡) + ∫0

𝜑(𝑠)

𝐹1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃).

(1)

The existence of solutions in the class of continuous functions 𝑥 ∈ 𝐶[0, 𝑇] and the continuous dependence on the delay function 𝜑 and the set of selections, of the setvalued functions 𝐹, 𝑆𝐹 are proved. As applications of (1) the non-linear Chandrasekhar set-valued functional integral equation 𝜑(𝑡) 𝑡 𝜑(𝑠) 𝑠 𝑥(𝑡) ∈ 𝑎(𝑡) + ∫0 𝐹1 (𝑏1 (𝑠)𝑥(𝑠), ∫0 𝑏2 (𝑠)𝑥(𝜃)𝑑𝜃, (2) 𝑡+𝑠

𝑠+𝜃

the delay Chandrasekhar quadratic functional integral equation 𝜑(𝑡)

𝑥(𝑡) = 𝑎(𝑡) + ∫0

𝑡 𝑏 (𝑠)𝑥(𝑠) 𝑡+𝑠 1

𝜙(𝑠)

⋅ (∫0

𝑠 𝑏 (𝑠)𝑥(𝜃) 𝑠+𝜃 2

𝑑𝜃)𝑑𝑠, 𝑡 ∈ 𝐼,

(3)

the set-valued fractional orders integral equation 𝑡 (𝑡−𝑠)𝛼−1

𝑥(𝑡) ∈ 𝑝(𝑡) + ∫0

Γ(𝛼)

𝑠 (𝑠−𝜃)𝛽−1

𝐹1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃)𝑑𝑠, 𝑡, 𝑠 ∈ 𝐼.

(4)

and the set-valued fractional-order integro-differential equations 𝑑𝑥(𝑡) 𝑑𝑡

∈ 𝐼 𝛼 𝐹1 (𝑡, 𝑥(𝑡), 𝐷𝛾 𝑥(𝑡)), 𝑡 ∈ (0, 𝑇],

𝑎𝑛𝑑 𝑥(0) = 𝑥𝑜 ,

(5)

where 𝛼, 𝛽, 𝛾 ∈ (0,1), will be considered. For properties and applications on differential inclusion (5) (see [5 - 7]) and reference therein. This paper is organized as follows: In Section 2, we recall some useful preliminaries. In Section 3, we investigate existence results for single-valued problem. In Section 4, we discuss the uniqueness of the solution for the VolterraStieltjes integral equation. Also, deals with the existence of continuous dependence of solutions for functional integral equations on delay function and functions 𝑔𝑖 , (𝑖 = 1,2), and some applications explain. While in Section 5, conditions are

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

221

added to our problem in order to obtain a new existence theorem with the application. In the last Section 6, we present the existence results for a set-valued problem with continuous dependence on the set 𝑆𝐹1 , we discuss some special cases of inclusion by presenting the existence of solutions for the Set-valued Chandrasekhar non-linear quadratic functional integral equation and Set-valued Volterra functional integral equation of fractional order, and, as an application, the set-valued fractional-order integro-differential equations will be considered. 2. PRELIMINARIES This section is devoted to providing the notation, definitions, and other auxiliary facts that will be needed in our further study. At the beginning, assume that 𝐸 is a Banach space with the norm ∥. ∥𝐸 . For an interval, 𝐼 = [0, 𝑇], where 𝑇 < ∞ and denote by 𝐶 = 𝐶(𝐼, 𝐸), the space consisting of all continuous functions defined on 𝐼 and taking values in the space 𝐸. This space will be furnished with the sup-norm. ∥ 𝑥 ∥𝐶 = sup|𝑥(𝑡)|. 𝑡∈𝐼

We will accept the following axiomatic definition and theorem of the concept of a set-valued map. Definition 2.1 Let 𝐹 be a set-valued map defined on a Banach space 𝐸, 𝑓 is called a selection of 𝐹 if 𝑓(𝑥) ∈ 𝐹(𝑥), for every 𝑥 ∈ 𝐸 and we denote by 𝑆𝐹 = {𝑓: 𝑓(𝑥) ∈ 𝐹(𝑥), 𝑥 ∈ 𝐸}, the set of all selections of 𝐹 (For the properties of the selection of 𝐹 see [8-10]). Definition 2.2 [9] A set-valued map 𝐹 from 𝐼 × 𝐸 to family of all non-empty closed subsets of 𝐸 is called Lipschitzian if there exists 𝑘 > 0 such that for all 𝑡 ∈ 𝐼 and all 𝑥1 , 𝑥2 ∈ 𝐸, we have ℎ(𝐹(𝑡, 𝑥1 ), 𝐹(𝑠, 𝑥2 )) ≤ 𝑘(|𝑡 − 𝑠| + |𝑥1 − 𝑥2 |),

(6)

Where ℎ(𝐴, 𝐵) is the Hausdorff distance between the two subsets 𝐴, 𝐵 ∈ 𝐼 × 𝐸 (properties of the Hausdorff distance (See [11])).

222

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

The following Theorem [11], Sect.9, Chap.1, Th.1] assumes the existence of Lipschitzian selection. Theorem 2.1 [12] Let 𝑀 be a metric space and 𝐹 be a Lipschitzian set-valued function from 𝑀 into the non-empty compact convex subsets of 𝑅𝑛 . Assume, moreover, that for some 𝜆 > 0, 𝐹(𝑥) ⊂ 𝜆𝐵 for all 𝑥 ∈ 𝑀 where 𝐵 is the unit ball on ℝ𝑛 . Then there exists a constant 𝑐 and a single-valued function 𝑓: 𝑀 → ℝ𝑛 , 𝑓(𝑥) ∈ 𝐹(𝑥) for 𝑥 ∈ 𝑀; this function is Lipschitzian with constant 𝑘. In what follows, we discuss a few auxiliary facts concerning the functions of bounded variation [13]. To this end, it assumes that x is a real function defined on a fixed interval [𝑎, 𝑏]. By the symbol ∨𝑏𝑎 𝑥 we will denote the variation of the function 𝑥 on the interval [𝑎, 𝑏]. In the case when ∨𝑏𝑎 𝑥 is finite we say that 𝑥 is of bounded variation on [𝑎, 𝑏]. In the case of a function 𝑢(𝑡, 𝑠) =: [𝑎, 𝑏] × [𝑐, 𝑑] → ℝ 𝑞 we can consider the variation ∨𝑡=𝑝 𝑢(𝑡, 𝑠) of the function 𝑡 → 𝑢(𝑡, 𝑠) (i.e., the variation of the function 𝑢(𝑡, 𝑠) with respect to the variable 𝑡) on the interval 𝑞 [𝑝, 𝑞] ⊂ [𝑎, 𝑏]. Similarly, we define the quantity ∨𝑠=𝑝 𝑢(𝑡, 𝑠). We will not discuss the properties of the various functions of bounded variation. We refer to [13] for the mentioned properties. Furthermore, assume that 𝑥 and 𝜙 are two real functions defined on the interval [𝑎, 𝑏]. Then, under some extra conditions (cf. [13]), we can define the Stieltjes integral (more precisely, the Riemann-Stieltjes integral) of the function 𝑥 with respect to the function 𝜙 on the interval [𝑎, 𝑏] which is denoted by the symbol 𝑏

∫𝑎 𝑥(𝑡)𝑑𝜙 (𝑡). In such a case, we say that 𝑥 is Stieltjes integrable on the interval [𝑎, 𝑏] with respect to 𝜙. In the relevant literature, we may encounter a lot of conditions guaranteeing the Stieltje's integrability [13, 10]. One of the most frequently exploited conditions requires that 𝑥 is continuous and 𝜙 is of bounded variation on [𝑎, 𝑏]. Next, we recall a few properties of the Stieltjes integral, which will be used in our considerations (cf. [13]). Lemma 2.2: [13] Assume that 𝑥 is Stieltjes integrable on the interval [𝑎, 𝑏] with respect to a function 𝜙 of bounded variation. Then

Uncertain Parameters

Advances in Special Functions of Fractional Calculus 𝑏

𝑏

223

𝑡

| ∫𝑎 𝑥(𝑡)𝑑𝜙 (𝑡)| ≤ ∫𝑎 |𝑥(𝑡)|𝑑(∨ 𝜙). 𝑎

Lemma 2.3: [13] Let 𝑥1 and 𝑥2 be Stieltjes integrable functions on the interval, [𝑎, 𝑏] with respect to a non-decreasing function 𝜙 such that 𝑥1 (𝑡) ≤ 𝑥2 (𝑡) for 𝑡 ∈ [𝑎, 𝑏]. Then the following inequality is satisfied: 𝑏

𝑏

∫𝑎 𝑥1 (𝑡)𝑑𝜙 (𝑡) ≤ ∫𝑎 𝑥2 (𝑡)𝑑𝜙 (𝑡). In the sequel, we will also consider the Stieltjes integrals of the form 𝑏

∫𝑎 𝑥(𝑠)𝑑𝑠 𝑔(𝑡, 𝑠), where 𝑔: [𝑎, 𝑏] × [𝑎, 𝑏] → ℝ and the symbol 𝑑𝑠 indicates the integration with respect to the variables. 3. SINGLE-VALUED PROBLEM Here we are consider with the non-linear single-valued delay functional integral equations of Volterra-Stieltjes type 𝜑(𝑡) 𝜑(𝑠) 𝑥(𝑡) = 𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃). (7) 3.1. Existence of Solutions I Consider the integral equation (7) under the following assumptions: (i) 𝜑: [0, 𝑇] → [0, 𝑇], 𝜑(𝑡) ≤ 𝑡 is continuous and increasing. (ii) 𝑎: [0, 𝑇] → [0, 𝑇] is continuous. (iii) a) 𝑓1 : [0, 𝑇] × [0, 𝑇] × ℝ × ℝ → ℝ is continuous and there exist continuous functions 𝑚𝑖 , 𝑘𝑖 : [0, 𝑇] × [0, 𝑇] → ℝ, i=1,2 such that |𝑓1 (𝑡, 𝑠, 𝑥, 𝑦)| ≤ 𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥| + |𝑦|). b) 𝑓2 : [0, 𝑇] × [0, 𝑇] × ℝ → ℝ is continuous such that |𝑓2 (𝑡, 𝑠, 𝑥)| ≤ 𝑚2 (𝑡, 𝑠) + 𝑘2 (𝑡, 𝑠)|𝑥|.

224

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

c) 𝑘 = sup𝑡∈[0,𝑇] 𝑘𝑖 (𝑡, 𝑠) and 𝑚 = sup𝑡∈[0,𝑇] 𝑚𝑖 (𝑡, 𝑠). (iv) 𝑔𝑖 : [0, 𝑇] × ℝ → ℝ, (𝑖 = 1,2) are continuous with 𝜇 = max{sup|𝑔𝑖 (𝑡, 𝜑(𝑡))| + sup|𝑔𝑖 (𝑡, 0)|, 𝑜𝑛 [0, 𝑇]}. (v) For all 𝑡1 , 𝑡2 ∈ 𝐼, 𝑡1 < 𝑡2 the functions, 𝑠 → 𝑔𝑖 (𝑡2 , 𝑠) − 𝑔𝑖 (𝑡1 , 𝑠) , are nondecreasing on [0, 𝑇]. (vi) 𝑔𝑖 (0, 𝑠) = 0, for any 𝑠 ∈ [0, 𝑇]. (vii) 𝑘𝜇 + 𝑘 2 𝜇2 < 1. Remark. (see [1]) Observe that the function 𝑠 → 𝑔(𝑡, 𝑠) is non-decreasing on the interval [0, 𝑇]. In fact, for 𝑠1 , 𝑠2 ∈ [0, 𝑇], with 𝑠1 < 𝑠2 , from assumptions (v) and (vi), we obtain 𝑔(𝑡, 𝑠2 ) − 𝑔(𝑡, 𝑠1 ) = [𝑔(𝑡, 𝑠2 ) − 𝑔(0, 𝑠2 )] − [𝑔(𝑡, 𝑠1 ) − 𝑔(0, 𝑠1 )] ≥ 0. Lemma 3.1 (see [1]) Assume that the function 𝑔 satisfies assumption (vi). Then for arbitrary 𝑠1 , 𝑠2 ∈ 𝐼, such that 𝑠1 < 𝑠2 , the function 𝑡 → 𝑔(𝑡, 𝑠2 ) − 𝑔(𝑡, 𝑠1 ) is nondecreasing on the interval 𝐼. In fact, take 𝑡1 , 𝑡2 ∈ [0, 𝑇] such that 𝑡1 < 𝑡2 . Then, by assumption (vi), we get [𝑔(𝑡2 , 𝑠2 ) − 𝑔(𝑡2 , 𝑠1 )] − [𝑔(𝑡1 , 𝑠2 ) − 𝑔(𝑡1 , 𝑠1 )] = [𝑔(𝑡2 , 𝑠2 ) − 𝑔(𝑡1 , 𝑠2 )] − [𝑔(𝑡2 , 𝑠1 ) − 𝑔(𝑡1 , 𝑠1 )] ≥ 0. For the existence of at least one solution of the quadratic integral equation (7), we have the following theorem. Theorem 3.2 Let the assumptions (i)-(vii) be satisfied, then the functional integral equation (7) has at least one solution 𝑥 ∈ 𝐶[0, 𝑇]. Proof. Define the operator 𝐴 by 𝜑(𝑡)

𝐴𝑥(𝑡) = 𝑎(𝑡) + ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

and let the set 𝑄𝑟 be defined by

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

(8)

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

225

𝑄𝑟 = {𝑥 ∈ 𝑅: |𝑥| ≤ 𝑟} ⊆ 𝐶[0, 𝑇], where 𝑎 + 𝑚𝜇 + 𝑘𝑚𝜇2 . 1 − [𝑘𝜇 + 𝑘 2 𝜇2 ]

𝑟=

It is clear that the set 𝑄𝑟 is a non-empty, bounded, closed, and convex set. Let 𝑥 ∈ 𝑄𝑟 , then 𝜑(𝑡)

|𝐴𝑥(𝑡)| = |𝑎(𝑡) + ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

≤ 𝑎 + |∫

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫

0

0

𝜑(𝑡)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑠)

≤𝑎+∫

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)| + | ∫

0

0

𝜑(𝑡)

≤𝑎+∫ 𝜑(𝑡)

𝜑( 𝑠)

+∫

𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)| + ∫

0

0

𝜑(𝑡)

≤𝑎+∫

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝜑(𝑠)

𝜑(𝑡)

𝑓(𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))

𝑚1 (𝑡, 𝑠)𝑑𝑠 𝑔1 (𝑡, 𝑠)

0

(𝑚2 (𝑠, 𝜃) + (𝑠, 𝜃)|𝑥(𝜃)|)𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)| + (𝑚2 + 𝑘2 𝑟) 𝜇)𝑑𝑠 𝑔1 (𝑡, 𝑠)

0

≤ 𝑎 + (𝑚 + 𝑘(𝑟 + (𝑚 + 𝑘𝑟) 𝜇))𝜇 ≤ 𝑟. This proves that the operator 𝐴 maps 𝑄𝑟 into itself, and the class of functions {𝐴𝑥} is uniformly bounded on 𝑄𝑟 . 𝜑.(𝑠)

Now, for 𝑥 ∈ 𝑄𝑟 and 𝑦(𝑠) = ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃), define the set

𝜃(𝛿) = sup {|𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠)) − 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))|: 𝑡1 , 𝑡2 ∈ [0, 𝑇], 𝑡1 < 𝑡2 , |𝑡2 − 𝑡1 | < 𝛿}, 𝑥∈𝑄𝑟

then from the uniform continuity of the function 𝑓1 : [0, 𝑇] × [0, 𝑇] × 𝑄𝑟 × 𝑄𝑟 → ℝ, assumption (iii), we deduce that 𝜃(𝛿) → 0, as 𝛿 → 0 independent of 𝑥 ∈ 𝑄𝑟 . Now, let 𝑡2 , 𝑡1 ∈ [0, 𝑇], |𝑡2 − 𝑡1 | < 𝛿, then we have

226

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

𝜑(𝑡2 )

𝜑(𝑠)

|𝐴𝑥(𝑡2 ) − 𝐴𝑥(𝑡1 )| = |𝑎(𝑡2 ) + ∫

𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), ∫

0

0

𝜑(𝑡 )

𝜑(𝑠)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) −𝑎(𝑡1 ) − ∫0 1 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), ∫0 ≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| 𝜑(𝑠) 𝜑(𝑡 ) 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) +| ∫0 2 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), ∫0 𝜑(𝑡1 )

− ∫0

𝜑(𝑠)

𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), ∫0

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| +

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡1 ) | ∫0 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠)

𝜑 (𝑡2 ) 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠) 1)

+ ∫𝜑(𝑡1

𝜑(𝑡 ) + ∫0 1 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠)|

− 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠)|

− 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡1 , 𝑠)|

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| φ(t1 )

+∫ 0

|(f1 (t 2 , s, x(s), y(s)) − 𝑓1 (t1 , s, x(s), y(s)))|ds g1 (t 2 , s) 𝜑(𝑡1 )

+ ∫0

𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 [𝑔1 (𝑡2 , 𝑠) − 𝑔1 (𝑡1 , 𝑠)]

𝜑 (𝑡 ) + ∫𝜑(𝑡1 )2 1

|𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))|𝑑𝑠 𝑔1 (𝑡2 , 𝑠)

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )|

𝜑(𝑡 ) + ∫0 1

𝜑(𝑡1 )

+ ∫0

𝜑(𝑡 )

𝜃(𝛿)𝑑𝑠 𝑔1 (𝑡2 , 𝑠) + ∫𝜑(𝑡 2) 𝑚1 (𝑠) + 𝑘1 (|𝑥(𝑠)| + |𝑦(𝑠)|)𝑑𝑠 𝑔1 (𝑡2 , 𝑠) 1

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑠)| + |𝑦(𝑠)|))𝑑𝑠 [𝑔1 (𝑡2 , 𝑠) − 𝑔1 (𝑡1 , 𝑠)].

The above inequality means that the class of functions {𝐴𝑥} is equicontinuous. Hence, from Arzela-Ascoli Theorem [14], 𝐴 is compact. Let {𝑥𝑛 } ⊂ 𝑄𝑟 , 𝑥𝑛 → 𝑥, then 𝜑(𝑡)

𝐴𝑥𝑛 (𝑡) = 𝑎(𝑡) + ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫0 𝜑(𝑡)

lim 𝐴𝑥𝑛 (𝑡) = lim (𝑎(𝑡) + ∫0

𝑛→∞

𝑛→∞

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃),

and from assumption (iii) (see [15]), we can get lim 𝐴𝑥𝑛 (𝑡)

𝜑(𝑡)

= 𝑎(𝑡) + ∫0

𝜑(𝑡)

= 𝑎(𝑡) + ∫0

= 𝑎(𝑡) +

𝑛→∞

𝜑(𝑠)

lim 𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫0

𝑛→∞

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, lim 𝑥𝑛 (𝑠), ∫0 𝑛→∞

𝜑(𝑡) 𝜑(𝑠) 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 ∫0

𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝑓2 (𝑠, 𝜃, lim 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝑛→∞

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃).

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

227

This proves that 𝐴𝑥𝑛 (𝑡) → 𝐴𝑥(𝑡) and 𝐴 is continuous. Now (see [15]) 𝐴 has at least one fixed point 𝑥 ∈ 𝑄𝑟 , and (7) has at least one solution 𝑥 ∈ 𝑄𝑟 ⊂ 𝐶[0, 𝑇]. This complete the proof. 4. UNIQUENESS OF THE SOLUTION To study the uniqueness of the solution of the functional integral equation (7), we replace the assumptions (iii) by: (iii) ∗ a) 𝑓1 : 𝐼 × 𝐼 × ℝ × ℝ → ℝ is continuous and meets the Lipschitz condition |𝑓1 (𝑡, 𝑠, 𝑥1 , 𝑦1 ) − 𝑓1 (𝑡, 𝑠, 𝑥2 , 𝑦2 )| ≤ 𝑘1 (|𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 |). b) 𝑓2 : 𝐼 × 𝐼 × ℝ → ℝ is continuous and meets the Lipschitz condition |𝑓2 (𝑡, 𝑠, 𝑥) − 𝑓2 (𝑡, 𝑠, 𝑦)| ≤ 𝑘2 |𝑥 − 𝑦|. From this assumption (𝑖𝑖𝑖)∗ , we have |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| − |𝑓1 (𝑡, 𝑠, 0,0)| ≤ |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠)) − 𝑓1 (𝑡, 𝑠, 0,0)| ≤ 𝑘1 (|𝑥| + |𝑦|), then |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| ≤ 𝑘1 (|𝑥| + |𝑦|) + |𝑓1 (𝑡, 𝑠, 0,0)| and |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| ≤ 𝑘1 (|𝑥| + |𝑦|) + 𝑚1 , where 𝑚1 = sup𝑡∈𝐼 |𝑓1 (𝑡, 𝑠, 0,0)| and |𝑓2 (𝑡, 𝑠, 𝑥(𝑠))| − |𝑓2 (𝑡, 𝑠, 0)| ≤ |𝑓2 (𝑡, 𝑠, 𝑥(𝑠)) − 𝑓2 (𝑡, 𝑠, 0)| ≤ 𝑘2 |𝑥|, then |𝑓2 (𝑡, 𝑠, 𝑥(𝑠))| ≤ 𝑘2 |𝑥| + |𝑓2 (𝑡, 𝑠, 0)|, and |𝑓2 (𝑡, 𝑠, 𝑥(𝑠))| ≤ 𝑘2 |𝑥| + 𝑚2 , where 𝑚2 = sup𝑡∈𝐼 |𝑓1 (𝑡, 𝑠, 0)|. Theorem 4.1: Let the assumptions (i)-(ii)-(𝑖𝑖𝑖)∗ -(iv)-(v)-(vi)-(vii) be satisfied, if 𝜇𝑘 + 𝑘 2 𝜇2 ≤ 1, then the solution of the functional integral equation (7) is unique solution 𝑥 ∈ 𝐶[0, 𝑇].

228

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

Proof. Let 𝑥1 , 𝑥2 be solutions of the (7), then |𝑥1 (𝑡) − 𝑥2 (𝑡)| 𝜑(𝑡) 𝜑(𝑠) = |𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥1 (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥1 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡)

−𝑎(𝑡) + ∫0 𝜑(𝑡)

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥2 (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥2 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

≤ ∫0

𝑘1 (|𝑥1 (𝑠) − 𝑥2 (𝑠)| 𝑑𝑠 𝑔1 (𝑡, 𝑠)

+ ∫0

|(𝑓2 (𝑠, 𝜃, 𝑥1 (𝜃)) − 𝑓2 (𝑠, 𝜃, 𝑥2 (𝜃)))| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑠)

≤ ≤

𝜑(𝑡) ∫0 𝜑(𝑡) ∫0

𝜑(𝑠)

𝑘1 (|𝑥1 (𝑠) − 𝑥2 (𝑠)| + ∫0

𝑘2 (|𝑥1 (𝜃) − 𝑥2 (𝜃)|) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝑘1 (|𝑥1 (𝑠) − 𝑥2 (𝑠)| +

𝑘2 (|𝑥1 (𝜃) − 𝑥2 (𝜃)|) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡) ∫0

𝜑(𝑠) ∫0

≤ 𝑘1 (|𝑥1 (𝑠) − 𝑥2 (𝑠)| + 𝑘2 ∥ 𝑥1 − 𝑥2 ∥ 𝜇)𝑑𝑠 𝑔1 (𝑡, 𝑠) ≤ 𝑘 ∥ 𝑥1 − 𝑥2 ∥ 𝜇 + 𝑘 2 ∥ 𝑥1 − 𝑥2 ∥ 𝜇2 , taking supremum over 𝑡 ∈ 𝐼 ∥ 𝑥1 − 𝑥2 ∥≤ (𝜇𝑘 + 𝑘 2 𝜇2 ) ∥ 𝑥1 − 𝑥2 ∥, and (1 − (𝜇 + 𝑘 2 𝜇2 )) ∥ 𝑥1 − 𝑥2 ∥≤ 0. Then 𝑥1 (𝑡) = 𝑥2 (𝑡) This completes the proof. 4.1. Continuous Dependence of the Solution In this section we are going to study the continuous dependence of the unique solution 𝑥 ∈ 𝐶[0, 𝑇] of the functional integral equation (7) on the delay function 𝜑(𝑡) and functions 𝑔𝑖 , 𝑖 = 1,2. 4.1.1. Continuous Dependence on the Delay Functions 𝝋(𝒕) Definition 4.1: The solutions of the functional integral equation (7), is depends continuously on the delay function 𝜑(𝑡) if ∀ 𝜖 > 0, ∃ 𝛿 > 0, such that |𝜑(𝑡) − 𝜑 ∗ (𝑡)| ≤ 𝛿 ⇒ ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝜖. Theorem 4.2: Let the assumptions (i)-(vii) of Theorem 4.1 be satisfied. Then the solution of the delay functional integral equation (7) depends continuously on the delay function 𝜑(𝑡).

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

229

Proof. Let 𝛿 > 0 be given such that |𝜑(𝑡) − 𝜑 ∗ (𝑡)| ≤ 𝛿 , ∀ 𝑡 ≥ 0, then |𝑥(𝑡) − 𝑥 ∗ (𝑡)| 𝜑(𝑡) 𝜑(𝑠) = |𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑∗ (𝑡)

−𝑎(𝑡) + ∫0 𝜑(𝑡)

≤ | ∫0

𝜑∗ (𝑠)

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝜑(𝑠)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡) 𝜑(𝑠) ∗ − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠) 𝜑(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) ∗ ∗ 𝜑 (𝑡) 𝜑 (𝑠) − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡) ≤ | ∫0 [𝑘(|𝑥(𝑠) − 𝑥 ∗ (𝑠)| 𝜑(𝑠) + |∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃)) − 𝜑(𝑠)

∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)|]𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡)

𝜑(𝑠)

+ ∫0

𝜑∗ (𝑡)

− ∫0

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝜑(𝑡)

≤ | ∫0

𝜑(𝑠)

+ |∫0

𝜑(𝑠)

∫0

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝜑∗ (𝑠)

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

[𝑘(|𝑥(𝑠) − 𝑥 ∗ (𝑠)| 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃)) −

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)|]𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡)

𝜑(𝑠)

+ ∫0

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

− ∫0

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

+ ∫0

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝜑(𝑡)

𝜑(𝑡) 𝜑∗ (𝑡)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝜑∗ (𝑠) 𝜑∗ (𝑠) 𝜑∗ (𝑠)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

− ∫0

𝑓2 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

≤ ∫0

[𝑘(|𝑥(𝑠) − 𝑥 ∗ (𝑠)|

+| ∫0

[𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) − 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) ]𝑑𝜃 𝑔2 (𝑠, 𝜃))|]𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡) 𝜑(𝑠)

𝜑(𝑡)

+ ∫0

𝜑∗ (𝑠)

∫0

𝜑(𝑡)

𝜑(𝑠)

[∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)𝑑𝑠 −

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)]𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑∗ (𝑠)

+ ∫𝜑∗ (𝑡) 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡)

≤ ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

[𝑘(|𝑥(𝑠) − 𝑥 ∗ (𝑠)|

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

230

Advances in Special Functions of Fractional Calculus 𝜑(𝑠)

El-Sayed et al.

+| ∫0

𝑘[|𝑥(𝜃) − 𝑥 ∗ (𝜃)|]𝑑𝜃 𝑔2 (𝑠, 𝜃)|)]𝑑𝑠 𝑔1 (𝑡, 𝑠)

+ ∫0

[∫𝜑∗ (𝑠) 𝑓(𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡)

𝜑(𝑠)

𝜑∗ (𝑠)

𝜑(𝑡)

+ ∫𝜑∗ (𝑡) 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥ +(𝑘𝑟 + 𝑚)𝜇[𝑔2 (𝑠, 𝜑(𝑠)) − 𝑔2 (𝑠, 𝜑∗ (𝑠))] +[𝑘𝑟 + 𝑚𝑘 2 𝑟 + 𝑚𝜇𝑘 + 𝑚][𝑔1 (𝑠, 𝜑(𝑠)) − 𝑔1 (𝑠, 𝜑∗ (𝑠))], taking supremum for 𝑡 ∈ 𝐼, we have ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥ +(𝑘𝑟 + 𝑚)𝜇[𝑔2 (𝑠, 𝜑(𝑠)) − 𝑔2 (𝑠, 𝜑∗ (𝑠))] +[𝑘𝑟 + 𝑚𝑘 2 𝑟 + 𝑚𝜇𝑘 + 𝑚][𝑔1 (𝑠, 𝜑(𝑠)) − 𝑔1 (𝑠, 𝜑∗ (𝑠))], and ∥ 𝑥 − 𝑥 ∗ ∥ [1 − (𝑘𝜇 + 𝑘 2 𝜇2 )] ≤ (𝑘𝑟 + 𝑚)𝜇[𝑔2 (𝑠, 𝜑(𝑠)) − 𝑔2 (𝑠, 𝜑 ∗ (𝑠))] +[𝑘𝑟 + 𝑚𝑘 2 𝑟 + 𝑚𝜇𝑘 + 𝑚][𝑔1 (𝑠, 𝜑(𝑠)) − 𝑔1 (𝑠, 𝜑∗ (𝑠))]. But from the continuity of 𝑔, we have |𝜑(𝑡) − 𝜑 ∗ (𝑡)| ≤ 𝛿 ⇒ |𝑔𝑖 (𝑡, 𝜑(𝑡)) − 𝑔𝑖 (𝑡, 𝜑 ∗ (𝑡))| ≤ 𝜖1 . Then ∥ 𝑥 − 𝑥∗ ∥ ≤

(𝑘𝑟+𝑚)𝜇𝜖1 +[𝑘𝑟+𝑚𝑘 2 𝑟+𝑚𝜇𝑘+𝑚]𝜖1 1−(𝑘𝜇+𝑘 2 𝜇2 )

= 𝜖.

This completes the proof.

4.1.2. Continuous Dependence on the Functions 𝒈𝒊 (𝒕, 𝒔) Definition 4.2: The solutions of the functional integral equation (7), is dependence continuously on the functions 𝑔𝑖 (𝑡, 𝑠), (𝑖 = 1,2. ) if ∀ 𝜖 > 0, ∃ 𝛿 > 0, such that |𝑔𝑖 (𝑡, 𝑠) − 𝑔𝑖∗ (𝑡, 𝑠)| ≤ 𝛿 ⇒ ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝜖. Theorem 4.3: Let the assumptions of Theorem 4.1 be satisfied. Then the solution of the delay functional integral equation (7) depends continuously on functions 𝑔𝑖 (𝑡, 𝑠), 𝑖 = 1,2.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

231

Proof. Let 𝛿 > 0 be given such that |𝑔𝑖 (𝑡, 𝑠) − 𝑔𝑖∗ (𝑡, 𝑠)| ≤ 𝛿 , ∀𝑡 ≥ 0, then |𝑥(𝑡) − 𝑥 ∗ (𝑡)| 𝜑(𝑡) 𝜑(𝑠) = |𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡)

−𝑎(𝑡) + ∫0 𝜑(𝑡)

≤ | ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃)

𝜑(𝑠)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑡) 𝜑(𝑠) − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠) 𝜑(𝑡) 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡) 𝜑(𝑠) − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓(𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃))𝑑𝑠 𝑔1∗ (𝑡, 𝑠)| 𝜑(𝑡) 𝜑(𝑠) ≤ ∫0 |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)) 𝜑(𝑠) −𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑡) 𝜑(𝑠) +| ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)) 𝜑(𝑠) 𝑓(𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃))𝑑𝑠 𝑔1∗ (𝑡, 𝑠)| −𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡) ≤ ∫0 𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| 𝜑(𝑠) + ∫0 |𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) − 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑡) 𝜑(𝑠) 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) +| ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡) 𝜑(𝑠) − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃) 𝜑(𝑡) 𝜑(𝑠) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃) 𝜑(𝑡) 𝜑(𝑠) − ∫0 𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓(𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃))𝑑𝑠 𝑔1∗ (𝑡, 𝑠)) 𝜑(𝑡)

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝜑(𝑠)

≤ ∫0

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0

+ ∫0

|𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝜑(𝑡)

𝜑(𝑠)

𝜑(𝑠)

−𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡)

𝜑(𝑠)

|𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

≤ ∫0

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0

𝜑(𝑡)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃))

+ ∫0

𝜑(𝑡)

𝑘2 |𝑥(𝜃)) − 𝑥 ∗ (𝜃)| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃)

𝜑(𝑠)

𝑘2 |𝑥(𝜃)) − 𝑥 ∗ (𝜃)| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠)

+ ∫0

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + | ∫0

− ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2∗ (𝑠, 𝜃)|)𝑑𝑠 𝑔1 (𝑡, 𝑠))

𝜑(𝑠)

𝜑(𝑡)

+ ∫0

𝜑(𝑠)

|𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2∗ (𝑠, 𝜃))|

× [𝑑𝑠 𝑔1 (𝑡, 𝑠) − 𝑑𝑠 𝑔1∗ (𝑡, 𝑠)] 𝜑(𝑡)

≤ ∫0

𝜑(𝑠)

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0

𝑘2 |𝑥(𝜃)) − 𝑥 ∗ (𝜃)| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

232

Advances in Special Functions of Fractional Calculus

El-Sayed et al. 𝜑(𝑠)

𝜑(𝑡)

+ ∫0 |𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))| 𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0 × [ 𝑑𝜃 𝑔2 (𝑠, 𝜃) − 𝑑𝜃 𝑔2∗ (𝑠, 𝜃)])𝑑𝑠 𝑔1 (𝑡, 𝑠)) 𝜑(𝑡)

+ ∫0

𝜑(𝑠)

𝑘1 (|𝑥 ∗ (𝑠)| + ∫0

|𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))|𝑑𝜃 𝑔2∗ (𝑠, 𝜃))

× [𝑑𝑠 𝑔1 (𝑡, 𝑠) − 𝑑𝑠 𝑔1∗ (𝑡, 𝑠)] 𝜑(𝑡)

≤ ∫0

𝜑(𝑠)

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0

𝑘2 |𝑥(𝜃)) − 𝑥 ∗ (𝜃)| 𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠)

𝜑(𝑡)

+ ∫0 𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0 [𝑚2 + 𝑘2 |𝑥 ∗ (𝜃)|] × [ 𝑑𝜃 𝑔2 (𝑠, 𝜃) − 𝑑𝜃 𝑔2∗ (𝑠, 𝜃)])𝑑𝑠 𝑔1 (𝑡, 𝑠)) 𝜑(𝑡)

+ ∫0

𝜑(𝑠)

𝑘1 (|𝑥 ∗ (𝑠)| + ∫0

[𝑚2 + 𝑘2 |𝑥 ∗ (𝜃)|]𝑑𝜃 𝑔2∗ (𝑠, 𝜃))

× [𝑑𝑠 𝑔1 (𝑡, 𝑠) − 𝑑𝑠 𝑔1∗ (𝑡, 𝑠)] ≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘[𝑚 + 𝑘𝑟]𝜇[𝑔2 (𝑠, 𝜑(𝑠)) − 𝑔2∗ (𝑠, 𝜑(𝑠))] +𝑘[𝑟 + 𝑘𝑟 + 𝑚]𝜇[𝑔1 (𝑡, 𝜑(𝑠)) − 𝑔1∗ (𝑡, 𝜑(𝑠))], taking supremum for 𝑡 ∈ 𝐼, we have ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +[𝑘𝑚 + 𝑘𝑟]𝜇𝛿

+𝑘[𝑟 + 𝑘𝑟 + 𝑚]𝜇𝛿 ∥ 𝑥 − 𝑥 ∗ ∥≤

(2𝑘𝑚+2𝑘𝑟+𝑘 2 𝑟)𝜇𝛿 (1−2𝑘𝜇+𝑘 2 𝜇 2 )

= 𝜖.

This completes the proof. 4.2. Application 1 Let the functions 𝑔𝑖 be defined by 𝑡+𝑠 𝑡ln , 𝑓𝑜𝑟 𝑡 ∈ (0, 𝑇], 𝑠 ∈ 𝐼 𝑡 𝑔1 (𝑡, 𝑠) = { 𝑓𝑜𝑟 𝑡 = 0, 𝑠 ∈ 𝐼 , 0, and 𝑠+𝜃 𝑠ln , 𝑓𝑜𝑟 𝑠 ∈ (0, 𝑇], 𝜃 ∈ 𝐼 𝑠 𝑔2 (𝑠, 𝜃) = { 0, 𝑓𝑜𝑟 𝑠 = 0, 𝜃 ∈ 𝐼 . Let 𝑓2 (𝑡, 𝑠, 𝑥(𝑠)) = 𝑏2 (𝑡)𝑥(𝑠) in Equation (7), then we obtain the non-linear Chandrasekhar functional integral equation in equation (7). It it clear that 𝑔1 , 𝑔2 satisfies our assumptions (𝑖𝑣) − (𝑣𝑖) (see [4]), and the integral equation (7) will be in the form. 𝜑(𝑡) 𝑡 𝜑(𝑠) 𝑠 𝑥(𝑡) = 𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑏2 (𝑠)𝑥(𝜃)𝑑𝜃. 𝑡+𝑠

𝑠+𝜃

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

233

5. EXISTENCE OF SOLUTIONS II Now we replace the assumptions (iii)-(a), (vii) of Theorem 3.2 by is continuous and there exist (𝑖𝑖𝑖)∗ 𝑎∗ ) 𝑓1 : [0, 𝑇] × [0, 𝑇] × ℝ × ℝ → ℝ continuous functions 𝑚𝑖 , 𝑘𝑖 : [0, 𝑇] × [0, 𝑇] → ℝ, i=1,2 such that |𝑓1 (𝑡, 𝑠, 𝑥, 𝑦)| ≤ 𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥|. |𝑦|). (𝑣𝑖𝑖 ∗ ) There exists a positive root 𝑙 of the algebraic equation 𝜇2 𝑘 2 𝑙2 + (𝑘𝑟𝑚 − 1)𝑙 + (𝑎 + 𝑚𝜇) = 0. Now we study the existence of the solutions of the integral equation (7) under the assumptions (i)-(ii),(𝑖𝑖𝑖 ∗ )-(𝑎∗ ) ,(vi),(𝑣𝑖𝑖 ∗ ). Theorem 5.1: Let the assumptions (i)-(ii),(𝑖𝑖𝑖 ∗ ) 𝑎∗ )-(vi),(𝑣𝑖𝑖 ∗ ) be satisfied. The functional integral equation (7) has at least one solution 𝑥 ∈ [0, 𝑇]. Proof. Define the operator 𝐴 by 𝜑(𝑡) 𝜑(𝑠) 𝐴𝑥(𝑡) = 𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) and the set 𝑄𝑙 by 𝑄𝑙 = {𝑥 ∈ ℝ: |𝑥| ≤ 𝑙} ⊆ 𝐶[0, 𝑇], Where, 𝑙 is a positive root of the algebraic equation 𝜇2 𝑘 2 𝑙2 + (𝑘𝑟𝑚 − 1) 𝑙 + (𝑎 + 𝑚𝜇) = 0. It is clear that the set 𝑄𝑙 is a non-empty, bounded, closed, and convex set. Now, let 𝑥 ∈ 𝑄𝑙 , then 𝜑(𝑡) 𝜑(𝑠) |𝐴𝑥(𝑡)| = |𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡)

≤ 𝑎 + | ∫0 ≤𝑎+

𝜑(𝑡) ∫0

≤𝑎 ×

+

𝜑(𝑠) ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑠)

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)| ⋅ | ∫0

𝜑(𝑡) ∫0

𝑓(𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)|

(𝑚2 (𝑠, 𝜃) + 𝑘2 (𝑠, 𝜃)|𝑥(𝜃)|) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡) (𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑡)| ⋅ (𝑚2 + 𝑘2 𝑙) ∫0

≤𝑎+ ≤ 𝑎 + (𝑚 + 𝑘(𝑙 ⋅ (𝑚 + 𝑘𝑙) 𝜇))𝜇 ≤ 𝑙.

𝜇)𝑑𝑠 𝑔1 (𝑡, 𝑠)

234

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

This proves that the operator 𝐴 maps 𝑄𝑙 into itself, and the class of functions {𝐴𝑥} is uniformly bounded on 𝑄𝑙 . 𝜑(𝑠)

Now, for 𝑥 ∈ 𝑄𝑟 and 𝑦(𝑠) = ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃), define the set

𝜃(𝛿) = sup {|𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠)) − 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))|: 𝑡1 , 𝑡2 ∈ [0, 𝑇], 𝑡1 < 𝑡2 , 𝑥∈𝑄𝑙

|𝑡2 − 𝑡1 | < 𝛿}, then from the uniform continuity of the function 𝑓1 : [0, 𝑇] × [0, 𝑇] × 𝑄𝑙 × 𝑄𝑙 → ℝ, and assumption (iii), we deduce that 𝜃(𝛿) → 0, as 𝛿 → 0 independent of 𝑥 ∈ 𝑄𝑙 . Now, let 𝑡2 , 𝑡1 ∈ [0, 𝑇], such that |𝑡2 − 𝑡1 | < 𝛿, then we have 𝜑(𝑡2 )

𝜑(𝑠)

|𝐴𝑥(𝑡2 ) − 𝐴𝑥(𝑡1 )| = |𝑎(𝑡2 ) + ∫

𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), ∫

0

0

𝜑(𝑡 )

𝜑(𝑠)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

−𝑎(𝑡1 ) − ∫0 1 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) ≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| 𝜑(𝑡 ) 𝜑(𝑠) +| ∫0 2 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡1 )

− ∫0

𝜑(𝑠)

𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), ∫0

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| +

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡1 ) 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠) | ∫0

𝜑 (𝑡2 ) 𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠) − 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠)| 1) 𝜑(𝑡1 ) + ∫0 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡2 , 𝑠)| − 𝑓1 (𝑡1 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 𝑔1 (𝑡1 , 𝑠)|

+ ∫𝜑(𝑡1

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| φ(t ) + ∫0 1 |(f1 (t 2 , s, x(s), y(s)) − 𝑓1 (t1 , s, x(s), y(s)))|ds g1 (t 2 , s) 𝜑(𝑡1 )

+ ∫0

𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))𝑑𝑠 [𝑔1 (𝑡2 , 𝑠) − 𝑔1 (𝑡1 , 𝑠)]

𝜑 (𝑡 ) + ∫𝜑(𝑡1 )2 1

|𝑓1 (𝑡2 , 𝑠, 𝑥(𝑠), 𝑦(𝑠))|𝑑𝑠 𝑔1 (𝑡2 , 𝑠)

≤ |𝑎(𝑡2 ) − 𝑎(𝑡1 )| 𝜑(𝑡1 )

𝜃(𝛿)𝑑𝑠 𝑔1 (𝑡2 , 𝑠) + ∫𝜑(𝑡 2) 𝑚1 (𝑠) + 𝑘1 (|𝑥(𝑠)| + |𝑦(𝑠)|)𝑑𝑠 𝑔1 (𝑡2 , 𝑠)

𝜑(𝑡1 )

(𝑚1 (𝑡, 𝑠) + 𝑘1 (𝑡, 𝑠)(|𝑥(𝑠)| + |𝑦(𝑠)|))𝑑𝑠 [𝑔1 (𝑡2 , 𝑠) − 𝑔1 (𝑡1 , 𝑠)].

+ ∫0 + ∫0

𝜑(𝑡 ) 1

The above inequality means that the class of functions {𝐴𝑥} is equicontinuous. Therefore, from Arzela-Ascoli Theorem [18], 𝐴 is compact. Let {𝑥𝑛 } ⊂ 𝑄𝑙 , 𝑥𝑛 → 𝑥, then

𝜑(𝑡)

𝐴𝑥𝑛 (𝑡) = 𝑎(𝑡) + ∫0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃),

Uncertain Parameters

Advances in Special Functions of Fractional Calculus 𝜑(𝑡)

𝜑(𝑠)

lim 𝐴𝑥𝑛 (𝑡) = lim (𝑎(𝑡) + ∫0

𝑛→∞

𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫0

𝑛→∞

235

𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃),

and from assumption (iii) (see [15]), we can get 𝜑(𝑡)

lim 𝐴𝑥𝑛 (𝑡) = 𝑎(𝑡) + ∫

𝑛→∞

𝜑(𝑡)

= 𝑎(𝑡) + ∫0 = 𝑎(𝑡) +

𝜑(𝑠)

lim 𝑓1 (𝑡, 𝑠, 𝑥𝑛 (𝑠), ∫

𝑛→∞

0

𝑓2 (𝑠, 𝜃, 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

0

𝜑(𝑠)

𝑓1 (𝑡, 𝑠, lim 𝑥𝑛 (𝑠), ∫0

𝑓2 (𝑠, 𝜃, lim 𝑥𝑛 (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝑛→∞ 𝜑(𝑡) 𝜑(𝑠) 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃). ∫0 𝑛→∞

This proves that 𝐴𝑥𝑛 (𝑡) → 𝐴𝑥(𝑡) and 𝐴 is continuous. Now (see [15]) 𝐴 has at least one fixed point 𝑥 ∈ 𝑄𝑟 , and (7) has at least one solution 𝑥 ∈ 𝑄𝑙 ⊂ 𝐶[0,1]. This complete the proof. 5.1. Application 2 Let the functions 𝑔𝑖 be defined by 𝑔1 (𝑡, 𝑠) = {

𝑡ln

𝑡+𝑠 𝑡

,

𝑓𝑜𝑟 𝑡 ∈ (0, 𝑇], 𝑠 ∈ 𝐼 ,

0,

𝑓𝑜𝑟 𝑡 = 0, 𝑠 ∈ 𝐼

and 𝑠ln 𝑔2 (𝑠, 𝜃) = { 0,

𝑠+𝜃 𝑠

,

𝑓𝑜𝑟 𝑠 ∈ (0, 𝑇], 𝜃 ∈ 𝐼 , 𝑓𝑜𝑟 𝑠 = 0, 𝜃 ∈ 𝐼 .

Let 𝑓2 (𝑡, 𝑠, 𝑥(𝑠)) = 𝑏2 (𝑡)𝑥(𝑠) and 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠)) = 𝑏1 (𝑠)𝑥(𝑠)𝑦(𝑠) in equation (7), then we can see that 𝑔1 , 𝑔2 satisfies our assumptions (𝑖𝑣) − (𝑣𝑖), (see [16]) and we obtain the nonlinear Chandrasekhar functional integral equation 𝑡

𝑥(𝑡) = 𝑎(𝑡) + ∫0

𝑡 𝑡+𝑠

𝑠

𝑏1 (𝑠)𝑥(𝑠) ⋅ (∫0

𝑠

𝑏 (𝑠)𝑥(𝜃)𝑑𝜃. 𝑠+𝜃 2

(9)

6. SET-VALUED PROBLEM Consider, now, the non-linear set-valued delay functional integral equations of Volterra-Stieltjes type (1) 𝜑(𝑡)

𝜑(𝑠)

𝑥(𝑡) ∈ 𝑎(𝑡) + ∫0 𝐹1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 under the additional condition

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃)

236

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

(i) 𝜑: [0, 𝑇] → [0, 𝑇], 𝜑(𝑡) ≤ 𝑡 is continuous and increasing. (ii) 𝑎: [0, 𝑇] → [0, 𝑇] is continuous. (𝑖𝑖𝑖) ∗ (a) 𝐹1 : 𝐼 × 𝐼 × ℝ × ℝ → 𝑃(ℝ) is a Lipschitzian set-valued map with a non-empty compact convex subset of 2ℝ . (b) 𝑓2 : [0, 𝑇] × [0, 𝑇] × ℝ → ℝ is continuous such that |𝑓2 (𝑡, 𝑠, 𝑥)| ≤ 𝑚2 (𝑡, 𝑠) + 𝑘2 (𝑡, 𝑠)|𝑥|. (c) 𝑘 = sup𝑡∈[0,𝑇] 𝑘𝑖 (𝑡, 𝑠) and 𝑚 = sup𝑡∈[0,𝑇] 𝑚𝑖 (𝑡, 𝑠). (iv) 𝑔𝑖 : [0, 𝑇] × ℝ → ℝ, 𝑖 = 1,2 are continuous with 𝜇 = max{sup|𝑔𝑖 (𝑡, 𝜑(𝑡))| + sup|𝑔𝑖 (𝑡, 0)|, 𝑜𝑛 [0, 𝑇]}. (v) For all 𝑡1 , 𝑡2 ∈ 𝐼, 𝑡1 < 𝑡2 the functions 𝑠 → 𝑔𝑖 (𝑡2 , 𝑠) − 𝑔𝑖 (𝑡1 , 𝑠) are nondecreasing on [0, 𝑇]. (vi) 𝑔𝑖 (0, 𝑠) = 0 for any 𝑠 ∈ [0, 𝑇]. (vii) 𝑘𝜇 + 𝑘 2 𝜇2 < 1. 6.1. Existence of Solution Theorem 6.1: Let the assumptions (i)-(ii)-(𝑖𝑖𝑖)∗∗ and (iv)-(viii) be satisfied. Then the functional integral inclusion (1) have at least one solution 𝑥 ∈ 𝐶(𝐼). Proof. It is clear that from Theorem 2.1 and assumption (𝑖𝑖𝑖)∗∗ , the set of Lipschitz selection of 𝐹1 is non empty. So, the solution of the single-valued integral equation (7) where 𝑓1 ∈ 𝑆𝐹1 , is a solution to the inclusion (1). It must be noted that the Lipschitz selection 𝑓1 : 𝐼 × 𝐼 × ℝ × ℝ → ℝ, satisfies the Lipschitz condition. |𝑓1 (𝑡, 𝑠, 𝑥1 , 𝑦1 ) − 𝑓1 (𝑡, 𝑠, 𝑥2 , 𝑦2 )| ≤ 𝑘1 (|𝑥1 − 𝑥2 | + |𝑦1 − 𝑦2 |). From this condition with 𝑚1 = sup𝑡∈𝐼 |𝑓1 (𝑡, 𝑠, 0,0)|, we have |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| − |𝑓1 (𝑡, 𝑠, 0,0)| ≤ |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠)) − 𝑓1 (𝑡, 𝑠, 0,0)| ≤ 𝑘1 (|𝑥| + |𝑦|),

then |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| ≤ 𝑘1 (|𝑥| + |𝑦|) + |𝑓1 (𝑡, 𝑠, 0,0)|

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

237

and |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠))| ≤ 𝑘1 (|𝑥| + |𝑦|) + 𝑚1 , i.e., assumption (iii) of Theorem 3.2 satisfies. So, all conditions of Theorem 3.2 hold. Observe that if 𝑥 ∈ 𝐶(𝐼, ℝ) is a solution of the functional integral equation (7), then 𝑥 is a solution to the functional integral inclusion (1). This complete the proof. 6.2. Continuous Dependence on the Set of Selection 𝑺𝑭𝟏 In this section, we study the continuous dependence of the solution on the set 𝑆𝐹1 of all selections of the set-valued function 𝐹1 . Definition 6.1: The solutions of the functional integral inclusion (1) dependence continuously on the set 𝑆𝐹1 of selections of the set-valued function 𝐹1 , if ∀ 𝜖 > 0, ∃ 𝛿 > 0, such that |𝑓1 (𝑡, 𝑠, 𝑥, 𝑦) − 𝑓1∗ (𝑡, 𝑠, 𝑥, 𝑦)| < 𝛿, 𝑓1 , 𝑓1∗ ∈ 𝑆𝐹1 , 𝑡 ∈ [0, 𝑇], then ∥ 𝑥 − 𝑥 ∗ ∥< 𝜖. Now, we have the following theorem Theorem 6.2: Let the assumptions Theorems 6.1 be satisfied with |𝑓2 (𝑡, 𝑠, 𝑥) − 𝑓2 (𝑡, 𝑠, 𝑦)| ≤ 𝑘2 |𝑥 − 𝑦|. Then the solution of the integral inclusion (1) depends continuously on the 𝑆𝐹1 of all Lipschitzian selections of 𝐹1 . Proof. For the two solutions 𝑥(𝑡) and 𝑥 ∗ (𝑡) of (7) corresponding to the two selections 𝑓1 , 𝑓1∗ ∈ 𝑆𝐹1 , we have |𝑥(𝑡) − 𝑥 ∗ (𝑡)| 𝜑(𝑡) 𝜑(𝑠) = |𝑎(𝑡) + ∫0 𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡)

−𝑎(𝑡) + ∫0 𝜑(𝑡)

≤ ∫0

𝜑(𝑠)

𝑓1∗ (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑠)

|𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝜑(𝑠) −𝑓1∗ (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃)) 𝑑𝜃 𝑔2 (𝑠, 𝜃))

𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))

238

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

𝜑(𝑠)

𝜑(𝑡)

≤ ∫0

|𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑠) −𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠) 𝜑(𝑡) 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝑠 𝑔(𝑡, 𝑠) + ∫0 |𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑠) −𝑓1∗ (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃) 𝜑(𝑡) 𝜑(𝑠) ≤ ∫0 |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃)) 𝜑(𝑠)

𝜑(𝑡)

−𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

𝑓2 (𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) + 𝛿 ∫0

𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑠) ≤ |𝑓1 (𝑡, 𝑠, 𝑥(𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃)) 𝜑(𝑠) −𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑠) 𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) +|𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0 𝜑(𝑡) ∫0

𝜑(𝑠)

𝜑(𝑡)

−𝑓1 (𝑡, 𝑠, 𝑥 ∗ (𝑠), ∫0

≤

𝜑(𝑡) ∫0 𝜑(𝑡)

𝑘1 |𝑥(𝑠) − 𝑥 𝑘1 | ∫

0

0

𝜑(𝑠)

𝜑(𝑡)

∗ (𝑠)|𝑑

𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑠)

+∫ − ∫0

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) + 𝛿 ∫0

𝑓2 (𝑠, 𝜃, 𝑥(𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠) 𝜑(𝑡)

𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))𝑑𝜃 𝑔2 (𝑠, 𝜃))|𝑑𝑠 𝑔1 (𝑡, 𝑠) + 𝛿 ∫0

𝑑𝑠 𝑔1 (𝑡, 𝑠)

≤ ∫0

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)|

+∫

|𝑓2 (𝑠, 𝜃, 𝑥(𝜃)) − 𝑓2 (𝑠, 𝜃, 𝑥 ∗ (𝜃))|𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑠)

0

𝜑(𝑡)

+𝛿 ∫

𝑑𝑠 𝑔1 (𝑡, 𝑠)

0

𝜑(𝑡)

≤ ∫0

𝜑(𝑠)

𝑘1 (|𝑥(𝑠) − 𝑥 ∗ (𝑠)| + ∫0

𝑘2 |𝑥(𝜃) − 𝑥 ∗ (𝜃)|𝑑𝜃 𝑔2 (𝑠, 𝜃))𝑑𝑠 𝑔1 (𝑡, 𝑠)

𝜑(𝑡)

+𝛿 ∫0 𝑑𝑠 𝑔1 (𝑡, 𝑠) ≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥ +𝛿𝜇. Taking supermum for 𝑡 ∈ 𝐼, we have ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝑘𝜇 ∥ 𝑥 − 𝑥 ∗ ∥ +𝑘 2 𝜇2 ∥ 𝑥 − 𝑥 ∗ ∥. Then ∥ 𝑥 − 𝑥 ∗ ∥≤

𝛿𝜇 1−(𝑘𝜇+𝑘 2 𝜇2 )

= 𝜖.

Thus from last inequality, we get ∥ 𝑥 − 𝑥 ∗ ∥≤ 𝜖.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

239

Which proves the continuous dependence of the solution on the set 𝑆𝐹1 of all Lipschitzian selections of 𝐹1 . This completes the proof. 6.3. Set-valued Chandrasekhar non-linear Quadratic Functional Integral Inclusion Now, as an application of the non-linear set-valued delay functional integral equations of Volterra-Stieltjes type (1), we have the following. Let the functions 𝑔𝑖 be defined by 𝑡+𝑠 𝑡ln , 𝑓𝑜𝑟 𝑡 ∈ (0, 𝑇], 𝑠 ∈ 𝐼 , 𝑡 𝑔1 (𝑡, 𝑠) = { 0, 𝑓𝑜𝑟 𝑡 = 0, 𝑠 ∈ 𝐼 and 𝑠+𝜃 𝑠ln , 𝑓𝑜𝑟 𝑠 ∈ (0, 𝑇], 𝜃 ∈ 𝐼 , 𝑠 𝑔2 (𝑠, 𝜃) = { 0, 𝑓𝑜𝑟 𝑠 = 0, 𝜃 ∈ 𝐼 . Let 𝑓2 (𝑡, 𝑠, 𝑥(𝑠)) = 𝑏2 (𝑠)𝑥(𝑠) and 𝐹1 (𝑡, 𝑠, 𝑥(𝑠), 𝑦(𝑠)) = 𝐹1 (𝑏1 (𝑠)𝑥(𝑠), 𝑦(𝑠)) where 𝑠 𝑠 𝑦(𝑠) = ∫0 𝑏2 (𝑠)𝑥(𝜃)𝑑𝜃, 𝑠+𝜃

in (1). Further, using the fact that functions 𝑔𝑖 satisfy assumptions (iv)-(vi) (see [4]), we obtain the Set-valued Chandrasekhar non-linear quadratic functional integral inclusion (2) 𝜑(𝑡) 𝑡 𝜑(𝑡) 𝑠 𝑥(𝑡) ∈ 𝑎(𝑡) + ∫0 𝐹1 (𝑏1 (𝑠)𝑥(𝑠), ∫0 𝑏2 (𝑠)𝑥(𝜃)𝑑𝜃)𝑑𝑠 𝑡, ∈ 𝐼. 𝑡+𝑠

𝑠+𝜃

Now, we can formulate the following existing result concerning with the the nonlinear Chandrasekhar functional integral inclusions (2). Theorem 6.3: Under the assumptions of Theorem 6.1, the fractional integral inclusions (2) has at least one continuous solution 𝑥 ∈ 𝐶[0, 𝑇]. 6.4. Volterra Integral Inclusion of Fractional Order In this subsection, we consider the fractional integral inclusion (4), which has the form 𝑡 (𝑡−𝑠)𝛼−1

𝑥(𝑡) ∈ 𝑝(𝑡) + ∫0

Γ(𝛼)

𝑠 (𝑠−𝜃)𝛽−1

𝐹1 (𝑠, 𝑥(𝑠), ∫0

Γ(𝛽)

𝑓2 (𝑠, 𝜃, 𝑥(𝜃)))𝑑𝜃)𝑑𝑠, 𝑡, 𝑠 ∈ [0, 𝑇].

240

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

Where 𝑡 ∈ 𝐼 = [0, 𝑇] and 𝛼, 𝛽 ∈ (0,1). Note, Γ(𝛼) denotes the gamma function. Let us mention that (4) represents the so-called non-linear Volterra integral inclusions of fractional orders. Recently, the inclusion of such a type was intensively investigated in the paper [17]. Consider the following assumptions: (𝑖) 𝜑: [0, 𝑇] → [0, 𝑇], 𝜑(𝑡) ≤ 𝑡 is continuous and increasing. (𝑖𝑖) 𝑎: [0, 𝑇] → [0, 𝑇] is continuous. (𝑖𝑖𝑖)∗∗ (a) 𝐹1 : 𝐼 × 𝐼 × ℝ × ℝ → 𝑃(ℝ) is a Lipschitzian set-valued map with a non-empty compact convex subset of 2ℝ . (b) 𝑓2 : [0, 𝑇] × [0, 𝑇] × ℝ → ℝ is continuous such that |𝑓2 (𝑡, 𝑠, 𝑥)| ≤ 𝑚2 (𝑡, 𝑠) + 𝑘2 (𝑡, 𝑠)|𝑥|. (c) 𝑘 = sup𝑡∈[0,𝑇] 𝑘𝑖 (𝑡, 𝑠) and 𝑚 = sup𝑡∈[0,𝑇] 𝑚𝑖 (𝑡, 𝑠). (𝑖𝑣) 𝑘𝜇 + 𝑘 2 𝜇2 < 1. Now, as in Theorem 3.2 and [17], the following Theorem can be proved. Theorem 6.4: Under the assumptions (i)-(iv), the fractional integral inclusion (4) has at least one continuous solution 𝑥 ∈ 𝐶[0, 𝑇]. Proof. Now, we show that the functional integral inclusion of fractional orders (4) can be treated as a particular case of the set-valued functional integral equation of Volterra-Stieltjes (1) studied in Section 3. Indeed, we can consider the functions 𝑔𝑖 (𝑤, 𝑧) = 𝑔𝑖 :△𝑖 → 𝑅, (𝑖 = 1,2) defined by the formula. 𝑔1 (𝑡, 𝑠) =

𝑡 𝛼 −(𝑡−𝑠)𝛼 Γ(𝛼+1)

, 𝑔2 (𝑠, 𝜃) =

𝑠 𝛽 −(𝑠−𝜃)𝛽 Γ(𝛽+1)

.

Note that the functions 𝑔𝑖 , (𝑖 = 1,2) satisfy assumptions (𝑖𝑣) − (𝑣𝑖) in Theorem 3.2, see [18, 19]. So, we can formulate the result for the existence of at least one continuous solution 𝑥 ∈ 𝐶[0, 𝑇] of Volterra integral inclusions of fractional order (4). This complete the proof.

Uncertain Parameters

Advances in Special Functions of Fractional Calculus

241

6.4.1. Differential Inclusion Consider now the initial value problem of the differential inclusions (5). Theorem 6.5: Let the assumptions of Theorem 6.4 be satisfied. Then the initial value problem (5) has at least one positive solution 𝑥 ∈ 𝐶([0,1]). 𝑑𝑥(𝑡) Proof. Let 𝑦(𝑡) = , then the inclusion (4), will be 𝑑𝑡

𝑦(𝑡) ∈ 𝐼 𝛼 𝐹1 (𝑡, 𝑥(𝑡), 𝐼1−𝜏 𝑦(𝑡)).

(10)

Letting 𝑓2 (𝑡, 𝑠, 𝑥(𝑠)) = 𝑥(𝑡) and 𝛽 = 1 − 𝜏. applying Theorem 6.4 on the functional inclusion (4), we deduce that there exists a continuous solution 𝑦 ∈ 𝐶[0, 𝑇] of the functional inclusion (5), and this solution depends on the set 𝑆𝐹1 . This implies that the existence of solution 𝑥 ∈ 𝐶[0, 𝑇], 𝑡

𝑥(𝑡) = 𝑥∘ + ∫0 𝑦(𝑠)𝑑𝑠 , of the initial-value problem (5). CONCLUSION In this investigation, two existence results for the non-linear Volterra-Stieltjes integral inclusion (1) are established. As applications, we discuss the non-linear Chandrasekhar set-valued functional integral equation (2) and a non-linear Chandrasekhar quadratic functional integral equation (3). Also, the set-valued fractional orders integral equation (4) and the set-valued fractional-order integrodifferential equations (5) will be considered. Additionally, continuous dependence of solutions for functional integral equation on delay function, 𝑔𝑖 , (𝑖 = 1,2), and on the set 𝑆𝐹1 , have been obtained. CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise.

242

Advances in Special Functions of Fractional Calculus

El-Sayed et al.

ACKNOWLEDGEMENT The authors thank the editors and the reviewers for their useful comments and remarks that helped to improve our manuscript. REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8] [9]

[10] [11] [12]

J. Banaś, and J.C. Mena, "Some properties of nonlinear Volterra-Stieltjes integral operators", Comput. Math. Appl., vol. 49, no. 9-10, pp. 1565-1573, 2005. http://dx.doi.org/10.1016/j.camwa.2004.05.016 J. Banaś, and J. Dronka, "Integral operators of volterra-stieltjes type, their properties and applications", Math. Comput. Model., vol. 32, no. 11-13, pp. 1321-1331, 2000. http://dx.doi.org/10.1016/S0895-7177(00)00207-7 A.M. El-Sayed, and S.M. Al-Issa, "On the existence of solutions of a set-valued functional integral equation of Volterra–Stieltjes type and some applications", Adv. Differ. Equ., vol. 2020, no. 1, p. 59, 2020. http://dx.doi.org/10.1186/s13662-020-2531-4 A.M.A. El-Sayed, Sh.M. Al-Issa, and Y. Omar, "On Chandrasekhar functional integral inclusion and Chandrasekhar quadratic integral equation via a non-linear Urysohn-Stieltjes functional integral inclusion", Adv. Differ. Equ., vol. 2021, no. 137, 2021. E. Ahmed, A.M.A. El-Sayed, A.E.M. El-Mesiry, and H.A.A. El-Saka, "Numerical solution for the fractional replicator equation", Int. J. Mod. Phys. C, vol. 16, no. 7, pp. 1017-1025, 2005. http://dx.doi.org/10.1142/S0129183105007698 A.M.A. El-Sayed, and F.M. Gaafar, "Fractional calculus and some intermediate physical processes", Appl. Math. Comput., vol. 144, no. 1, pp. 117-126, 2003. http://dx.doi.org/10.1016/S0096-3003(02)00396-X S.Z. Rida, A.M.A. El-Sayed, and A.A.M. Arafa, "Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model", J. Stat. Phys., vol. 140, no. 4, pp. 797-811, 2010. http://dx.doi.org/10.1007/s10955-010-0007-8 A. Cellina, and S. Solimini, "Continuous extension of selection", Bull. Pol. Acad. Sci. Math., vol. 35, no. 9, 1978. A.M.A. El-Sayed, and A.G. Ibrahim, "Multivalued fractional differential equations", Appl. Math. Comput., vol. 68, no. 1, pp. 15-25, 1995. http://dx.doi.org/10.1016/0096-3003(94)00080-N K. Kuratowski, and C. Ryll-Nardzewski, Ageneral theorem on selectors”Bull. de Íacademic polonaise des sciences, S ér., vol. 13, Sci-math. Astron-phys, 1995. J.P. Aubin, and A. Cellina, Differential inclusions: set-valued maps and viability theory., vol. 264, Springer ScienceBusiness Media, 2012. A.M.A. El-Sayed, and A.G. Ibrahim, "Set-valued integral equations of fractional-orders", Appl. Math. Comput., vol. 118, no. 1, pp. 113-121, 2001.

Uncertain Parameters

[13]

[14] [15]

[16]

[17]

[18]

[19]

Advances in Special Functions of Fractional Calculus

243

http://dx.doi.org/10.1016/S0096-3003(99)00087-9 J. Appell, J. Bana's, and N. Merentes, Bounded variation and around., vol. 17, Series in Nonlinear Analysis and Applications: De Gruyter, Berlin, 2013. http://dx.doi.org/10.1515/9783110265118 A.N. Kolmogorov, and S.V. Fomin, Introductory Real Analysis, Dovor Publ Inc.: New York, 1975. K. Goebel, and W.A. Kirk, Topics in Metric Fixed point theory., Cambridge University Press: Cambridge, 1990. http://dx.doi.org/10.1017/CBO9780511526152 J. Banaś, and T. Zaja̧c, "A new approach to the theory of functional integral equations of fractional order", J. Math. Anal. Appl., vol. 375, no. 2, pp. 375-387, 2011. http://dx.doi.org/10.1016/j.jmaa.2010.09.004 A.M. El-Sayed, and S.M. Al-Issa, "On the existence of solutions of a set-valued functional integral equation of Volterra–Stieltjes type and some applications", Adv. Differ. Equ., vol. 2020, no. 1, p. 59, 2020. http://dx.doi.org/10.1186/s13662-020-2531-4 J. Banaś, and B. Rzepka, "Nondecreasing solutions of a quadratic singular Volterra integral equation", Math. Comput. Model., vol. 49, no. 3-4, pp. 488-496, 2009. http://dx.doi.org/10.1016/j.mcm.2007.10.021 A.M.A. El-Sayed, and S.M. Al-Issa, "On a set-valued functional integral equation of Volterra-Stieltjes type", J. Math. Com. Sci., vol. 21, no. 4, pp. 273-285, 2020. http://dx.doi.org/10.22436/jmcs.021.04.01

244

Advances in Special Functions of Fractional Calculus, 2023, 244-258

CHAPTER 13

Certain Saigo Fractional Derivatives of Extended Hypergeometric Functions S. Jain1,*, R. Goyal2, P. Agarwal2,3 and S. Momani3,4 1

Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India

2

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India

3

Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

4

Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan Abstract: This article aims to establish Saigo fractional derivatives of extended hypergeometric functions. Some special cases of these integrals are also derived.

Keywords: Beta function, Gamma Function, Saigo-fractional integral operator, Saigo-fractional derivative operator, Reimann-Liouville fractional integral operator, Reimann-Liouville fractional derivative operator, Hadamard product, Gauss hypergeometric function, Confluent hypergeometric function. 1. INTRODUCTION In recent years, many extensions and generalizations of special functions witnessed a significant evolution. This modification in the theory of special functions offers an analytic foundation for the many problems in mathematical physics and engineering sciences, which have been solved and have various practical applications. The theory of special functions revolves around the two most important basic special functions, i.e., the beta function and the Gamma function because most of the special functions are expressed either in terms of the beta function or the Gamma function.

*Corresponding

author S. Jain: Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India; Tel: 9928279174; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

245

The classical Euler beta function is defined as follows [1]: 1

𝐵(𝑥1 , 𝑥2 ) = ∫0 𝑡 𝑥1 −1 (1 − 𝑡)𝑥2 −1 𝑑𝑡,

ℜ(𝑥1 ), ℜ(𝑥1 ) > 0.

(1)

Gamma function is defined as follows [1]: ∞

Γ(𝑥1 ) = ∫0 𝑡 𝑥1 −1 𝑒 −𝑡 𝑑𝑡,

ℜ(𝑥1 ) > 0.

(2)

Further, the mathematical and physical applications of hypergeometric functions are found in various areas of applied mathematics, mathematical physics, and engineering. The Gauss hypergeometric function is a solution of a homogenous second-order differential equation which is called the hypergeometric differential equation, and it is given by 𝑧(1 − 𝑧)

𝑑2𝑤 𝑑𝑧 2

+ (𝑐 − (𝑎 + 𝑏 + 1)𝑧)

The Gauss hypergeometric function 2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧)

2 𝐹1

𝑑𝑤 𝑑𝑧

− 𝑎𝑏𝑤 = 0.

(3)

is defined as [2]:

= 𝐹(𝑎, 𝑏, 𝑐; 𝑧) = ∑∞ 𝑘=0

(𝑎)𝑘 (𝑏)𝑘 𝑧 𝑘 (𝑐)𝑘

𝑘!

,

(4)

where (𝑢)𝑘 represents the Pochhammer symbol defined below: (𝑢)𝑘 =

Γ(𝑢 + 𝑘) 1 𝑘 = 0; 𝑢 ∈ ℂ/{0}, ={ 𝑢(𝑢 + 1). . . . . (𝑢 + 𝑘 − 1) 𝑘 ∈ ℕ; 𝑢 ∈ ℂ. Γ(𝑢) 𝑧

Later Kummer replaced the parameter 𝑧 by d taking limit 𝑏 → ∞ in the equation 𝑏 (3), then the hypergeometric differential equation becomes a confluent hypergeometric differential equation or Kummer's equation. 𝑧

𝑑2𝑤 𝑑𝑧 2

+ (𝑐 − 𝑧)

𝑑𝑤 𝑑𝑧

− 𝑎𝑤 = 0.

(5)

The confluent hypergeometric function is the solution of the above differential equation (5). A confluent hypergeometric function is defined as [2]:

246

Advances in Special Functions of Fractional Calculus 1 𝐹1 (𝑎, 𝑐; 𝑧)

= Φ(𝑎, 𝑐; 𝑧) = ∑∞ 𝑘=0

Jain et al. (𝑎)𝑘 𝑧 𝑘 (𝑐)𝑘 𝑘!

,

(6)

Very recently, Goyal et al. [3] introduced an extension of the beta function using the Wiman function, thus studying various properties and relationships of that function: (𝑢)

−1

1

𝐵(𝑢1 ,𝑢2 ) (𝑦1 , 𝑦2 ) = ∫0 𝑡 𝑦1 −1 (1 − 𝑡)𝑦2−1 𝐸𝑢1 ,𝑢2 (−𝑢(𝑡(1 − 𝑡)) ) 𝑑𝑡,

(7)

where, min{ℜ(𝑦1 ), ℜ(𝑦2 )} > 0, ℜ(𝑢1 ) > 0, ℜ(𝑢2 ) > 0, 𝑢 ≥ 0, and 𝐸𝑢1 ,𝑢2 (𝑧) is a 2-parameter Mittag-Leffler function given by [4]. Motivated by the above work, Jain et al. [5] extended Gauss hypergeometric function, and confluent hypergeometric function by using the above-extended beta function and studied various properties of these extended functions. They also studied the increasing or decreasing nature (monotonicity), log-concavity, and logconvexity of the extended beta function defined in [3]. (𝑠) 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑧)

(𝑠)

=

∑∞ 𝑘=0

(𝑞0 )𝑘 𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑘,𝑞2 −𝑞1 ) 𝑧 𝑘 1 2 , 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑘!

(8) (𝑠)

Where, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, |𝑧| < 1 and d 𝐵(𝑠1,𝑠2) (𝑤1 , 𝑤2 ) is the extended beta function. The extended confluent hypergeometric function is defined as [5]: (𝑠) Φ(𝑠1 ,𝑠2 ) (𝑞1 , 𝑞2 ; 𝑧)

(𝑠)

=

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑘,𝑞2 −𝑞1 ) 𝑧 𝑘 1 2 ∑∞ , 𝑘=0 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑘!

(9) (𝑠)

where, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, and 𝐵(𝑠1 ,𝑠2 ) (𝑤1 , 𝑤2 ) is the extended beta function. The concept of Hadamard product(convolution) of the functions 𝑓 and 𝑔 is very important for our results. Hadamard product of 𝑓 and 𝑔 defined as follows [6]:

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

247

(10) The introduction of fractional calculus is a very important development in the field of calculus because it has to be widely applicable in many fields of mathematical, physical and applied sciences. The Fractional calculus operators having various special functions, have been used for modelling of system in turbulence and fluid dynamics, stochastic dynamic system, thermonuclear fusion, image processing, nonlinear biological system, and in quantum mechanics, due to which many researchers [7-21] are continuously working on them for many years. Reimann-Liouville fractional integral operators defined as follows [21]: 𝑎 𝑓(𝑡)) (𝑥) = (𝐼0,𝑥

𝑥 1 ∫ (𝑥 Γ(𝑎) 0

− 𝑡)𝑎−1 𝑓(𝑡)𝑑𝑡,

(11)

And 𝑎 (𝐼𝑥,∞ 𝑓(𝑡)) (𝑥) =

∞ 1 ∫ (𝑡 Γ(𝑎) 𝑥

− 𝑥)𝑎−1 𝑓(𝑡)𝑑𝑡,

(12)

Here, 𝑥 > 0, 𝑎 ∈ ℂ and ℜ(𝑎) > 0. Reimann-Liouville fractional derivative operators are defined as follows [21]: 𝑎 (𝐷0,𝑥 𝑓(𝑡)) (𝑥) = (

=

𝑥 1 𝑑 𝑘 ∫ (𝑥 − 𝑡)𝑘−𝑎−1 𝑓(𝑡)𝑑𝑡, ) 𝑑𝑥 Γ(𝑘 − 𝑎) 0

𝑘

𝑑

𝑘−𝑎 ( ) (𝐼0,𝑥 𝑓(𝑡)) (𝑥) 𝑑𝑥

(𝑘 = [ ℜ(𝑎)] + 1)

(13)

And 𝑎 (𝐷𝑥,∞ 𝑓(𝑡)) (𝑥) = (−1)𝑘 (

𝑑

𝑘

∞ 𝑑 𝑘 1 ∫ (𝑡 − 𝑥)𝑘−𝑎−1 𝑓(𝑡)𝑑𝑡, ) 𝑑𝑥 Γ(𝑘 − 𝑎) 𝑥

𝑘−𝑎 = (−1)𝑘 ( ) (𝐼𝑥,∞ 𝑓(𝑡)) (𝑥) 𝑑𝑥

(𝑘 = [ ℜ(𝑎)] + 1)

(14)

248

Advances in Special Functions of Fractional Calculus

Jain et al.

Here, 𝑥 > 0, 𝑎 ∈ ℂ and ℜ(𝑎) > 0 and [x] denotes the greatest integer less than or equal to the real number x. Saigo fractional integral operators defined as follows [23]: 𝑎,𝑎′,𝑏 (𝐼0,𝑥 𝑓(𝑡)) (𝑥)

𝑥 𝑎−𝑎′ Γ(𝑎)

𝑥

∫0 (𝑥 − 𝑡)𝑎−1

2 𝐹1 (𝑎

𝑥 1 ∫ 𝑡 −𝑎−𝑎′ (𝑡 Γ(𝑎) 0

− 𝑥)𝑎−1

𝑡

(15)

+ 𝑎′, −𝑏, 𝑎; 1 − ) 𝑓(𝑡)𝑑𝑡, 𝑥

And 𝑎,𝑎′,𝑏 (𝐼𝑥,∞ 𝑓(𝑡)) (𝑥) =

2 𝐹1 (𝑎

𝑥

+ 𝑎′, −𝑏, 𝑎; 1 − ) 𝑓(𝑡)𝑑𝑡, 𝑡

Here, , 𝑥 > 0, 𝑎, 𝑎′, 𝑏 ∈ ℂ and ℜ(𝑎) > 0 and hypergeometric function

2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧)

(16) is Gauss

defined in [2]. Saigo fractional derivative operators are defined as follows [23]: 𝑎,𝑎′,𝑏 −𝑎,−𝑎′,𝑎+𝑏 𝑓(𝑡)) (𝑥) = (𝐼0,𝑥 𝑓(𝑡)) (𝑥), (𝐷0,𝑥

=

𝑑

𝑘

−𝑎+𝑘,−𝑎′−𝑘,𝑎+𝑏−𝑘 ( ) (𝐼0,𝑥 𝑓(𝑡)) (𝑥)(ℜ(𝑎) > 0, 𝑘 = [ ℜ(𝑎)] + 1) 𝑑𝑥

(17)

And 𝑎,𝑎′,𝑏 −𝑎,−𝑎′,𝑎+𝑏 (𝐷𝑥,∞ 𝑓(𝑡)) (𝑥) = (𝐼𝑥,∞ 𝑓(𝑡)) (𝑥), 𝑑

𝑘

−𝑎+𝑘,−𝑎′−𝑘,𝑎+𝑏 = (−1)𝑘 ( ) (𝐼𝑥,∞ 𝑓(𝑡)) (𝑥) 𝑑𝑥

(ℜ(𝑎) > 0, 𝑘 = [ ℜ(𝑎)] + 1) (18)

Here, 𝑥 > 0, 𝑎, 𝑎′, 𝑏 ∈ ℂ and ℜ(𝑎) > 0, and [x] denotes the greatest integer less than or equal to the real number x. We observe that Saigo fractional derivative operators contain Reimann- Liouville fractional derivative operators. Remark: If we set 𝑎′ = −𝑎 in the equations (17) and (18), then Saigo fractional derivative operators reduce to Reimann-Liouville fractional derivative operators (13) and (14), respectively.

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

𝑎,−𝑎,𝑏 𝑎 (𝐷0,𝑥 𝑓(𝑡)) (𝑥) = (𝐷0,𝑥 𝑓(𝑡)) (𝑥)

249

(19)

And 𝑎,−𝑎,𝑏 𝑎 (𝐷0,𝑥 𝑓(𝑡)) (𝑥) = (𝐷𝑥,∞ 𝑓(𝑡)) (𝑥)

(20)

The following lemmas proved in Kilbas, and Sebast in [23, 24] are useful to prove our main results. Lemma 1. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > 𝑚𝑎𝑥[0, ℜ(𝑎′ − 𝑏)] then following formulas holds true: 𝑎,𝑎′,𝑏 𝑐−1 (𝐼0,𝑥 𝑡 )(𝑥 ) =

Γ(𝑐)Γ(𝑐+𝑏−𝑎′) Γ(𝑐−𝑎′)Γ(𝑎+𝑏+𝑐)

𝑥 𝑐−𝑎′−1 ,

(21)

And 𝑎,𝑎′,𝑏 𝑐−1 (𝐼𝑥,∞ 𝑡 )(𝑥) =

Γ(𝑎′−𝑐+1)Γ(𝑐−𝑏+1) Γ(1−𝑐)Γ(𝑎+𝑏+𝑎′−𝑐+1)

𝑥 𝑐−𝑎′−1 ,

(22)

Here, ℜ(𝑐) < 1 + 𝑚𝑖𝑛[ℜ(𝑎′), ℜ(𝑏)]. Lemma 2. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 be such that ℜ(𝑎) > 0, then following formulas holds true: ′

𝑎,𝑎 ,𝑏 𝑐−1 (𝐷0,𝑥 𝑡 ) (𝑥) =

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 , Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′)

(ℜ(𝑐) > −𝑚𝑖𝑛{0, ℜ(𝑎 + 𝑎′ + 𝑏) })

(23)

And ′

𝑎,𝑎 ,𝑏 𝑐−1 (𝐷𝑥,∞ 𝑡 ) (𝑥) =

Γ(1 − 𝑐 − 𝑎′)Γ(1 − 𝑐 + 𝑎 + 𝑏) 𝑐+𝑎′−1 𝑥 , Γ(1 − 𝑐)Γ(1 − 𝑐 + 𝑏 − 𝑎′)

(ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(𝑎 + 𝑏)} 𝑎𝑛𝑑 𝑛 = [ℜ(𝑎)] + 1) (24) 1.1 Main Results In this section, we establish Saigo fractional derivatives of the extended hypergeometric functions defined in (8) and (9) by using Lemma (2).

250

Advances in Special Functions of Fractional Calculus

Jain et al.

Theorem 1. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑚𝑖𝑛{0, ℜ(𝑎 + 𝑎′ + 𝑏) } 𝑎𝑛𝑑 ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended Gauss hypergeometric function defined in (8), is given by: ′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥 (𝑠)

× 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

2 𝐹2 (𝑐, 𝑐

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′)

+ 𝑏 + 𝑎 + 𝑎′, 𝑐 + 𝑏, 𝑐 + 𝑎′; 𝑥)

(25)

Where, 2 F2 a, b, c, d ; z is a special case of the generalized hypergeometric

function p Fq a1 ,.........., a p , b1 ,..........aq ; z when 𝑝 = 𝑞 = 2 and defined in [2]. Proof: (𝑠)

By taking 𝑓(𝑡) = 𝑡 𝑐−1 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡) and using the definition of extended Gauss hypergeometric function (8), we get: ′

(𝑠)

′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑎,𝑎 ,𝑏 𝑐−1 ∞ ∑𝑛=0 (𝐷0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥 𝑡

(𝑞0 )𝑛 𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑞1 ) 𝑡 𝑛 1 2 ) (𝑥) 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑛!

(26)

Then interchanging the order of derivative and summation, which is valid under the conditions of Theorem (1), we have: (𝑠)

′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 (𝐷0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = ∑∞ 𝑛=0

(𝑞0 )𝑛 𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑞1 ) 1 𝑎,𝑎′ ,𝑏 𝑐+𝑛−1 1 2 (𝐷0,𝑥 𝑡 ) (𝑥) ) 𝐵(𝑞1 ,𝑞2 −𝑞1 𝑛!

(27)

Then by using (21), we have: ′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) (𝐷0,𝑥

=

(𝑠)

∑∞ 𝑛=0

(𝑞0 )𝑛 𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑞1 ) 1 Γ(𝑐+𝑛)Γ(𝑐+𝑛+𝑏+𝑎+𝑎′) 1 2 𝑥 𝑐+𝑛+𝑎′−1 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑛! Γ(𝑐+𝑛+𝑏)Γ(𝑐+𝑛+𝑎′)

After simplification, the above equation reduces to

(28)

Hypergeometric Functions ′

Advances in Special Functions of Fractional Calculus

251

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥 (𝑠)

𝑥 𝑐+𝑎′−1 ∑∞ 𝑛=0

(𝑞0 )𝑛 𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑞1 ) 1 Γ(𝑐+𝑛)Γ(𝑐+𝑛+𝑏+𝑎+𝑎′) 1 2 𝑥𝑛 𝑛! Γ(𝑐+𝑛+𝑏)Γ(𝑐+𝑛+𝑎′) 𝐵(𝑞1 ,𝑞2 −𝑞1 )

After using the (𝑎)𝑛 = ′

Γ(𝑎+𝑛) Γ(𝑎)

(29)

, the above equation reduces to the following form:

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 (𝐷0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) ∞

=𝑥

𝑐+𝑎′−1

(𝑠)

(𝑞0 )𝑛 𝐵(𝑠1,𝑠2) (𝑞1 + 𝑛, 𝑞2 − 𝑞1 ) (𝑐)𝑛 (𝑐 + 𝑏 + 𝑎 + 𝑎′)𝑛 𝑥 𝑛 Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) ∑ Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′) (𝑐 + 𝑏)𝑛 (𝑐 + 𝑎′)𝑛 𝑛! 𝐵(𝑞1 , 𝑞2 − 𝑞1 ) 𝑛=0

then by interpreting the above equation with the help of the concept of the Hadamard Product given by (10), we get our desired result. ′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥 (𝑠)

× 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

2 𝐹2 (𝑐, 𝑐

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′)

+ 𝑏 + 𝑎 + 𝑎′, 𝑐 + 𝑏, 𝑐 + 𝑎′; 𝑥).

Theorem 2. Consider 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(𝑎 + 𝑏)} 𝑎𝑛𝑑 𝑛 = [ℜ(𝑎)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended Gauss hypergeometric function defined in (8), is given by: Γ(1 − 𝑐 − 𝑎′)Γ(1 − 𝑐 + 𝑎 + 𝑏) 𝑐+𝑎′−1 1 𝑎,𝑎′ ,𝑏 𝑐−1 (𝑠) (𝐷𝑥,∞ 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; )) (𝑥) = 𝑥 𝑡 Γ(1 − 𝑐)Γ(1 − 𝑐 + 𝑏 − 𝑎′) × 𝐹(𝑠(𝑠)1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

2 𝐹2 (1

− 𝑐 − 𝑎′, 1 − 𝑐 + 𝑎 + 𝑏, 1 − 𝑐, 1 − 𝑐 + 𝑏 − 𝑎′; 𝑥)

(29a)

Where, 2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined in [2]. Theorem 3. Consider 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑚𝑖𝑛 {0, ℜ(𝑎 + 𝑎′ + 𝑏) } 𝑎𝑛𝑑 ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0,

252

Advances in Special Functions of Fractional Calculus

Jain et al.

𝑠 ≥ 0, then Saigo fractional derivative of extended confluent hypergeometric function defined in (9), is given by ′

(𝑠)

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 Φ(𝑠1 ,𝑠2) (𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥

(𝑠)

× Φ(𝑠1,𝑠2) (𝑞1 , 𝑞2 ; 𝑥 ) ∗

2 𝐹2 (𝑐, 𝑐

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′)

+ 𝑏 + 𝑎 + 𝑎′, 𝑐 + 𝑏, 𝑐 + 𝑎′; 𝑥) (30)

Where, 2 F2 a, b, c, d ; z is a special case of the generalized hypergeometric

function p Fq a1 ,.........., a p , b1 ,..........aq ; z when 𝑝 = 𝑞 = 2 and defined in [2].

Theorem 4. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(𝑎 + 𝑏)} 𝑎𝑛𝑑 𝑛 = [ℜ(𝑎)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended confluent hypergeometric function defined in (9), is given by 1 Γ(1 − 𝑐 − 𝑎′)Γ(1 − 𝑐 + 𝑎 + 𝑏) 𝑐+𝑎′−1 𝑎,𝑎′ ,𝑏 𝑐−1 (𝑠) 𝑥 (𝐷𝑥,∞ 𝑡 Φ(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; )) (𝑥) = 𝑡 Γ(1 − 𝑐)Γ(1 − 𝑐 + 𝑏 − 𝑎′) (𝑠)

× Φ(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

2 𝐹2 (1

− 𝑐 − 𝑎′, 1 − 𝑐 + 𝑎 + 𝑏, 1 − 𝑐, 1 − 𝑐 + 𝑏 − 𝑎′; 𝑥)

(31)

Where, 2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined in [2]. The proofs of Theorems (2), (3) and (4) are the same as those of Theorem (1). 1.2 Some Special Cases of the above Fractional Integral Formulas By substituting the suitable values to the parameters involved in the results established in Theorems 1-4, we have the following special cases. By putting 𝑎′ = −𝑎, the Saigo hypergeometric fractional derivative operators reduce to RiemannLiouville fractional derivative operators, then the results in (23), (29), (30) and (31) reduce to the following form. Corollary 1. Let 𝑎, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > 0 , ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0 and 𝑠 ≥ 0, then the following result holds true:

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus (𝑠)

𝑎 𝑐−1 (𝐷0,𝑥 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝑠)

× 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

1 𝐹1 (𝑐, 𝑐

253

Γ(𝑐) 𝑥 𝑐+𝑎−1 Γ(𝑐 + 𝑎)

+ 𝑎; 𝑥)

(32)

Where, 1 𝐹1 (𝑎, 𝑏 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧), when 𝑝 = 𝑞 = 1 and defined in [2]. Corollary 2. Consider 𝑎, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(−𝑎′)} 𝑎𝑛𝑑 𝑛 = [ℜ(−𝑎′)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, , ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended Gauss hypergeometric function defined in (8), is given by 1 Γ(1 − 𝑐 − 𝑎) 𝑐+𝑎−1 (𝑠) 𝑎 (𝐷𝑥,∞ 𝑡 𝑐−1 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; )) (𝑥) = 𝑥 𝑡 Γ(1 − 𝑐) (𝑠)

× 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

1 𝐹1 (1

− 𝑐 − 𝑎, 1 − 𝑐; 𝑥)

(33)

Where, 1 𝐹1 (𝑎, 𝑏 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 1 and defined in [2]. Corollary 3. Consider 𝑎, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > 0 𝑎𝑛𝑑 ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended confluent hypergeometric function defined in (9), is given by (𝑠)

𝑎 𝑐−1 (𝐷0,𝑥 𝑡 Φ(𝑠1 ,𝑠2 ) (𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝑠)

Γ(𝑐) 𝑥 𝑐+𝑎−1 Γ(𝑐 + 𝑎)

× Φ(𝑠1 ,𝑠2) (𝑞1 , 𝑞2 ; 𝑥) ∗

1 𝐹1 (𝑐, 𝑐

+ 𝑎; 𝑥)

(34)

Where, 1 𝐹1 (𝑎, 𝑏; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 1 and defined in [2]. Corollary 4. Let 𝑎, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(−𝑎′)} 𝑎𝑛𝑑 𝑛 = [ℜ(−𝑎′)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of extended confluent hypergeometric function defined in (9), is given by

254

Advances in Special Functions of Fractional Calculus

Jain et al.

1 Γ(1 − 𝑐 − 𝑎) 𝑐+𝑎−1 (𝑠) 𝑎 (𝐷𝑥,∞ 𝑡 𝑐−1 Φ(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; )) (𝑥) = 𝑥 𝑡 Γ(1 − 𝑐) (𝑠)

× Φ(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

1 𝐹1 (1

− 𝑐 − 𝑎, 1 − 𝑐; 𝑥)

(35)

Where, 1 𝐹1 (𝑎, 𝑏 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 1 and defined in [2]. If we 𝑠1 = 𝑠2 = 1and 𝑠 = 0, then a new extension of Gauss hypergeometric function (8) and a new extension of confluent hypergeometric function (9) reduce to Gauss hypergeometric function (4) and confluent hypergeometric function (6), then from the formulae established in (23), (29), (30) and (31), we have the following results. ℜ(𝑐) > Corollary 5. Assume 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, −𝑚𝑖𝑛{0, ℜ(𝑎 + 𝑎′ + 𝑏) } 𝑎𝑛𝑑 ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, then Saigo fractional derivative Gauss hypergeometric function defined in (4), is given by ′

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 (𝐷0,𝑥

× Where, function in [2].

2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡)) (𝑥)

2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥)

∗

2 𝐹2 (𝑐, 𝑐

=

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′)

+ 𝑏 + 𝑎 + 𝑎′, 𝑐 + 𝑏, 𝑐 + 𝑎′; 𝑥)

(36)

2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑

; 𝑧) is a special case of the generalized hypergeometric 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined

Corollary 6. Consider 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(𝑎 + 𝑏)} 𝑎𝑛𝑑 𝑛 = [ℜ(𝑎)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0 then Saigo fractional derivative of Gauss hypergeometric function defined in (4), is given by ′

𝑎,𝑎 ,𝑏 𝑐−1 (𝐷𝑥,∞ 𝑡

×

2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ;

2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥)

∗

2 𝐹2 (1

1 Γ(1 − 𝑐 − 𝑎′)Γ(1 − 𝑐 + 𝑎 + 𝑏) 𝑐+𝑎′−1 𝑥 )) (𝑥) = 𝑡 Γ(1 − 𝑐)Γ(1 − 𝑐 + 𝑏 − 𝑎′) − 𝑐 − 𝑎′, 1 − 𝑐 + 𝑎 + 𝑏, 1 − 𝑐, 1 − 𝑐 + 𝑏 − 𝑎′; 𝑥)

(37)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

255

Where, 2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑 ; 𝑧) is a special case of the generalized hypergeometric function 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined in [2]. Corollary 7. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑚𝑖𝑛{0, ℜ(𝑎 + 𝑎′ + 𝑏) } 𝑎𝑛𝑑 ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of confluent hypergeometric function defined in (6), is given by ′

𝑎,𝑎 ,𝑏 𝑐−1 𝑡 Φ(𝑞1 , 𝑞2 ; 𝑡)) (𝑥) = (𝐷0,𝑥

× Φ(𝑞1 , 𝑞2 ; 𝑥) ∗ Where, function in [2].

2 𝐹2 (𝑐, 𝑐

Γ(𝑐)Γ(𝑐 + 𝑏 + 𝑎 + 𝑎′) 𝑐+𝑎′−1 𝑥 Γ(𝑐 + 𝑏)Γ(𝑐 + 𝑎′) + 𝑏 + 𝑎 + 𝑎′, 𝑐 + 𝑏, 𝑐 + 𝑎′; 𝑥)

(38)

2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑

; 𝑧) is a special case of the generalized hypergeometric , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined 𝐹 (𝑎 𝑝 𝑞 1

Corollary 8. Let 𝑎, 𝑎′, 𝑏, 𝑐 ∈ ℂ be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + 𝑚𝑖𝑛{ℜ(−𝑎′ − 𝑛), ℜ(𝑎 + 𝑏)} 𝑎𝑛𝑑 𝑛 = [ℜ(𝑎)] + 1, ℜ(𝑞2 ) > ℜ(𝑞1 ) > 0, ℜ(𝑠1 ) > 0, ℜ(𝑠2 ) > 0, 𝑠 ≥ 0, then Saigo fractional derivative of confluent hypergeometric function defined in (6), is given by 1 Γ(1 − 𝑐 − 𝑎′)Γ(1 − 𝑐 + 𝑎 + 𝑏) 𝑐+𝑎′−1 𝑎,𝑎′ ,𝑏 𝑐−1 (𝐷𝑥,∞ 𝑡 Φ (𝑞0 , 𝑞1 , 𝑞2 ; )) (𝑥) = 𝑥 Γ(1 − 𝑐)Γ(1 − 𝑐 + 𝑏 − 𝑎′) 𝑡 × Φ(𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗

Where, function in [2].

2 𝐹2 (1

− 𝑐 − 𝑎′, 1 − 𝑐 + 𝑎 + 𝑏, 1 − 𝑐, 1 − 𝑐 + 𝑏 − 𝑎′; 𝑥)

(39)

2 𝐹2 (𝑎, 𝑏, 𝑐, 𝑑

; 𝑧) is a special case of the generalized hypergeometric 𝑝 𝐹𝑞 (𝑎1 , . . . . . . . . . . , 𝑎𝑝 , 𝑏1 , . . . . . . . . . . 𝑎𝑞 ; 𝑧) when 𝑝 = 𝑞 = 2 and defined

CONCLUSION Motivated by the demonstrated usages and the potential for applications of the many operators in fractional calculus and also the considerably large spectrum of a special function in mathematical, physical and engineering, we have introduced here new results for the Saigo fractional derivative involving extended hypergeometric functions. Our results have been expressed as the Hadamard

256

Advances in Special Functions of Fractional Calculus

Jain et al.

product of the extended hypergeometric functions and generalized hypergeometric functions. Some special cases of our main results have also been derived. CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENTS The authors would like to thank the anonymous referees for their valuable comments and suggestions. Shilpi Jain is very thankful to SERB (project number: MTR/2017/000194) for providing the necessary facilities. Rahul Goyal and Praveen Agarwal are thankful to the (NBHM (R.P)/RD II/7867) for providing the necessary facilities. REFERENCES [1] [2] [3]

[4]

[5]

[6] [7]

[8]

E.D. Rainville, Special Functions., Macmillan: New York, NY, USA, 1960. W.J.F. Olver, W.D. Lozier, F.R. Boisvert, and W.C. Clark, NIST Handbook of Mathematical Functions., Cambridge University Press: New York, NY, USA, 2010. R. Goyal, S. Momani, P. Agarwal, and M.T. Rassias, "An Extension of Beta Function by Using Wiman’s Function", Axioms, vol. 10, no. 3, p. 187, 2021. http://dx.doi.org/10.3390/axioms10030187 A. Wiman, "Über den Fundamentalsatz in der Teorie der Funktionen Ea(x)", Acta Math., vol. 29, no. 0, pp. 191-201, 1905. http://dx.doi.org/10.1007/BF02403202 S. Jain, R. Goyal, P. Agarwal, A. Lupica, and C. Cesarano, "Some results of extended beta function and hypergeometric functions by using Wiman’s function", Mathematics, vol. 9, no. 22, p. 2944, 2021. http://dx.doi.org/10.3390/math9222944 T. Pohlen, The Hadamard Product and Universal Power Series, Universit_at Trier Trier Germany, Germany, 2009. P. Agarwal, "Fractional integration of the product of two H-functions and a general class of polynomials", Asian J. Appl. Sci., vol. 5, no. 3, pp. 144-153, 2012. http://dx.doi.org/10.3923/ajaps.2012.144.153 G. Singh, P. Agarwal, S. Araci, and M. Acikgoz, "Certain fractional calculus formulas involving extended generalized Mathieu series", Adv. Di_erence Equ, no. 1, pp. 1-30, 2018.

Hypergeometric Functions

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18] [19]

[20]

[21]

Advances in Special Functions of Fractional Calculus

257

P. Agarwal, F. Qi, M. Chand, and G. Singh, "Some fractional differential equations involving generalized hypergeometric functions", Journal of Applied Analysis, vol. 25, no. 1, pp. 3744, 2019. http://dx.doi.org/10.1515/jaa-2019-0004 P. Agarwal, T.M. Rassias, G. Singh, and S. Jain, Certain Fractional Integral and Differential Formulas Involving the Extended Incomplete Generalized Hypergeometric Functions, Mathematical Analysis and Applications. Springer Optimization and Its Applications., vol. Vol. 154, Springer: Cham, 2019. P. Agarwal, G. Wang, and M. Al-Dhaifallah, "Fractional calculus operators and their applications to thermal systems", Adv. Mech. Eng., vol. 10, no. 6, 2018. http://dx.doi.org/10.1177/1687814018782028 I.O. Kymaz, P. Agarwal, and S. Jain, "Cetinkaya, A.,On a new extension of Caputo fractional derivative operator", Advances in Real and Complex Analysis with Applications, pp. 261275, 2017. İ.O. Kıymaz, A. Çetinkaya, and P. Agarwa, "An extension of Caputo fractional derivative operator and its applications", J. Nonlinear Sci. App., vol. 9, no. 6, pp. 3611-3621, 2016. http://dx.doi.org/10.22436/jnsa.009.06.14 P. Agarwal, and J. Choi, "Fractional calculus operators and their image formulas", Journal of the Korean Mathematical Society, vol. 53, no. 5, pp. 1183-1210, 2016. http://dx.doi.org/10.4134/JKMS.j150458 J. Choi, and P. Agarwal, "A note on fractional integral operator associated with multiindex Mittag-Leffler functions", Filomat, vol. 30, no. 7, pp. 1931-1939, 2016. http://dx.doi.org/10.2298/FIL1607931C P. Agarwal, J. Choi, K.B. Kachhia, J.C. Prajapati, and H. Zhou, "Some integral transforms and frac-tional integral formulas for the extended hypergeometric functions", Communications of the Korean Mathematical Society, vol. 31, no. 3, pp. 591-601, 2016. http://dx.doi.org/10.4134/CKMS.c150213 J. Choi, P. Agarwal, and S. Jain, "Certain fractional integral operators and extended generalized Gauss hypergeometric functions", Kyungpook mathematical journal, vol. 55, no. 3, pp. 695-703, 2015. http://dx.doi.org/10.5666/KMJ.2015.55.3.695 H.m. Srivastava, and P. Agarwal, "Certain Fractional Integral Operators and the Generalized Incom-plete Hypergeometric Functions", Appl. Appl. Math., vol. 8, no. 2, 2013.333345 P. Agarwal, S. Jain, M. Chand, S.K. Dwivedi, and S. Kumar, "Bessel functions associated with Saigo-Maeda fractional derivative operators", J. Fract. Calc. Appl., vol. 5, no. 2, pp. 102-112, 2014. D. Kumar, P. Agarwal, and S.D. Purohit, "Generalized fractional integration of the Hfunction involv-ing general class of polynomials", Walailak J. Sci. Technol., vol. 11, no. 12, pp. 1019-1030, 2014. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Appli-cations., Gordon and Breach: New York, NY, USA, 1993.

258

[22] [23] [24]

Advances in Special Functions of Fractional Calculus

Jain et al.

I.N. Sneddon, The use in mathematical physics of Erdlyi-Kober operators and of some of their generalizations. M. Saigo, "A remark on integral operators involving the Gauss hypergeometric functions", Math. Rep. Kyushu Univ., vol. 11, pp. 135-143, 1978. A.A. Kilbas, and N. Sebastian, "Generalized fractional integration of Bessel function of the first kind", Integr. Transforms Spec. Funct., vol. 19, no. 12, pp. 869-883, 2008. http://dx.doi.org/10.1080/10652460802295978

Advances in Special Functions of Fractional Calculus, 2023, 259-269

259

CHAPTER 14

Some Erdélyi-Kober Fractional Integrals of the Extended Hypergeometric Functions S. Jain1,*, R. Goyal2, P. Agarwal2,3, Clemente Cesarano4 and Juan L.G. Guirao5 1

Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India

2

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India

3

Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

4

Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy 5

Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Hospital de Marina, Murcia, 30203, Spain Abstract: This paper aims to establish some new formulas and results related to the Erdélyi-Kober fractional integral operator applied to the extended hypergeometric functions. The results are expressed as the Hadamard product of the extended and confluent hypergeometric functions. Some special cases of our main results are also derived.

Keywords: Gamma Function, Beta function, Erdélyi-Kober fractional integral operators, Hadamard product, Gauss hypergeometric function, Confluent hypergeometric function, extended hypergeometric functions. 1. INTRODUCTION AND PRELIMINARIES In the last few years, many generalizations of special functions with different kernels witnessed a significant evolution. This modification in the theory of special functions offers an analytic foundation for the many scientific problems in mathematical physics, biology, and engineering sciences, which have been solved and have many practical uses. *Corresponding

author S. Jain: Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India; Tel: 9928279174; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

260

Advances in Special Functions of Fractional Calculus

Jain et al.

Beta and Gamma functions are two important members of a class of special functions which play a vital role in the theory of special functions. Many special functions are expressed either in terms of the beta function or the Gamma function. The gamma function is defined as follows [1]. ∞

Γ(𝑦1 ) = ∫0 𝑡 𝑦1 −1 𝑒 −𝑡 𝑑𝑡, ℜ(𝑦1 ) > 0.

(1.1)

The classical Euler beta function is defined as follows [1]. 1

𝐵(𝑦1 , 𝑦2 ) = ∫0 𝑡 𝑦1 −1 (1 − 𝑡)𝑦2 −1 𝑑𝑡, ℜ(𝑦1 ), ℜ(𝑦2 ) > 0.

(1.2)

Further, the mathematical and physical uses of hypergeometric functions are found in various areas of applied mathematics, mathematical physics, and engineering. The Gauss hypergeometric function is a solution of a homogenous second-order differential equation which is called the hypergeometric differential equation, and it is given by: 𝑧(1 − 𝑧)

𝑑2𝑤 𝑑𝑧 2

+ (𝑐 − (𝑎 + 𝑏 + 1)𝑧) 2 𝐹1

The Gauss hypergeometric function 2 𝐹1 (𝑎, 𝑏, 𝑐; 𝑧)

𝑑𝑤 𝑑𝑧

− 𝑎𝑏𝑤 = 0 (1.3)

is defined as [2]:

= 𝐹(𝑎, 𝑏, 𝑐; 𝑧) = ∑∞ 𝑘=0

(𝑎)𝑘 (𝑏)𝑘 𝑧 𝑘 (𝑐)𝑘

𝑘!

,

(1.4)

where (𝑢)𝑘 represents the Pochhammer symbol defined below: (𝑢)𝑘 : =

Γ(𝑢+𝑘) Γ(𝑢)

={

1 𝑢(𝑢 + 1) ⋯ (𝑢 + 𝑘 − 1)

𝑘 = 0; 𝑢 ∈ ℂ\{0}, 𝑘 ∈ ℕ; 𝑢 ∈ ℂ.

Series representation and integral representation of Gauss hypergeometric function 2 𝐹1 is defined as [2]: 𝐹(𝑟0 , 𝑟1 , 𝑟2 ; 𝑧) = ∑∞ 𝑛=0

𝐵(𝑟1 +𝑛,𝑟2 −𝑟1 ) 𝐵(𝑟1 ,𝑟2 −𝑟1 )

(𝑟0 )𝑛

𝑧𝑛 𝑛!

,

(1.5)

where ℜ(𝑟2 ) > ℜ(𝑟1 ) > 0 and |𝑧| < 1. 𝐹(𝑟0 , 𝑟1 , 𝑟2 ; 𝑧) =

1 𝐵(𝑟1 ,𝑟2 −𝑟1

1

∫ 𝑡 𝑟1 −1 (1 − 𝑡)𝑟2 −𝑟1 −1 (1 − 𝑧𝑡)−𝑟0 𝑑𝑡. ) 0

(1.6)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

261

𝑧

Then later, Kummer changes the parameter z by and taking limit 𝑏 → ∞ in the 𝑏 (1.3), then the hypergeometric differential equation becomes a confluent hypergeometric differential equation or Kummer’s equation. 𝑧

𝑑2𝑤 𝑑𝑧 2

+ (𝑐 − 𝑧)

𝑑𝑤 𝑑𝑧

− 𝑎𝑤 = 0

(1.7)

The confluent hypergeometric function is the solution of the above differential equation (1.7). A confluent hypergeometric function is defined as [2]: 1 𝐹1 (𝑎, 𝑐; 𝑧)

= Φ(𝑎, 𝑐; 𝑧) = ∑∞ 𝑘=0

(𝑎)𝑘 𝑧 𝑘 (𝑐)𝑘 𝑘!

.

(1.8)

Series representation and integral representation of a confluent hypergeometric function are defined as [2]: Φ(𝑟1 , 𝑟2 ; 𝑧) = ∑∞ 𝑛=0

𝐵(𝑟1 +𝑛,𝑟2 −𝑟1 ) 𝑧 𝑛 𝐵(𝑟1 ,𝑟2 −𝑟1 )

𝑛!

,

(1.9)

where ℜ(𝑟2 ) > ℜ(𝑟1 ) > 0; Φ(𝑟1 , 𝑟2 ; 𝑧) =

1 1 ∫ 𝐵(𝑟1 ,𝑟2 −𝑟1 ) 0

𝑡 𝑟1 −1 (1 − 𝑡)𝑟2 −𝑟1−1 𝑒 𝑧𝑡 𝑑𝑡.

(1.10)

Recently, Goyal et al. [3] studied a new extension of the beta function using the 2parameter Mittag-Leffler function as a kernel and derived some important results for extended beta functions. (𝑎)

1

𝐵(𝑎1,𝑎2) (𝑥1 , 𝑥2 ) = ∫0 𝑡 𝑥1 −1 (1 − 𝑡)𝑥2 −1 𝐸𝑎1,𝑎2 (−𝑎(𝑡(1 − 𝑡))−1 )𝑑𝑡,

(1.11)

where min{ℜ(𝑥1 ), ℜ(𝑥2 )} > 0, ℜ(𝑎1 ) > 0, ℜ(𝑎2 ) > 0, 𝑎 ≥ 0 and 𝐸𝑎1 ,𝑎2 (𝑧) is a 2-parameter Mittag-Leffler function given by [4]. Then later, Jain et al. [5] introduced new extensions of the Gauss hypergeometric function and confluent hypergeometric function by using the above-extended beta function and studied various properties of these extended functions. They also have inequalities of the extended beta function defined in [3]. The extended confluent hypergeometric function is defined as [5]: (𝑎) Φ(𝑎1 ,𝑎2) (𝑏1 , 𝑏2 ; 𝑧)

(𝑎)

=

∑∞ 𝑟=0

𝐵(𝑎 ,𝑎 ) (𝑏1 +𝑟,𝑏2 −𝑏1 ) 𝑧 𝑟 1 2 , 𝑟! 𝐵(𝑏1 ,𝑏2 −𝑏1 )

(1.12)

262

Advances in Special Functions of Fractional Calculus

Jain et al.

where (ℜ(𝑏2 ) > ℜ(𝑏1 ) > 0, ℜ(𝑎1 ) > 0, ℜ(𝑎2 ) > 0 and 𝑎 ≥ 0). The extended Gauss hypergeometric function is defined as [5]: (𝑎)

𝐹(𝑎1,𝑎2) (𝑏0 , 𝑏1 , 𝑏2 ; 𝑧) = ∑∞ 𝑟=0

(𝑎) 𝐵(𝑎 ,𝑎 ) (𝑏1 +𝑟,𝑏2 −𝑏1 ) 𝑧𝑟 1 2 (𝑏0 )𝑟 , 𝐵(𝑏1 ,𝑏2 −𝑏1 ) 𝑟!

Where ℜ(𝑏2 ) > ℜ(𝑏1 ) > 0, ℜ(𝑎1 ) > 0, ℜ(𝑎2 ) > 0, (𝑎) 𝐵(𝑎1,𝑎2) (𝑥1 , 𝑥2 ) is the extended beta function.

𝑎 ≥ 0,

(1.13) |𝑧| < 1

and

Fractional calculus is an area of mathematics study that grows out of the traditional definitions and results of calculus integral and derivative operators in much the similar way fractional exponents is an outgrowth of exponents with an integer value. The concept of fractional calculus (fractional derivatives and fractional integral) is very old. In 1695, L’Hospital asked the question as to the meaning of 𝑑𝑛𝑦 𝑑𝑥 𝑛

1

1

if 𝑛 = ; that is "what if n is fractional ?" Leibniz replied that "𝑑𝑥 2 will be equal 2

to 𝑥√𝑑𝑥: 𝑥" It is generally known that integer-order derivatives and integrals have clear physical and geometric interpretations. However, in the case of fractionalorder integration and differentiation, which shows a rapidly growing area both in theory and in applications to real world problems, it is not so. Since the appearance of the idea of differentiation and integration of arbitrary (not necessarily an integer) order, there was not any acceptable geometric and physical interpretation of these operations for more than 300 years. The Fractional calculus operators having many special functions have been used for modelling the system in turbulence and fluid dynamics, stochastic dynamic system, thermonuclear fusion, image processing, nonlinear biological system and in quantum mechanics, due to which many scientists and researchers [6- 19] are continuously working on them for many years. Erdélyi-Kober fractional integrals are defined as follows [20, 21]: 𝑎,𝑏 (𝐾0,𝑥 𝑓(𝑡))(𝑥) =

𝑥 −𝑎−𝑏 Γ(𝑎)

𝑥

∫0 (𝑥 − 𝑡)𝑎−1 𝑡 𝑏 𝑓(𝑡)𝑑𝑡,

(1.14)

and 𝑎,𝑏 (𝐾𝑥,∞ 𝑓(𝑡))(𝑥) =

𝑥𝑏

∞ ∫ (𝑡 Γ(𝑎) 𝑥

− 𝑥)𝑎−1 𝑡 −𝑎−𝑏 𝑓(𝑡)𝑑𝑡,

(1.15)

here, 𝑥 > 0, 𝑎, 𝑏 ∈ 𝐶 and ℜ(𝑎) > 0. In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g) that expresses how the shape of one is modified by

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

263

the other. The concept of Hadamard product (convolution) of the functions 𝑓1 and 𝑓2 is very important to prove our main results. Hadamard product of 𝑓1 and 𝑓2 defined as follows [22]: 𝑛 (𝑓1 ∗ 𝑓2 )(𝑧) = ∑∞ 𝑛=0 𝑎𝑛 𝑏𝑛 𝑧 = (𝑓2 ∗ 𝑓1 )(𝑧)

(1.16)

The following lemmas proved in Kilbas, and Sebast in [23, 24] are useful to prove our main results. Lemma 1.1. Let 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 be such that ℜ(𝑎) > 0, then the following formulas hold true: 𝑎,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 )(𝑥) = 𝑥 𝑐−1

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

(ℜ(𝑐) > −ℜ(𝑏))

(1.17)

and 𝑎,𝑏 𝑐−1 (𝐾𝑥,∞ 𝑡 )(𝑥) = 𝑥 𝑐−1

Γ(1−𝑐+𝑏) Γ(1−𝑐+𝑏+𝑎)

(ℜ(𝑐) < 1 + ℜ(𝑏))

(1.18)

2. MAIN RESULTS In this section, we established Erdélyi-Kober fractional integrals of the extended hypergeometric functions, defined in (1.13) and (1.12) by using the Lemma (1.1). 𝑎,𝑏 Theorem 2.1. The Erdélyi-Kober fractional integrals 𝐾0,𝑥 (𝑎, 𝑏, 𝑐 ∈ 𝐶) of the (𝑠)

extended Gauss hypergeometric function 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡) is given by: (𝑠)

𝑎,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

(𝑠)

(2.1)

× 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (𝑐 + 𝑏, 𝑐 + 𝑏 + 𝑎; 𝑥), where 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 ,ℜ(𝑎) > 0, ℜ(𝑐) > −ℜ(𝑏) and confluent hypergeometric function hypergeometric function.

1 𝐹1 (𝑎, 𝑏; 𝑥)

is

(𝑠)

Proof. By taking 𝑓(𝑡) = 𝑡 𝑐−1 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡) and using the definition of extended Gauss hypergeometric function (1.13), we get: (𝑠)

𝑎,𝑏 𝑐−1 𝑎,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = (𝐾0,𝑥 𝑡 (∑∞ 𝑛=0

(𝑠)

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑝1 ) 𝑧𝑛 1 2 (𝑞0 )𝑛 )). (2.2) 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑛!

264

Advances in Special Functions of Fractional Calculus

Jain et al.

Then interchanging the order of integration and summation, which is valid under the conditions of Theorem (2.1), we have: (𝑠)

(𝑠)

𝑎,𝑏 𝑐−1 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = ∑∞ (𝐾0,𝑥 𝑛=0 (𝑞0 )𝑛

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑝1 ) 1 𝑎,𝑏 𝑐+𝑛−1 1 2 𝑡 )(𝑥). (𝐾0,𝑥 𝐵(𝑞1 ,𝑞2 −𝑞1 ) 𝑛!

(2.3)

Then by using (1.17), we have: 𝑎,𝑏 𝑐−1 (𝑠) (𝐾0,𝑥 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥)

×

Γ(𝑐+𝑏+𝑛) 𝑥 𝑐+𝑛−1 Γ(𝑐+𝑏+𝑎+𝑛)

𝑛!

(𝑠)

∑∞ 𝑛=0

=

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑝1 ) (𝑞0 )𝑛 1 2 𝐵(𝑞1 ,𝑞2 −𝑞1 )

(2.4)

.

After simplification, the above equation reduces to 𝑎,𝑎′,𝑏 𝑐−1 (𝑠) 𝑡 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) (𝐾0,𝑥

×

Γ(𝑐+𝑏+𝑛) 𝑥 𝑛 Γ(𝑐+𝑏+𝑎+𝑛) 𝑛!

(𝑠)

=

𝑥 𝑐−1 ∑∞ 𝑛=0

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑝1 ) (𝑞0 )𝑛 1 2 𝐵(𝑞1 ,𝑞2 −𝑞1 )

(2.5)

.

After using the (𝑎)𝑛 =

Γ(𝑎+𝑛) Γ(𝑎)

, the above equation reduces to the following form: (𝑠)

𝑎,𝑎′,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

(2.6)

(𝑠)

× ∑∞ 𝑛=0 (𝑞0 )𝑛

𝐵(𝑠 ,𝑠 ) (𝑞1 +𝑛,𝑞2 −𝑝1 ) (𝑐+𝑏)𝑛 𝑥 𝑛 1 2 . (𝑐+𝑏+𝑎)𝑛 𝑛! 𝐵(𝑞1 ,𝑞2 −𝑞1 )

then by interpreting the above equation with the help of the concept of the Hadamard Product given by (1.16), we get our desired result. (𝑠)

𝑎,𝑎′,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1

×

(𝑠) 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗1

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

(2.7)

𝐹1 (𝑐 + 𝑏, 𝑐 + 𝑏 + 𝑎; 𝑥),

𝑎,𝑏 Theorem 2.2. The Erdélyi-Kober fractional integrals 𝐾𝑥,∞ (𝑎, 𝑏, 𝑐 ∈ 𝐶) of the (𝑠) extended Gauss hypergeometric function 𝐹(𝑠1 ,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡) is given by: (𝑠)

1

Γ(1−𝑐+𝑏)

𝑡

Γ(1−𝑐+𝑏+𝑎)

𝑎,𝑏 𝑐−1 (𝐾𝑥,∞ 𝑡 𝐹(𝑠1,𝑠2) (𝑞0 , 𝑞1 , 𝑞2 ; ))(𝑥) = 𝑥 𝑐−1 (𝑠)

× 𝐹(𝑠1 ,𝑠2 ) (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (1 − 𝑐 + 𝑏, 1 − 𝑐 + 𝑏 + 𝑎; 𝑥),

(2.8)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

where 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 ,ℜ(𝑎) > 0, ℜ(𝑐) < 1 + ℜ(𝑏) and confluent hypergeometric function hypergeometric function.

1 𝐹1 (𝑎, 𝑏; 𝑥)

265

is

𝑎,𝑏 Theorem 2.3. The Erdélyi-Kober fractional integrals 𝐾0,𝑥 (𝑎, 𝑏, 𝑐 ∈ 𝐶) of the (𝑠)

extended confluent hypergeometric function 𝛷(𝑠1 ,𝑠2 ) (𝑞1 , 𝑞2 ; 𝑡) is given by: (𝑠)

𝑎,𝑏 𝑐−1 (𝐾0,𝑥 𝑡 Φ(𝑠1,𝑠2) (𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1

×

(𝑠) Φ(𝑠1 ,𝑠2) (𝑞1 , 𝑞2 ; 𝑥) ∗1

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

(2.9)

𝐹1 (𝑐 + 𝑏, 𝑐 + 𝑏 + 𝑎; 𝑥),

where 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 ,ℜ(𝑎) > 0, ℜ(𝑐) > −ℜ(𝑏) and confluent hypergeometric function hypergeometric function.

1 𝐹1 (𝑎, 𝑏; 𝑥)

is

𝑎,𝑏 Theorem 2.4. The Erdélyi-Kober fractional integrals 𝐾𝑥,∞ (𝑎, 𝑏, 𝑐 ∈ 𝐶) of the (𝑠) extended confluent hypergeometric function 𝛷(𝑠1 ,𝑠2 ) (𝑞1 , 𝑞2 ; 𝑡) is given by: (𝑠)

1

Γ(1−𝑐+𝑏)

𝑡

Γ(1−𝑐+𝑏+𝑎)

𝑎,𝑏 𝑐−1 (𝐾𝑥,∞ 𝑡 Φ(𝑠1,𝑠2) (𝑞1 , 𝑞2 ; ))(𝑥) = 𝑥 𝑐−1

×

(𝑠) Φ(𝑠1 ,𝑠2 ) (𝑞1 , 𝑞2 ; 𝑥) ∗1

(2.10)

𝐹1 (1 − 𝑐 + 𝑏, 1 − 𝑐 + 𝑏 + 𝑎; 𝑥),

where 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑥 > 0 ,ℜ(𝑎) > 0, ℜ(𝑐) > −ℜ(𝑏) and confluent hypergeometric function hypergeometric function.

1 𝐹1 (𝑎, 𝑏; 𝑥)

is

The proofs of Theorems (2.2),(2.3) and (2.4) are similar to those of Theorem (2.1). 3. SOME SPECIAL CASES OF THE ABOVE FRACTIONAL INTEGRAL FORMULAS If we put 𝑠1 = 𝑠2 = 1 and 𝑠 = 0, then a new extension of Gauss hypergeometric function (1.13) and a new extension of confluent hypergeometric function (1.12) reduce to Gauss hypergeometric function (1.4) and confluent hypergeometric function (1.8) then from the formulae established in (2.1), (2.8), (2.9) and (2.10), we have the following results. Corollary 3.1. Consider 𝑎, 𝑏, 𝑐 ∈ 𝐶 be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑅(𝑏) and ℜ(𝑞2 ) > 𝑅(𝑞1 ) > 0, then Erdélyi-Kober fractional integrals of Gauss hypergeometric function defined in (1.4), is given by:

266

Advances in Special Functions of Fractional Calculus

Jain et al.

𝑎,𝑏 𝑐−1 𝑡 2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1 (𝐾0,𝑥

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

×2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (𝑐 + 𝑏, 𝑐 + 𝑏 + 𝑎; 𝑥).

(3.1)

Corollary 3.2. Let 𝑎, 𝑏, 𝑐 ∈ 𝐶 be such that ℜ(𝑎) > 0, ℜ(𝑐) < 1 + ℜ(𝑏), then Erdélyi-Kober fractional integrals of Gauss hypergeometric function defined in (1.4), is given by: 1

Γ(1−𝑐+𝑏)

𝑡

Γ(1−𝑐+𝑏+𝑎)

𝑎,𝑏 𝑐−1 (𝐾𝑥,∞ 𝑡 2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; ))(𝑥) = 𝑥 𝑐−1

×2 𝐹1 (𝑞0 , 𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (1 − 𝑐 + 𝑏, 1 − 𝑐 + 𝑏 + 𝑎; 𝑥),

(3.2)

Corollary 3.3. Let 𝑎, 𝑏, 𝑐 ∈ 𝐶 be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑅(𝑏) and ℜ(𝑞2 ) > 𝑅(𝑞1 ) > 0, then Erdélyi-Kober fractional integrals of confluent hypergeometric function defined in (1.8), is given by: 𝑎,𝑏 𝑐−1 𝑡 1 𝐹1 (𝑞1 , 𝑞2 ; 𝑡))(𝑥) = 𝑥 𝑐−1 (𝐾0,𝑥

Γ(𝑐+𝑏) Γ(𝑐+𝑏+𝑎)

×1 𝐹1 (𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (𝑐 + 𝑏, 𝑐 + 𝑏 + 𝑎; 𝑥),

(3.3)

Corollary 3.4. Consider 𝑎, 𝑏, 𝑐 ∈ 𝐶 be such that ℜ(𝑎) > 0, ℜ(𝑐) > −𝑅(𝑏) and ℜ(𝑞2 ) > 𝑅(𝑞1 ) > 0, then Erdélyi-Kober fractional integrals confluent hypergeometric function defined in (1.8), is given by: 1

Γ(1−𝑐+𝑏)

𝑡

Γ(1−𝑐+𝑏+𝑎)

𝑎,𝑏 𝑐−1 (𝐾𝑥,∞ 𝑡 1 𝐹1 (𝑞1 , 𝑞2 ; ))(𝑥) = 𝑥 𝑐−1

×1 𝐹1 (𝑞1 , 𝑞2 ; 𝑥) ∗1 𝐹1 (1 − 𝑐 + 𝑏, 1 − 𝑐 + 𝑏 + 𝑎; 𝑥),

(3.4)

CONCLUSION We have been inspired by applications of many operators in fractional calculus and a considerably large spectrum of special functions in mathematical, physical and engineering sciences. We have introduced here new formulas for the Erdélyi-Kober fractional integral involving extended hypergeometric functions. Our results have been expressed as the Hadamard product of the extended hypergeometric functions and confluent hypergeometric functions. Some special cases of our results have also been derived.

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

267

CONSENT FOR PUBLICATON Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest. ACKNOWLEDGEMENTS The authors would like to thank the anonymous referees for their valuable comments and suggestions. Shilpi Jain is very thankful to SERB (project number: MTR/2017/000194) for providing the necessary facility, and Rahul Goyal and Praveen Agarwal are thankful to the (NBHM (R.P)/RD II/7867) for providing the necessary facility. REFERENCES [1] [2] [3]

[4]

[5]

[6] [7]

[8] [9]

E.D. Rainville, Special Functions., Macmillan: New York, NY, USA, 1960. W.J.F. Olver, W.D. Lozier, F.R. Boisvert, and W.C. Clark, NIST Handbook of Mathematical Functions., Cambridge University Press: New York, NY, USA, 2010. R. Goyal, S. Momani, P. Agarwal, and M.T. Rassias, "An Extension of Beta Function by Using Wiman’s Function", Axioms, vol. 10, no. 3, p. 187, 2021. http://dx.doi.org/10.3390/axioms10030187 A. Wiman, "Über den Fundamentalsatz in der Teorie der Funktionen Ea(x)", Acta Math., vol. 29, no. 0, pp. 191-201, 1905. http://dx.doi.org/10.1007/BF02403202 S. Jain, R. Goyal, P. Agarwal, A. Lupica, and C. Cesarano, "Some results of extended beta function and hypergeometric functions by using Wiman’s function", Mathematics, vol. 9, no. 22, p. 2944, 2021. http://dx.doi.org/10.3390/math9222944 T. Pohlen, The Hadamard Product and Universal Power Series, Universitát Trier Trier Germany, Germany, 2009. P. Agarwal, "Fractional integration of the product of two H-functions and a general class of polynomials", Asian J. Appl. Sci., vol. 5, no. 3, pp. 144-153, 2012. http://dx.doi.org/10.3923/ajaps.2012.144.153 G. Singh, P. Agarwal, S. Araci, and M. Acikgoz, "Certain fractional calculus formulas involving extended generalized Mathieu series", Adv. Di_erence Equ, no. 1, pp. 1-30, 2018. P. Agarwal, F. Qi, M. Chand, and G. Singh, "Some fractional differential equations involving generalized hypergeometric functions", J. App. Anal., vol. 25, no. 1, pp. 37- 44, 2019.

268

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18] [19]

[20]

[21] [22]

Advances in Special Functions of Fractional Calculus

Jain et al.

http://dx.doi.org/10.1515/jaa-2019-0004 P. Agarwal, T.M. Rassias, G. Singh, and S. Jain, Certain Fractional Integral and Differential Formulas Involving the Extended Incomplete Generalized Hypergeometric Functions, Mathematical Analysis and Applications. Springer Optimization and Its Applications., vol. Vol. 154, Springer: Cham, 2019. P. Agarwal, G. Wang, and M. Al-Dhaifallah, "Fractional calculus operators and their applications to thermal systems", Adv. Mech. Eng., vol. 10, no. 6, 2018. http://dx.doi.org/10.1177/1687814018782028 I.O. Kymaz, P. Agarwal, and S. Jain, "Cetinkaya, A.,On a new extension of Caputo fractional derivative operator", Advances in Real and Complex Analysis with Applications, pp. 261275, 2017. İ.O. Kıymaz, A. Çetinkaya, and P. Agarwa, "An extension of Caputo fractional derivative operator and its applications", J. Nonlinear Sci. Appl., vol. 9, no. 6, pp. 3611-3621, 2016. http://dx.doi.org/10.22436/jnsa.009.06.14 P. Agarwal, and J. Choi, "Fractional calculus operators and their image formulas", Journal of the Korean Mathematical Society, vol. 53, no. 5, pp. 1183-1210, 2016. http://dx.doi.org/10.4134/JKMS.j150458 J. Choi, and P. Agarwal, "A note on fractional integral operator associated with multiindex Mittag-Leffler functions", Filomat, vol. 30, no. 7, pp. 1931-1939, 2016. http://dx.doi.org/10.2298/FIL1607931C P. Agarwal, J. Choi, K.B. Kachhia, J.C. Prajapati, and H. Zhou, "Some integral transforms and frac-tional integral formulas for the extended hypergeometric functions", Communications of the Korean Mathematical Society, vol. 31, no. 3, pp. 591-601, 2016. http://dx.doi.org/10.4134/CKMS.c150213 J. Choi, P. Agarwal, and S. Jain, "Certain fractional integral operators and extended generalized Gauss hypergeometric functions", Kyungpook mathematical journal, vol. 55, no. 3, pp. 695-703, 2015. http://dx.doi.org/10.5666/KMJ.2015.55.3.695 H.m. Srivastava, and P. Agarwal, "Certain Fractional Integral Operators and the Generalized Incom-plete Hypergeometric Functions", Appl. Appl. Math., vol. 8, no. 2, 2013.333345 P. Agarwal, S. Jain, M. Chand, S.K. Dwivedi, and S. Kumar, "Bessel functions associated with Saigo-Maeda fractional derivative operators", J. Fract. Calc. Appl., vol. 5, no. 2, pp. 102-112, 2014. D. Kumar, P. Agarwal, and S.D. Purohit, "Generalized fractional integration of the Hfunction involv-ing general class of polynomials", Walailak J. Sci. Technol., vol. 11, no. 12, pp. 1019-1030, 2014. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Appli-cations., Gordon and Breach: New York, NY, USA, 1993. I.N. Sneddon, The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations. In Fractional Calculus and Its Applications (West Haven, CT, USA, 15–16 June 1974); Ross, B., Ed.; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1975; 457, 37–79.

Hypergeometric Functions

[23] [24]

Advances in Special Functions of Fractional Calculus

269

M. Saigo, "A remark on integral operators involving the Gauss hypergeometric functions", Math. Rep. Kyushu Univ., vol. 11, pp. 135-143, 1978. A.A. Kilbas, and N. Sebastian, "Generalized fractional integration of Bessel function of the first kind", Integr. Transforms Spec. Funct., vol. 19, no. 12, pp. 869-883, 2008. http://dx.doi.org/10.1080/10652460802295978

270

Advances in Special Functions of Fractional Calculus, 2023, 270-283

CHAPTER 15

On Solutions of the Kinetic Model by Sumudu Transform Esra Karatas Akgül2,*, Fethi Bin Muhammed Belgacem3 and Ali Akgül 1,2,4 1

Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon 2

Department of Mathematics, Siirt University, Art and Science Faculty, 56100 Siirt, Turkey

3

Department of Mathematics, Faculty of Basic Education, PAAET, Al-Ardhiya, Kuwait

4

Near East University, Mathematics Research Center, Department of Mathematics, Near East Boulevard, PC: 99138, Nicosia /Mersin 10 – Turkey Abstract: This paper investigates the kinetic model with four different fractional derivatives. We obtain the solutions of the models by Sumudu transform and demonstrate our results with some figures. We prove the accuracy of the Sumudu transform by some theoretical results and applications.

Keywords: Sumudu transforms, Fractional derivatives, Kinetic model. 1. INTRODUCTION Mathematical modeling formulas have been used to predict the growth of microorganisms, the spread of epidemics, and drying kinetic models are some realworld problems. Modeling of mass transfers is created with Fick’s law. For the fractional version of Fick’s law, see [1] and the references in this work. Although the classical Lewis model defines the exponential behavior of diffusion, the fractional Lewis model can also use the non-exponential behavior of diffusion [2]. The speed of the grain drying operation varies with the supply, such as temperature, airflow and mechanical drying systems. Additionally, drying kinetic models express the time required for optimum moisture loss. We consider [3]:

*Corresponding

author Esra Karatas Akgül: Department of Mathematics, Siirt University, Art and Science Faculty, 56100 Siirt, Turkey; E-mail: [email protected]

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus 𝑑𝑁 𝑑𝑡

= 𝐷∇2 𝑁,

271

(1.1)

Here, 𝐷 is the diffusion coefficient, and 𝑁 is the moisture content of the food. We present the Lewis model [4] as: 𝑑𝑁 𝑑𝑡

= −𝑘(𝑁(𝑡) − 𝑁𝑒 ),

(1.2)

where 𝑁𝑒 is equilibrium moisture content, and 𝑘 is a constant of drying has dimension 𝑚𝑖𝑛−1 . Then, we get: 𝑁(𝑡) = (𝑁0 − 𝑁𝑒 )exp(−𝑘𝑡) + 𝑁𝑒 ,

(1.3)

Here, 𝑁0 is the initial moisture content inside the food. Fractional analysis has gained a lot of attention recently. The most crucial reason for this is that it has implementations in real-world problems and gives better results in comparison. Fractional calculus has also been used to model physical and engineering processes. Recently, fractional calculus has played a huge role in various fields, such as mechanics, electricity, chemistry, biology, and economics, especially in control theory. Atangana et al. [5] studied the generalized mass transport equation. The analysis of new trends in the fractional differential equation has been investigated in [6, 7]. Abdeljawad et al. [8] surveyed fractional differences and integration by parts. Some classes of ordinary differential equations have been worked on by Jarad et al. [9]. Afshari et al. [10] examined a new fixed point theorem with an implementation of a coupled fractional differential equations system. KÃrt et al. [11] studied a certain bivariate Mittag-Leffler function in 2020. Integral transforms and some special functions were investigated by Saxena et al. [12] and Fernandez et al. [13] also conducted various studies on this subject. The purpose of this study, because of the usefulness and significance of the fractional differential equations in certain physical problems, is to give numerical results using the Sumudu transformation for the kinetic model (1.2). 2. MATHEMATICAL BACKGROUND The Sumudu transformation was first introduced by Watagula [14]. Weerakoon [15] investigated the complex inverse formula for Sumudu transform. Asiru has studied the implementation of the Sumudu transform [16], discrete dynamical system equations [17] and more [18]. Sumudu transform is a theoretical dual of the Laplace transform. Sumudu transforms to convolution-type integral equation was applied by Belgacem et al. [19]. Additionally, Belgacem et al. [20] mentioned many

272

Advances in Special Functions of Fractional Calculus

Jain et al.

features of the Sumudu transform. For more details, we refer readers to [21-26]. Purohit et al. [27] examined an application of Sumudu transform for a fractional kinetic equation. Nisar et al. [28-30] also studied these types of equations. Definition 2.1 Over the set of functions, 𝐴 = {𝑔(𝑡)|∃𝑁, 𝜏1 , 𝜏2 > 0, |𝑔(𝑡)| < 𝑁exp(|𝑡|/𝜏𝑗 , if 𝑡 ∈ (−1)𝑗 × [0, ∞)},

(2.1)

the Sumudu transform is defined as [20]: ∞

𝐺(𝑢) = 𝑆[𝑔(𝑡)] = ∫0 𝑓(𝑢𝑡)exp(−𝑡)𝑑𝑡, 𝑢 ∈ (−𝜏1 , 𝜏2 ).

(2.2)

The Sumudu transform changes domain size and shape, not units, unlike Laplace transform. ODEs solved by the Laplace transform can be solved by Sumudu transform and almost vice versa, except for some possibly artificially manufactured examples. The primal thing is that Sumudu is more natural and easier to understand, so there are lots of Sumudu transform’s applications in the literature. Actually, this transform is linear, protects linear functions, and hence especially does not change units [14]. Definition 2.2 Let 𝜂, 𝜁: [0, ∞) → ℜ, then the convolution of 𝜂, 𝜁 is [31] 𝑡

(𝜂 ∗ 𝜁) = ∫0 𝜂(𝑡 − 𝑢)𝜁(𝑢)𝑑𝑢

(2.3)

and assume that 𝜂, 𝜁: [0, ∞) → ℜ, then we have 𝑆{(𝜂 ∗ 𝜁)(𝑡)} = 𝑢𝑆{𝜂(𝑡)}𝑆{𝜁(𝑡)}.

(2.4)

Definition 2.3 The Caputo fractional derivative is presented as follows [32]; 𝐶 𝛼 𝑎 𝐷 𝑔(𝑡)

=

𝑡 1 ∫ (𝑡 Γ(𝑛−𝛼) 𝑎

− 𝑧)𝑛−𝛼−1 𝑔(𝑛) (𝑧)𝑑𝑧

(2.5)

where 𝛼 ∈ ℂ, 𝑅𝑒(𝛼) > 0, 𝑛 = [𝑅𝑒(𝛼)] + 1. Lemma 2.4 The Sumudu transform of Caputo fractional derivative is defined by [31] 𝑆[𝐶0 𝐷𝑡𝛼 𝑔(𝑡)] =

𝐺[𝑢]−𝑔(0) 𝑢𝛼

,

(2.6)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

273

here 𝐺[𝑢] = 𝑆[𝑔(𝑡)]. Definition 2.5 The Caputo-Fabrizio fractional derivative by [32]: 𝐶𝐹𝐶 𝛼 𝑎 𝐷 𝑔(𝑡)

=

𝑀(𝛼) 1−𝛼

𝑡

∫𝑎 𝑔′(𝑧)exp(−𝜆(𝑡 − 𝑧))𝑑𝑧

where 0 < 𝛼 < 1, 𝑀(𝛼) is a normalization function and 𝜆 =

𝛼 1−𝛼

(2.7)

.

Lemma 2.6 The Sumudu transform of CFC fractional derivative is found as [31]: 𝛼 𝑆[𝐶𝐹𝐶 0 𝐷𝑡 𝑔(𝑡)] =

𝑀(𝛼)

𝐺[𝑢]

−

𝑔(0)

𝑀(𝛼)

.

(2.8)

∫ 𝑔′(𝑧)𝐸𝛼 (−𝜆(𝑡 − 𝑧)𝛼 )𝑑𝑧 1−𝛼 𝑎

(2.9)

𝛼 𝑢) 1−𝛼

(1−𝛼) (1+

𝛼 𝑢) 1−𝛼

(1−𝛼) (1+

Definition 2.7 The Atangana-Baleanu fractional derivative as [32]: 𝐴𝐵𝐶 𝛼 𝑎 𝐷 𝑔(𝑡)

=

𝐵(𝛼)

𝑡

where 0 < 𝛼 < 1, 𝐵(𝛼) is a normalization function and 𝜆 =

𝛼 1−𝛼

.

Lemma 2.8 The Sumudu transform of ABC fractional derivative is obtained as [31]: 𝑆[0𝐴𝐵𝐶 𝐷𝑡𝛼 𝑔(𝑡)] =

𝐵(𝛼)

𝐺[𝑢] 𝛼 𝛼 𝑢 ) 1−𝛼

(1−𝛼) (1+

−

𝐵(𝛼)

𝑔(0) 𝛼 𝛼 𝑢 ) 1−𝛼

(1−𝛼) (1+

.

(2.10)

Definition 2.9 We present the constant proportional Caputo (CPC) derivative [33] as: 𝐶𝑃𝐶 𝛼 0 𝐷𝑡 𝑔(𝑡)

=

𝑡 1 ∫ Γ(1−𝛼) 0

(𝑘1 (𝛼)𝑔(𝜏) + 𝑘0 (𝛼)𝑔′(𝜏))(𝑡 − 𝜏)−𝛼 𝑑𝜏

(2.11)

Lemma 2.10 The Sumudu transform of constant proportional Caputo (CPC) derivative is acquired as [33]: 𝛼 1−𝛼 𝑆{𝐶𝑃𝐶 + 𝑘0 (𝛼)[𝐺[𝑢] − 𝑔(0)]𝑠 −𝛼 0 𝐷𝑡 𝑔(𝑡)} = 𝑘1 (𝛼)𝐺[𝑢]𝑠

(2.12)

Definition 2.11 The classical Mittag-Leffler function which has one parameter 𝐸𝛼 (𝑧) is given as [32]:

274

Advances in Special Functions of Fractional Calculus

𝐸𝛼 (𝑧) = ∑∞ 𝑘=0

𝑧𝑘 Γ(𝛼𝑘+1)

Jain et al.

(𝑧 ∈ ℂ, 𝑅𝑒(𝛼) > 0),

(2.13)

also, the Mittag-Leffler kernel is as follows: 𝐸𝛼,𝛽 (𝑧) = ∑∞ 𝑘=0

𝑧𝑘 Γ(𝛼𝑘+𝛽)

(𝑧, 𝛽 ∈ ℂ, 𝑅𝑒(𝛼) > 0, 𝑅𝑒(𝛽) > 0),

(2.14)

in here, 𝐸𝛼,𝛽 (𝑧) coincides with the Mittag-Leffler function Eq.(2.12) when 𝛽 = 1. Lemma 2.12 The Sumudu transform of the Mittag-Leffler function as [31]: 𝑺[𝑬𝜶 (−𝝀𝒕𝜶 )] =

𝟏 𝟏+𝝀𝒖𝜶

𝑆[1 − 𝐸𝛼 (−𝜆𝑡 𝛼 )] =

,

𝜆𝑢𝛼 1+𝜆𝑢𝛼

(2.15) .

(2.16)

3. MAIN RESULTS We take into consideration the Lewis model (see Equation (1.2)) with a power law, exponential decay and Mittag-Leffler kernels in this section. Fractional derivatives are important for real-world problems, which are physical, engineering, and biological problems. 3.1. Lewis Drying Kinetic Model in Fractional Cases We present the model (Equation (1.2)) in Caputo, Caputo-Fabrizio, AtanganaBaleanu and constant proportional Caputo cases, so analytical solutions can be found using Sumudu transforms. 3.1.1. Caputo Fractional Derivative for Drying Kinetic We consider 𝐶 𝛼 𝑎 𝐷 𝑁(𝑡)

= −𝑘 𝛼 (𝑁(𝑡) − 𝑁𝑒 ), 0 < 𝛼 ≤ 1, 𝑁(0) = 1.

(3.1) (3.2)

Implementing the Sumudu transform to the above equation and utilizing the initial condition, we will have 𝑆{𝐶𝑎 𝐷𝛼 𝑁(𝑡)} = 𝑆{−𝑘 𝛼 (𝑁(𝑡) − 𝑁𝑒 )}.

(3.3)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

275

If the expression Eq.(2.6) is utilized then we find as below 𝑆{𝑁(𝑡)}−𝑁(0) 𝑢𝛼

= −𝑘 𝛼 𝑆{𝑁(𝑡)} + 𝑘 𝛼 𝑆{𝑁𝑒 },

𝑆{𝑁(𝑡)} =

1+𝑁𝑒 𝑘 𝛼 𝑢𝛼 1+𝑘 𝛼 𝑢𝛼

(3.4)

,

(3.5)

by taking the inverse Sumudu transform of Eq. (3.5), we can acquire 𝑁(𝑡) = 𝑆 −1 {

1 1+𝑘 𝛼 𝑢

−1 { 𝛼 } + 𝑁𝑒 𝑆

𝑘 𝛼 𝑢𝛼 1+𝑘 𝛼 𝑢𝛼

}.

(3.6)

If we use the relations in Lemma (2.12), the following solution is obtained: 𝑁(𝑡) = 𝐸𝛼 (−𝑘 𝛼 𝑡 𝛼 ) + 𝑁𝑒 [1 − 𝐸𝛼 (−𝑘 𝛼 𝑡 𝛼 )].

(3.7)

3.1.2. Caputo-Fabrizio Fractional Derivative for Drying Kinetic We take into consideration: 𝐶𝐹𝐶 𝛼 𝑎 𝐷 𝑁(𝑡)

= −𝑘 𝛼 (𝑁(𝑡) − 𝑁𝑒 ), 0 < 𝛼 ≤ 1, 𝑁(0) = 1.

(3.8) (3.9)

By applying the Sumudu transform and using the initial condition, we get 𝛼 𝛼 𝑆{𝐶𝐹𝐶 𝑎 𝐷 𝑁(𝑡)} = −𝑆{𝑘 (𝑁(𝑡) − 𝑁𝑒 )},

(3.10)

Utilising the relation Eq.(2.8), we can obtain 𝑀(𝛼) 𝑆{𝑁(𝑡)} 𝛼

1−𝛼 1+ 𝑢 1−𝛼

−

𝑀(𝛼)

𝑁(0) 𝛼 𝑢) 1−𝛼

(1−𝛼) (1+

𝑆{𝑁(𝑡)} =

= −𝑘 𝛼 𝑆{𝑁(𝑡)} + 𝑘 𝛼 𝑆{𝑁𝑒 }, 𝑁𝑒 𝑘 𝛼

+

𝑀(𝛼) 𝛼 (1−𝛼)(1+ 𝑢) 1−𝛼

𝑘𝛼+

𝑀(𝛼) 1 (1−𝛼)(1+ 𝛼 𝑢) 1−𝛼

.

𝑀(𝛼) 𝛼 𝑢) (1−𝛼)(1+ 1−𝛼

𝑘𝛼+

(3.11)

(3.12)

When we implement the inverse Sumudu transform, we can find 𝑁(𝑡) = 𝑆 −1 {

𝑁𝑒 𝑘 𝛼 𝑘𝛼+

𝑀(𝛼)

𝛼 (1−𝛼)(1+ 𝑢) 1−𝛼

} + 𝑆 −1 {

𝑀(𝛼) 1 (1−𝛼)(1+ 𝛼 𝑢) 1−𝛼 𝑀(𝛼) 𝑘𝛼+ 𝛼 𝑢) (1−𝛼)(1+ 1−𝛼

},

(3.13)

276

Advances in Special Functions of Fractional Calculus

Jain et al.

and the solution is obtained as follows: 𝑁(𝑡) = 𝑁𝑒 −

𝑀(𝛼) 𝑀(𝛼)+𝑘 𝛼 (1−𝛼)

−𝑘 𝛼 𝛼𝑡

exp (

𝑀(𝛼)+𝑘 𝛼 (1−𝛼)

) [𝑁𝑒 − 1].

3.2. Atangan-Baleanu Fractional Derivative for Drying Kinetic We consider

𝐴𝐵𝐶 𝛼 𝑎 𝐷 𝑁(𝑡)

= −𝑘 𝛼 (𝑁(𝑡) − 𝑁𝑒 ), 0 < 𝛼 ≤ 1,

𝑁(0) = 1.

(3.14)

(3.15)

If we use the relation Eq. (2.10) and using the initial condition, we obtain 𝐵(𝛼)

𝑆{𝑁(𝑡)} 𝛼 𝛼 𝑢 ) 1−𝛼

(1−𝛼) (1+

−

𝐵(𝛼)

1 𝛼 𝛼 𝑢 ) 1−𝛼

(1−𝛼) (1+

𝑆{𝑁(𝑡)} =

= −𝑘 𝛼 𝑆{𝑁(𝑡)} + 𝑘 𝛼 𝑆{𝑁(𝑒)}. 𝑁𝑒 𝑘 𝛼 (1−𝛼)

𝐵(𝛼)+𝑘 𝛼 (1−𝛼+𝛼𝑢𝛼 )

+

𝐵(𝛼) 𝐵(𝛼)+𝑘 𝛼 (1−𝛼+𝛼𝑢𝛼 )

Applying the inverse Sumudu transform, we can get 𝑆{𝑁(𝑡)} = 𝑆 −1 {

+𝑆 −1 {

+𝑆 −1 {

𝑁𝑒 𝑘 𝛼 (1−𝛼)

1 𝑘𝛼 𝛼𝑢𝛼 𝐵(𝛼)+𝑘𝛼 (1−𝛼)

𝐵(𝛼)+𝑘 𝛼 (1−𝛼) 1+

𝑁𝑒 𝑘 𝛼

𝛼𝑢𝛼

𝐵(𝛼)+𝑘 𝛼 (1−𝛼)

1+𝐵(𝛼)+𝑘𝛼 (1−𝛼)

𝑘𝛼 𝛼𝑢𝛼

𝐵(𝛼)

1 𝑘𝛼 𝛼𝑢𝛼 𝐵(𝛼)+𝑘𝛼 (1−𝛼)

𝐵(𝛼)+𝑘 𝛼 (1−𝛼) 1+

}

},

From the Sumudu transformation features, we have: 𝑁(𝑡) =

𝑁𝑒 𝑘 𝛼 (1−𝛼) 𝐵(𝛼)+𝑘 𝛼 (1−𝛼)

𝐸𝛼 (−

𝑘 𝛼 𝛼𝑡 𝛼 𝐵(𝛼)+𝑘 𝛼 (1−𝛼)

)

}

(3.16) . (3.17)

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

+𝑁𝑒 [1 − 𝐸𝛼 (− +

𝐵(𝛼) 𝐵(𝛼)+𝑘 𝛼 (1−𝛼)

𝑘 𝛼 𝛼𝑡 𝛼 𝐵(𝛼)+𝑘 𝛼 (1−𝛼))

𝐸𝛼 (−

277

)]

𝑘 𝛼 𝛼𝑡 𝛼 𝐵(𝛼)+𝑘 𝛼 (1−𝛼)

).

3.2.1. Constant Proportional Caputo Fractional Derivative for Drying Kinetic We take into consideration: 𝐶𝑃𝐶 𝛼 0 𝐷𝑡 𝑁(𝑡)

= −𝑘 𝛼 (𝑁(𝑡) − 𝑁𝑒 ), 0 < 𝛼 ≤ 1, 𝑁(0) = 1.

(3.18) (3.19)

Applying the Sumudu transformation to the above equation, we will get 𝛼 𝛼 𝛼 𝑆{𝐶𝑃𝐶 0 𝐷𝑡 𝑁(𝑡)} = −𝑘 𝑆{𝑁(𝑡)} + 𝑘 𝑆{𝑁(𝑒)}.

(3.20)

If the expression of the Equation (2.12) is utilized, then we find: 𝑘1 (𝛼)𝑆{𝑁(𝑡)}𝑠 1−𝛼 + 𝑘0 (𝛼)[𝑆{𝑁(𝑡)} − 𝑁(0)]𝑠 −𝛼 = −𝑘 𝛼 𝑆{𝑁(𝑡)} + 𝑘 𝛼 𝑁(𝑒).

(3.21)

and 𝑆{𝑁(𝑡)} = +

𝑘 𝛼 𝑁𝑒 𝑘1

(𝛼)𝑠 1−𝛼 +𝑘

0 (𝛼)𝑠

−𝛼 +𝑘 𝛼

𝑘0 (𝛼)𝑠 −𝛼 1−𝛼 𝑘1 (𝛼)𝑠 +𝑘0 (𝛼)𝑠 −𝛼 +𝑘 𝛼

= 𝑁𝑒 [1 − + [1 −

−𝑘1 (𝛼)𝑠 1−𝛼 −𝑘0 (𝛼)𝑠 −𝛼 𝑘𝛼

−1

]

−𝑘1 (𝛼)𝑠−𝑘 𝛼 𝑠 𝛼 −1 𝑘0 (𝛼)

]

−𝑘1 (𝛼)𝑠 1−𝛼 −𝑘0 (𝛼)𝑠 −𝛼

= 𝑁𝑒 ∑∞ 𝑐=0 [

𝑘𝛼

𝑐

]

−𝑘1 (𝛼)𝑠−𝑘 𝛼 𝑠 𝛼 𝑐

+ ∑∞ 𝑐=0 [

= 𝑁𝑒 ∑∞ 𝑐=0

𝑘0 (𝛼) 1 𝑘 𝛼𝑐

]

𝑐 ∑𝑐𝑑=0 ( ) [−𝑘1 (𝛼)𝑠1−𝛼 ]𝑐−𝑑 [−𝑘0 (𝛼)𝑠 −𝛼 ]𝑑 𝑑

278

Advances in Special Functions of Fractional Calculus

+ ∑∞ 𝑐=0

1 𝑘0

(𝛼)𝑑

Jain et al.

𝑐 ∑𝑐𝑑=0 ( ) [−𝑘1 (𝛼)𝑠]𝑐−𝑑 [−𝑘 𝛼 𝑠 𝛼 ]𝑑 𝑑

𝑐 𝑐 = 𝑁𝑒 ∑∞ 𝑐=0 ∑𝑑=0 (−1) 𝑐 𝑐 + ∑∞ 𝑐=0 ∑𝑑=0 (−1)

𝑘1 (𝛼)𝑐−𝑑 𝑘0 (𝛼)𝑑 𝑘 𝛼𝑐

𝑘1 (𝛼)𝑐−𝑑 𝑘 𝛼𝑑 𝑘0

(𝛼)𝑐

𝑐 ( ) 𝑠 (1−𝛼)(𝑐−𝑑)−𝛼𝑑 𝑑

𝑐 ( ) 𝑠 𝑐−𝑑+𝛼𝑑 , 𝑑

Then, we apply the inverse Sumudu transform and obtain: 𝑐 𝑐 𝑁(𝑡) = 𝑁𝑒 ∑∞ 𝑐=0 ∑𝑑=0 (−1) 𝑐 𝑐 + ∑∞ 𝑐=0 ∑𝑑=0 (−1)

𝑘1 (𝛼)𝑐−𝑑 𝑘0 (𝛼)𝑑 𝑘 𝛼𝑐

𝑘1 (𝛼)𝑐−𝑑 𝑘 𝛼𝑑 𝑘0 (𝛼)𝑐

𝑐 𝑡 (1−𝛼)(𝑐−𝑑)−𝛼𝑑 ( ) 𝑑 Γ((1−𝛼)(𝑐−𝑑)−𝛼𝑑+1)

𝑐 𝑡 (𝑐−𝑑)+𝛼𝑑 ( ) , 𝑑 Γ((𝑐−𝑑)+𝛼𝑑+1)

If we take 𝑏 = 𝑐 − 𝑑, we will get: ∞ 𝑁(𝑡) = 𝑁𝑒 ∑∞ 𝑑=0 ∑𝑏=0 ∞ + ∑∞ 𝑑=0 ∑𝑏=0

(𝑑+𝑏)! (−𝑘1 (𝛼))𝑏 (−𝑘0 (𝛼))𝑑 𝑘 𝛼 (𝑏+𝑑)

𝑟!𝑏!

(𝑑+𝑏)! (−𝑘1 (𝛼))𝑏 (−𝑘 𝛼𝑑 ) 𝑘0 (𝛼)𝑏+𝑑

𝑑!𝑏!

𝑡 (1−𝛼)𝑏−𝛼𝑑 Γ((1−𝛼)𝑏−𝛼𝑑+1)

𝑡 𝑏+𝛼𝑑 Γ(𝑏+𝛼𝑑+1)

,

𝑁(𝑡) = 𝑑 −𝑘 (𝛼) 𝑏 1 ∞ (𝑑+𝑏)! −𝑘0 (𝛼) −𝛼 𝑁𝑒 ∑∞ ] [ 1𝛼 𝑡 1−𝛼 ] 𝑑=0 ∑𝑏=0 𝑑!𝑏! [ 𝑘 𝛼 𝑡 𝑘 Γ((1−𝛼)𝑏−𝛼𝑑+1) ∞ + ∑∞ 𝑑=0 ∑𝑏=0

(𝑑+𝑏)! 𝑑!𝑏!

−𝑘 𝛼

[

𝑘0 (𝛼)

𝑑 −𝑘 (𝛼) 1

𝑡𝛼] [

𝑘0 (𝛼)

𝑏

𝑡]

1 Γ(𝑏+𝛼𝑑+1)

We can write this series as [34]: −𝑘1 (𝛼) 1−𝛼 −𝑘0 (𝛼) −𝛼 𝑡 , 𝛼 𝑡 ) 𝑘𝛼 𝑘

1 𝑁(𝑡) = 𝑁𝑒 𝐸1−𝛼,−𝛼,1 ( −𝑘1 (𝛼)

1 +𝐸1,𝛼,1 (

𝑘0 (𝛼)

𝑡,

−𝑘 𝛼 𝑘0 (𝛼)

𝑡 𝛼 ).

,

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

279

4. NUMERICAL RESULTS We compared the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives. We used the parameters which are 𝛼 = 0.5,0.8,0.9,1 order and 𝑘 = 0.109725, 𝑁𝑒 = 0.07 during our paper.

Fig. (1). Numerical simulation with Caputo (red), CFC (blue) and ABC (black) for 𝛼 = 0.5.

Fig. (2). Numerical simulation with Caputo (red), CFC (blue) and ABC (black) for 𝛼 = 0.8.

280

Advances in Special Functions of Fractional Calculus

Jain et al.

Fig. (3). Numerical simulation with Caputo (red), CFC (blue) and ABC (black) for 𝛼 = 0.9.

Fig. (4). Numerical simulation with Caputo (red), CFC (blue) and ABC (black) for 𝛼 = 1.

CONCLUSION We investigated the kinetic model in detail by Sumudu transform in this paper. As a result, we kept in mind the Lewis model used for the grain drying process in regard to Caputo, Caputo Fabrizio, Atangana-Baleanu and constant proportional Caputo fractional derivatives. The exact solutions obtained by the Sumudu transform gave good results. This proved the efficiency of our proposed method. We demonstrated the results with some (Figs. 1-4) for different values of the fractional order.

Hypergeometric Functions

Advances in Special Functions of Fractional Calculus

281

CONSENT FOR PUBLICATION Not applicable. CONFLICT OF INTEREST The authors declare no conflict of interest, financial or otherwise. ACKNOWLEDGEMENT Declared none. REFERENCES [1]

[2] [3]

[4]

[5]

[6]

[7] [8] [9]

[10]

R. Metzler, and J. Klafter, "The random walk’s guide to anomalous diffusion: A fractional dynamics approach", Phys. Rep., 2000, pp. 339-177. http://dx.doi.org/10.1016/S0370-1573(00)00070-3 S. Fomin, V. Chugunov, and T. Hashida, Mathematical modeling of anomalous diffusion in porous media., Fract. Differ. Calc, 2011, pp. 1-28. R. Ozarslan, and E. Bas, "Kinetic Model for Drying in Frame of Generalized Fractional Derivatives", Fractal ve Fractional, 2020, pp. 4-17. http://dx.doi.org/10.3390/fractalfract4020017 W. K. Lewis, "The rate of drying of solid materials", Ind. Eng. Chem, vol. 13, 1921, pp. 427432. http://dx.doi.org/10.1021/ie50137a021 A. Atangana, and A. Kilicman, On the generalized mass transport equation to the concept of variable fractional derivative, 2014. http://dx.doi.org/10.1155/2014/542809 A. Atangana, and Z. Hammouch, "Fractional calculus with power law: The cradle of our ancestors", Eur. Phys. J. Plus, vol. 134, no. 9, p. 429, 2019. http://dx.doi.org/10.1140/epjp/i2019-12777-8 A. Atangana, A. AkgÃ1/4l, “Analysis of New Trends of Fractional Differential Equations,” Fractional Order Analysis: Theory., Methods and Applications, 2020, pp. 91-111. T. Abdeljawad, and D. Baleanu, "Fractional Differences and Integration by Parts", J. Comput. Anal. Appl., vol. 13, no. 3, 2011. F. Jarad, T. Abdeljawad, and Z. Hammouch, "On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative", Chaos Solitons Fractals, vol. 117, pp. 16-20, 2018. http://dx.doi.org/10.1016/j.chaos.2018.10.006 H. Afshari, F. Jarad, and T. Abdeljawad, "On a new fixed point theorem with an application on a coupled system of fractional differential equations", Adv. Differ. Equ., vol. 2020, no. 1, p. 461, 2020.

282

[11] [12]

[13]

[14] [15] [16]

[17]

[18]

[19]

[20]

[21] [22] [23]

[24]

Advances in Special Functions of Fractional Calculus

Jain et al.

http://dx.doi.org/10.1186/s13662-020-02926-0 PMID: 32226454 C. Kürt, A. Fernandez, and M. A. Ã–zarslan, "On a certain bivariate Mittag-Leffler function analysed from a fractional-calculus point of view", Math Meth Appl Sci, 2020, pp. 1-21. R.K. Saxena, S.L. Kalla, and R. Saxena, Multivariate analogue of generalized Mittag-Leffler function, 2011. http://dx.doi.org/10.1080/10652469.2010.533474 A. Fernandez, T. Abdeljawad, and D. Baleanu, "Relations between fractional models with three-parameter Mittag-Leffler kernels", Adv. Differ. Equ., vol. 2020, no. 1, p. 186, 2020. http://dx.doi.org/10.1186/s13662-020-02638-5 PMID: 32226454. G. K. Watugala, "Sumudu transform-a new integral transform to solve differential equations and control engineering problems", Math probs in Engg Indust-6, 1998, pp. 319-329. S. Weerakoon, "Complex inversion formula for Sumudu transform", Int. J. Math. Educ. Sci. Technol., vol. 29, no. 4, pp. 618-621, 1998. M.A. Asiru, "Sumudu transform and the solution of integral equations of convolution type", Int. J. Math. Educ. Sci. Technol., vol. 32, no. 6, pp. 906-910, 2001. http://dx.doi.org/10.1080/002073901317147870 M.A. Asiru, "Further properties of the Sumudu transform and its applications", Int. J. Math. Educ. Sci. Technol., vol. 33, no. 3, pp. 441-449, 2002. http://dx.doi.org/10.1080/002073902760047940 M.A. Asiru, "Classroom note: Application of the Sumudu transform to discrete dynamic systems", Int. J. Math. Educ. Sci. Technol., vol. 34, no. 6, pp. 944-949, 2003. http://dx.doi.org/10.1080/00207390310001615499 F.B.M. Belgacem, A.A. Karaballi, and S.L. Kalla, "Analytical investigations of the Sumudu transform and applications to integral production equations", Math. Probl. Eng., vol. 2003, no. 3, pp. 103-118, 2003. http://dx.doi.org/10.1155/S1024123X03207018 F.B.M. Belgacem, and A.A. Karaballi, "Sumudu transform fundamental properties investigations and applications", J. Appl. Math. Stochastic Anal., vol. 2006, pp. 1-23, 2006. http://dx.doi.org/10.1155/JAMSA/2006/91083 F.B.M. Belgacem, "Introducing and analysing deeper Sumudu properties", Nonlinear Stud. J., vol. 13, no. 1, pp. 23-41, 2006. F.B.M. Belgacem, "Sumudu transform applications to Besselâ€™s Functions and Equations", Appl. Math. Sci., vol. 4, no. 74, pp. 3665-3686, 2010. A. Akgül, "A novel method for a fractional derivative with non-local and non-singular kernel", Chaos Solitons Fractals, vol. 114, pp. 478-482, 2018. http://dx.doi.org/10.1016/j.chaos.2018.07.032 E.K. Akgül, "Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives", Chaos, vol. 29, no. 2, 2019.023108. http://dx.doi.org/10.1063/1.5084035 PMID: 30823731

Hypergeometric Functions

[25]

[26]

[27]

[28] [29]

[30]

[31]

[32]

[33]

[34]

Advances in Special Functions of Fractional Calculus

283

K.M. Owolabi, A. Atangana, and A. Akgül, "Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model", Alex. Eng. J., vol. 59, no. 4, pp. 2477-2490, 2020. http://dx.doi.org/10.1016/j.aej.2020.03.022 A. Atangana, A. Akgül, and K.M. Owolabi, "Analysis of fractal fractional differential equations", Alex. Eng. J., vol. 59, no. 3, pp. 1117-1134, 2020. http://dx.doi.org/10.1016/j.aej.2020.01.005 S.D. Purohit, and F. Uçar, "An application of q-Sumudu transform for fractional q-kinetic equation", Turk. J. Math., vol. 42, no. 2, pp. 726-734, 2018. http://dx.doi.org/10.3906/mat-1703-7 K. S. Nisar, F. B. M. Belgacem, and M. S. Abouzaid, "On fractional kinetic equations and their Sumudu transform multiparameter Struve functions based solutions", arXiv, 2017. K.S. Nisar, and F.B.M. Belgacem, "Dynamic k-Struve Sumudu solutions for fractional kinetic equations", Adv. Differ. Equ., vol. 2017, no. 1, p. 340, 2017. http://dx.doi.org/10.1186/s13662-017-1397-6 K.S. Nisar, A. Shaikh, G. Rahman, and D. Kumar, "Solution of fractional kinetic equations involving class of functions and Sumudu transform", Adv. Differ. Equ., vol. 2020, no. 1, p. 39, 2020. http://dx.doi.org/10.1186/s13662-020-2513-6 A. Atangana, and A. Akgül, "Can transfer function and Bode diagram be obtained from Sumudu transform", Alex. Eng. J., vol. 59, no. 4, pp. 1971-1984, 2020. http://dx.doi.org/10.1016/j.aej.2019.12.028 B. Acay, E. Bas, and T. Abdeljawad, "Fractional economic models based on market equilibrium in the frame of different type kernels", Chaos Solitons Fractals, vol. 130, 2020.109438. http://dx.doi.org/10.1016/j.chaos.2019.109438 D. Baleanu, A. Fernandez, and A. Akgül, "On a Fractional Operator Combining Proportional and Classical Differintegrals", Mathematics, vol. 8, no. 3, p. 360, 2020. http://dx.doi.org/10.3390/math8030360 A. Fernandez, and C Kürt, "M. A. Ã–zarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators.

284

Advances in Special Functions of Fractional Calculus, 2023, 284-288

SUBJECT INDEX A Adaptive 3, 4, 13, 24 anti-synchronization 24 control methods 3, 4 synchronization 4, 13 Algorithm 74, 144 efficient 74 Analysis of fractional order dynamical systems 2 Analytic 113, 244, 259 foundation 244, 259 mapping 113 Anti-periodic BVP 173 Anti-synchronization 1, 3, 4, 6, 7, 9, 23, 24, 33 behavior 23 error 9 Arzela-Ascoli theorem 204, 226, 234 Ascoli’s theorem 180, 194 Atangana-Baleanu 273, 279 derivatives 279 fractional 273 Autonomous 1, 9, 10, 11, 23 chaotic systems 1, 9, 10, 11, 23 system 11, 23

B Baleanu derivatives 157 Banach contraction theorem 159 Bernstein 73, 74, 75, 77, 83, 84, 85, 86, 87, 88, 89 Bézier techniques 74 polynomials (BP) 73, 74, 75, 77, 83, 84, 85, 86, 87, 88, 89 Bessel functions 98, 100 Bezier coefficients 74 Bohmann-Korovkin-type approximation theorem 74 Bohnentblust-Karlin theorem 170 Boundary conditions 118, 146, 160, 174, 175

anti-periodic 174 periodic 160 Bounded variation 114, 222 Brownian motion 186

C Caputo 1, 2, 5, 136, 137, 138, 139, 140, 141, 143, 154, 157, 158, 273, 274, 275, 279, 280 definition 138, 139 derivatives 2 Fabrizio and Atangana-Baleanu derivatives 279 Fabrizio fractional derivative 273, 275 fractional 136, 138 implementing 136 sense 1 time-fractional 136, 137, 138, 139, 140, 141, 143 Carathèodory 185, 187, 191, 196, 199, 206 conditions 187, 191, 199, 206 theorem 185, 191 Cardiac tissues 2 CFC fractional 273 Chaotic dynamical 2 system mimics 2 systems 2 Chaotic systems 2, 3, 4, 6, 33 anti-synchronize 4 integer-order 33 synchronizing 2 Compactness 159, 168, 170 condition 159, 170 weakening 159 Computations, numerical 154 Conditions 58, 59, 64, 116, 118, 159, 161, 196, 211, 220, 222, 235, 236 growth 118, 196 Confluent hypergeometric function 244, 245, 246, 254, 255, 259, 261, 265, 266 Conformal mapping 112

Praveen Agarwal & Shilpi Jain (Eds.) All rights reserved-© 2023 Bentham Science Publishers

Subject Index

Constant proportional caputo (CPC) 273, 274, 277, 280 Control theory, nonlinear 4 Convergence 74, 79, 112, 113, 195 analysis 79 theorem 74, 112, 113 dominated 195 Convex 158, 166, 171 nonempty 158 subsets 166, 171

D Darbo’s fixed point theorem 169 Derivative(s) 1, 4, 13, 24, 73, 122, 124, 126, 129, 138, 145, 244, 247, 248, 252, 262 fractional order time 1, 4 integer-order 262 operators 122, 124, 126, 129, 138, 244, 247, 248, 252, 262 orders 145 Differential equations 39, 43, 45, 47, 48, 49, 99, 100, 157, 158, 185, 245, 260, 261, 271 system 271 Differential operator 2, 5 Diffusion 146, 270, 271 coefficient 271 Diffusive processes 189 Dirac-delta function 42 Dirichlet 112, 116, 138 boundary conditions 138 series 112, 116 Drive system 6, 9, 11, 13, 16, 19, 22, 23, 24, 27, 30, 33 Drying kinetic 274, 275, 276, 277 Dynamical 2, 33 behavior 2 systems 2 nonlinear 2, 33 Dynamics 1, 2, 247, 262 fluid 247, 262 nonlinear 2

Advances in Special Functions of Fractional Calculus 285

E Efficiency 73, 74, 83, 136, 145, 280 computational 73 Electric circuit analysis 39 Electrodynamics 213 Equation 8, 23, 125, 126, 129, 131, 138, 139, 140, 141, 142, 143, 144, 145, 146, 158, 187, 213, 214, 232, 233, 251, 264, 271, 272, 274, 277 algebraic 233 analytical solution of 146 dimensional linear diffusion 145 discrete dynamical system 271 fractional kinetic 272 hybrid differential 187, 213 linear 140 perturbed differential 158 Erdelyi-Kober 186, 188, 189, 190 operators 186, 188, 189 sense of 186, 189 Erdélyi-Kober fractional 189, 265, 266 diffusion 189 integrals of gauss hypergeometric function 265, 266 Expansion 64, 75, 76, 83, 89 asymptotic 64 binomial 75, 76

F Fick’s law 270 Fixed point theorem 158, 169 celebrated 158 for monotone operators 169 Fluid mechanics 2 Formula, mathematical modeling 270 Fourier inversion formula 113 Fractional 1, 2, 5, 39, 40, 41, 43, 47, 48, 49, 137, 138, 145, 146, 157, 158, 189, 190, 239, 240, 247, 252, 262, 265 270, 271, 274 boundary conditions 145, 146 calculus operators 247, 262

286 Advances in Special Functions of Fractional Calculus

Agarwal and Jain

coupled 271 derivatives 1, 5, 41, 262, 270, 274 exponents 39, 262 integral formulas 252, 265 linear 39 partial differential equations (FPDEs) 39, 40, 49, 137, 145 Fractional integration 186, 188 operation 188 operator 186 Fractional order 1, 2, 4, 5, 9, 10, 11, 187, 191, 196, 197, 198, 205, 206, 214, 219, 220, 239, 240, 241 derivatives 4 dynamical systems 2 functional quadratic equation of 197 integral inclusions of 240 quadratic integral equation of 191, 196, 197, 198, 214 set-valued 219, 220, 241 Functional analysis 187, 262 nonlinear 187 Function projective synchronization 3

Gould-Hopper 122, 123, 128 Genocchi polynomials (GHGP) 128 polynomials (GHP) 122, 123 Grain drying 270, 280 operation 270 process 280

G

Image processing 247, 262 Integral equation 185, 186, 196, 197, 198, 199, 205, 206, 207, 211, 212, 219, 220, 223, 224, 236, 240, 241 set-valued functional 219, 220, 240, 241 single-valued 236 Iterative methods 136, 137, 144, 145, 147, 154

Galerkin and collocation methods 74 Gauss 244, 245, 248, 250, 254, 259, 260, 265, 266 hypergeometric function 244, 245, 248, 250, 254, 259, 260, 265, 266 Gauss-Seidel 136, 137 iteration 137 methods 136 General fractional order 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213 Generalized hypergeometric function 64, 94, 98, 251, 252, 253, 254, 255, 256 Genocchi 121, 122, 123, 124, 126, 128, 130, 132, 134 numbers and Genocchi polynomials 123 polynomials 121, 122, 123, 124, 126, 128, 130, 132, 134

H Half-sweep preconditioned gauss-Seidel 143 Hausdorff 168, 169, 221 distance 221 measure of noncompactness 168, 169 HSPGS method 137, 143, 144, 145, 154 Hybrid functional 212 Hypergeometric functions 62, 63, 64, 65, 67, 69, 71, 245, 247, 249, 251, 253, 254, 255, 260, 261 derivative gauss 254 Hypergeometric series 62, 100

I

K Kakutani theorem 167, 172 Koebe function 95 maps 95 Kummer’s equation 245, 261

L Laplace transform method 113 Lax equivalence theorem 141 Linear 42, 78, 170, 214

Subject Index

change 42 combinatory 78 functional argument 214 operator 170 Linear system 136, 140, 143, 154 preconditioned 143 Lipschitz condition 211, 227, 236 Lommel functions 98, 99 Lorenz system 11 Lyapunov 4, 8, 9 function 8 stability method 4 stability theory 4, 9

M

Advances in Special Functions of Fractional Calculus 287

Plancherel theorem 118 Pochhammer symbol 54, 245, 260 Properties, fundamental 74

Q Quadratic integral equations (QIEs) 185, 214 R Riemann-Liouville fractional 55, 58, 138, 190, 252 calculus 58 integration 190

S Mittag-Leffler function 43, 53, 274 Monotone 161, 182 mapping 161 single/mutivalued operators 182 Monotone self 161, 162 mapping 161 operators 162 Monotonicity 202, 246 Moulton method 4, 33

N Neutron transport 185 Nonlinear 7, 214, 219 control function 7 functional arguments 214 Non-linear functional integral equation 219

O Operational techniques 121

P PDEs, nonlinear 137 Perron’s condition 211 Picard and Adomian methods 198

Stability analysis 141 Stieltje’s integrability 222 Stochastic processes 189 Struve functions 98 Sumudu transformation 271, 277 Sumudu transform 273 of ABC fractional 273 of CFC fractional 273 Synchronization 2, 3, 4, 11, 13 behavior 11 error vectors 13 method 4 of chaos 2 of chaotic dynamical systems 2 of chaotic systems 3, 4 Systems 1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 24, 74, 247, 262, 270 chaotic communication 2 drive and response 3, 13, 24 hyperchaotic 4 mechanical drying 270

T TBS technique 90 Time-fractional orders 150, 151, 152, 153 Transcendental functions 73

288 Advances in Special Functions of Fractional Calculus

V Viscoelasticity 2 Visual mathematical analysis 133 Volterra 240 non-linear 240

W Weierstrass theorem 80 Whittaker transform 69 Wiman function 246

Agarwal and Jain