General fractional derivatives: theory, methods, and applications 9781138336162, 1138336165, 9780429284083

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GENERAL FRACTIONAL DERIVATIVES Theory, Methods and Applications Xiao-Jun Yang

General Fractional Derivatives Theory, Methods and Applications

General Fractional Derivatives Theory, Methods and Applications

Xiao-Jun Yang

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 c 2019 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-33616-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a notfor-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Yang, Xiao-Jun (Mathematician), author. Title: General fractional derivatives : theory, methods, and applications / Xiao-Jun Yang. Description: Boca Raton : CRC Press, Taylor & Francis Group, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2019011932| ISBN 9781138336162 (hardback : alk. paper) | ISBN 9780429284083 (ebook) Subjects: LCSH: Fractional calculus. | Calculus. Classification: LCC QA314 .Y36 2019 | DDC 515/.83--dc23 LC record available at https://lccn.loc.gov/2019011932 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To the 110th Anniversary of the 1909 Founding of China University of Mining and Technology.

Contents

Preface

xiii

Author

xvii

1 Introduction 1.1 History of fractional calculus . . . . . . . . . . . . . . . . . . 1.1.1 The contribution for fractional calculus and applications . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The contribution for generalized fractional calculus and applications . . . . . . . . . . . . . . . . . . . . . . . 1.2 History of special functions . . . . . . . . . . . . . . . . . . 1.3 Special functions with respect to another function . . . . . . 2 Fractional Derivatives of Constant Order and Applications 2.1 Fractional derivatives within power-law kernel . . . . . . . . 2.2 Riemann-Liouville fractional calculus . . . . . . . . . . . . . 2.2.1 Riemann-Liouville fractional integrals . . . . . . . . . 2.2.2 Riemann-Liouville fractional derivatives . . . . . . . . 2.2.3 Riemann-Liouville fractional derivatives of a purely imaginary order . . . . . . . . . . . . . . . . . . . . . 2.3 Liouville-Sonine-Caputo fractional derivatives . . . . . . . . 2.3.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Liouville-Sonine-Caputo fractional derivatives . . . . 2.4 Liouville-Gr¨ unwald–Letnikov fractional derivatives . . . . . 2.4.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Liouville-Gr¨ unwald-Letnikov fractional derivatives . 2.4.3 Kilbas-Srivastava-Trujillo fractional derivatives . . . . 2.5 Tarasov type fractional derivatives . . . . . . . . . . . . . . . 2.5.1 Tarasov type fractional derivatives . . . . . . . . . . . 2.5.2 Extended Tarasov type fractional derivatives . . . . . 2.6 Riesz fractional calculus . . . . . . . . . . . . . . . . . . . . 2.7 Feller fractional calculus . . . . . . . . . . . . . . . . . . . . 2.8 Richard fractional calculus . . . . . . . . . . . . . . . . . . . 2.9 Erd´elyi-Kober type fractional calculus . . . . . . . . . . . . 2.9.1 Erd´elyi-Kober type operators of fractional integration and fractional derivative . . . . . . . . . . . . . . . .

1 1 1 23 24 31 39 39 52 52 52 54 63 63 66 70 70 72 73 74 74 75 78 79 80 81 81 vii

viii

Contents 2.9.2

2.10

2.11

2.12 2.13

2.14 2.15

Fractional integrals and fractional derivatives of the Erd´elyi-Kober-Riesz, Erd´elyi-Kober-Feller and Erd´elyi-Kober-Richard types . . . . . . . . . . . . . . Katugampola fractional calculus . . . . . . . . . . . . . . . . 2.10.1 Katugampola fractional integrals and Katugampola fractional derivatives . . . . . . . . . . . . . . . . . . 2.10.2 Katugampola type fractional integrals and Katugampola type fractional derivatives involving the exponential function . . . . . . . . . . . . . . . . . . . . . . . Hadamard fractional calculus . . . . . . . . . . . . . . . . . . 2.11.1 Hadamard fractional integrals and fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.2 Hadamard type fractional integrals and fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . Marchaud fractional derivatives . . . . . . . . . . . . . . . . Tempered fractional calculus . . . . . . . . . . . . . . . . . . 2.13.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . 2.13.2 Tempered fractional derivatives . . . . . . . . . . . . 2.13.3 Tempered fractional derivatives with respect to another function . . . . . . . . . . . . . . . . . . . . . 2.13.4 Tempered fractional derivatives of a purely imaginary order . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.5 Tempered fractional integrals . . . . . . . . . . . . . 2.13.6 Tempered fractional integrals of a purely imaginary order . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.7 Tempered fractional derivatives in sense of LiouvilleSonine and Liouville-Sonine-Caputo types . . . . . . 2.13.8 Tempered fractional derivatives involving power-sine and power-cosine functions . . . . . . . . . . . . . . . 2.13.9 Tempered fractional calculus involving powerKohlrausch-Williams-Watts function . . . . . . . . . 2.13.9.1 Tempered fractional derivative in the LiouvilleSonine-Caputo type involving the kernel of the power-Kohlrausch-Williams-Watts function . . . . . . . . . . . . . . . . . . . . . . 2.13.9.2 Tempered fractional integral involving the kernel of the power-Kohlrausch-WilliamsWatts function . . . . . . . . . . . . . . . . 2.13.10 Sabzikar-Meerschaert-Chen tempered fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.11 Torres tempered fractional derivatives . . . . . . . . . Tempered fractional derivatives involving Mittag-Leffler function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liouville-Weyl fractional calculus . . . . . . . . . . . . . . .

86 87 87

89 90 90 91 95 96 96 96 98 99 100 101 102 105 107

107

108 115 116 122 123

Contents 2.16 Kilbas-Srivastava-Trujillo fractional calculus with respect to another function . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Sousa-de Oliveira fractional derivative with respect to another function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Liouville-Weyl fractional calculus with respect to another function in the Sense of Riesz, Feller and Richard types . . . . . 2.18.1 Liouville-Weyl fractional integrals with respect to another function in the sense of Riesz, Feller and Richard types . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2 Liouville-Weyl fractional derivatives with respect to another function in the sense of Riesz, Feller and Richard types . . . . . . . . . . . . . . . . . . . . . . 2.19 Hilfer derivatives . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1 Hilfer derivatives . . . . . . . . . . . . . . . . . . . . 2.19.2 Sousa-de Oliveira fractional derivatives with respect to another function in the sense of Hilfer type . . . . . . 2.19.3 Riesz, Feller and Richard fractional derivatives with respect to another function via Hilfer fractional derivative . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20.1 Relaxation equations within fractional derivatives . . 2.20.2 Rheological models within fractional derivative . . .

ix

124 126 127

127

127 131 131 134

135 136 136 140

3 General Fractional Derivatives of Constant Order and Applications 145 3.1 General fractional derivatives involving the kernel of MittagLeffler function . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.1.1 General fractional derivatives involving the kernel of Mittag-Leffler function without power law . . . . . . 145 3.1.2 General fractional derivatives involving the kernel of Mittag-Leffler-Gauss function with power law . . . . 147 3.1.3 General fractional derivatives involving the kernel of Mittag-Leffler function with power law . . . . . . . . 149 3.1.4 General fractional derivatives involving the kernel of Mittag-Leffler function with the negative power law . 154 3.2 General fractional derivatives involving the kernel of MittagLeffler function in the sense of Riesz, Feller and Richard types 157 3.3 General fractional derivatives involving the kernel of MittagLeffler function with respect to another function . . . . . . . 162 3.4 General fractional derivatives involving the kernel of KohlrauschWilliams-Watts function . . . . . . . . . . . . . . . . . . . . 164 3.4.1 Special cases: the kernel of Gauss function . . . . . . 166 3.4.2 Special cases: the kernel of Gaussian-like function . . 168 3.5 General fractional derivatives in the Miller-Ross kernel . . . 169

x

Contents 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20

Hilfer type general fractional derivatives in the Miller-Ross kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General fractional derivatives in the one-parametric LorenzoHartley kernel . . . . . . . . . . . . . . . . . . . . . . . . . . Hilfer type general fractional derivatives involving the oneparametric Lorenzo-Hartley kernel . . . . . . . . . . . . . . . General fractional derivatives in the subcosine kernel via Mittag-Leffler function . . . . . . . . . . . . . . . . . . . . . Hilfer type general fractional derivatives within the subcosine kernel via Mittag-Leffler function . . . . . . . . . . . . . . . General fractional derivatives in the subsine kernel via MittagLeffler function . . . . . . . . . . . . . . . . . . . . . . . . . . Hilfer type general fractional derivatives within the subsine kernel via Mittag-Leffler function . . . . . . . . . . . . . . . . . General fractional derivatives in the two-parametric LorenzoHartley kernel . . . . . . . . . . . . . . . . . . . . . . . . . . General fractional derivatives in the Gorenflo-Mainardi kernel via Wiman Function . . . . . . . . . . . . . . . . . . . . . . . Hilfer type general fractional derivatives involving the kernel of the Gorenflo-Mainardi function . . . . . . . . . . . . . . . . . General fractional derivatives in the generalized Prabhakar kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General fractional derivatives involving the generalized Prabhakar kernel with respect to another function . . . . . . . . . Hilfer type general fractional derivatives within the Prabhakar kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilfer type general fractional derivatives within the Prabhakar kernel with respect to another function . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.1 Relaxation models within general fractional derivatives 3.20.2 Rheological models within general fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . .

175 177 180 181 187 189 195 197 203 210 211 218 220 222 223 223 227

4 Fractional Derivatives of Variable Order and Applications 235 4.1 Fractional derivatives of variable order involving the kernel of the singular power function . . . . . . . . . . . . . . . . . . . 235 4.2 Riesz, Feller and Richard types fractional derivatives of variable order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.3 Hilfer type fractional derivatives of variable order involving the kernel of the singular power function . . . . . . . . . . . . . 244 4.4 Tempered fractional derivatives of variable order involving the kernel of the weak power-law function . . . . . . . . . . . . . 246 4.5 General fractional derivatives of variable order involving the kernel of Mittag-Leffler function . . . . . . . . . . . . . . . . 247

Contents General fractional derivatives of variable order involving the kernel of Wiman function . . . . . . . . . . . . . . . . . . . . 4.7 General fractional derivatives of variable order involving the one-parametric Lorenzo-Hartley function . . . . . . . . . . . 4.8 General fractional derivatives of variable order involving the Miller-Ross function . . . . . . . . . . . . . . . . . . . . . . 4.9 General fractional derivatives of variable order involving the Prabhakar function . . . . . . . . . . . . . . . . . . . . . . . 4.10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.1 Relaxation models within variable-order fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Rheological models within variable-order fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . .

xi

4.6

250 253 255 257 259 259 262

5 Fractional Derivatives of Variable Order with Respect to Another Function and Applications 267 5.1 Fractional derivatives of variable order with respect to another function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.2 Riesz type fractional derivatives of variable order with respect to another function . . . . . . . . . . . . . . . . . . . . . . . 272 5.3 Hilfer type fractional derivatives of variable order with respect to another function . . . . . . . . . . . . . . . . . . . . . . . 275 5.4 Tempered fractional derivatives of variable order with respect to another function . . . . . . . . . . . . . . . . . . . . . . . 277 5.5 General fractional derivatives of variable order via MittagLeffler function with respect to another function . . . . . . . 279 5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 5.6.1 Relaxation models based on variable-order fractional derivatives with respect to another function . . . . . 280 5.6.2 Rheological models in variable-order fractional derivatives with respect to another function . . . . . . . . 284 A Laplace Transforms of the Functions

289

B Fourier Transforms of the Functions

291

C Mellin Transforms of the Functions

293

D The Special Functions and Their Expansions D.1 The one-parametric special functions via Mittag-Leffler functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 The one-parametric special functions via Plotnikov functions D.3 The one-parametric special functions via Miller-Ross functions D.4 The one-parametric special functions via Rabotnov functions D.5 The one-parametric special functions via one-parametric Lorenzo-Hartley functions . . . . . . . . . . . . . . . . . . . .

295 295 299 300 302 305

xii

Contents D.6 The three-parametric special functions via Prabhakar functions D.7 The two-parametric special functions via Wiman functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.8 The two-parametric special functions via generalized Wiman functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.9 The two-parametric special functions via two-parametric Lorenzo-Hartley functions . . . . . . . . . . . . . . . . . . . . D.10 The two-parametric special functions via two-parametric Gorenflo-Mainardi functions . . . . . . . . . . . . . . . . . .

307 322 325 328 332

Bibliography

335

Index

361

Preface

Modern developments in theoretical and applied science have widely depended on knowledge of the derivatives and integrals of the positive integer order, the properties of the mathematical functions, from the gamma and beta functions to the special functions, and the integro-differential operators where the integrals are of the convolution type and exist the singular, weakly singular and non-singular kernels of power-law type, which exhibit the derivatives and integrals of the variable order due to the appearance of the complexity in the natural phenomena. The aim of this monograph is mainly to investigate the fractional calculus and general fractional calculus of constant order, fractional calculus and general factional calculus of variable order, and their extended versions with respect to another function. Another aim is to present the anomalous relaxation and rheological models in light of the complexity in nature. The topics are important and interesting for scientists and engineers in the fields of mathematics, physics, chemistry and engineering. In view of the above-mentioned avenues of their potential application to describe the numerous widespread real-world phenomena, we systematically investigate the special functions, the fractional calculus, fractional calculus with respect to another function, variable-order fractional calculus, variable-order fractional calculus with respect to another function, general fractional calculus, general fractional calculus with respect to another function, variable-order general fractional calculus, variable-order general fractional calculus with respect to another function. More specifically, we have clearly illustrated the basic theory of the general fractional calculus, the Fourier and Laplace transforms of general fractional integrals and general fractional derivatives and some illustrative examples. The book is divided into five chapters with four appendices. Chapter 1 investigates the history of fractional calculus of constant and variable orders, general fractional calculus of constant and variable orders and general calculus, for example, the Riemann-Liouville fractional derivative, Liouville-Gr¨ unwald fractional derivative, Marchaud fractional derivative, Hadamard fractional derivative, Hille-Tamarkin fractional derivative, Riesz fractional derivative, Hilfer fractional derivative, Erd´elyi-Kober fractional derivative, Liouville-Sonine-Caputo fractional derivative and the corresponding inverse operators. Meanwhile, the special functions containing the Gamma function, Gauss hypergeometric function, Bessel function, Kummer function, Kohlrausch-Williams-Watts function, Bessel-Clifford function,

xiii

xiv

Preface

Macdonald function, Mittag-Leffler function, Wiman function, Wright function, Meijer G function, MacRobert E function, Rabotnov function, Agarwal function, Erd´elyi-Magnus-Oberhettinger-Tricomi function, Fox H-function, Prabhakar function, Miller-Ross function, Gorenflo-Mainardi function and two-parametric Lorenzo-Hartley function. The special functions with respect to another function are proposed for the first time. In Chapter 2 we report the theory of function space containing a family of the Lebesgue measurable functions and absolutely continuous functions. In view of the introductions of the fractional calculus and generalized fractional calculus in first chapter, we directly introduce the concepts of the fractional derivatives and fractional integrals of constant order within and without singular and non-singular; for example, Riemann-Liouville fractional calculus, Liouville-Sonine and Liouville-Sonine-Caputo fractional derivatives LiouvilleGr¨ unwald–Letnikov fractional derivatives, Tarasov type fractional derivatives, Riesz fractional calculus, Feller fractional calculus, Richard fractional calculus, Erd´elyi-Kober type fractional calculus, Katugampola fractional calculus, Hadamard fractional calculus, Marchaud fractional derivatives, tempered fractional calculus within power-exponential, power-sine, power-cosine, power-Kohlrausch-Williams-Watts and power-Mittag-Leffler functions, tempered fractional calculus with respect to another function, tempered fractional derivatives of a purely imaginary order, Sabzikar-Meerschaert-Chen tempered fractional calculus, Torres tempered fractional derivatives, LiouvilleWeyl fractional calculus, Kilbas-Srivastava-Trujillo fractional calculus, Sousade Oliveira fractional derivative with respect to another function, LiouvilleWeyl fractional calculus with respect to another function in the sense of Riesz, Feller and Richard types, Hilfer derivatives, Riesz, Feller and Richard fractional derivatives with respect to another function, fractional calculus involving the exponential function, fractional derivatives involving the kernel of Prabhakar function, fractional derivatives involving the kernel of Wiman function and so on. The integral transforms of the fractional derivatives and fractional integrals of the constant order are discussed in detail. Some applications in engineering are also considered. Chapter 3 addresses the general fractional derivatives of constant order involving the kernel of the special functions, such as the Mittag-Leffler function, Kohlrausch-Williams-Watts function, Wiman function, Prabhakar function, Gorenflo-Mainardi function, Miller-Ross function, Lorenzo-Hartley function, subcosine function and subsine function. The general fractional derivatives and the general fractional integrals in the complex and real orders are discussed in detail, and the Laplace transforms of the corresponding differential and integral operators are also given. The general fractional derivatives and the general fractional integrals with respect to another function are also presented. The general fractional derivatives are also considered to describe the relaxation and rheological models in complex phenomena. Chapter 4 is devoted to the variable-order fractional calculus and the variable-order general fractional calculus involving the kernel of the special

Preface

xv

functions, such as the Mittag-Leffler function, weak power-law function, Lorenzo-Hartley function, Wiman function, Miller-Ross function and Prabhakar function. The definitions of the fractional derivatives and the general fractional integrals are presented. The Hilfer type fractional derivatives of variable order, the Riesz type fractional derivatives of variable order, the Feller type fractional derivatives of variable order and the Richard type fractional derivatives of variable order are also considered. The general fractional derivatives and general fractional integrals of the variable order involving the special functions are also discussed. The mathematical models for the relaxation and rheological models are addressed in detail. In Chapter 5, we present the variable-order fractional calculus with respect to another function and the variable-order general fractional calculus involving the Mittag-Leffler function with respect to another function. The variable-order general fractional derivatives containing the kernel of the special functions, such as the Mittag-Leffler function and weak power-law function are presented. The definitions of the variable-order fractional derivatives and the variable-order general fractional integrals with respect to another function are introduced. The Hilfer type fractional derivatives of variable order with respect to another function and the Riesz type fractional derivatives of variable order with respect to another function are also proposed. The mathematical models for the relaxation and rheological models are discussed in detail. It is my pleasure to express my grateful thanks to my friends and colleagues around the world who offered their suggestions of the preparation of the book. My special thanks go to my supervisor, Professor Feng Gao, and my colleagues, Professor Fu-Bao Zhou, Professor Yang Ju, Professor Hong-Wen Jing, Professor Zhan-Guo Ma and Professor Ting Li, in China University of Mining and Technology, who helped at various stages of the preparation of the book, and the financial support of the Yue-Qi Scholar of the China University of Mining and Technology (Grant No.04180004), the 333 Project of Jiangsu Province (Grant No.BRA2018320) and the State Key Research Development Program of the People’s Republic of China (Grant No. 2016YFC0600705). Finally, I also wish to express my special thanks to Mrs. Aastha Sharma, senior acquisitions editor, Mrs. Shikha Garg, editorial assistant, and several staff members of CRC Press-Taylor & Francis Group for their cooperation in the production process of this book. Xiao-Jun Yang Professor State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and Technology Xuzhou City, Jiangsu Province, China

Author

Dr. Xiao-Jun Yang is a full Professor of Applied Mathematics and Mechanic, at State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, China. His scientific interests include: Viscoelasticity, Nonlinear Dynamics, Mathematical Physics, Analytical, Approximate, Numerical and Exact Solutions for ODEs and PDEs, Integral Transforms and Their Applications, Continuous Mechanics, Rock mechanics, Fluid Mechanics, Heat Transfer, and Traffic Flow, Wavelets, Signal Processing, Biomathematics, General Calculus and Applications, Fractional Calculus and Applications, Local Fractional Calculus and Applications, General Fractional Calculus and Applications, Variable Order Fractional Calculus and Applications and Variable Order Fractional Calculus and Applications with Respect to Another Function. He has published over 160 journal articles and 5 books, as well as monographs, and edited volumes and 10 chapters. In 2017 and 2018, he was awarded the Elsevier Most Cited Chinese Researchers in Mathematics and the 2017 Chinese 100 Best Impact International Academic Paper. In 2018, he was selected as one of the “333” Program Talents of Jiangsu Province, China. He is currently an editor of several scientific journals, such as Fractals-Complex Geometry, Patterns, and Scaling in Nature and Society, Applied Numerical Mathematics, Mathematical Modelling and Analysis, and Thermal Science. He is also the referee of articles and books from Springer Nature, Elsevier, World Scientific and CRC Press publishers.

xvii

Symbol Description R C N N0

The set of real numbers Re (α) The set of complex numbers The set of natural numbers Im (α) The set of positive integer numbers i

The real part of the complex number–α The imaginary part of the complex number–α Imaginary unit

Chapter 1 Introduction

In this chapter, we introduce the histories of the fractional calculus of constant and variable orders, fractional calculus of constant and variable orders with respect to another function, general fractional calculus of constant and variable orders and general fractional calculus of constant and variable orders with respect to another function. Meanwhile, the special functions contain the Gamma function, Gauss hypergeometric function, Bessel function, Kummer function, Kohlrausch-Williams-Watts function, Bessel-Clifford function, Macdonald function, Mittag-Leffler function, Wiman function, Wright function, Meijer G function, MacRobert E function, Rabotnov function, Agarwal function, Erd´elyi-Magnus-Oberhettinger-Tricomi function, Fox H function, Prabhakar function, Miller-Ross function, Gorenflo-Mainardi function, two-parametric Lorenzo-Hartley function, and so on. The special functions with respect to another function are proposed for the first time.

1.1

History of fractional calculus

The consideration for generalizing the notion of differentiation dn y/dxn of order n from an integer number to a non-integer number was started at birth. The non-integer order, denoted by n, can be considered as follows: fractional, irrational, or complex, fractal dimension, real function or complex functions [1, 2, 3]. There are the fractional calculus (fractional-order calculus) of constant and variable orders, general fractional calculus, local fractional calculus (also called the fractal calculus), and general calculus, investigated by the researchers based on the differences of the kernel functions. To start with the ideas, we introduce their histories.

1.1.1

The contribution for fractional calculus and applications

On September 30, 1695, Leibniz replied to L’Hopital in the letter [4]: “You can see by that, sir, that one can express by an infinite series a quantity such as d1/2 xy ord1:2 xy. Although infinite series and geometry are distant relations, infinite series admits only the use of exponents that are positive and negative 1

2

General Fractional Derivatives: Theory, Methods and Applications

integers, and does not, as yet, know the use of fractional exponents.” Later, in the same letter, √ Leibniz continues prophetically: “Thus it follows that d1/2 x will be equal tox dx : x. This is an apparent paradox from which, one day, useful consequences will be drawn.” On December 28, 1695, in the correspondence with Johann Bernoulli, Leibniz mentioned derivatives of “general order.” [5]. On May 28, 1697, in the Leibniz’s correspondence with John Wallis, in which Wallis’s infinite product for 12 π is discussed, Leibniz states that differential calculus might have been used to achieve this result. He uses the 1 notation d 2 y to denote the derivative of order [6]. In 1738, Euler took the first step that where he wrote: “When n is a positive integer, the ratio dn p, p is a function of x, to dxn can always be expressed algebraically. Now it is asked: what kind of ratio can be made if n be a fraction? If n is a positive integer, dn can be found by continued differentiation. Such a way, however, is not evident if n is a fraction. But the matter may be expedited with the help of the interpolation of series as explained earlier in this dissertation.” (See [7]). He gave the definition of the fractional derivative as follows: Daα+ f (x) =

Γ (1 − β) 1 . Γ (1 − α − β) (x − a)β+α

(1.1)

In 1772, Lagrange indirectly contributed to the law of exponents (indices) for differential operators of integer order by the expression [8] dm dn dm+n · y = y, dxm dxn dxm+n

(1.2)

which is also true for the fractional calculus and proved by Letnikov in 1868 [9]. In 1812, Laplace proposed the idea of differentiation of non-integer order for the functions, which is represented by an integral [10]: Z T (t)t−x dt. (1.3) In 1819, Lacroix used the Euler idea, given by [6] dn xa /dxn , and obtained by [11] 1 d 2 νa Γ (1 + α) a− 1
ν 2 , = (1.4) 1 2 Γ 1 + 12 dν which reduced for α = 1 to

1

d2 ν

√ 2 ν = √ . π

(1.5) 1 dν 2 In 1822, Fourier contributed to the interesting results by the expression [12]: dα f (x) 1 = dxα 2π

+∞ Z Z α λ dλ −∞

+∞

−∞

f (t) cos (λx − tx + pπ/2) dt

(1.6)

Introduction

3

in order to suggest the derivative for non-integer order. In 1823, Abel considered the integral in the form [13, 14] Zx

f (t) α dt = m (x) , x > a, 0 < α < 1 (x − t)

(1.7)

a

to solve the tautochrone problem. In 1832, Liouville derived the formula [15, 16] Dα f (x) =

∞ X

ak x Ck aα , ke

(1.8)

k=0 ∞ P

where f (x) =

k=0

D

−α

f (x) =

ak x Ck aα and the fractional integration was defined as [15] ke

Z∞

1 α

(−1) Γ (α)

f (x + t) tα−1 dt, x ∈ R, Re (α) > 0,

(1.9)

0

the fractional derivative as a limit of a difference quotient [16] ∆α hf , h→0 hα

Dα f = lim

(1.10)

where ∆α h f is a difference of fractional order, which is related to the sum of exponential functions, and the fractional derivative formula [15] α

D f (x) =

Z∞

1 α

(−1) Γ (α)

dn f (x + t) α−1 t dt, x ∈ R, Re (α) > 0, dxn

(1.11)

0

and the fractional differential formula [15] 1 √ 2g

Zh 0

1 df (t) p dt = m (t) , x ∈ R, (x − t) dt

(1.12)

though not quite rigorously from the modern point of view. In 1835, Liouville considered the fractional derivative of the function f (x), denoted by Dα f (x), of order α , which is positive, negative, real and imaginary [17]. In 1847, Riemann derived the formula [18] D

−α

1 f (x) = Γ (α)

Zx

α−1

(x − t)

f (t) dt,

c

which can be linked with the fractional integral formula of Liouville.

(1.13)

4

General Fractional Derivatives: Theory, Methods and Applications

In 1869, Sonine called the Riemann-Liouville definition starting with the Cauchy’s integral formula for the first time [19]. In 1867, Gr¨ unwald developed the Liouville’s idea of the sum of the exponential functions [17], and proposed the following formula [20]: ∆α h f (x) , h→0 hα

Dα f (x) = lim

(1.14)

where ∆α h f is a difference of fractional order, given by ∆α hf =

∞ X

i





α i

(−1)

i=0

f (x − hi).

(1.15)

This operator is called the Liouville-Gr¨ unwald fractional operator [21, 22]. In 1868, Letnikov considered the above formula [23] ∆α h f (x) , h→0 hα

Dα f (x) = lim

(1.16)

where ∆α h f is a difference of fractional order, whose appropriate interpretation of the fractional difference is related with Riemann’s and Liouville’s expressions. This operator is also called the Gr¨ unwald-Letnikov fractional operator [3]. In 1872, Sonine introduced the following fractional derivative, given as [24] Zx

1 D f (x) = Γ (p − α + 1) α

df (t) p−α (x − t) dt. dt

(1.17)

a

In 1880, Sonine introduced the kernel of the form [25] Θ (t) = eλt ,

(1.18)

which leads to the following expressions 1 M (t) = Γ (−α)

Z∞

e−λt f (t) dt

(1.19)

0

and Zb M (t) =

1 tn+1

e− 2 (t− t ) f (t) dt. λ

1

(1.20)

a

In 1884, Sonine proposed a solution of the Abel type equation [26] Zx

Zx g (x − t) Φ (x) dt =

a

Φ (x) dt, a

(1.21)

Introduction

5

where Zh ψ (h − t) ϕ (t) dt = 1

g (h) =

(1.22)

0

with ψ (t) = t−p

∞ X i=0

and ϕ (t) = t

−q

∞ X i=0

n

(ty) Γ (n + 1) Γ (n − p + 1)

(1.23)

n

(−ty) Γ (n + 1) Γ (n − q + 1)

(1.24)

for p + q = 1, which are called the Sonine conditions (for more details on the Sonine condition, see [3, 27, 28]), and introduced the kernel of the form Θ (t) =

tα eλt , Γ (1 − α)

(1.25)

which leads to the following expression Z∞

1 M (t) = Γ (−α)

e−xt f (t) dt. t1+α

(1.26)

0

In 1890, Sonine introduced the fractional multiple integral of the form [29] M (t) = e−c1 t

Rx

α

(x − t1 ) 1 eb1 t1 dt1 . . .

a

...

tn−1 R

ti−1 R

α

(xi−1 − ti ) i ebi ti dti

a

f (tn ) (xn−1 − tn )

αn bn tn

e

(1.27)

dti

a

In 1892, Hadamarod proposed the following fractional integral [30] zα I f (x) = Γ (α) α

Z1

f (zu)

1−α du.

0

(1 − u)

(1.28)

In 1896, Volterra introduced the general Abel’s integral equation [31] Zx

κ (x, t) f (t) α dt = g (x) , (x − t)

0 < α < 1,

(1.29)

a

where κ (x, t) is a continuous function. In 1909, Bocher considered that Abel’s equation [32] 1 Γ (α)

Zx

f (t)

1−α dt

a

(x − t)

= g (x) ,

0 < α < 1,

(1.30)

6

General Fractional Derivatives: Theory, Methods and Applications

is solvable in the set of all Lebesgue measurable functions if and only if there are Zx 1 g (t) h (x) = (1.31) 1−α dt Γ (1 − α) (x − t) a

and h (a) = 0, where

Z

(1.32)

x

h (t) = C +

g (x) dx

(1.33)

a

and Z

b

|g (x)| dx < ∞.

(1.34)

a

In 1917, Weyl derived the fractional integrals by [33]: α I+ f

Zx

1 (x) = Γ (α)

f (t)

1−α dt,

(x − t)

−∞

α I− f (x) =

1 Γ (α)

Z∞

f (t)

1−α dt,

(x − t)

x

(1.35)

(1.36)

which are called the Weyl fractional integrals. In 1927, Marchaud introduced the fractional differentiation of the form [34]: Z∞ l 1 ∆t f (x) Dα f (x) = C f (t) dt, (1.37) Γ (α) tα+1 0

∆lt f

where (x) is the finite difference of order l for l > α and l ∈ N. This operator is now named Marchaud fractional derivatives. When ∆lt f (x), it is called the Weyl type finite difference for l = 1 and 1 > α > 0. In 1927, Hadamard introduced the fractional integral expressed as [34] Ia(α) f

1 (x) = Γ (α)

Zx a

f (t) dt
1−α , 0 < α < 1, t ln xt

(1.38)

which is called the Hadamard fractional integral, and the fractional derivative was given by Da(α) f

1 d (x) = x Γ (1 − α) dx

Zx 0

f (t) dt
α , 0 < α < 1, t ln xt

which is called the Hadamard fractional derivative.

(1.39)

Introduction

7

In 1928, Tonelli introduced the following result [35]: If Z x g (x) dx f (t) = C +

(1.40)

a

and Z

b

|g (x)| dx < ∞,

(1.41)

a

where Abel’s equation with 0 < α < 1 , 1 Γ (α)

Zx

f (t)

1−α dt

(x − t)

a

= g (x) ,

(1.42)

has the solution [35]   Z x (1) 1 f (a) f (t) f (x) = + α dt , Γ (1 − α) xα a (x − t)

(1.43)

which is called the Tonelli theorem. In 1930, Tamarkin gave the expression as follows [36]: The Abel’s equation [36] 1 Γ (α)

Zx

f (t)

1−α dt

a

(x − t)

= g (x) ,

0 < α < 1,

(1.44)

is solvable in the set of all Lebesgue measurable functions if and only if there are [36] Zx 1 g (t) h (x) = (1.45) 1−α dt Γ (1 − α) (x − t) a

and h (a) = 0, where

Z

(1.46)

x

h (t) = C +

g (x) dx

(1.47)

a

and Z

b

|g (x)| dx < ∞.

(1.48)

a

This was called the Tamarkin theorem. In 1930, Hille and Tamarkin proposed the Abel type integral equation of the second kind [37] λ g (x) − Γ (α)

Zx

g (t)

1−α dt

0

(x − t)

= f (x) ,

0 < α < 1,

(1.49)

8

General Fractional Derivatives: Theory, Methods and Applications

with the solution (also called the Hille-Tamarkin fractional derivative, see [37]) d g (x) = dx

Zx

α

Eα [λ (x − t) ] f (t) dt,

(1.50)

0

the integral equation of the form Zx κ (x − t, λ) g (t) dt,

f (x) = g (x) − λ

(1.51)

0

where κ (x − t, λ) = eλ(t−x) ,

(1.52)

and the integral equation in the form [37] Zx f (x) = g (x) − λ

κ (x − t, λ) g (t) dt,

(1.53)

0

where

d E1−α [λΓ (1 − α) xα ] . (1.54) dx In 1931, Watanabe considered the Leibniz’ rule for analytic functions [38]: κ (x, λ) =

Daα [f (x) g (x)] =

 ∞  X α Daα−β−k [f (x)] Daβ+k [g (x)]. β+k

(1.55)

k=−∞

In 1935, Sato considered an Abel type equation with a monotone increasing function, denoted by ϕ (x), which is given as [39]: Z∞

f (u) du 1

x

= Ω (x) , x > 0.

(1.56)

(ϕ (u) − ϕ (x)) 2

In 1938, Love introduced the convergent fractional integral by [40]: α I+ f

1 lim (x) = Γ (α) N →∞

ZN

f (x − t) dt, t1−α

(1.57)

0

and Love and Young investigated the formula for fractional integration by parts in the form [41] Zb a

(Daα+ f

Zb (t)) g (t) dt = a

f (t) (Dbα− g (t)) dt

(1.58)

Introduction

9

and the formula for the fractional integration by parts Zb

(Iaα+ f

Zb

f (t) (Ibα− g (t)) dt,

(t)) g (t) dt =

(1.59)

a

a

which is called the Love-Young formula for the fractional derivative. In 1939, Hille introduced the following fractional differential operator [42] d R (α, λ) g (x) = λ dx

Zx

α

Eα [λ (t − x) ] f (t) dt.

(1.60)

0

In 1940, Erd´elyi and Kober derived the fractional integrals [43] 2x−2(α+β) Γ (α)

Zx

x2 − t 2


α−1

t2β+1 f (x) dt

(1.61)

t1−2β−2β f (x) dt,

(1.62)

0

and 2x−2β Γ (α)

Z∞

x2 − t2


α−1

x

which are called the Erd´elyi-Kober fractional integrals. In 1941, Cossar reported the following fractional integral [44] 1 d − lim Γ (1 − α) N →∞ dx

ZN

f (t) α dt. (t − x)

(1.63)

x

In 1949, Riesz considered the fractional form as [45]  α α IRn f (x) , Reα > 0, −α (−∆) 2 f (x) = F −1 |x| F f (x) = −α DR n f (x) , Re < 0, where F is the Fourier transform operator and Z 1 f (t) α IRn f (x) = n−α dt, ϑn (α) |x − t|

(1.64)

(1.65)

Rn

and the fractional derivative as 1 dα f (x) d2 α = ϑn (α) dx2 d |x| with

1 = ϑn (α) 2 cos

Z∞

1−α

|x − t|

f (t) dt

(1.66)

1


.

(1.67)

−∞

πα 2

Γ (2 − α)

10

General Fractional Derivatives: Theory, Methods and Applications

In 1951, Erd´elyi introduced the fractional integrals by [46] Zx −η−mα+m−1 x m η,α Rm f (x) = 1−α f (t) dt Γ (α) (xm − tm )

(1.68)

0

and η,α Sm f

mx−η−mα+m−1 (x) = Γ (α)

Zx

tη (xm − tm )

0

1−α f

(t) dt,

(1.69)

where α > 0, η > − 12 and m > 0. In 1952, as the general version of the Riesz fractional calculus, Feller proposed [47] α F Iϑ f

(t) =

sin((α−ϑ)π/2) 1 sin(πϑ) Γ(α)

Rt −∞

f (τ ) dτ (τ −t)1−α

+

sin((α+ϑ)π/2) 1 sin(πϑ) Γ(α)

R∞ t

f (τ ) dτ , (t−τ )1−α

(1.70) and α F Dϑ f

Zt

1 d sin ((α + ϑ) π/2) (t) = − sin (πϑ) Γ (1 − α) dt

−∞ Z∞

−

sin ((α − ϑ) π/2) 1 d sin (πϑ) Γ (1 − α) dt

f (τ ) α dτ (τ − t) f (τ ) α dτ . (t − τ )

(1.71)

t

In 1957, Hille and Phillips introduced the integral of the form [48] Z∞ λα+1 (α) Iλ f (t) = tα eλ(x−t) f (t) dt. (1.72) Γ (α + 1) 0

In 1961, Chen proposed the fractional derivatives by [49] Zx 1 d (f (t) − f (a)) α D f (x) = n−α dt, Γ (n − α) dx |x − t|

(1.73)

a

and 1 D f (x) = Γ (n − α) α

Zx

1 df (t) dt, α |x − t| dt

(1.74)

a

and the fractional integrals by 1 I f (x) = Γ (α) α

Zx (t − x)

α−1

(f (t) − f (x)) dt

(1.75)

α−1

(1.76)

−∞

and 1 I f (x) = Γ (α) α

Zx

|x − t| a

f (t) dt.

Introduction

11

In 1961, Peters considered the fractional integrals by [50] Qη,α k f

Zx (x) =

o n p
α t1−η Jα k x2 − t2 x2 − t2 2 f (t) dt

(1.77)

n p o
α tη+1 Iα k x2 − t2 x2 − t2 2 f (t) dt,

(1.78)

0

and Pkη,α f

Zx (x) = 0

 √ which are called the Peters fractional operators, where Jα k x2 − t2 and  √ Iα k x2 − t2 are the Bessel function and the modified Bessel function, respectively. In 1962, Burlak introduced the fractional integral in the kernel of the Bessel function of the first kind as [51] Zx g (x) =

n p o
α Jα k x2 − t2 x2 − t2 2 f (t) dt

(1.79)

0

and the fractional derivative in the kernel of the modified Bessel function of the first kind as Zx n p o
α+1 f (x) = k t x2 − t2 2 I−(α+1) k x2 − t2 g (t) dt, (1.80) 0

which are called the Burlak fractional integral and derivative operators, respectively. In 1962, Sneddon proposed the integral equation with trigonometric kernels [52] Z∞ f (x) = t−1 cos (xt) g (t) dt. (1.81) 0

In 1963, Srivastava introduced the integral equations with the kernel of the cosine function by [53] Z∞ f (x) =

t2k−1 cos (xt) g (t) dt,

(1.82)

0

Z∞ f (x) =

t2k−1 sin (xt) g (t) dt,

(1.83)

0

Z∞ m (x) =

cos (xt) g (t) dt 0

(1.84)

12

General Fractional Derivatives: Theory, Methods and Applications

and

Z∞ m (x) =

sin (xt) g (t) dt,

(1.85)

0

where m (x) = 0. In 1964, Srivastava proposed the fractional integral in the kernel of the confluent hypergeometric function by [54] Ia(α) f (x) =

Zx

α−1

(x − t) Γ (α)

1 F1

(β; α; x − t) f (t) dt,

(1.86)

0

where 1 F1 (β; α; x − t) is the confluent hypergeometric function. In 1965, Cooke introduced the fractional integral by [55]  Rb t2(η+1)−1  2x−2(η+α)  f (t) t2η−1 dt, α > 0,  Γ(α)  (x2 −t2 )1−α  a η,α f (x) , α = 0, Ia;σ f (x) =  
α  2x−2(η+α)−1 d Rb 2   x − t2 t2η+1 f (t) dt, −1 < α < 0, Γ(1+α) dx

(1.87)

a

which is called the Cooke fractional operator. In 1965, the solution of Abel’s equation of the more general equation of the form [56] 1 Γ (α)

Zx

m (x − t) f (t) 1−α

(x − t)

a

dt = g (x) ,

0 < α < 1,

(1.88)

was considered by Sakalyuk, where m (x − t) is a polynomial. In 1966, Lowndes proposed the integral equations with the kernel of the trigonometric function [57] Z∞ f (x) =

t−1 cos (xt) g (t) dt

(1.89)

t−1 sin (xt) g (t) dt.

(1.90)

0

and

Z∞ f (x) = 0

In 1967, Saxena introduced the fractional integral within the kernel of the Gauss hypergeometric function by [58] Ia(α) f

x−γ−1 (x) = Γ (α)

Zx 2 F1 0

(1 − α, β + m; β; t/x) f (t) tγ dt,

(1.91)

Introduction

13

where 2 F1 (1 − α, β + m; β; t/x) is the Gauss hypergeometric function. In 1967, Kalisch [59] proposed the fractional derivative of purely imaginary order α, where α = iθ, namely, Zx

1 (x) = Γ (1 + iθ)

(iθ) Ia+ f

iθ

(1.92)

iθ

(1.93)

(x − t) f (t) dt a

and (iθ) Ib− f

1 (x) = Γ (1 + iθ)

Zb

(t − x) f (t) dt. x

In 1967, Parashar reported the fractional integral in the kernel of the Meijer G function (see [60]). In 1967, Caputo introduced the fractional derivative of the form [61] 1 D f (x) = Γ (n − α) α

Zx

1 ∂ n f (t) dt, α ∂tn (x − t)

(1.94)

0

and did not report the results from Liouville and Sonine (see the report in Chapter 2). In 1967, Bosanquet proposed that ψ (t) = tα−1 /Γ (α) and ϕ (t) = −α t /Γ (1 − α) in the theory of fractional calculus with the Sonine condition [62]: Zx ψ (x − t) ϕ (t) dt = 1.

(1.95)

0

In 1967, Dzherbashyan used the generalization of Hadamarod’s idea and gave the following fractional integral [63]: Z1

1 I f (x) = Γ (α) α

f (zu)

1−α du.

0

(− ln u)

(1.96)

In 1968, Srivastava proposed the fractional operator of the form [64]

(−) Kζ,α,n f

n ζ (x) = x Γ (α)

Z∞

α−1

(un − xn )

u−ζ−nα+n−1 f (u) du, x > 0,

(1.97)

x

which is related to the generalized Whittaker transform. In 1968, Flett suggested the Hadamard fractional integral with the principal value [65].

14

General Fractional Derivatives: Theory, Methods and Applications In 1968, Dzhrbashyan and Nersesyan proposed the fractional derivative [66] 1 D f (x) = Γ (n − α) α

Z∞

∂ n f (t) dt. n−α ∂tn (x − t) 1

0

(1.98)

In 1969, Kalla introduced the integral operators involving Fox’s H-function (see [67]). In 1970, Smit and De Vries introduced the fractional derivative which is the same as the Caputo definition and did not report the results from Liouville, Sonine and Caputo. In 1970, Lowndes proposed the fractional integral in the kernel of the Bessel function and modified Bessel function of the first kind in the following form [69]: g (x) = 2

−α −2η α+1

x

Zx

k

t2(η+α)+1 x2 − t2


α+1 2

n p o I−(α+1) k x2 − t2 f (t) dt,

0

(1.99) and α −2(η+α)−1 −α

f (x) = 2 t

k

d dx

Zx

t1+2η x2 − t2


α2

o n p I−α k x2 − t2 g (t) dt.

0

(1.100) In 1970, Osler derived the fractional derivative with another function h (z) in the complex domain by [70, 71] Z 1 f (z) h(1) (z) (α) Dh(z) f (z) = (1.101) α+1 dz. Γ (α) (h (z) − h (z)) L

In 1970, Osler introduced the fractional derivative with respect to the continuously differentiable, increasing function in the form [72, 73, 74, 75] (α) Da;κ f

1 1 d (x) = (1) Γ (1 − α) κ (x) dx

Zx

f (t) (1) (t) dt, 0 < α < 1, ακ (κ (x) − κ (t))

a

(1.102) where a = κ−1 (0), and, in 1971, the fractional integral of the form (α) Ia;κ f (x) =

1 Γ (α)

Zx a

f (t)

(1) 1−α κ

(κ (x) − κ (t))

(t) dt, 0 < α < 1,

(1.103)

and (α) Da;κ f (x) =

1 f (t) α Γ (1 − α) (κ (x) − κ (t))α Γ (1 − α)

Zx a

f (x) − f (t)

(1) 1+α κ

(κ (x) − κ (t))

(t) dt,

(1.104) where 0 < α < 1.

Introduction

15

In 1971, Love considered the fractional derivative of purely imaginary order as [76, 77] Zx 1 d f (t) Da(iα) f (x) = dt, (1.105) iα Γ (1 − iα) dx (x − t) a

and the fractional integral of pure imaginary order as [75, 76] I

(iα)

1 f (t) = Γ (iα)

Z∞

f (t) 1−iα

0

(x − t)

dt.

(1.106)

In 1971, Rafal’son introduced the Bessel fractional integration in the form (see [78]) Z∞ 1 (α) α−1 x−t I− f (t) = (x − t) e f (t) dt, (1.107) Γ (α) x

and the Bessel fractional derivative in the form (α)

D+ f (t) =

1 Γ (α)

Z∞

α−1 x−t (α)

(x − t)

e

f

(t) dt,

(1.108)

x

which is called the Rafal’son type Bessel fractional derivative. In 1972, Prabhakar introduced the fractional integral in the kernel of the Humbert function by [79] Ia(α) f

Zx (x) =

α−1

(x − t) Γ (α)

 Θ1

 t α, b; c; 1 − ; λ (x − t) f (t) dt, x

(1.109)

0


where Θ1 α, b; c; 1 − xt ; λ (x − t) is the Humbert function. In 1973, based on the Sonine condition and Bosanquet’s results, Choudhary proposed the following relationship [80] Zt f (t) = a

   d Zt  ψ (t − x) − ϕ (x − u) f (u) du dx,  dt 

(1.110)

u

where a > 0 and a > t. It is called the Choudhary formula. In 1975, Sneddon introduced the fractional integral by [81] η,α Ia;σ f

σx−σ(η+α) (x) = Γ (α)

Zx

tσ(η+1)−1

1−α f

a

(xσ − tσ )

which is called the Sneddon fractional integral.

(t) dt,

(1.111)

16

General Fractional Derivatives: Theory, Methods and Applications

In 1976, Malovichko reported the fractional integral in the kernel of the Bessel function [82]. In 1977, Krasnov presented the fractional derivative of a function with respect to another function in terms of fractional differences (see [83]). In 1978, Saigo introduced the fractional integral operator involving the Gauss hypergeometric function in the form [84]

Ixα,β,η f

x−α−β (x) = Γ (α)

Zx

α−1

(x − t)

(α + β, −η; α; 1 − t/x) f (t) dt,

2 F1

0

(1.112) where 2 F1 (α + β, −η; α; 1 − t/x) is the Gauss hypergeometric function. In 1978, Srivastav introduced the fractional integral in the kernel of the Bessel function of the first kind in the following form [85] Qη,α k f

α −2(α+η)

(x) = 2 x

k

1−α 2

Zx

t2η+1 x2 − t2


1−α 2

o n p Iα−1 k x2 − t2 f (t) dt,

0

(1.113) which is called the Srivastav fractional integral. In 1979, Gearhart reported the fractional integral [86] (α) I− f

1 (t) = Γ (α)

Z∞

α−1 λ(x−t)

(x − t)

e

f (t) dt,

(1.114)

0

which is called the Rafal’son-Gearhart type Bessel fractional integral. In 1979, McBride introduced the expression with respect to the continuously differentiable, increasing function by [87] 1 Γ (α)

Zx

f (t)

(1) 1−α κ

(κ (x) − κ (t))

0

(t) dt,

(1.115)

which yields the following (α) Im f

m (x) = Γ (α)

Zx 0

f (t) (xm

1−α x

− tm )

m−1

dt,

(1.116)

and the fractional integral in the form Ia(α) f

Zx (x) =

α−1

(xm − tm ) Γ (α)

  tm mtm−1 f (t) dt, 2 F1 α, b; c; 1 − m x

0

where 2 F1 α, b; c; 1 −

t x




is the Gauss hypergeometric function.

(1.117)

Introduction

17

In 1980, Skornik introduced the tempered fractional integral by [88, 89] Zx α−1 2 t2 (x − t) (α) − x4 Ia f (x) = e f (t) e 4 dt, 0 < α < 1, (1.118) Γ (α) a

which is called the Skornik fractional integral, the tempered fractional derivative in the form Zx t2 x2 e4 e− 4 dn (α) Da f (x) = (1.119) α f (t) dt, 0 < α < 1, Γ (1 − α) dxn (x − t) a

and Da(α) Ia(α) f (x) = f (x) . (1.120) In 1985, Estrada and Kanwal derived the integral equations including singular function with Cauchy kernel, given as (see [90]) Z∞ f (u) du (1.121) α−1 = Ω (x) , x > 0. (h (x) − h (x)) x

In 1986, Vu Kim Tuan reported the fractional integral in the kernel of the Meijer G function (see [91]). In 1987, Samko, Kilbas and Marichev introduced (see [3])   Zx α−1 (h (x) − h (t)) h (t) (α) Ia f (x) = Θ1 α, b; c; 1 − ; λ (h (x) − h (t)) Γ (α) h (x) 0

f (t) h(1) (t) dt,

I

(α)

(1.122)

1 f (x) = Γ (α)

Zx

1−α

a

D(m+α) f (x) =

1 Γ (α)

Zx

f (t) − f (a) (x − t)

f (m) (t) − f (m) (a) 1−α

a

and

(x − t)

dt,

0 0)

0

and

Z∞ Γ (x, a) =

tx−1 e−t dt (a ≥ 0; Re (x) > 0 when a = 0) .

a

For more details of the history investigation of the Euler gamma functions, see [190, 191, 192, 193, 194, 195, 241, 242, 243, 244]. The properties of the incomplete gamma functions are presented as follows (see [233]): 1. For Re (x) > −1 there is the difference equation (see [190]; also see [245]) Γ (x + 1) = Γ (x) x. 2. For x ∈ C there is the Euler reflection formula (see [192]; also see [246]) π Γ (1 − x) Γ (x) = . sin (πx) 3. For x ∈ C there is (see [245, 248], also see [249]) ∞  x −x 1 = xe−γx Π 1 + e n, Γ (x + 1) n n=1 where γ is Euler’s constant. 4. For x, p ∈ C, Re (x) ≥ 0 and Re (p) ≥ 0, there is [250] Z∞

e−pt tx−1 dt =

Γ (p) . px

0

5. For x, p ∈ C, Re (x) ≥ 0 and Re (p) ≥ 0, there are (for the real parameters, see [232, 238, 250]) Z∞ e

−tp

 dt = Γ

 1 +1 , p

0

Z∞ e

−tp x−1

t

1 dt = Γ p

  x p

0

and

Z∞ e 0

−pt2 x−1

t

x p− 2  x  Γ . dt = 2 2

26

General Fractional Derivatives: Theory, Methods and Applications 6. For p, q ∈ C, Re (q) ≥ 0 and Re (p) ≥ 0, there is (for real parameters, see [232, 251]; also see [181]) Z∞ e

−ptq

dt = p

− q1

  1 . Γ 1+ q

0

7. For x, p ∈ C, Re (q) ≥ 0, Re (x) ≥ 0 and Re (p) ≥ 0, there is [212] Z∞

x

e

−ptq x−1

t

p− q dt = Γ q

  x . q

0

8. For x ∈ C there is (see [252]; also see [253]) Z 1 1 = s−x es ds, Γ (x) 2πi E

where the contour of integration E is the Hankel contour. 9. For x ∈ C and Re (x) > 1, there is (see [253, 254, 255]) 1 Γ (x) = cos (2πx)

Z∞

1 1 tx−1 cos tdt, − + n < x < + n, n ∈ N0 . 2 2

0

10. For x ∈ C and Re (x) > 1, there is (see [253, 254, 255]) Γ (x) =

1 sin (2πx)

Z∞

tx−1 sin tdt, −1 + n < x < 1 + n, n ∈ N0 .

0

• Euler beta function In 1772, Euler gave the beta function defined as [247] Z1 B (x, y) =

y−1

e−t tx−1 (1 − t)

dt, Re (x) > 0, Re (y) > 0,

(1.177)

0

which is called the Euler beta function. In 1763, Bayes considered the incomplete beta function defined by [256] Zm B (x, y; m) = 0

y−1

tx−1 (1 − t)

dt, x > 0, y > 0.

(1.178)

Introduction

27

For more details of the history investigation of the incomplete gamma functions, see [257]. The property of the Euler beta function is presented as follows [253]: B (x, y) =

Γ (x) Γ (y) . Γ (x + y)

• The Gauss hypergeometric function The major development of the theory of the hypergeometric function was carried out by Gauss and published in 1812 (see [196]). In 1812, Gauss introduced the series by [196] 1+

∞ X (a)n (b)n tn a (a + 1) b (b + 1) t2 ab t+ + ··· = , c c (c + 1) 2! (c)n n! n=0

(1.179)

which is called the Gauss hypergeometric function (GHF), where the Pochhammer symbol, which was introduced by Pochhammer in 1870, is given as [197] Γ (a + n) (a)n = , n = 0, 1, 2, · · · . (1.180) Γ (a) • The Bessel function In 1824, Bessel proposed the function (see [198]; also see [199, 200, 201]) Jp (t) =

∞ X

n

2n−p

(−1) (t/2) Γ (k + p + 1) n! n=0

,

(1.181)

which is called the Bessel function of the first kind of order. • The Kummer function In 1836, Kummer proposed the confluent hypergeometric function defined as follows [202] ∞ X (a)n tn , (1.182) M (a; c; t) = (c)n n! n=0 which is called the Kummer function or Kummer confluent hypergeometric function. • The Kohlrausch-Williams-Watts function In 1854, Kohlrausch introduced the stretched exponential function defined as [203] α eα (t) = e−t , (1.183) which is called the Kohlrausch function, Kohlrausch-Williams-Watts function or the Williams-Watts function due to the repeated work in 1970 [204].

28

General Fractional Derivatives: Theory, Methods and Applications • The Bessel-Clifford function

In 1882, Clifford introduced a Bessel type function in the form [205] Jα (t) =


k −t2 /4 , (α + 1)k Γ (k + 1)

∞ X k=0

(1.184)

which is called the Bessel-Clifford function. • The Macdonald function In 1898, Macdonald introduced the function by [206] Kν (t) =

π (I−ν (t) − Iν (t)) , 2 sin (νπ)

(1.185)

which is called the Macdonald function, where Iν (t) =

∞ X

ν+2m

(t/2) . n!Γ (n + 1 + ν) n=0

(1.186)

• The Mittag-Leffler function In 1903, the Mittag-Leffler function, introduced by Swedish mathematician Gosta Mittag-Leffler, is defined as [207] Eν (t) =

∞ X

tn . Γ (nν + 1) n=0

(1.187)

• The Wiman function In 1905, Wiman introduced the function by [208] Eν,υ (t) =

∞ X

tn , Γ (nν + υ) n=0

(1.188)

which is called the Wiman function. In 1907, Barnes reported the hypergeometric function [209] 2 F1

(a1 , a2 ; c1 ; t) =

∞ P n=0

(a1 )n (a2 )n tn (c1 )n n!

=

Γ(c) 1 2πi Γ(a)Γ(b)

R L

Γ(a+s)Γ(b+s)Γ(−s) Γ(c+s)

s

(−t) ds,

(1.189) where the numbers 2 and 1 refer to the number of the numerator and denominator parameters, respectively. • The Wright function

Introduction

29

In 1933, Maitland Wright introduced the function defined by [210] Wν,µ (t) =

∞ X

1 tn , ν > −1, µ ∈ C, Γ (nν + µ) n! n=0

(1.190)

which is called the Wright function. • The generalized Bailey hypergeometric function In 1935, Bailey introduced the generalized hypergeometric function by [211] (a1, a2 , · · · , ap ; c1 , c2 , ·· · , cq ; t) a1 , a1 , · · · , ap ; t =p Fq · · · , cq  c1 , c1 ,  ap =p Fq t bq ∞ P (a1 )n ···(ap )n tn = (c1 ) ···(cq ) n! ,

p Fq

n=0

n

(1.191)

n

which is called the generalized Bailey hypergeometric function. • The Meijer G-function In 1936, Meijer addressed the function [212]   Z a , · · · , ap 1 m,n Gp,q t 1 = b1 , · · · , bq 2πi L

m

n

j=1 p

j=1 q

Π Γ (bj − s) Π Γ (1 − aj + s)

Π

j=n+1

Γ (aj − s)

Π

j=m+1

ts ds, Γ (1 − bj + s) (1.192)

which is called the Meijer G-function function. • The MacRobert E-function In 1939, MacRobert reported the function [213]  p   Π Γ(aj )  ap  j=1 −1  Fq −t (p = q + 1/and/ |t| > 1) ,  q  bq  Π Γ(bj )  j=1  p  p      Π Γ(aj −ah ) ap j=1 Γ (ah ) tah E t = q bq  Π Γ(bj −ah )  j=1     p−q   a , 1 + a − b , · · · , 1 + a − b ; (−1) t h h 1 h q   ×p+1 Fq−1   1 + ah − a1 , · · · , 1 + ah − ap   (p = q + 1 and |t| < 1) , (1.193) which is called the MacRobert E-function. Remark that the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer’s G-function was able to include those as particular cases as well. Meijer’s G-function is actually a special case of Meijer’s G-function (see [214, 215, 216]).

30

General Fractional Derivatives: Theory, Methods and Applications • The Rabotnov function

In 1948, Rabotnov introduced the fractional exponential function by [217] 0)

(1.204)

0

and the upper Gamma function with respect to another function as Z∞ Γ (γ (x) , a) =

tγ(x)−1 e−t dt (a ≥ 0; Re (γ (x)) > 0 when a = 0) , (1.205)

a

respectively. As the direct results, we have Γ (γ (x) + 1) = Γ (γ (x)) γ (x)

(1.206)

χ (γ (x) , a) + Γ (γ (x) , a) = Γ (γ (x)) .

(1.207)

and The Pochhammer symbol with respect to another function is defined as (γ (x))κ =

Γ (γ (x) + κ) . Γ (γ (x))

(1.208)

The corresponding lower Srivastava-Chaudhry-Agarwal symbol with respect to another function is defined as (γ (x) ; a)κ =

χ (γ (x) + κ, a) , Γ (γ (x))

(1.209)

where x ∈ C, κ ∈ C and a ≥ 0, and the upper Srivastava-Chaudhry-Agarwal symbol with respect to another function as [γ (x) ; a]κ =

Γ (γ (x) + κ, a) (x ∈ C, γ ∈ C, a ≥ 0) . Γ (γ (x))

(1.210)

The Gauss hypergeometric function with respect to another function is defined as ∞ X (a (x))κ (b (x))κ tκ , 2 F1 (a (x) , b (x) ; c (x) ; t) = (c (x))κ κ! κ=0

(1.211)

Introduction

33

which can be represented in the form: 2 F1 (a (x) , b (x) ; c (x) ; t) R Γ(a(x)+s)Γ(b(x)+s)Γ(−s) s Γ(c(x)) 1 = 2πi (−t) ds, Γ(a(x))Γ(b(x)) Γ(c(x)+s)

(1.212)

L

where the Pochhammer symbol with respect to another is denoted as (a (x))κ =

Γ (a (x) + κ) , κ = 0, 1, 2, · · · . Γ (a (x))

(1.213)

The Bessel function with respect to another function of the first kind of order is defined as Jp(x) (t) =

∞ X

κ

2κ−p(x)

(t/2) (−1) Γ (k + p (x) + 1) κ! κ=0

, p (x) ≥ 0, −∞ < x < ∞.

(1.214) The Kummer confluent hypergeometric function with respect to another function is defined as M (a (x) ; c (x) ; t) =

∞ X (a (x))κ tκ , −∞ < x < ∞. (c (x))κ κ! κ=0

(1.215)

The Bessel-Clifford function with respect to another function is defined as Jα(x) (t) =

∞ X k=0


k −t2 /4 . (α (x) + 1)k Γ (k + 1)

(1.216)

The Macdonald function with respect to another function is defined as Kν(x) (t) =


π I−ν(x) (t) − Iν(x) (t) , 2 sin (ν (x) π)

where Iν(x) (t) =

∞ X

(1.217)

ν(x)+2κ

(t/2) . κ!Γ (κ + 1 + ν (x)) κ=0

(1.218)

The Mittag-Leffler function with respect to another function is defined as Eν(x) (t) =

∞ X

tκ , Γ (κν (x) + 1) κ=0

(1.219)

where t, ν (x) ∈ C, < (ν (x)) , < (υ (x)) ∈ R+ 0 , κ ∈ N and Γ (·) is the gamma function with respect to another function. The Wiman function with respect to another function is defined as Eν(x),υ(x) (t) =

∞ X

tκ , Γ (κν (x) + υ (x)) κ=0

(1.220)

34

General Fractional Derivatives: Theory, Methods and Applications

where t, ν (x) , υ (x) ∈ C, < (ν (x)) , < (υ (x)) ∈ R+ 0 and κ ∈ N. In particular, when υ (x) = υ, the Wiman function with respect to another function is defined as Eν(x),υ (t) =

∞ X

tκ , Γ (κν (x) + υ) κ=0

(1.221)

where t, ν (x) , υ ∈ C, < (ν (x)) , < (υ) ∈ R+ 0 and κ ∈ N. The Wright function with respect to another function is defined as Wν(x),µ(x) (t) =

∞ X

tκ 1 , Γ (κν (x) + µ (x)) κ! κ=0

(1.222)

where ν (x) > −1 and µ (x) ∈ C. In particular, the Wright function with respect to another function is defined as ∞ X 1 tκ Wν(x),µ (t) = , (1.223) Γ (κν (x) + µ) κ! κ=0 where ν (x) > −1 and µ ∈ C. The Rabotnov function with respect to another function is defined as 0, Re (β) > 0, t ∈ [a, b] and f (t) ∈ Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have (see [3, 75]; also see [152])     α+β α β Ia+ Ia+ f (t) = Ia+ f (t) (2.155) and



   α+β α β Ib− Ib− f (t) = Ib− f (t) .

(2.156)

(2) Let Re (β) > Re (α) > 0 , t ∈ [a, b] and f (t) ∈ Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have (see [166]; also see [3, 75, 152])     β β−α α Da+ Ia+ f (t) = Ia+ f (t) (2.157) and



   β−α α β Db− Ib− f (t) = Ib− f (t) .

(2.158)

(3) Let Re (α) > 0, t ∈ [a, b] and f (t) ∈ Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have (see [3, 75]; also see [152])
α α Da+ Ia+ f (t) = f (t) (2.159) and
α α Db− Ib− f (t) = f (t) .

(2.160)

(4) Let Re (α) > 0 , κ ∈ N and t ∈ [a, b]. Then we have (see [3, 75]; also see [152])

α+κ κ Dα Da+ f (t) = Da+ f (t) (2.161) and

κ α+κ κ Dκ Db− f (t) = (−1) Db− f (t) .

(2.162) g−α
(5) Let Re (α) > 0, g = [Re (α)] + 1, t ∈ [a, b] and let fg−α (t) = I a+ f (t). α α α If f (t) ∈ Ia+ (Lκ ), where Ia+ (Lκ ) = f : f = Ia+ l, l ∈ Lκ (a, b) , then we have (see [3, 75]; also see [152])   β α Ia+ Da+ f (t) = f (t) . (2.163)

60

General Fractional Derivatives: Theory, Methods and Applications g−α
(6) Let Re (α) > 0, g = [Re (α)] + 1, t ∈ [a, b] and let fg−α (t) = I a+ f (t). α α α If f (t) ∈ Ib− (Lκ ), where Ib− (Lκ ) = f : f = Ib− l, l ∈ Lκ (a, b) , then we have (see [3, 75]; also see [152])
α α Ib− Db− f (t) = f (t) . (2.164)
κ−α (7) Let Re (α) > 0, κ = [Re (α)] + 1, t ∈ [a, b] and let fκ−α (t) = Ia+ f (t). If fκ−α (t) ∈ AC κ [a, b], then we have (see [3, 75]; also see [152]) κ−1 (κ−j) X fκ−α
(a) α−j α α Ia+ Da+ f (t) = f (t) − (t − a) . Γ (α − 1 − j) j=0

(2.165)

Theorem 2.7 (Tamarkin Theorem) (see [166]; also
see [3, 75, 152]) Let α > κ−α 0, κ = [α]+1, t ∈ [a, b] and let fκ−α (t) = Ia+ f (t). If fκ−α (t) ∈ AC κ [a, b], then we have (see [166]; also see [3, 75, 152]) κ−1 (κ−j) X fκ−α
(a) α−j α α (t − a) . Ia+ Da+ f (t) = f (t) − Γ (α − 1 − j) j=0

(2.166)

In particular, if 1 > Re (α) > 0, then we have the following
result: 1−α Let 1 > Re (α) > 0, t ∈ [a, b] and let f1−α (t) = Ia+ f (t). Then, we have (see [166]; also see [3, 75, 152]) α−1
(t − a) α α Ia+ Da+ f (t) = f (t) − f1−α (a) , (2.167) Γ (α)
1−α where f1−α (a) = Ia+ f (t). In particular, we have the following result. Let Re (α) > κ, t ∈ [a, b] and f (t) ∈ Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have (see [3, 75]; also see [152])

α−κ κ Dα Ia+ f (t) = Ia+ f (t) (2.168)

and

α−κ κ Dκ Ib− f (t) = Ib− f (t) .

(2.169)

Property 2.5 (see [3, 75]; also see [152]) Let α > 0, β > 0, g > α > g −1 (g ∈ N), κ > β > κ−1 (κ ∈ N), α+β > g and fκ−α (t) ∈ AC κ [a, b]. Then we have 

κ  −α−j     X (t − a) (β−j) β α+β α Da+ Da+ f (t) = Da+ f (t)− Da+ f (a+) . (2.170) Γ (1 − a − j) j=1

For the more details of the semigroup property of the fractional derivative and integration operators, see [166, 3, 75]; also see [152, 154]. We now consider the rules for fractional integration by parts (see [3, 75]; also see [152]).

Fractional Derivatives of Constant Order and Applications

61

Property 2.6 (see [3, 75]; also see [152]) Let α > 0, g1 ≥ 1, g2 ≥ 1, and (1/g1 ) + (1/g2 ) ≤ 1 + α. If f1 (t) ∈ Lg1 (a, b) and f2 (t) ∈ Lg2 (a, b), then we have Zb f1 (τ )

α Ia+ f2




Zb (τ ) dτ =

a


α f2 (τ ) Ib− f1 (τ ) dτ .

(2.171)

a

Property 2.7 (see [3, 75]; also see [152]) α Let α > 0, g1 ≥1, g2 ≥ 1, and (1/g1 )+(1/g ∈ Ia+ (Lκ ), 2 ) ≤ 1+α. If f1 (t) α α α where Ia+ (Lκ ) = f : f = Ia+ l, l ∈ Lκ (a, b) , and f2 (t) ∈ Ib− (Lκ ), where  α α Ib− (Lκ ) = f : f = Ib− l, l ∈ Lκ (a, b) , then we have Zb f1 (τ )

α Da+ f2

Zb




(τ ) dτ =

a


α f2 (τ ) Db− f2 (τ ) dτ .

(2.172)

a

Let 0 < Re (α) < 1. Then we have α I+ f




Zt

1 (t) = Γ (α)

f (τ )

1−α dτ ,

−∞

α I− f




1 (t) = Γ (α)

Z∞

f (τ )

1−α dτ ,

t


α D+ f (t) =

(t − τ )

(t − τ ) Zt

1 d Γ (1 − α) dt

(2.173)

(2.174)

f (1) (τ ) α dτ (t − τ )

(2.175)

f (1) (τ ) α dτ . (t − τ )

(2.176)

−∞

and α D− f




1 d (t) = Γ (1 − α) dt

Z∞ t

Property 2.8 (see [3, 75]) If 0 < Re (α) < 1 and f (t) ∈ H1 (R), then the following relations (see [3, 75])   f (ω) β F I+ f (t) (ω) = α (−iω) and   f (ω) β F I− f (t) (ω) = α (iω) α

α

hold, where (∓iω) = |ω| e∓απi

sgn(α)/2

and F {f (t)} (ω) = f (ω).

62

General Fractional Derivatives: Theory, Methods and Applications

Property 2.9 (see [3, 75]) If 0 < Re (α) < 1, then the following relations (see [3, 75])
α α F D+ f (t) (ω) = (−iω) f (ω) and
α α F D− f (t) (ω) = (iω) f (ω) α

α

hold, where (∓iω) = |ω| e∓απi

sgn(α)/2

and F {f (t)} (ω) = f (ω) .

Let 0 < Re (α) < 1. Then we have (see [152]) α I0+ f




1 (t) = Γ (α)

Zt 0

f (τ )

and α D0+ f




1−α dτ ,

(2.177)

f (1) (τ ) α dτ . (t − τ )

(2.178)

(t − τ )

1 d (t) = Γ (1 − α) dt

Zt 0

Property 2.10 (see [150]) If 0 < Re (α) < 1, then the following relations (see [150])   (f ) (s) β (2.179) L I+ f (t) (s) = sα and
α L D+ f (t) (ω) = sα (f ) (s) , (2.180) hold, where L {f (t)} (s) = f (s). Property 2.11 (see [150]) If κ < Re (α) < 1 + κ, then we have (see [150])
κ−α L D+ f (t) (ω) = sκ−α (f ) (s) , (2.181) where L {f (t)} (s) = f (s).

α I0+ f (t) |t=0 = 0, there is (see [150])  α
L D0+ f (t) = sα f (s) ,    α,λ f (t) |t=0 6= 0, we have (see [150]) and when Dκ I0+ Find that, when Dκ

L



α D0+ f




κ−1 X  κ−α 
α (t) = s f (s) − sκ−µ−1 Dµ I0+ f (+0) .

(2.182)

(2.183)

µ=0

In particular, when κ = 1, there is (see [150])  α 1−α L D0+ f (t) = sα f (s) − I0+ f (+0) ,    α,λ and when κ = 1 and Dκ I0+ f (t) |t=0 = 0, we have (see [150])  α L D0+ f (t) = sα f (s) .

(2.184)

(2.185)

Fractional Derivatives of Constant Order and Applications

63

Property 2.12 (see [150]) For 0 < Re (α) and κ = [Re (α)] + 1, there exists (see [150]) α M {D0+ f (t)} =

κ−1 X Γ (1 + j − $)  $−κ−1 κ−α
∞ Γ (1 + α − $) f ($ − α)+ t I0+ f (t) 0 . Γ (1 − $) Γ (1 − $) j=0

(2.186)

In particular, when 0 < Re (α) < 1 , there is (see [150])  α Γ (1 + α − $) 
∞ 1−α f ($ − α) + t$−1 I0+ M D0+ f (t) = f (t) 0 . (2.187) Γ (1 − $) For more results on the Riemann-Liouville fractional integrals and fractional derivatives on a finite interval of the real line, see (see [149, 150, 152, 153]).

2.3 2.3.1

Liouville-Sonine-Caputo fractional derivatives Motivations

The left-sided fractional differential operator is defined as (see [152]) " # κ−1

Rt P f (j) (a) j 1 d κ 1 α f (t) − dτ , LC Da+ f (t) = Γ(κ−α) dt Γ(1+j) (t − a) (τ −t)α−κ+1 j=0

a

(2.188) where Re (α) ≥ 0, t > a and κ = [Re (α)] + 1, and the right-sided fractional differential operator as (see [152]) α LC Db− f




(t) =

1 Γ(κ−α)


d κ − dt

Rb t

" 1 (t−τ )α−κ+1

f (t) −

κ−1 P j=0

(−1)j f (j) (b) Γ(1+j)

# j

(b − t)

dτ ,

(2.189)

where Re (α) ≥ 0, t < b and κ = [Re (α)] + 1, which are due to the KilbasSrivastava-Trujillo ideas (see [152]). The operators are called as the left-sided and right-sided Liouville-Sonine-Caputo fractional derivatives, respectively. Liouville introduced the fractional derivative operator in the form [15] α L D0+ f

(t) =

1

Z∞

α

(−1) Γ (α)

dκ f (t + τ ) α−1 τ dτ , dtκ

(2.190)

0

where Re (α) > 0 and t ∈ R, which, by taking dκ f (t + τ ) dκ f (t + τ ) = κ , κ dt d (t + τ )

(2.191)

64

General Fractional Derivatives: Theory, Methods and Applications

can be rewritten as α L D0+ f

(t) =

Z∞

1 α

(−1) Γ (α)

dκ f (t + τ ) α−1 dτ . κ τ d (t + τ )

(2.192)

0

Setting t + τ = h, which leads to d (t + τ ) = dτ = dh,

(2.193)

we have from Eqs. (2.192) and (2.193) that α L Dt+ f

Z∞

1

(t) =

α

(−1) Γ (α)

dκ f (h) α−1 (h − t) dh, dhκ

(2.194)

t

which yields α L Dt+ f (t) =

1

Z∞

α

dκ f (h) dhκ α dh

(h − t)

(−1) Γ (α)

=

1

Z∞

α

(−1) Γ (α)

t

f (κ) (h) α dh, (h − t)

t

(2.195) From Eq. (2.194), we have [3, 75, 261] Γ (z + κ) = (z)κ Γ (z) which leads to [75] α L Dt+ f (t) =

1 Γ (κ − α)

Z∞

f (κ) (h)

α−κ+1 dh,

(h − t)

t

(2.196)

which can be used to consider the problems in mathematical physics (see [264, 265, 266, 267, 268]). Liouville introduced the special case of the fractional derivative represented in the form [15] Zh λ ¯ 0

1

df (t) p dt = λ ¯ (h − t) dt

Zh 0

f (1) (t)

  1 1 2 ¯Γ L D0+ f (t) , 1 dt = λ 2 2 (h − t)

(2.197)

√ where λ ¯ = 1/ 2g. 1 When g = 2π , we easily obtain that 1 √ π

Zh 0

1

df (t) 1
p dt = Γ 12 (h − t) dt

Zh

1

p 0

1 df (t) 2 dt = L D0+ f (t) . (2.198) (h − t) dt

Fractional Derivatives of Constant Order and Applications

65

Sonine introduced the fractional derivative presented as follows (see [24]): α S Da+ f




1 (t) = Γ (p + 1 − α)

Zt

df (t) p−α (h − t) dt, dt

(2.199)

a

where Re (p) < α < Re (p + 1), which can be rewritten as follows: α S Da+ f




1 (t) = Γ (p + 1 − α)

Zt

df (t) 1 dt, dt (h − t)α−p

a

(2.200)

which, by taking p = 0, becomes α S Da+ f




1 (t) = Γ (1 − α)

Zt

df (t) 1 dt. dt (h − t)α

(2.201)

a

Caputo introduced the fractional derivative in the form (see [61]) κ+α C D0+ f

1 (x) = Γ (1 − α)

Zt

1 dκ+1 f (t) dt, α dtκ+1 (h − t)

(2.202)

0

which is, by taking p = κ + 1 and h = κ + α, given as h C D0+ f

1 (x) = Γ (p − h)

Zt

dp f (t) dt. h−p+1 dtp (h − t) 1

0

(2.203)

Thus, we get [61] α C D0+ f

(x) =

1 Γ(κ−α)

Rt 0

dκ f (t) 1 dt (h−t)α−κ+1 dtκ

1 Γ(κ−α)

=

Rt 0

f (κ) (t) dt. (h−t)α−κ+1

(2.204) Thus, we have α S Da+ f

α (x) =C Da+ f (x) =

1 Γ(1−α)

Rt 0

f (1) (t) (h−t)α dt,

(2.205)

where κ = 1 and a = 0, 1 2

1

S Da+ f

2 (x) =C Da+ f (x) =

Rt

1 Γ(

1 2

)

0

f (1) (t) 1 (h−t) 2

1 2 dt =L D0+ f (t) ,

(2.206)

where κ = 1 and α = 12 , α S D+∞ f

(x) =

1 Γ(1−α)

R∞ f (1) (t)

α

t

(h−t)

1 dt = − Γ(1−α)

Rt ∞

f (1) (t) (h−t)α dt

α = −C D+∞ f (x) ,

(2.207)

66

General Fractional Derivatives: Theory, Methods and Applications

where a = −∞, α L D−∞ f

α LC Da+ f

(x) = lim




(t)

(2.208)

α (t) = −C D0+ f (x) .

(2.209)

a→−∞

and α LC Da+ f

lim

a→0




Thus, we have α LS Da+ f

1 (x) = Γ (1 − α)

Zt

f (1) (τ ) α dτ , (t − τ )

(2.210)

a

which is due to Liouville and Sonine, 1 (x) = Γ (κ − α)

α LSC D+∞ f

Zt

f (κ) (t)

α−κ+1 dt,

∞

(h − t)

(2.211)

which is due to Liouville (for the case a = ∞), Sonine (for the case κ = 1) and Gerasimov [269, 265], 1 (x) = Γ (κ − α)

α LSC D+∞ f

Zt

f (κ) (t)

α−κ+1 dt,

0

(h − t)

(2.212)

which is due to Liouville [15], Sonine (for the case κ = 1), Caputo (for the case a = 0, see [61, 270, 271]) and Smit and De Vries [68] α LSC Da+ f

1 (x) = Γ (κ − α)

Zt

f (κ) (t)

α−κ+1 dt,

(h − t)

a

(2.213)

which is due to the detailed information of the books by Podlubny [150], Kilbas, Srivastava, Trujillo [152] and Diethelm [154].

2.3.2

Liouville-Sonine-Caputo fractional derivatives

Definition 2.10 Let Re (α) ≥ 0 and κ = [Re (α)] + 1. The left-sided and right-sided fractional derivatives are defined as α LSC Da+ f

(x) =

α Ia+

h i f (κ) (x) =

1 Γ (κ − α)

Zt

f (κ) (t)

α−κ+1 dt,

a

(h − t)

(2.214)

and α LSC Db− f

(x) =

α Ib−

h

f

(κ)

i (x) =

κ

(−1) Γ (κ − α)

Zb

f (κ) (t)

α−κ+1 dt,

t

(t − h)

(2.215)

Fractional Derivatives of Constant Order and Applications

67

which are called the left-sided and right-sided Liouville-Sonine-Caputo fractional derivatives (due to the honors of Liouville, Sonine and Caputo), respectively. See that many authors called them as the Liouville-Caputo fractional derivatives [265, 272, 273] or Caputo fractional derivatives [150, 152, 154, 274, 275]. In particular, when 0 < Re (α) < 1, Eq.(2.214) and Eq.(2.215) can be represented in the forms [150, 152, 154]: α LS Da+ f

(x) =

α Ia+

h

f

(1)

i (x) =

1 Γ (1 − α)

Zt

f (1) (t) α dt, (h − t)

(2.216)

a

and α LS Db− f

(x) =

α Ib−

h i f (1) (x) = −

1 Γ (1 − α)

Zb

f (1) (t) α dt, (t − h)

(2.217)

t

which are called as the left-sided and right-sided Liouville-Sonine fractional derivatives in honor of Liouville and Sonine. The relationships among the Riemann-Liouville fractional derivatives and Liouville-Sonine-Caputo fractional derivatives are presented as follows [152]. Property 2.13 Let Re (α) ≥ 0, κ =
[Re (α)] + 1 and f
(t) ∈ AC κ (a, b). α α Then fractional derivatives LSC Da+ f (t) and LSC Db− f (t) exist almost everywhere on (a, b) and can be represented as follows (see [150, 151, 154]): α LSC Da+ f

(x) =

κ−1 X j=0


f (j) (a) j−α α (t − a) + Da+ f (t) Γ (1 + j − α)

(2.218)

and α LSC Db− f

(x) =

κ−1 X j=0

κ


(−1) f (j) (b) j−α α (b − t) + Db− f (t) , Γ (1 + j − α)

(2.219)

respectively. κ Property 2.14 Let α ≥ 0,
κ = α + 1 and f (t)
∈ AC (a, b). Then fracα α tional derivatives LSC Da+ f (t) and LSC Db− f (t) exist almost everywhere on (a, b) and can be represented as follows (see [150, 151, 154]): α LSC Da+ f (x) =

κ−1 X j=0


f (j) (a) j−α α (t − a) + Da+ f (t) Γ (1 + j − α)

(2.220)

and α LSC Db− f (x) =

κ−1 X j=0

respectively.

κ


(−1) f (j) (b) j−α α (b − t) + Db− f (t) , Γ (1 + j − α)

(2.221)

68

General Fractional Derivatives: Theory, Methods and Applications

Property
2.15 Let α ≥ 0
and κ = α + 1. Then fractional derivatives α α LSC D0+ f (t) and LSC D+ f (t) can be represented as follows (see [152]): α LSC D0+ f

(x) =

κ−1 X j=0


f (j) (0) α tj−α + D0+ f (t) , Γ (1 + j − α)

and α LSC D+ f

1 (x) = Γ (κ − α)

Zt

f (κ) (t)

α−κ+1 dt.

−∞

(h − t)

(2.222)

(2.223)

The relations among the Riemann-Liouville fractional derivatives and Liouville-Sonine-Caputo fractional derivatives are given as follows (see [152]): Property 2.16 Let 0 < Re (α) < 1 and f (t) ∈ AC (a, b). Then
the α Riemann-Liouville and Liouville-Sonine fractional derivatives LS Da+ f (t)
α and LS Db− f (t) exist almost everywhere on (a, b) and can be represented as follows (see [150, 151, 154]): −α
(t − a) α α D f (x) = D f (t) − LS a+ a+ Γ (1 − α)

(2.224)

−α
(b − t) α (x) = Db− f (t) − , Γ (1 − α)

(2.225)

and α LS Db− f

respectively. Property 2.17 Let 0 < α 0 and h > 0, the left-sided and right-sided fractional differences are given as follows (see [3, 20, 152]): ∆α hf

(t) =

∞ X

i



(−1)

i=0

∆α −h f

(t) =

∞ X

i

(−1)

i=0



α i



α i



f (t − hi),

(2.250)

f (x + hi) .

(2.251)

Property 2.24 (see [3, 152]) If α > 0 and β > 0, then the semigroup properties (see [3, 152])   β ∆α ∆ f (t) = ∆α+β f (t) (2.252) h h h and



 β α+β ∆α −h ∆−h f (t) = ∆−h f (t) ,

(2.253)

hold for any bounded function f (t). Property 2.25 (see [22, 152]) If α > 0 and f (t) ∈ L1 (R), then the Fourier transform of the fractional difference ∆α h f (t) is represented in the form (see [22, 152]):
ith α (F ∆α (F f ) (ω) . (2.254) h f (t)) (ω) = 1 − e

72

General Fractional Derivatives: Theory, Methods and Applications

2.4.2

Liouville-Gr¨ unwald-Letnikov fractional derivatives

Definition 2.12 Following the above results, the left- and right-sided fracα α tional difference derivatives, denoted by d D+ f (t) and d D− f (t), are defined as follows: ∆α α h f (t) (α > 0) , (2.255) d D+ f (t) = lim h→+0 hα and ∆α −h f (t) α (α > 0) , (2.256) d D− f (t) = lim h→+0 hα respectively, which, in honor of Liouville, Gr¨ unwald and Letnikov, are called the Liouville-Gr¨ unwald-Letnikov fractional derivatives [3, 22, 75, 152]. Definition 2.13 When  fι (t) =

f (t) , t ∈ [a, b] , 0, t ∈ / [a, b] ,

(2.257)

f (t − hi) (t ∈ R, h > 0, α > 0)

(2.258)

we have ∆α h,+a f

(t) =

t−a [X h ]

i



α i



α i



(−1)

i=0

and ∆α −h,−b f

(t) =

b−t [X h ]

i



(−1)

i=0

f (x + hi) (t ∈ R, h > 0, α > 0) , (2.259)

such that the left-sided and right-sided Liouville-Gr¨ unwald-Letnikov fractional derivatives on the interval [a, b] can be expressed as follows: ∆α h,+a f (t) h→+0 hα

(2.260)

∆α −h,−b f (t) , h→+0 hα

(2.261)

α d Da+ f (t) = lim

and α d Db− f

(t) = lim

respectively. Definition 2.14 (see [279]) Let f (t) ∈ Lκ (R). Then we have α d Da+ f (t) =

α Γ (1 − α)

Zt

f (t) − f (τ ) 1+α

a

(τ − t)

dτ

(2.262)

dτ .

(2.263)

and α d Db− f (t) =

α Γ (1 − α)

Zb

1+α

t

where 0 < α < 1 and 1 < κ < 1/α.

f (t) − f (τ ) (t − τ )

Fractional Derivatives of Constant Order and Applications

2.4.3

73

Kilbas-Srivastava-Trujillo fractional derivatives

Definition 2.15 (see [152]) For α > 0 there are −α

α Ma+ f (t) =

f (t) (t − a) Γ (1 − α)

and

α +d Da+ f (t)

(2.264)

−α

f (t) (b − t) α +d Db− f (t) , (2.265) Γ (1 − α) α α where Ma+ f (t) and M−b f (t) are the left-sided and right-sided KilbasSrivastava-Trujillo fractional derivatives, respectively. α M−b f (t) =

Definition 2.16 (see [3, 75]) For α > 0 there are   Zt 1 α f (t) t−α + α f (t) − f (τ ) dτ  M0+ f (t) = 1+α Γ (1 − α) (τ − t)

(2.266)

0

and   Z0 1 f (t) − f (τ ) −α α f (t) (−t) + α  M−0 f (t) = 1+α dτ Γ (1 − α) (t − τ )

(2.267)

t

α α where M0+ f (t) and M−0 f (t) are the left-sided and right-sided KilbasSrivastava-Trujillo fractional derivatives, respectively.

Property 2.26 (see [279]) For 1 > α > 0 there are −α

α d Da+ l

α α = 0, d Db− l = 0, Ma+ l=

l (t − a) , Γ (1 − α)

(2.268)

−α

α M−b l=

l (b − t) , Γ (1 − α)

(2.269)

where l is any constant. Property 2.27 (see [279]) For 1 > α > 0 there are ∆α h f (t) α h→+0 |h|

(2.270)

(t) = lim

∆α −h f (t) , α h→+0 |h|

(2.271)

∆α f (t)+∆α f (t) 1 lim h |h|α −h 2cos(απ/2) h→+0 " # Rt f (t)−f (τ ) R∞ f (t)−f (τ ) α dτ + (t−τ )1+α dτ . 2Γ(1−α)cos(απ/2) (τ −t)1+α −∞ t

(2.272)

α d D+ f

and α d D− f

(t) = lim

which leads to α T a Dt f

=

(t) =

For the information of the Liouville-Gr¨ unwald-Letnikov fractional derivatives, see [20, 279].

74

General Fractional Derivatives: Theory, Methods and Applications

2.5

Tarasov type fractional derivatives

We introduce the fractional derivative of Liouville–Gr¨ unwald–Letnikov-Riesz type, introduced by Tarasov (see [279]), which is called the Tarasov fractional derivative in honor of Tarasov.

2.5.1

Tarasov type fractional derivatives

Definition 2.17 Let 1 > α > 0. The fractional derivative of Liouville– Gr¨ unwald–Letnikov-Riesz type given as (see [279])  t  Z Z∞ f (t) − f (τ ) f (t) − f (τ ) α α   T a Dt f (t) = 1+α dτ + 1+α dτ , 2Γ (1 − α) cos (απ/2) (τ − t) (t − τ ) −∞

t

(2.273) is called the Tarasov fractional derivative, which is a family of the fractional derivative of the Gr¨ unwald–Letnikov–Riesz type. Since there are (see [279]) α d D+ f

∆α h f (t) α h→+0 |h|

(t) = lim

=

α Γ(1−α)

Rt −∞

f (t)−f (τ ) dτ (τ −t)1+α

=

α Γ(1−α)

R∞ f (τ )−f (t−τ ) 0

τ 1+α

dτ

(2.274) and α d D− f

∆α −h f (t) |h|α h→+0

(t) = lim

=

α Γ(1−α)

R∞ f (t)−f (τ ) t

(t−τ )1+α

dτ =

α Γ(1−α)

R∞ f (τ )−f (t+τ ) 0

τ 1+α

dτ ,

(2.275) we have (see [279]) α T a Dt f

α (t) = − 2Γ (1 − α) cos (απ/2)

Z∞

f (τ ) − f (t + τ ) dτ . τ 1+α

(2.276)

0

Making use of sin (απ) α = Γ (1 + α) Γ (1 − α) π

(2.277)

sin (απ) = 2 sin (απ/2) cos (απ/2) ,

(2.278)

and the Tarasov fractional derivative can be rewritten as follows: R∞ f (τ )−f (t+τ ) α α dτ T a Dt f (t) = − 2Γ(1−α) cos(απ/2) τ 1+α 0

=

Γ(1+α) sin(απ/2) π

R∞ f (t+τ )−f (τ ) 0

τ 1+α

(2.279) dτ .

Fractional Derivatives of Constant Order and Applications

2.5.2

75

Extended Tarasov type fractional derivatives

Definition 2.18 Let 1 > α > 0. The extended Tarasov type fractional derivative of Liouville–Gr¨ unwald–Letnikov-Feller type is defined as FV

∆α ∆α −h f (t) h f (t) + H− (ϑ, α) × lim , α α h→+0 h→+0 |h| |h| (2.280)

α Dt;ϑ f (t) = H+ (ϑ, α) × lim

where two components can be expressed as follows: H− (ϑ, α) =

sin ((α − ϑ) π/2) sin (πϑ)

(2.281)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.282)

sin (απ) α = Γ (1 + α) , Γ (1 − α) π

(2.283)

and

With the use of

we obtain ∆α h f (t) + H− (ϑ, α) α h→+0 |h| ∞ R (t−τ ) × Γ(1+α)πsin(πα) × f (τ )−f dτ = sin((α+ϑ)π/2) sin(πϑ) τ 1+α 0 ∞ R (t+τ ) − sin((α−ϑ)π/2) × Γ(1+α)πsin(πα) f (τ )−f dτ . sin(πϑ) τ 1+α 0 FV

α Dt;ϑ f (t) = H+ (ϑ, α) × lim

∆α −h f (t) |h|α h→+0

× lim

(2.284) Thus, the extended Tarasov type fractional derivative of Liouville–Gr¨ unwald– Letnikov-Feller type can be expressed as follows: ∆α h f (t) + H− (ϑ, α) α h→+0 |h| t R Γ(1+α) sin((α+ϑ)π/2) (τ ) = sin(πα) × f(τ(t)−f dτ sin(πϑ) × π −t)1+α ∞ Rb (t)−f (τ ) Γ(1+α) sin((α−ϑ)π/2) + sin(πα) × f(t−τ dτ . sin(πϑ) × π )1+α ∞ FV

α Dt;ϑ f (t) = H+ (ϑ, α) × lim

∆α −h f (t) |h|α h→+0

× lim

(2.285) Definition 2.19 Let 1 > α > 0. The extended Tarasov type fractional derivative of Liouville–Gr¨ unwald–Letnikov-Richard type is defined as  t  Z Z∞ α f (t) − f (τ ) f (t) − f (τ )  α  L Dτ f (t) = 1+α dτ + 1+α dτ . 2Γ (1 − α) sin (απ/2) (τ − t) (t − τ ) −∞

t

(2.286)

76

General Fractional Derivatives: Theory, Methods and Applications

The relations among the Tarasov type fractional derivatives of the Liouville–Gr¨ unwald–Letnikov-Riesz, Liouville–Gr¨ unwald–Letnikov-Feller and Liouville–Gr¨ unwald–Letnikov-Richard types in the sense of Tarasov type are presented as follows: (1) When ϑ = 0, we have (see [168]) H+ (0, α) = H− (0, α) =

1 , 2cos (απ/2)

(2.287)

which yields from Eq. (5.63) and Eq. (5.64) that the Tarasov fractional derivative of two forms: ∆α f (t)

α h Dt;0 f (t) = H+ (0, α) × lim |h| + H− (0, α) × lim α h→+0 h→+0   α α ∆−h f (t) ∆h f (t) 1 = 2cos(απ/2) × lim |h|α + lim |h|α FV

h→+0

∆α −h f (t) |h|α

h→+0

= T a Dτα f (t) . (2.288) (2) When ϑ = 1, we have (see [279]) H+ (1, α) = H− (1, α) =

1 , 2sin (απ/2)

(2.289)

such that ∆α f (t)

α h Dt;1 f (t) = H+ (1, α) × lim |h| + H− (1, α) × lim α h→+0 h→+0   α ∆−h f (t) ∆α 1 h f (t) = 2sin(απ/2) × lim |h|α + lim |h|α FV

h→+0

∆α −h f (t) |h|α

h→+0

= L Dτα f (t) . (2.290) Analogous to the above considerations, we have the new family of the Tarasov type fractional derivatives of the Liouville–Gr¨ unwald–LetnikovRiesz, Liouville–Gr¨ unwald–Letnikov-Feller and Liouville–Gr¨ unwald–LetnikovRichard types as follows. Definition 2.20 Let 1 > α > 0. The Tarasov type fractional derivatives of the Liouville–Gr¨ unwald–Letnikov-Riesz on [a, b] is defined by α T a Dt;[a,b] f

(t) =

α ∆α 1 h,+a f (t) + ∆−h,−b f (t) lim . α 2cos (απ/2) h→+0 |h|

(2.291)

Definition 2.21 Let 1 > α > 0. The Tarasov type fractional derivatives of the Liouville–Gr¨ unwald–Letnikov-Feller on [a, b] is defined by FV

∆α h,+a f (t) |h|α h→+0

α Dt;[a,b];ϑ f (t) = H+ (ϑ, α) × lim

∆α −h,−b f (t) . |h|α h→+0

+ H− (ϑ, α) × lim

(2.292)

Fractional Derivatives of Constant Order and Applications

77

Definition 2.22 Let 1 > α > 0. The Tarasov type fractional derivatives of the Liouville–Gr¨ unwald–Letnikov-Richard on [a, b] is defined by α L Dt;[a,b] f

(t) =

α ∆α 1 h,+a f (t) +∆−h,−b f (t) . lim α 2sin (απ/2) h→+0 |h|

(2.293)

We now give the expressions as follows ∆α f (t)+∆α f (t) 1 lim h,+a |h|α −h,−b 2cos(απ/2) h→+0 " # Rb f (t)−f (τ ) Rt f (t)−f (τ ) Γ(1+α) sin(απ/2) × dτ + (t−τ )1+α dτ , π (τ −t)1+α t a

α T a Dt;[a,b] f

=

(t) =

∆α

(2.294)

∆α

f (t)

α + H− (ϑ, α) × lim −h,−b Dt;[a,b];ϑ f (t) = H+ (ϑ, α) × lim h,+a α |h|α h→+0 # |h| h→+0  t b R f (t)−f (τ ) R f (t)−f (τ ) α × sin((α+ϑ)π/2) = Γ(1−α) dτ + sin((α−ϑ)π/2) dτ . sin(πϑ) sin(πϑ) (τ −t)1+α (t−τ )1+α

f (t)

FV

a

t

(2.295) and α T a Dt;[a,b] f

= =

∆α f (t)+∆α f (t) 1 lim h,+a |h|α −h,−b 2sin(απ/2) h→+0 # " Rt f (t)−f (τ ) Rb f (t)−f (τ ) α dτ + (t−τ )1+α dτ Γ(1−α) × (τ −t)1+α a t

(t) =

1 2sin(απ/2)

×

Γ(1+α) cos(απ/2) π

×

" Rt a

f (t)−f (τ ) dτ (τ −t)1+α

+

Rb t

(2.296)

# f (t)−f (τ ) dτ (t−τ )1+α

.

In particular, we take the following forms: ∆α

f (t)

α Dt;[a,b];0 f (t) = H+ (0, α) × lim h,+a + H− (0, α) × lim |h|α h→+0 h→+0 " # t b R R f (t)−f (τ ) (t)−f (τ ) 1 α = 2cos(απ/2) × Γ(1−α) × dτ + f(t−τ dτ (τ −t)1+α )1+α a t " # Rt f (t)−f (τ ) Rb f (t)−f (τ ) × dτ + dτ = Γ(1+α) sin(απ/2) 1+α π (τ −t) (t−τ )1+α

FV

a

∆α −h,−b f (t) |h|α

t

α =T a Dt;[a,b] f (t)

(2.297) and ∆α

f (t)

α + H− (ϑ, α) × lim f (t) = H+ (ϑ, α) × lim h,+a Dt;[a,b];1 |h|α h→+0 h→+0 " # Rt f (t)−f (τ ) Rb f (t)−f (τ ) 1 α = 2sin(απ/2) × Γ(1−α) × dτ + (t−τ )1+α dτ (τ −t)1+α a t " # t b R R f (t)−f (τ ) f (t)−f (τ ) Γ(1+α) cos(απ/2) × dτ + (t−τ )1+α dτ = π (τ −t)1+α FV

a

∆α −h,−b f (t) |h|α

t

α =L Dt;[a,b] f (t)

(2.298)

78

General Fractional Derivatives: Theory, Methods and Applications

α α α Note that T a Dtα f (t), T a Dt;[a,b] f (t), L Dtα f (t), L Dt;[a,b] f (t), F V Dt;ϑ f (t) α and F V Dt;[a,b];ϑ f (t) are the family of the Tarasov type fractional derivatives.

2.6

Riesz fractional calculus

The definition of the Riesz fractional derivative in the one-dimensional and three-dimensional spaces are given as follows [44, 67]. Definition 2.23 The Riesz fractional integration of functions of many variables as potential type operators is defined as [44, 168]  α α −α Rz IRn f (t) , Re (α) > 0, (2.299) (−∆) 2 f (t) = F −1 |t| F f (t) = α Rz DRn f (t) , Re (α) > 0. where Rz IRαn f (t) (Re (α) > 0) is the Riesz fractional integral in the threedimensional space and Rz DRαn f (t) is the Riesz fractional derivative in the three-dimensional space. Definition 2.24 The Riesz fractional integral in the one-dimensional space is defined as [44, 168] α Rz IR f

(t) =

(L I+α f +L I−α f )(t) 2 cos(πα/2)

=

1 2Γ(α) cos(πα/2)

R∞ −∞

f (τ ) dτ . |τ −t|1−α

(2.300)

Definition 2.25 The Riesz fractional derivative in the one-dimensional space is defined as [44, 168] α Rz DR f

(t) =

α f )(t) (L D+α f +L D−

2 cos(πα/2)

=

Γ(1+α) sin(πα/2) π

R∞ f (t+τ )−2f (t)+f (t−τ )

×

τ 1−α

0

dτ ,

(2.301) where α L D+ f

Zt

α (t) = Γ (1 − α)

1+α

−∞

and α L D− f (t) =

f (t) − f (τ )

α Γ (1 − α)

Z∞

(t − τ )

f (t) − f (τ ) 1+α

t

(τ − t)

dτ

(2.302)

dτ .

(2.303)

With the help of cos (πα/2) =

eiπα/2 + e−iπα/2 , 2

(2.304)

Fractional Derivatives of Constant Order and Applications we have α Rz IR f

(t) =

and α Rz DR f

2.7

(t) =

(L I+α f +L I−α f )(t)

=

2 cos(πα/2)

α f )(t) (L D+α f +L D−

2 cos(πα/2)

=

79

(L I+α f +L I−α f )(t)

(2.305)

eiπα/2 +e−iπα/2

α f )(t) (L D+α f +L D−

eiπα/2 +e−iπα/2

(2.306)

.

Feller fractional calculus

The conceptions of the Feller fractional derivative and Feller fractional integral are given as follows [47]. Definition 2.26 Let 1 > Re (α) > 0. We define the Feller fractional integral by (see [47]; for real number of the order, see [168]) α F IR,ϑ f

α α f (t) f (t) + H+ (ϑ, α)L I− (t) = H− (ϑ, α)L I+ t R sin((α−ϑ)π/2) f (τ ) sin((α+ϑ)π/2) 1 = × Γ(α) dτ + × sin(πϑ) sin(πϑ) (τ −t)1−α −∞

1 Γ(α)

R∞ t

f (τ ) dτ , (t−τ )1−α

(2.307) where two components can be expressed as follows: H− (ϑ, α) =

sin ((α − ϑ) π/2) , sin (πϑ)

(2.308)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.309)

and

Definition 2.27 Let 1 > Re (α) > 0. The fractional Feller derivative can be represented in the form (see [47]; for real number of the order, see [168]):
α α α F DR,ϑ f (t) = − H+ (ϑ, α)L D+ f (t) + H− (ϑ, α)L D− f (t) Rt f (τ ) R∞ f (τ ) sin((α−ϑ)π/2) 1 d 1 d = − sin((α+ϑ)π/2) × × α dτ − sin(πϑ) Γ(1−α) dt (τ −t) sin(πϑ) Γ(1−α) dt (t−τ )α dτ . −∞

t

(2.310) The formulations are called the Feller fractional calculus in honor of Feller. For 1 > Re (α) > 0, we have, by taking (see [168]) H+ (0, α) = H− (0, α) =

1 , 2cos (απ/2)

(2.311)

that α α F IR,0 f (t) = H− (0, α)L I+ f (t) α α (L I f +L I− f )(t) = 2+cos(πα/2) =Rz IRα f (t)

α + H+ (0, α)L I− f (t)

(2.312)

80

General Fractional Derivatives: Theory, Methods and Applications

and α α F DR,0 f (t) = − H+ (0, α)L D+ f (t) α α D f (t)+ D f (t) L − =Rz DRα f (t) . = − L +2cos(απ/2)

2.8


α + H− (0, α)L D− f (t)

(2.313)

Richard fractional calculus

In 2011, Richard proposed the fractional derivative and fractional integral (see [280]; also see [168]). For 1 > Re (α) > 0, we have, by taking H+ (1, α) = H− (1, α) =

1 , 2sin (απ/2)

(2.314)

that (see [280]) α F IR,1 f

α α (t) = H− (1, α)L I+ f (t) + H+ (1, α)L I− f (t) =

(L I+α f +L I−α f )(t) 2sin(απ/2)

(2.315) and α F DR,1 f (t) = −

α
(L D+α f +L D− f )(t) α α f (t) = f (t)+H− (1, α)L D− H+ (1, α)L D+ , 2sin(απ/2) (2.316)

which are deduced in 2011 by Richard (see [168]). Definition 2.28 For 1 > Re (α) > 0, Richard defines (see [168]) α Ri IR f

(t) =

(L I+α f +L I−α f )(t) 2sin(απ/2)

=

1 2Γ(α) sin(πα/2)

R∞ −∞

f (τ ) dτ |τ −t|1−α

(2.317)

and α Ri DR f

(t) =

α f )(t) (L D+α f +L D−

2sin(απ/2)

=

Γ(1+α) cos(πα/2) π

×

R∞ f (t+τ )−f (t−τ ) 0

τ 1−α

dτ ,

(2.318) which are called the Richard fractional integral and the Richard fractional derivative, respectively. In fact, according to the expression of the Riesz fractional derivative, we write Richard fractional derivative as α Ri DR f

(t) =

α f )(t) (L D+α f +L D−

2sin(απ/2)

=

Γ(1+α) cos(πα/2) π

×

R∞ f (t+τ )−2f (t)+f (t−τ ) 0

τ 1−α

dτ .

(2.319) They are called the Richard fractional calculus in honor of Richard.

Fractional Derivatives of Constant Order and Applications

2.9 2.9.1

81

Erd´ elyi-Kober type fractional calculus Erd´ elyi-Kober type operators of fractional integration and fractional derivative

The conceptions of the Erd´elyi-Kober type operators of fractional integration and fractional derivative, as the extensions of the Riemann-Liouville left-sided and right-sided fractional integrals, are given as follows [3, 105, 181]. Let M = (a, b) (−∞ ≤ a < b ≤ ∞) be a finite interval on the real axis R. Definition 2.29 Let α ∈ C, Re (α) > 0, σ > 0, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional integrals as (see [3, 152]) Zt σ(η+1)−1
τ f (τ ) σt−σ(α+η) α (2.320) Ia+;σ,η f (t) = 1−α dτ σ σ Γ (α) (t − τ ) a

and α Ib−;σ,η f




σt−ση (t) = Γ (α)

Zb

τ σ(1−α−η)−1 f (τ ) (τ σ − tσ )

t

1−α

dτ ,

(2.321)

respectively. Definition 2.30 When a = −∞ and b = −∞, we should present the following notations (see [3, 152]) α E I+;σ,η f




Zt

σt−σ(α+η) (t) = Γ (α)

−∞

and α E I−;σ,η f




σt−ση (t) = Γ (α)

Z∞

τ σ(η+1)−1 f (τ ) (tσ − τ σ )

τ σ(1−α−η)−1 f (τ ) 1−α

(τ σ − tσ )

t

dτ

(2.322)

dτ ,

(2.323)

1−α

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals. In particular, when σ = 1, we should obtain the following notations in the forms: Zt (η+1)−1
t−(α+η) τ f (τ ) α Ia+;1,η f (t) = dτ (2.324) 1−α Γ (α) (t − τ ) a

and α Ib−;1,η f




t−η (t) = Γ (α)

Zb

τ −(α+η) f (τ ) 1−α

t

(τ − t)

dτ ,

(2.325)

82

General Fractional Derivatives: Theory, Methods and Applications

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals, respectively. In particular, when σ = 1, a = −∞ and b = ∞ we represent (see [3, 152]) t−(α+η) Γ(α)



+ α Iη,α f (t) = I0+;1,η f (t) = and

− α Kη,α f (t) = I−;1,η f (t) =

t−η Γ(α)

Rt 0

τ η f (τ ) dτ (t−τ )1−α

R∞ τ −(α+η) f (τ ) t

dτ ,

(τ −t)1−α

(2.326)

(2.327)

which are known as the Kober or Erd´elyi-Kober integral operators. In particular, when η = 0, we should obtain the following notations in the forms: Zt
σt−σα τ σ−1 f (τ ) α Ia+;σ,0 f (t) = (2.328) 1−α dτ Γ (α) (tσ − τ σ ) a

and
α Ib−;σ,0 f (t) =

σ Γ (α)

Zb

τ σ(1−α)−1 f (τ ) 1−α

(τ σ − tσ )

t

dτ ,

(2.329)

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals, respectively. In particular, when η = 0, a = −∞ and b = −∞, we should present the following notations α Y I+;σ,0 f




Zt

σt−σα (t) = Γ (α)

τ σ−1 f (τ )

1−α dτ

−∞

and
α Y I−;σ,0 f (t) =

σ Γ (α)

Z∞

(tσ − τ σ )

τ σ(1−α)−1 f (τ ) 1−α

(τ σ − tσ )

t

dτ ,

(2.330)

(2.331)

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals, respectively. In particular, when η = 0 and σ = 1, we should obtain the following notations in the forms (see [280]): α Ia+;1,0 f




t−α (t) = Γ (α)

Zt

f (τ )

1−α dτ

a

(t − τ )

(2.332)

and α Ib−;1,0 f




1 (t) = Γ (α)

Zb

τ −α f (τ )

1−α dτ ,

t

(τ − t)

(2.333)

Fractional Derivatives of Constant Order and Applications

83

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals, respectively. In particular, when η = 0 , σ = 1, a = −∞ and b = −∞, we should obtain the following notations in the forms (see [280]): α L I+;1,0 f




Zt

t−α (t) = Γ (α)

f (τ ) (t − τ )

−∞

and α L I−;1,0 f




1 (t) = Γ (α)

Z∞

1−α dτ

τ −α f (τ )

1−α dτ ,

t

(τ − t)

(2.334)

(2.335)

which are called the left-sided and the right-sided Erd´elyi-Kober type fractional integrals, respectively. Let M = (a, b) (−∞ ≤ a < b ≤ ∞) be a finite interval on the real axis R. Definition 2.31 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, σ > 0, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])

κ
1 α α (2.336) Da+;σ,η f (t) = t−ση σtσ−1 D tσ(α+η) Ia+;σ,η+α f (t) and 

 α f (t) = t−σ(η+α) Db−;σ,η

1 σtσ−1

D


κ

  α tσ(κ+η−α) Ib−;σ,η+α−κ f (t) , (2.337)

respectively, where D = d/dt. Definition 2.32 Let α ∈ C, 1 > Re (α) > 0, σ > 0, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])


1 α α (2.338) f (t) = t−ση σtσ−1 Da+;σ,η f (t) D tσ(α+η) Ia+;σ,η+α and 

 α Db−;σ,η f (t) = t−σ(η+α)

1 σtσ−1 D




  α tσ(1+η−α) Ib−;σ,η+α−1 f (t) , (2.339)

respectively, where D = d/dt. Definition 2.33 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, σ = 1, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])


κ α α (2.340) f (t) Da+;1,η f (t) = t−η σ1 D tα+η Ia+;1,η+α and 

 α Db−;1,η f (t) = t−(η+α)

respectively, where D = d/dt.


κ κ+η−α 1 t σD



 α Ib−;1,η+α−κ f (t) ,

(2.341)

84

General Fractional Derivatives: Theory, Methods and Applications

Definition 2.34 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, σ = 1, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])  κ

1 α α D+;1,η f (t) = t−η D tα+η I+;1,η+α f (t) (2.342) σ and α D−;1,η f




(t) = t

−(η+α)



1 D σ

κ


α tκ+η−α I−;1,η+α−κ f (t) ,

(2.343)

respectively, where D = d/dt. Definition 2.35 Let α ∈ C, 1 > Re (α) > 0, σ = 1, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])

α α Da+;1,η f (t) = t−η (D) tα+η Ia+;1,η+α f (t) (2.344) and 

 α Db−;σ,η f (t) = t−σ(η+α)

1 σtσ−1 D




  α tσ(1+η−α) Ib−;σ,η+α−1 f (t) , (2.345)

respectively, where D = d/dt. Definition 2.36 Let α ∈ C, 1 > Re (α) > 0, σ = 1, t > 0 and η ∈ C. We define the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives as (see [3, 152])

α α D+;1,η f (t) = t−η (D) tα+η I+;1,η+α f (t) (2.346) and
α D−;σ,η f (t) = t−σ(η+α)




α D tσ(1+η−α) I−;σ,η+α−1 f (t) , σ−1 σt 1

(2.347)

respectively, where D = d/dt. In particular, we have the following cases: (1) When κ = [Re (α)] + 1, a = −∞ and b = −∞, we should present the following notations (see [3, 152])  κ

1 α −ση α D f (t) = t D tσ(α+η) I+;σ,η+α f (t) (2.348) E +;σ,η σ−1 σt and α E D−;σ,η f




(t) = t

−σ(η+α)



1 σtσ−1

κ D


α tσ(κ+η−α) I−;σ,η+α−κ f (t) . (2.349)

Fractional Derivatives of Constant Order and Applications

85

(2) When κ = [Re (α)] + 1, η = 0, there are the following notations (see [3, 152]) κ 

1 α α tσα Ia+;σ,α D f (t) (2.350) Da+;σ,0 f (t) = σ−1 σt and
α Db−;σ,0 f (t) = t−σα



1

κ

D σtσ−1


α tσ(κ−α) Ib−;σ,α−κ f (t) .

(2.351)

(3) When κ = [Re (α)] + 1 and η = 0, there are the following notations (see [3, 152])  κ

1 α α tσα I+;σ,α D+;σ,0 f (t) = D f (t) (2.352) σ−1 σt and
α D−;σ,0 f (t) = t−σα



1

D σtσ−1

κ


α tσ(κ−α) I−;σ,α−κ f (t) .

(2.353)

(4) When 0 < Re (α) < 1 and η = 0, there are the following notations (see [3, 152])  

1 α α Da+;σ,0 f (t) = D tσα Ia+;σ,α f (t) (2.354) σ−1 σt and
α Db−;σ,0 f (t) = t−σα




α D tσ(1−α) Ib−;σ,α−1 f (t) . σ−1 σt 1

(2.355)

(5) When 0 < Re (α) < 1, η = 0, a = −∞ and b = −∞, there are the following notations (see [3, 152])  

1 α α D+;σ,0 f (t) = D tσα I+;σ,α f (t) (2.356) σ−1 σt and
α D−;σ,0 f (t) = t−σα




α D tσ(1−α) I−;σ,α−1 f (t) . σ−1 σt 1

(2.357)

(6) When 0 < Re (α) < 1 , η = 0 and σ = 1, there are the following notations (see [3, 152])

α α Da+;1,0 f (t) = (D) tα Ia+;1,α f (t) (2.358) and

α α Db−;1,0 f (t) = t−α (D) t1−α Ib−;1,α−1 f (t) .

(2.359)

(7) When 0 < Re (α) < 1, η = 0, σ = 1, a = −∞ and b = −∞, there are the following notations (see [3, 152])

α α D+;1,0 f (t) = (D) tα I+;1,α f (t) (2.360)

86

General Fractional Derivatives: Theory, Methods and Applications

and

α α D−;1,0 f (t) = t−α (D) t1−α I−;1,α−1 f (t) .

(2.361)

The above operators are also called the left-sided and the right-sided Erd´elyi-Kober type fractional derivatives. Property 2.28 (see [3, 152]) Let Re (α) > 0 and −∞ ≤ a < b ≤ ∞. Then we have
α α Da+;σ,η Ia+;σ,η f (t) = f (t) , (2.362)
α α Db−;σ,η Ib−;σ,η f (t) = f (t) , (2.363)
α α D+;σ,η I+;σ,η f (t) = f (t) ,

(2.364)


α α D−;σ,η I−;σ,η f (t) = f (t) .

(2.365)

and

With the motivation of the Riesz idea, we define the following fractional integrals and fractional derivatives of the Erd´elyi-Kober-Riesz, Erd´elyi-KoberFeller and Erd´elyi-Kober-Richard types as follows.

2.9.2

Fractional integrals and fractional derivatives of the Erd´ elyi-Kober-Riesz, Erd´ elyi-Kober-Feller and Erd´ elyi-Kober-Richard types

Definition 2.37 Let α ∈ C, σ ∈ C, a < t < b and Re (α) > 0. The fractional integral in the sense of the Erd´elyi-Kober-Riesz type on R as α Rw IR f

=

(t) = −ση

α α f )+(E I−;σ,η f ))(t) ((E I+;σ,η

σt 2 cos(πα/2)Γ(α)

×

2 cos(πα/2) Rt τ σ(η+1)−1 f (τ ) t−σα dτ (tσ −τ σ )1−α −∞

+

R∞ τ σ(1−α−η)−1 f (τ ) t

(τ σ −tσ )1−α

! dτ

,

(2.366) and the fractional derivative in the sense of the Erd´elyi-Kober-Riesz type on R as α Rw DR f (t) α α f )f +(E D−;σ,η f ))(t) ((E D+;σ,η = 2 cos(πα/2) 
κ
−ση t 1 α = 2 cos(πα/2) × σtσ−1 D tσ(α+η) Ia+;σ,η+α f (t)   i
κ 1 α +t−σα σtσ−1 D tσ(κ+η−α) Ib−;σ,η+α−κ f (t) ,

+ t−(η+α)


κ 1 σD (2.367)

respectively.

Fractional Derivatives of Constant Order and Applications

87

Definition 2.38 Let α ∈ C, σ ∈ C, a < t < b and Re (α) > 0. The fractional integral in the sense of the Erd´elyi-Kober-Feller type on R defined as α Rv IR f

=

(t) =

α α f )+(E I−;σ,η f ))(t) ((E I+;σ,η

−ση

σt 2 sin(πα/2)Γ(α)

×

2 sin(πα/2) Rt τ σ(η+1)−1 f (τ ) dτ t−σα (tσ −τ σ )1−α −∞

+

R∞ τ σ(1−α−η)−1 f (τ ) t

(τ σ −tσ )1−α

! dτ

(2.368) and the fractional derivative in the sense of the Erd´elyi-Kober-Feller type on R as α

α

((E D+;σ,η f )f +(E D−;σ,η f ))(t) α Rv DR f (t) =
 1 2 sin(πα/2)
κ −ση t α = 2 sin(πα/2) f (t) × σtσ−1 D tσ(α+η) Ia+;σ,η+α   i
κ 1 α +t−σα σtσ−1 D tσ(κ+η−α) Ib−;σ,η+α−κ f (t) ,

+ t−(η+α)


κ 1 σD (2.369)

respectively. Definition 2.39 Let α ∈ C, σ ∈ C, a < t < b and Re (α) > 0. The fractional integral in the sense of the Erd´elyi-Kober-Richard type on R as α Ro IR f

=

(t) =

α α f )+(E I−;σ,η f ))(t) ((E I+;σ,η

σt−ση 2 sin(πα/2)Γ(α)

×

2 sin(πα/2) Rt τ σ(η+1)−1 f (τ ) t−σα dτ (tσ −τ σ )1−α −∞

+

R∞ τ σ(1−α−η)−1 f (τ ) t

(τ σ −tσ )1−α

! dτ

,

(2.370) and the fractional derivative in the sense of the Erd´elyi-Kober-Richard type on R as α α f )f +(E D−;σ,η f ))(t) ((E D+;σ,η α Ro DR f (t) = 2 sin(πα/2)  1
κ
t−ση α = 2 sin(πα/2) × σtσ−1 D tσ(α+η) Ia+;σ,η+α f (t)  i 
κ 1 α f (t) , D tσ(κ+η−α) Ib−;σ,η+α−κ +t−σα σtσ−1

+ t−(η+α)


κ 1 σD (2.371)

respectively.

2.10 2.10.1

Katugampola fractional calculus Katugampola fractional integrals and Katugampola fractional derivatives

We present the definitions and properties of the fractional integrals and fractional derivatives involving the Riemann-Liouville and the Hadamard fractional calculus, introduced by Katugampola in 2011 and in 2014, respectively.

88

General Fractional Derivatives: Theory, Methods and Applications

Definition 2.40 The generalized versions of the well-known RiemannLiouville and the Hadamard fractional integrals are defined by (see [281]) α K Ia+;σ f




Zt

σ 1−α (t) = Γ (α)

τ σ−1 f (τ )

1−α dτ

a

(tσ − τ σ )

(2.372)

and (see [281])
σ 1−α α K Ib−;σ f (t) = Γ (α)

Zb

τ σ−1 f (τ ) (τ σ − tσ )

t

1−α dτ ,

(2.373)

which are called the left-sided and the right-sided Katugampola fractional integrals in honor of Katugampola, where α ∈ C, σ ∈ C, a < t < b and Re (α) > 0. In fact, we rewrite Eq. (2.373) as α K Ia+;σ f




σ −α (t) = Γ (α)

Zt a

f (τ ) (tσ

1−α

(dτ σ ).

(2.374)

1−α

(dτ σ ),

(2.375)

− τ σ)

In a similar way, we also rewrite Eq. (2.374) as α Ia+;σ,0 f




t−σα (t) = Γ (α)

Zt a

f (τ ) (tσ

− τ σ)

which, compared between Eq. (2.374) and Eq. (2.375), implies that

α −α σα α t Ia+;σ,0 f (t) . K Ia+;σ f (t) = σ

(2.376)

Definition 2.41 The fractional derivatives involving the generalized version of the well-known Riemann-Liouville and the Hadamard fractional integrals defined as (see [282])


α 1−σ d κ α (2.377) K Da+;σ f (t) = t K Ia+;σ f (t) dt and



α K Db−;σ f



d (t) = t1−σ dt


κ 

α K Ib−;σ f



(t) ,

(2.378)

which are called the left-sided and the right-sided Katugampola fractional derivatives, respectively, where α ∈ C, σ ∈ C, a < t < b and κ = [Re (α)] + 1. The properties of the Katugampola fractional derivatives are given as follows.

Fractional Derivatives of Constant Order and Applications

89

Property 2.29 (see [282]) Let α ∈ R, σ ∈ C, λ ∈ C and 0 < α < 1. Then, for f (t) ∈ Lκ (a, b) (−∞ < a < b < ∞) we have (see [282])
α α (2.379) K Da+;σ,λ K Ia+;σ,λ f (t) = f (t) , α α K Db−;σ,λ K Ib−;σ,λ f




α α K D+;σ,λ K I+;σ,λ f




α α K D−;σ,λ K I−;σ,λ f




(t) = f (t) ,

(2.380)

(t) = f (t) ,

(2.381)

(t) = f (t) ,
(t) = f (t)

(2.382)

α α K D+;σ,λ f K I+;σ f α α K D−;σ,λ K I−;σ f

(t) = f (t) .

(2.384)

(2.383)

and


By the motivation of the idea, we have the following results. Definition 2.42 Let α ∈ C, σ ∈ C,λ ∈ C, a < t < b and Re (α) > 0. The left-sided fractional integral
involving the exponential function in the interval α [a, b], denoted by Kv Ia+;σ f (t), is defined as α Kv Ia+;σ,λ f




Zt

eλt σ 1−α (t) = Γ (α)

τ σ−1 e−λτ

1−α f

a

(tσ − τ σ )

(τ ) dτ ,

(2.385)

and the right-sided fractional  integralinvolving the exponential function in the α f (t), as interval [a, b], denoted by Kv Ib−;σ

α Kv Ib−;σ,κ f




eλt σ 1−α (t) = Γ (α)

Zb

τ σ−1 e−λτ f (τ ) 1−α

t

(τ σ − tσ )

dτ ,

(2.386)

where κ = [Re (α)] + 1.

2.10.2

Katugampola type fractional integrals and Katugampola type fractional derivatives involving the exponential function

Definition 2.43 Let α ∈ C, σ ∈ C, λ ∈ C, a < t < b, Re (α) > 0 and κ = [Re (α)] + 1. The left-sided Katugampola type fractionalderivative involving  α the exponential function in the interval [a, b], denoted by Kv Da+;σ,λ f (t), is defined as α Kv Da+;σ,λ f




σ α−κ+1 eλt (t) = Γ (κ − α)



1−σ

t

d dt

κ Z t

τ σ−1 e−λτ f (τ )

α−κ+1 dτ ,

a

(tσ − τ σ )

(2.387)

90

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Katugampola type fractional derivative involving the expo  α nential function in the interval [a, b] denoted by Kv Db−;σ,λ f (t), as

α Kv Db−;σ,λ f




σ α−κ+1 eλt (t) = Γ (κ − α)

 κ Zb σ−1 −λτ τ e f (τ ) 1−σ d dτ . t α−κ+1 σ σ dt (τ − t )

(2.388)

t

Property 2.30 Let α ∈ R, σ ∈ C,λ ∈ C and 0 < α < 1. Then, for f (t) ∈ Lκ (a, b) (−∞ < a < b < ∞) we have
α α (2.389) Kv Da+;σ,λ Kv Ia+;σ,λ f (t) = f (t) , α α Kv Db−;σ,λ Kv Ib−;σ,λ f




α α Kv D+;σ,λ Kv I+;σ,λ f




α α Kv D−;σ,λ Kv I−;σ,λ f




(t) = f (t) ,

(2.390)

(t) = f (t) ,

(2.391)

(t) = f (t) ,
α α Kv D+;σ,λ f Kv I+;σ f (t) = f (t)

(2.392)

α α Kv D−;σ,λ Kv I−;σ f

(2.394)

(2.393)

and

2.11 2.11.1




(t) = f (t) .

Hadamard fractional calculus Hadamard fractional integrals and fractional derivatives

The definitions and properties of the Hadamard fractional integrals and fractional derivatives are introduced as follows [3, 30, 152]. Let M = (a, b) (0 ≤ a < b ≤ ∞) be a finite interval on the real axis R+ . Definition 2.44 Let α ∈ C and Re (α) > 0. The left-sided and the right-sided Hadamard type fractional integrals are defined as (see [3, 152])
α H Ia+ f (t) =

1 Γ (α)

Zt 

σ

log

t τ

dτ , τ

(2.395)

log

τ σ dτ f (τ ) , t τ

(2.396)

f (τ )

a

and
α H Ib− f (t) =

1 Γ (α)

Zb  t

respectively.

Fractional Derivatives of Constant Order and Applications Definition 2.45 Let α ∈ C, Re (α) > 0 and δ = tD, where D = and right-sided Hadamard fractional derivatives are defined as α H Da+ f

=

1 Γ(α)




(t) =

d t dt


κ Rt

1 Γ(α)

log

a

κ

(tD)

Rt

log

a


t σ τ


t σ τ

d dt .

91 The left-

f (τ ) dτ τ (2.397)

f (τ ) dτ τ ,

and α H Db− f

=




(t) =


d κ t dt

1 Γ(α)

Rb

1 Γ(α)

log

t

(tD)

κ

Rb

log

t


τ σ t


τ σ t

f (τ ) dτ τ (2.398)

f (τ )

dτ τ ,

respectively. Property 2.31 (see [152]) Let 0 ≤ a < b ≤ ∞, Re (α) > 0 and Re (β) > 0. Then we have β !  β+α−1  Γ (1 + β) t t α (t) = log , (2.399) I log H a+ a Γ (α + β) a α H Ib−



b log t

β ! (t) =

Γ (1 + β) Γ (α + β)

 log

b t

β+α−1 ,

 β !  β−α−1 t Γ (1 + β) t α (t) = log H Da+ a Γ (β − α) a and α H Db−



b log t

β !

Γ (1 + β) (t) = Γ (β − α)



b log t

(2.400)

(2.401)

β−α−1 .

(2.402)

Property 2.32 (see [152]) Let 0 ≤ a < b ≤ ∞, Re (α) > 0 and Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have
α α (2.403) H Da+ H Ia+ (t) = f (t) and α α H Db− H Ib−

2.11.2




(t) = f (t) .

(2.404)

Hadamard type fractional integrals and fractional derivatives

d Let α ∈ C, Re (α) > 0 and `λ = e−λt D, where λ ∈ C and D = dt . Following the motivation of the idea, we give the definitions of the Hadamard type fractional derivatives as follows:

92

General Fractional Derivatives: Theory, Methods and Applications

Definition 2.46 Let α ∈ C , Re (α) > 0 and κ = [Re (α)] + 1. The left-sided and the right-sided Hadamard type fractional derivatives are defined as α Hv Da+ f




(t) =

1 Γ(α)

d e−λt dt

1 Γ(α)

d e−λt dt


κ Rt

log


t σ τ

f (τ ) dτ τ ,

(2.405)

log


τ σ t

f (τ ) dτ τ ,

(2.406)

a

and α Hv Db− f




(t) =


κ Rb t

respectively. Definition 2.47 Let α ∈ C and 1 > Re (α) > 0. The left-sided and the right-sided Hadamard type fractional derivatives are defined as α Hv Da+ f




(t) =

1 Γ(α)

d e−λt dt

1 Γ(α)

d e−λt dt


Rt

log


t σ τ

f (τ ) dτ τ ,

(2.407)

log


τ σ t

f (τ ) dτ τ ,

(2.408)

a

and α Hv Db− f




(t) =


Rb t

respectively. Definition 2.48 Let α ∈ C , Re (α) > 0 and κ = [Re (α)] + 1. We define the left-sided and the right-sided Hadamard type fractional integrals as α Hv Ia+ f




(t) =

1 Γ(α)

d t dt

1 Γ(α)

d t dt


κ Rt

log


t σ τ

eκλτ f (τ ) dτ τ ,

(2.409)

log


τ σ t

eκλτ f (τ ) dτ τ ,

(2.410)

a

and α Hv Ib− f




(t) =


κ Rb t

respectively. Definition 2.49 Let α ∈ C and 1 > Re (α) > 0. We define the left-sided Hadamard type fractional integral as α Hv Ia+ f




(t) =

1 Γ(α)

d t dt


Rt

log

a


t σ τ

eλτ f (τ ) dτ τ ,

(2.411)

and the right-sided Hadamard type fractional integral as α Hv Ib− f

respectively.




(t) =

1 Γ(α)

d t dt


κ Rb t

log


τ σ t

eλτ f (τ ) dτ τ ,

(2.412)

Fractional Derivatives of Constant Order and Applications Property 2.33 Let 0 ≤ a < b ≤ ∞, 1 > Re (α) > we have  β !  t Γ (1 + β) −λt α (t) = e log Hv Ia+ log a Γ (α + β)

93

0 and Re (β) > 0. Then t a

β+α−1 ,

β !  β+α−1  b b Γ (1 + β) −λt e log , log (t) = t Γ (α + β) t  β !  β−α−1 t Γ (1 + β) t α −λt (t) = t log Hv Da+ e a Γ (β − α) a

α Hv Ib−

(2.413)

(2.414)

(2.415)

and α −λt Hv Db− e



b log t

β !

 β−α−1 b Γ (1 + β) t log (t) = . Γ (β − α) t

(2.416)

Property 2.34 Let 0 ≤ a < b ≤ ∞, Re (α) > 0 and Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have
α α (2.417) Hv Da+ Hv Ia+ (t) = f (t) and α α Hv Db− Hv Ib−




(t) = f (t) .

(2.418)

Definition 2.50 Let α ∈ C , Re (α) > 0 and κ = [Re (α)] + 1. We define the left-sided fractional derivative involving the exponential function as α Sv Da+ f




(t) =

1 Γ(α)

d e−λt dt


κ Rt a

f (τ ) dτ , (τ −t)α−κ+1

(2.419)

and the right-sided fractional derivative involving the exponential function as α Sv Db− f




(t) =

1 Γ(α)

d e−λt dt


κ Rb t

f (τ ) dτ , (t−τ )α−κ+1

(2.420)

respectively. In particular, when α ∈ C and 1 > Re (α) > 0, we define the left-sided and the right-sided fractional derivatives involving the exponential function as α Sv Da+ f




(t) =

1 Γ(α)

d e−λt dt

1 Γ(α)

d e−λt dt


Rt a

f (τ ) dτ , (τ −t)α−κ+1

(2.421)

f (τ ) dτ , (t−τ )α−κ+1

(2.422)

and α Sv Db− f

respectively.




(t) =


Rb t

94

General Fractional Derivatives: Theory, Methods and Applications

Definition 2.51 Let α ∈ C , Re (α) > 0 and κ = [Re (α)] + 1. We define the left-sided fractional derivative involving the kernel of the power-exponential function as α Sv Ia+ f




1 (t) = Γ (α)

Zt

eλτ

α−κ+1 f

(τ − t)

a

(τ ) dτ ,

(2.423)

and the right-sided fractional derivative involving the kernel of the powerexponential function as α Sv Ib− f




1 (t) = Γ (α)

Zb

eλτ

α−κ+1 f

t

(t − τ )

(τ ) dτ .

(2.424)

In particular, when α ∈ C and 1 > Re (α) > 0, we define the left-sided fractional integral involving the kernel of the power-exponential function as α Sv Ia+ f




1 (t) = Γ (α)

Zt

eλτ α f (τ ) dτ , (τ − t)

(2.425)

a

and the right-sided fractional integral involving the kernel of the powerexponential function as α Sv Ib− f




1 (t) = Γ (α)

Zb

eλτ α f (τ ) dτ . (t − τ )

(2.426)

t

Property 2.35 Let 0 ≤ a < b ≤ ∞, 1 > Re (α) > 0 and Re (β) > 0. Then we have !! β−1 β+α−1 (t − a) α −λτ (t − a) (t) = , (2.427) Sv Ia+ e Γ (β) Γ (β + α) !! β−1 β−α−1 (t − a) α −λτ (t − a) (t) = , (2.428) Sv Ib− e Γ (β) Γ (β − α) ! β−1 β+α−1 (t − a) (b − t) α (2.429) (t) = e−λt Sv Da+ Γ (β) Γ (β + α) and α (t Sv Db−

β−1

− a) Γ (β)

!

β−α−1

(t) = e−λt

(b − t) . Γ (β − α)

(2.430)

Property 2.36 Let 0 ≤ a < b ≤ ∞, Re (α) > 0 and Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have
α α (2.431) Sv Da+ Sv Ia+ (t) = f (t) , and α α Sv Db− Sv Ib−




(t) = f (t) .

(2.432)

Fractional Derivatives of Constant Order and Applications

2.12

95

Marchaud fractional derivatives

We introduce the concepts and properties of the Marchaud fractional derivatives in honor of Marchaud [3, 75]. Definition 2.52 Let 1 > Re (α) > 0. The left-sided Marchaud fractional derivative is defined by (for real number of order, see [3, 75]): α M D+ f




(t) =

α Γ(1−α)

R∞ f (τ )−f (t−τ ) 0

τ α+1

dτ =

α Γ(1−α)

Rt −∞

f (t)−f (τ ) dτ , (t−τ )α+1

and the right-sided Marchaud fractional derivative by
R∞ f (τ )−f (t+τ ) R∞ f (t)−f (τ ) α α α dτ = Γ(1−α) dτ , M D− f (t) = Γ(1−α) τ α+1 (t−τ )α+1

(2.433)

(2.434)

t

0

where −∞ < t < ∞. In fact, for 1 > Re (α) > 0 the left-sided Marchaud fractional derivative is defined as Rt f (t)−f (τ ) R∞ f (t−τ ) R∞ f (t)−f (t−τ ) d a 1 α dτ = dτ , dτ = α+1 α Γ(1−α) Γ(1−α) dt τ Γ(1−α) τ α+1 (t−τ ) −∞

0

0

(2.435) and the right-sided Marchaud fractional derivative as R∞ f (t)−f (τ ) R∞ f (t+τ ) R∞ f (t)−f (t+τ ) α 1 d a dτ = dτ = dτ . α+1 α Γ(1−α) Γ(1−α) dt τ Γ(1−α) τ α+1 (t−τ ) t

0

0

(2.436) In particular, when 1 > α > 0, the left-sided Marchaud fractional derivative is defined by (see [3, 75]): α M D+ f




(t) =

α Γ(1−α)

R∞ f (τ )−f (t−τ ) 0

τ α+1

dτ =

α Γ(1−α)

Rt −∞

f (t)−f (τ ) dτ , (t−τ )α+1

and the right-sided Marchaud fractional derivative as
R∞ f (τ )−f (t+τ ) R∞ f (t)−f (τ ) α α α dτ = dτ . α+1 M D− f (t) = Γ(1−α) τ Γ(1−α) (t−τ )α+1

(2.437)

(2.438)

t

0

For 1 > α > 0 we have (see [3, 75]) α Γ(1−α)

Rt −∞

f (t)−f (τ ) dτ (t−τ )α+1

=

d 1 Γ(1−α) dt

R∞ f (t−τ ) 0

τα

dτ =

a Γ(1−α)

R∞ f (t)−f (t−τ ) 0

τ α+1

dτ ,

(2.439) and α Γ(1−α)

R∞ f (t)−f (τ ) t

(t−τ )

α+1

dτ =

1 d Γ(1−α) dt

R∞ f (t+τ ) 0

τα

dτ =

a Γ(1−α)

R∞ f (t)−f (t+τ ) 0

τ α+1

dτ . (2.440)

96

General Fractional Derivatives: Theory, Methods and Applications

2.13

Tempered fractional calculus

2.13.1

Motivations

In 1697, Bernoulli was of great interest to finding the following number given as (see [283])  κ 1 e = lim 1 + , (2.441) κ→∞ κ which is now known as e, called as the Euler number. In 1748, Euler represented the exponential function in the form (see [284, 285, 286])  κ t et = lim 1 + . (2.442) κ→∞ κ Euler also gives the following expression (see [284]) eiθ = cos θ + i sin θ,

(2.443)

which leads to (see [284]) cos θ =

eiθ + e−iθ 2

(2.444)

sin θ =

eiθ − e−iθ . 2i

(2.445)

and

2.13.2

Tempered fractional derivatives

The conceptions of the tempered fractional calculus containing the Bessel fractional integrals and Bessel fractional derivatives are given as follows. Definition 2.53 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville type is defined as 

RL α,λ Cp Da+ f



1 d (t) = Γ (1 − α) dt

Zt

f (τ ) −λ(t−τ ) dτ , αe (t − τ )

(2.446)

a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Db− f



−1 d (t) = Γ (1 − α) dt

Zb t

respectively.

f (τ ) −λ(τ −t) dτ , αe (τ − t)

(2.447)

Fractional Derivatives of Constant Order and Applications

97

Definition 2.54 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville type is defined as 

RL α,λ Cp Da+ f



(t) =

dκ 1 Γ(κ−α) dtκ

Rt

f (τ ) e−λ(t−τ ) dτ , (t−τ )α−κ+1

a

(2.448)

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Db− f



κ

κ

(−1) d (t) Γ(κ−α) dtκ

Rb t

f (τ ) e−λ(τ −t) dτ , (τ −t)α−κ+1

(2.449)

respectively. In particular, when α ∈ R, 1 > α > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of Riemann-Liouville type is defined as 

RL α,λ Cp Da+ f



1 d (t) = Γ (1 − α) dt

Zt

f (τ ) −λ(t−τ ) dτ , αe (t − τ )

(2.450)

a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Db− f



1 d (t) = − Γ (1 − α) dt

Zb

f (τ ) −λ(τ −t) dτ . αe (τ − t)

(2.451)

t

Similarly, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of RiemannLiouville type is defined as 

RL α,λ Cp Da+ f



(t) =

1 dκ Γ(κ−α) dtκ

Rt a

f (τ ) e−λ(t−τ ) dτ , (t−τ )α−κ+1

(2.452)

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Db− f



(t) =

(−1)κ dκ Γ(κ−α) dtκ

Rb t

f (τ ) e−λ(τ −t) dτ . (τ −t)α−κ+1

(2.453)

Definition 2.55 Let α ∈ R, κ < α < κ + 1, a = −∞, b = ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville type on the real line R is defined as 

RL α,λ Cp D+ f



(t) =

1 dκ Γ(κ−α) dtκ

Rt −∞

f (τ ) e−λ(t−τ ) dτ , (t−τ )α−κ+1

(2.454)

98

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type on the real line R as   ∞ (−1)κ dκ R f (τ ) RL α,λ e−λ(τ −t) dτ . (2.455) Cp D− f (t) = Γ(κ−α) dtκ (τ −t)α−κ+1 t

In particular, when α ∈ R,1 > α > 0, a = −∞, b = ∞ and λ ∈ C, the left -sided tempered fractional derivative in the sense of Riemann-Liouville type on the real line R is defined as   Rt f (τ ) −λ(t−τ ) 1 d RL α,λ D f (t) = Γ(1−α) dτ , (2.456) Cp + dt (t−τ )α e −∞

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type on the real line R as   R∞ f (τ ) −λ(τ −t) −1 d RL α,λ dτ . (2.457) Cp D− f (t) = Γ(1−α) dt (τ −t)α e t

2.13.3

Tempered fractional derivatives with respect to another function

Definition 2.56 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville
type with respect to another function φ (t) h(1) (t) > 0; t ∈ [a, b] is defined as   Rt e−λ(h(t)−h(τ )) h(1) (τ ) 1 d 1 RL α,λ (2.458) D f (t) = f (τ ) dτ , (1) Cp a+;h Γ(1−α) h (t) dt (h(t)−h(τ ))α a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function as   Rb e−λ(h(τ )−h(t)) h(1) (τ ) d 1 −1 RL α,λ (2.459) D f (τ ) dτ , f (t) = Γ(1−α) Cp b−;h (h(τ )−h(t))α h(1) (t) dt t

respectively. Definition 2.57 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville type with respect to another function is defined as  κ Rt −λ(h(t)−h(τ )) (1)   e h (τ ) 1 d 1 RL α,λ f (τ ) dτ , (2.460) D f (t) = (1) Cp a+ Γ(κ−α) h (t) dt (h(t)−h(τ ))α a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function as    κ Rb −λ(h(t)−h(τ )) (1) e h (τ ) (−1)κ 1 d RL α,λ df (τ ) τ , (2.461) Cp Db− f (t) = Γ(κ−α) h(1) (t) dt (h(τ )−h(t))α−κ+1 t

respectively.

Fractional Derivatives of Constant Order and Applications

99

In particular, when α ∈ R, 1 > α > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of Riemann-Liouville type with respect to another function is defined as   Rt e−λ(h(t)−h(τ )) h(1) (τ ) 1 1 d RL α,λ (2.462) D f (t) = Γ(1−α) f (τ ) dτ , Cp a+ h(1) (t) dt (h(t)−h(τ ))α−κ+1 a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function as 

RL α,λ Cp Db− f



1 1 d (t) = − Γ(1−α) h(1) (t) dt

Rb t

e−λ(h(τ )−h(t)) h(1) (τ ) f (h(τ )−h(t))α

(τ ) dτ .

(2.463)

Similarly, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function is defined as    κ Rt −λ(h(t)−h(τ )) (1) e h (τ ) 1 d 1 RL α,λ D f (t) = f (τ ) dτ , (2.464) (1) Cp a+ Γ(κ−α) h (t) dt (h(t)−h(τ ))α−κ+1 a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function as 

RL α,λ Cp Db− f



(t) =

(−1)κ Γ(κ−α)



1 h(1) (t)

d dt

κ Rb t

e−λ(h(τ )−h(t)) h(1) (τ ) f (h(τ )−h(t))α−κ+1

(τ ) dτ .

(2.465)

Definition 2.58 Let α ∈ R, κ < α < κ + 1, a = −∞, b = ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Riemann-Liouville type with respect to another function on the real line R is defined as   κ Rt −λ(h(t)−h(τ )) (1)  e h (τ ) d 1 1 RL α,λ f (τ ) dτ , Cp D+ f (t) = Γ(κ−α) h(1) (t) dt (h(t)−h(τ ))α−κ+1 −∞

(2.466) and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function on the real line R as    κ R∞ −λ(h(τ )−h(t)) (1) (−1)κ e h (τ ) 1 d RL α,λ f (τ ) dτ . Cp D− f (t) = Γ(κ−α) h(1) (t) dt (h(τ )−h(t))α−κ+1 t

(2.467)

2.13.4

Tempered fractional derivatives of a purely imaginary order

In particular, when α ∈ R, 1 > α > 0, a = −∞, b = ∞ and λ ∈ C, the leftsided tempered fractional derivative in the sense of Riemann-Liouville type with respect to another function on the real line R is defined as   Rt e−λ(h(t)−h(τ )) h(1) (τ ) 1 1 d RL α,λ D f (t) = f (τ ) dτ , (2.468) (1) Cp + Γ(1−α) h (t) dt (h(t)−h(τ ))α−κ+1 −∞

100

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type with respect to another function on the real line R as 

RL α,λ Cp D− f



−1 1 d Γ(1−α) h(1) (t) dt

(t) =

R∞ e−λ(h(τ )−h(t)) h(1) (τ ) (h(τ )−h(t))α−κ+1

t

f (τ ) dτ .

(2.469)

In particular, when α = iχ, χ > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional derivatives of a purely imaginary order in the sense of Riemann-Liouville type are defined as 

RL iχ,λ Cp Da+ f



(t) =

1 d Γ(1−iχ) dt

and 

RL iχ,λ Cp D+ f



(t) =

1 d Γ(1−iχ) dt

Rt a

Rt −∞

f (τ ) e−λ(t−τ ) dτ , (t−τ )iχ

f (τ ) e−λ(t−τ ) dτ , (t−τ )iχ

(2.470)

(2.471)

and the right-sided tempered fractional derivatives of a purely imaginary order in the sense of Riemann-Liouville type as RL iχ,λ Cp Db− f



RL iχ,λ Cp D− f





(t) =

−1 d Γ(1−iχ) dt

and 

2.13.5

(t) =

−1 d Γ(1−iχ) dt

Rb t

R∞ t

f (τ ) e−λ(t−τ ) dτ , (τ −t)iχ

(2.472)

f (τ ) e−λ(t−τ ) dτ . (τ −t)iχ

(2.473)

Tempered fractional integrals

Definition 2.59 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional integral of Riemann-Liouville type is defined as 

α,λ Cp Ia+ f



Zt (t) = a

1 (t − τ )

1−α α+1 E1,−α

(−λ (t − τ )) f (τ ) dτ ,

(2.474)

and the right-sided tempered fractional integral of Riemann-Liouville type as 

α,λ Cp Ib− f



Zb (t) = t

1

1−α α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ ,

(2.475)

1−α where E1,−α (t) is the Prabhakar function (for more details of the special functions, see Chapter 1 and Appendix D).

Fractional Derivatives of Constant Order and Applications

101

Generally, when α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C, the left-sided tempered fractional integral of Riemann-Liouville type is defined as 

RL α,λ Cp Ia+ f



Zt (t) = a

1

κ−α α+1 E1,−α

(t − τ )

(−λ (t − τ )) f (τ ) dτ ,

(2.476)

and the right-sided tempered fractional integral of Riemann-Liouville type as 

RL α,λ Cp Ib− f



Zb (t) = t

1

κ−α α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ ,

(2.477)

α−1 where E1,υ (t) is the Prabhakar function.

2.13.6

Tempered fractional integrals of a purely imaginary order

In particular, when α = iχ, χ > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided and right-sided tempered fractional integrals of a purely imaginary order in the sense of Riemann-Liouville type are defined as 



RL iθ,λ Cp Ia+ f

Zt (t) = a





RL iθ,λ Cp Ib− f

Zb (t) = t



iθ,λ Cp I+ f



Zt (t) = −∞

1 iθ+1

1−iθ E1,−iθ (−λ (t − τ )) f (τ ) dτ ,

(2.478)

iθ+1

1−iθ E1,−iθ (−λ (τ − t)) f (τ ) dτ ,

(2.479)

iθ+1

1−iθ E1,−iθ (−λ (t − τ )) f (τ ) dτ ,

(2.480)

(t − τ ) 1 (τ − t) 1

(t − τ )

and 

iθ,λ Cp I− f



Z∞ (t) =

1 iθ+1

t

(τ − t)

1−iθ E1,−iθ (−λ (τ − t)) f (τ ) dτ .

(2.481)

In a similar manner, when α ∈ R, κ < α < κ + 1, a = −∞, b = ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of RiemannLiouville type is defined as 

RL α,λ Cp Ia+ f



Zt (t) = a

1

κ−α α+1 E1,−α

(t − τ )

(−λ (t − τ )) f (τ ) dτ ,

(2.482)

102

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Ib− f



Zb (t) = t

1

κ−α α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ .

(2.483)

In particular, when α ∈ R, 0 < α < 1, a = −∞, b = ∞ and λ ∈ C, the left-sided tempered fractional derivative in the sense of Riemann-Liouville type is defined as 

RL α,λ Cp Ia+ f



Zt (t) = a

1

1−α α+1 E1,−α

(t − τ )

(−λ (t − τ )) f (τ ) dτ ,

(2.484)

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp Ib− f



Zb (t) = t

1

1−α α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ .

(2.485)

Analogous to the expressions of Eq.(2.484) and Eq.(2.485), when α ∈ C,κ+ 1 > Re (α) > κ and λ ∈ C, the left-sided tempered fractional derivative in the sense of Riemann-Liouville type is defined as 

RL α,λ Cp I+ f



Zt (t) = −∞

1 (t − τ )

κ−α α+1 E1,−α

(−λ (t − τ )) f (τ ) dτ ,

(2.486)

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type as 

RL α,λ Cp I− f



Zb (t) = t

1

κ−α α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ ,

(2.487)

respectively.

2.13.7

Tempered fractional derivatives in the sense of Liouville-Sonine and Liouville-Sonine-Caputo types

We now consider the left-sided and right-sided tempered fractional derivatives in the sense of Liouville-Sonine and Liouville-Sonine-Caputo types. Definition 2.60 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C.

Fractional Derivatives of Constant Order and Applications

103

The left-sided tempered fractional derivative in the sense of Liouville-Sonine type is defined as 

LS α,λ Cp Da+ f



1 (t) = Γ (1 − α)

Zt

e−λ(t−τ ) (1) (τ ) dτ , αf (t − τ )

(2.488)

a

and the right-sided tempered fractional derivative on the interval [a, b] in the sense of Liouville-Sonine type as 

LS α,λ Cp Db− f



−1 (t) = Γ (1 − α)

Zb

e−λ(τ −t) (1) (τ ) dτ , αf (τ − t)

(2.489)

t

respectively. Definition 2.61 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Liouville-Sonine-Caputo type is defined as 

LSC α,λ Cp Da+ f



(t) =

1 Γ(κ−α)

Rt a

e−λ(t−τ ) f (κ) (t−τ )α−κ+1

(τ ) dτ ,

(2.490)

and the right-sided tempered fractional derivative in the sense of LiouvilleSonine-Caputo type as 

LSC α,λ Cp Db− f



(t) =

(−1)κ Γ(κ−α)

Rb t

e−λ(τ −t) f (κ) (τ −t)α−κ+1

(τ ) dτ ,

(2.491)

respectively. Definition 2.62 Let α ∈ R, κ < α < κ + 1, a = −∞, b = ∞ and λ ∈ C. The left-sided tempered fractional derivative in the sense of Liouville-SonineCaputo type on the real line R is defined as 

LSC α,λ Cp D+ f



(t) =

1 Γ(κ−α)

Rt −∞

e−λ(t−τ ) f (1) (t−τ )α−κ+1

(τ ) dτ ,

(2.492)

and the right-sided tempered fractional derivative in the sense of LiouvilleSonine-Caputo type on the real line R as 

LSC α,λ Cp D− f



(t) =

(−1)κ Γ(κ−α)

R∞ t

e−λ(τ −t) f (κ) (τ −t)α−κ+1

(τ ) dτ ,

(2.493)

respectively. The properties of the tempered fractional derivatives and tempered fractional integrals are as follows:

104

General Fractional Derivatives: Theory, Methods and Applications

Property 2.37 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ AC [a, b]. Then we have 

LS α,λ Cp Da+ f



(t) =





(t) =



RL α,λ Cp Da+ f



(t) +

e−λ(t−a) 1 f (a) , Γ (1 − α) (t − a)α

(2.494)



(t) −

e−λ(b−t) 1 f (b) . Γ (1 − α) (b − t)α

(2.495)

and 

LS α,λ Cp Db− f

RL α,λ Cp Db− f

Property 2.38 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. The we have 

LSC α,λ Cp Da+ f



(t) =



RL α,λ Cp Da+ f



(t) +

κ−1 X

j−α

e−λ(t−a) (t − a) Γ (1 + j − α)

j=0

f (j) (a), (2.496)

and 

LSC α,λ Cp Db− f



(t) =



RL α,λ Cp Db− f



(t) +

κ−1 X j=0

j

j−α

(−1) e−λ(t−a) (b − t) Γ (1 + j − α)

f (j) (b). (2.497)

Property 2.39 If Re (α) > 0 and Re (β) > 0, then we have     RL α+β,λ RL α,λ RL β,λ D f (t) I f (t) = I Cp a+ Cp a+ Cp a+ 

and

RL α,λ RL β,λ Cp Ib− Cp Ib− f



(t) =





RL α,λ RL β,λ Cp I+ Cp I+ f





RL α,λ RL β,λ Cp I− Cp I− f



(2.498)

RL α+β,λ f Cp Db−



(t) ,

(2.499)

(t) =



RL α+β,λ f Cp D+



(t) ,

(2.500)

(t) =



RL α+β,λ f Cp D−



(t) ,

(2.501)

where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞). Property 2.40 If Re (α) > 0, then we have   RL α,λ RL α,λ Cp Da+ Cp Ia+ f (t) = f (t) 

and

(2.502)

RL α,λ RL α,λ Cp Db− Cp Ib− f



(t) = f (t) ,

(2.503)



RL α,λ RL α,λ Cp D+ Cp I+ f



(t) = f (t) ,

(2.504)



RL α,λ RL α,λ Cp D− Cp I− f



(t) = f (t) ,

(2.505)

where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞).

Fractional Derivatives of Constant Order and Applications Property 2.41 If Re (α) > 0, then we have n  o
α−κ α α,λ L RL s f (s) , D f (t) = s−1 λ + s Cp 0+

105

(2.506)

n

o
α−κ α s f (s) , (2.507) (t) = s−1 λ + s   κ−1 n o X
α−κ α,λ −1 L LSC sα−κ sκ f (s) − λ+s f (κ−j) (0) , Cp D0+ f (t) = s L

RL α,λ Cp I0+ f

j=0

(2.508) and L

n

LS α,λ Cp D0+ f

o
α−1 α−1 (t) = s−1 λ + 1 s (sf (s) − f (0)) .

Find that, when Dκ L and when Dκ

L

n

RL α,λ Cp D0+ f







RL α,λ Cp I0+ f

n

RL α,λ Cp D0+ f

RL α,λ Cp I0+ f







(2.509)

 (t) |t=0 = 0, we have

o
α−κ α (t) = 1 + s−1 λ s f (s) ,

(2.510)

 (t) |t=0 6= 0,

κ−1 i o  α−κ X κ−µ−1  µ hRL α,λ sα f (s)− s D Cp I0+ f (+0) . (t) = 1 + s−1 λ µ=0

(2.511)

In particular, when κ = 1, there is n o
α−1 α α,λ α,λ −1 L RL λ s f (s) − RL Cp D0+ f (t) = 1 + s Cp I0+ f (+0) , and when κ = 1 and D L

2.13.8

n



RL α,λ Cp I0+ f

RL α,λ Cp D0+ f





(2.512)

 (t) |t=0 = 0,

o
α−1 α (t) = 1 + s−1 λ s f (s) .

(2.513)

Tempered fractional derivatives involving power-sine and power-cosine functions

Definition 2.63 Let α ∈ R, −∞ < a < b < ∞ and γ > 0. The left-sided tempered fractional derivative in the Liouville-Sonine-Caputo type involving the kernel of the cosine function is defined as
LSC α,γ Cb Da+ f (t)

1 = Γ (κ − α)

Zt

cos (−γ (t − τ )) α−κ+1

a

(t − τ )

f (κ) (τ ) dτ ,

(2.514)

106

General Fractional Derivatives: Theory, Methods and Applications

the right-sided tempered fractional derivative in the sense of Liouville-SonineCaputo type involving the kernel of the cosine function as κ


LSC α,γ Cb Db− f (t)

(−1) = Γ (κ − α)

Zb

cos (−γ (τ − t)) α−κ+1

(τ − t)

t

f (κ) (τ ) dτ ,

(2.515)

the left-sided tempered fractional derivative in the sense of Liouville-SonineCaputo type involving the kernel of the sine-power function as
LSC α,γ Cc Da+ f (t)

1 = Γ (κ − α)

Zt

sin (−γ (t − τ )) α−κ+1

(t − τ )

a

f (κ) (τ ) dτ ,

(2.516)

the right-sided tempered fractional derivative in the sense of Liouville-SonineCaputo type involving the kernel of the sine-power function as κ


LSC α,γ Cc Db− f (t)

(−1) = Γ (κ − α)

Zb

sin (−γ (τ − t)) α−κ+1

(τ − t)

t

f (κ) (τ ) dτ ,

(2.517)

the left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the cosine function is defined as
RL α,γ Cb Da+ f (t)

=

1 dκ Γ (κ − α) dtκ

Zt

cos (−γ (t − τ )) α−κ+1

a

(t − τ )

f (τ ) dτ ,

(2.518)

the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the cosine function as κ


RL α,γ Cb Db− f (t)

(−1) dκ = Γ (κ − α) dtκ

Zb

cos (−γ (τ − t)) α−κ+1

t

(τ − t)

f (τ ) dτ ,

(2.519)

the left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the sine-power function as
RL α,γ Cc Da+ f (t)

1 dκ = Γ (κ − α) dtκ

Zt

sin (−γ (t − τ )) α−κ+1

a

(t − τ )

f (τ ) dτ ,

(2.520)

the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the sine-power function as κ


RL α,γ Cc Db− f (t)

(−1) dκ = Γ (κ − α) dtκ

Zb

sin (−γ (τ − t)) α−κ+1

t

(τ − t)

f (τ ) dτ ,

(2.521)

Fractional Derivatives of Constant Order and Applications

107

the left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the cosine function is defined as
RL α,γ Cb Da+ f (t)

1 d = Γ (1 − α) dt

Zt

cos (−γ (t − τ )) f (τ ) dτ , α (t − τ )

(2.522)

a

the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the cosine function as
RL α,γ Cb Db− f (t)

−1 d = Γ (1 − α) dt

Zb

cos (−γ (τ − t)) f (τ ) dτ , α (τ − t)

(2.523)

t

the left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the sine-power function as
RL α,γ Cc Da+ f (t)

1 d = Γ (1 − α) dt

Zt

sin (−γ (t − τ )) f (τ ) dτ , α (t − τ )

(2.524)

a

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the sine-power function as
RL α,γ Cc Db− f (t)

−1 d = Γ (1 − α) dt

Zb

sin (−γ (τ − t)) f (τ ) dτ . α (τ − t)

(2.525)

t

2.13.9

Tempered fractional calculus involving Kohlrausch-Williams-Watts function

power-

The conceptions of the fractional derivatives and fractional integrals with the weak singular power-Kohlrausch-Williams-Watts function kernel can be given as follows. 2.13.9.1

Tempered fractional derivative in the Liouville-SonineCaputo type involving the kernel of the powerKohlrausch-Williams-Watts function

Definition 2.64 The left-sided tempered fractional derivative in the LiouvilleSonine-Caputo type involving the kernel of the power-Kohlrausch-WilliamsWatts function is defined as 

LSC α,λ,γ V v Da+ f



1 (t) = Γ (κ − α)

Zt

γ

e−λ(t−τ )

α−κ+1 f

a

(t − τ )

(κ)

(τ ) dτ ,

(2.526)

108

General Fractional Derivatives: Theory, Methods and Applications

the right-sided tempered fractional derivative involving the kernel of the powerKohlrausch-Williams-Watts function in the sense of Liouville-Sonine-Caputo type as 

LSC α,λ,γ V v Db− f



κ

(−1) (t) = Γ (κ − α)

Zb

γ

e−λ(τ −t)

α−κ+1 f

(κ)

(τ − t)

t

(τ ) dτ ,

(2.527)

the left-sided tempered fractional derivative in the Liouville-Sonine type involving the kernel of the power-Kohlrausch-Williams-Watts function is defined as 

LSC α,λ,γ V v Da+ f



1 (t) = Γ (1 − α)

Zt

γ

e−λ(t−τ ) (1) (τ ) dτ , α f (t − τ )

(2.528)

a

the right-sided tempered fractional derivative in the sense of Liouville-Sonine type involving the kernel of the power-Kohlrausch-Williams-Watts function as 

LSC α,λ,γ V v Db− f



−1 (t) = Γ (1 − α)

Zb

γ

e−λ(τ −t) (1) (τ ) dτ , α f (τ − t)

(2.529)

t

the left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the power-Kohlrausch-Williams-Watts function as 

RL α,λ,γ V v Da+ f



1 dκ (t) = Γ (κ − α) dtκ

Zt a

γ

e−λ(t−τ ) (t − τ )

α−κ+1 f

(τ ) dτ ,

(2.530)

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the power-Kohlrausch-Williams-Watts function as 

RL α,λ,γ V v Db− f



κ

(−1) dκ (t) = Γ (κ − α) dtκ

Zb

γ

e−λ(τ −t)

α−κ+1 f

t

(τ − t)

(τ ) dτ ,

(2.531)

where α ∈ R,−∞ < a < b < ∞, γ ≥ 0 and λ ≥ 0. 2.13.9.2

Tempered fractional integral involving the kernel of the power-Kohlrausch-Williams-Watts function

Definition 2.65 The left-sided tempered fractional derivative in the sense of Riemann-Liouville type involving the kernel of the power-KohlrauschWilliams-Watts function is defined as 

LS α,λ,γ V v Ia+ f



1 (t) = Γ (α)

Zt a

γ

e−λ(t−τ ) α f (τ ) dτ , (t − τ )

(2.532)

Fractional Derivatives of Constant Order and Applications

109

and the right-sided tempered fractional derivative in the sense of RiemannLiouville type involving the kernel of the power-Kohlrausch-Williams-Watts function as Zb −λ(τ −t)γ   e 1 LS α,λ,γ (2.533) f (t) = α f (τ ) dτ , V v Ib− Γ (α) (τ − t) t

where α ∈ R, −∞ < a < b < ∞, γ ≥ 0 and λ ≥ 0. Definition 2.66 The left-sided tempered fractional derivative in the sense of Riesz type, introduced by Chen and Deng in 2017, is defined as (see [287]) 

LSC α,λ,γ V cd Da+ f



1 (t) = Γ (α − 1)

Zt 0

γ

e−λ(t−τ ) (t − τ )

β 2−α Rzl D[a,b] f

(t) dτ (t > a) (2.534)

where the fractional derivative, being coupled with Riesz fractional derivative in the sense of Riemann-Liouville type, is defined as (see [5])   β β L Da+ f +L Db− f (t) β (2.535) Rzl D[a,b] f (t) = − 2 cos (πβ/2) with the conditions β L Da+ f

(t) =



β Da+ f



 (t) =

d dt

2 

 β Ia+ f (t) (1 < β < 2)

and β L Db− f

2      d β β Ib− (t) = Db− f (t) (1 < β < 2) . f (t) = − dt

This is the left-sided tempered fractional derivative, being coupled with Riesz fractional derivative in the sense of Liouville-Sonine-Caputo type. In this case, we also write a generalization of Eq. (2.534) as 

LSC α,λ,γ V cd Da+ f



(t) =

1 Γ (κ − α)

Zt

e−λ(t−τ )

β D[a,b] f (t) dτ ,

α−κ+1

a

where

(t − τ ) 

β Rzh D[a,b] f

γ

(t) = −

β L Da+ f

(2.536)

Rzh

 β +L Db− f (t)

2 cos (πβ/2)

(2.537)

with the conditions  κ     d β β β D f (t) = D f (t) = Ia+ f (t) (κ < β < κ + 1) L a+ a+ dt

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General Fractional Derivatives: Theory, Methods and Applications

and β L Db− f

(t) =



β Db− f



 (t) =

d − dt

κ 

 β Ib− f (t) (κ < β < κ + 1) .

This is the generalized version of Eq. (2.534) in the sense of Riemann-Liouville type. As the generalized version, we also define the left-sided tempered fractional derivative, being coupled with Riesz fractional derivative in the sense of Liouville-Sonine-Caputo type, as 

LSC α,λ,γ V cd Da+ f



1 (t) = Γ (κ − α)

Zt

γ

e−λ(t−τ ) (t − τ )

a

where

 β Rzd D[a,b] f

(t) = −

β D[a,b] f (t) dτ ,

α−κ+1

β Lr Da+ f

(2.538)

Rzd

 β +Lr Db− f (t)

2 cos (πβ/2)

(2.539)

with the conditions β Lr Da+ f

h i β β (t) =LSC Da+ f (x) = Ia+ f (κ) (x) (κ < β < κ + 1)

β Lr Db− f

h i β β (t) =LSC Db− f (x) = Ib− f (κ) (x) (κ < β < κ + 1) .

and

In fact, we also define the left-sided and right-sided tempered fractional derivative, being coupled with Liouville-Sonine-Caputo fractional derivative, as 

LSC α,λ,γ Hh Da+ f



1 (t) = Γ (κ − α)

Zt a

γ

e−λ(t−τ ) (t − τ )

h i β (κ) D f (x) dτ , a+ α−κ+1

(2.540)

the right-sided tempered fractional derivative, being coupled with LiouvilleSonine-Caputo fractional derivative, as 

LSC α,λ,γ Hh Db− f



κ

(−1) (t) = Γ (κ − α)

Zb t

γ

e−λ(t−τ )

h i β (κ) D f (x) dτ , α−κ+1 b− (t − τ )

(2.541)

the left-sided tempered fractional derivative, being coupled with the left-sided Riemann-Liouville fractional derivative, as 

RL α,λ Hv Da+ f



(t) =

1 Γ (κ − α)

Zt

e−λ(t−τ ) α−κ+1

a

(t − τ )



d dt

κ 

  β Ia+ f (t) dτ , (2.542)

Fractional Derivatives of Constant Order and Applications

111

and the right-sided tempered fractional derivative, being coupled with the right-sided Riemann-Liouville fractional derivative, as 

RL α,λ Hv Db− f



κ

(−1) (t) = Γ (κ − α)

Zb

e−λ(τ −t) α−κ+1

t

(τ − t)

 κ    d β − Ib− f (t) dτ , dt

(2.543) where −∞ < a < b < ∞, κ + 1 > α > κ and κ < β < κ + 1. Being motivated by the idea, the left-sided tempered fractional derivative, being coupled with the right-sided Liouville-Sonine-Caputo fractional derivative, is defined as 

LSC α,λ,γ Y L Da+ f



1 (t) = Γ (κ − α)

Zt

γ

e−λ(t−τ ) (t − τ )

a

β α−κ+1 Da+

h

i f (κ) (x) dτ ,

(2.544)

and the right-sided tempered fractional derivative, being coupled with the right-sided Liouville-Sonine-Caputo fractional derivative, as 

LSC α,λ,γ Y L Db− f



κ

(−1) (t) = Γ (κ − α)

Zb

γ

e−λ(τ −t)

β α−κ+1 Db−

h

(τ − t)

t

i f (κ) (x) dτ ,

(2.545)

where −∞ < a < b < ∞, κ + 1 > α > κ and κ < β < κ + 1. In particular, when −∞ < a < b < ∞, 1 > α > 0 and 0 < β < 1, the left-sided tempered fractional derivative, being coupled with the left-sided Liouville-Sonine fractional derivative, is defined as 

LSC α,λ,γ Y L Da+ f



1 (t) = Γ (1 − α)

Zt

γ i e−λ(t−τ ) β h (1) (x) dτ , α Da+ f (t − τ )

(2.546)

a

and the left-sided tempered fractional derivative, being coupled with the rightsided Liouville-Sonine fractional derivative, as 

LSC α,λ,γ Y L Db− f



−1 (t) = Γ (1 − α)

Zb

γ

i e−λ(τ −t) β h (1) D f (x) dτ . α b− (τ − t)

(2.547)

t

Similarly, the left-sided tempered fractional derivative, being coupled with the left-sided Riemann-Liouville fractional derivative, is defined as 

RL α,λ,γ Y V Da+ f



1 (t) = Γ (κ − α)

Zt

γ

e−λ(t−τ )

α−κ+1

a

(t − τ )



d dt

κ 

β Ia+ f



 (t) dτ , (2.548)

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General Fractional Derivatives: Theory, Methods and Applications

and the left-sided tempered fractional derivative, being coupled with the rightsided Riemann-Liouville fractional derivative, as 

RL α,λ,γ Y V Db− f



κ

(−1) (t) = Γ (κ − α)

Zb

e−λ(τ −t)

γ

α−κ+1

(τ − t)

t

 κ    d β − Ib− f (t) dτ , dt

(2.549) where −∞ < a < b < ∞ and κ < β < κ + 1. In particular, when −∞ < a < b < ∞, 1 > α > 0 and 0 < β < 1, the left-sided tempered fractional derivative, being coupled with the left-sided Riemann-Liouville fractional derivative, is defined as 

RL α,λ,γ Y V Da+ f



Zt

1 (t) = Γ (1 − α)

γ

e−λ(t−τ ) α (t − τ )



 d  β  I f (t) dτ , dt a+

(2.550)

a

and the left-sided tempered fractional derivative, being coupled with the rightsided Riemann-Liouville fractional derivative, as 

RL α,λ,γ Y V Db− f



κ

(−1) (t) = Γ (1 − α)

Zb

γ

e−λ(τ −t) α (τ − t)

  d  β  − I f (t) dτ , dt b−

(2.551)

t

where −∞ < a < b < ∞ and κ < β < κ + 1. We notice that Pollard and Widder considered the solution of the heat equation given as follows (see [288]): 1 g (t) = 4π

Zb

1

e− 4 (τ −t)

−1

3

t

f (τ ) dτ .

(τ − t) 2

The left-sided tempered fractional integral on the interval [a, b] is defined as (see [3, 75]; also see [289]): 

LS α,λ Dv Ia+ f



1 (t) = Γ (α)

Zt

e−λ(t−τ ) α f (τ ) dτ , (t − τ )

(2.552)

a

and the right-sided tempered fractional integral on the interval [a, b] as (see [3, 75]; also see [289]) 

LS α,λ Dv Ib− f



1 (t) = Γ (α)

Zb t

where λ ∈ C.

e−λ(τ −t) α f (τ ) dτ , (τ − t)

(2.553)

Fractional Derivatives of Constant Order and Applications

113

From the expressions Eq.(2.552) and Eq.(2.553), the left-sided tempered fractional integral involving the cosine function on the interval [a, b] is defined as 

LS α,λ Dvc Ia+ f



1 (t) = Γ (α)

Zt

cos (−γ (t − τ )) f (τ ) dτ , α (t − τ )

(2.554)

a

the right-sided tempered fractional integral involving the cosine function as 

LS α,λ Dvc Ib− f



(t) =

1 Γ (α)

Zb

cos (−γ (τ − t)) f (τ ) dτ , α (t − τ )

(2.555)

t

the left-sided tempered fractional integral involving the sine function as 

LS α,λ Dvs Ia+ f



1 (t) = Γ (α)

Zt

sin (−γ (t − τ )) f (τ ) dτ , α (τ − t)

(2.556)

a

and the right-sided tempered fractional integral involving the sine function as 

LS α,λ Dvs Ib− f



1 (t) = Γ (α)

Zb

sin (−γ (τ − t)) f (τ ) dτ , α (τ − t)

(2.557)

t

where γ ∈ C. The left-sided tempered fractional derivative on the interval [a, b] is represented as (for λ = 1, see [3, 75]; also see [289]) 

LS α,λ Dv Da+ f



Zt

e−λt dκ (t) = Γ (κ − α) dtκ

e−λτ

α−κ+1 f

(t − τ )

a

(τ ) dτ ,

(2.558)

and the right-sided tempered derivative on the interval [a, b] as (for λ = 1, see [3, 75]; also see [289]) 

LS α,λ Dv Db− f



κ

(−1) e−λt dκ (t) = Γ (κ − α) dtκ

Zb

e−λτ

α−κ+1 f

t

(τ − t)

(τ ) dτ ,

(2.559)

where λ ∈ C. For λ ∈ C, the left-sided tempered fractional derivative on the interval [a, b] is represented as (for λ = 1, see [3, 75]; also see [289]) 

LS α,λ Dv Da+ f



e−λt (t) = Γ (κ − α)

Zt

eλτ α−κ+1

a

(t − τ )



d +λ dτ

κ f (τ ) dτ ,

(2.560)

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General Fractional Derivatives: Theory, Methods and Applications

which yields 

LS α,λ Dv Da+ f

1 (t) = Γ (κ − α)



Zt

e−λ(t−τ )



α−κ+1

a

(t − τ )

d +λ dτ

κ f (τ ) dτ ,

(2.561)

and the right-sided tempered derivative on the interval [a, b] as (for λ = 1, see [3, 75]; also see [289]) 

LS α,λ Dv Db− f



κ

(−1) eλt (t) = Γ (κ − α)

Zb

e−λτ



α−κ+1

(τ − t)

t

κ d − λ f (τ ) dτ , dτ

(2.562)

which leads to 

LS α,λ Dv Db− f



κ

(t) =

(−1) Γ (κ − α)

Zb

e−λ(τ −t) α−κ+1

t

(τ − t)

κ

(−1)



κ d − λ f (τ ) dτ , dτ (2.563)

where  and

d +λ dt

κ

f (t) = e−λt


dκ λt e f (t) κ dt

κ
d dκ − λ f (t) = eλt κ e−λt f (t) . dt dt For λ ∈ C, the left-sided tempered fractional derivative within the powerexponential function on the interval [a, b] is represented as (for λ = 1, see [3, 75]; also see [289])
Zt   e−λt eλτ d eλτ f (τ ) LS α,λ dτ , (2.564) α Dv Da+ f (t) = Γ (1 − α) dτ (t − τ ) 

a

which reduces to 

LS α,λ Dv Da+ f



(t) =

1 Γ (1 − α)

Zt


eλ(t−τ ) d eλτ f (τ ) dτ , α dτ (t − τ )

(2.565)

a

and the right-sided tempered derivative on the interval [a, b] as (for λ = 1, see [3, 75]; also see [289]) 

LS α,λ Dv Db− f



(t) =

−e−λt Γ (1 − α)

Zb


eλτ d e−λτ f (τ ) dτ , α dτ (τ − t)

(2.566)


e−λ(τ −t) d e−λτ f (τ ) dτ . α dτ (τ − t)

(2.567)

t

which can be written as 

LS α,λ Dv Db− f



−1 (t) = Γ (1 − α)

Zb t

Fractional Derivatives of Constant Order and Applications

2.13.10

115

Sabzikar-Meerschaert-Chen tempered fractional calculus

Definition 2.67 The left-sided tempered fractional derivative within the power-exponential function, introduced by Sabzikar, Meerschaert and Chen in 2015, is defined as (see [290]) 

α,λ LS Sme D+ f



α (t) = Γ (1 − α)

Z∞

e−λτ (f (t) − f (t − τ )) dτ , τ 1+α

(2.568)

0

and the right-sided tempered fractional derivative within the power-exponential function as 

α,λ LS Sme D− f



α (t) = Γ (1 − α)

Z∞

e−λτ (f (t) − f (t + τ )) dτ , τ 1+α

(2.569)

0

where λ ∈ C, which are called the left-sided and right-sided SabzikarMeerschaert-Chen tempered fractional derivatives (in honor of Sabzikar, Meerschaert and Chen) (see [290]). In particular, with the use of the same idea in Eq. (2.444) and Eq. (2.445), we have e−iγ(t−τ ) = cos (−γ (t − τ )) + i sin (−γ (t − τ )) (2.570) such that there are from Eq. (2.568) and Eq. (2.569) α,γ
LS Smec D+ f (t)

α = Γ (1 − α)

Z∞

cos (−γτ ) (f (t) − f (t − τ )) dτ , τ 1+α

(2.571)

cos (−γτ ) (f (t) − f (t + τ )) dτ , τ 1+α

(2.572)

sin (−γτ ) (f (t) − f (t − τ )) dτ , τ 1+α

(2.573)

sin (−γτ ) (f (t) − f (t + τ )) dτ , τ 1+α

(2.574)

0

α,γ
LS Smec D− f (t)

α = Γ (1 − α)

Z∞ 0

α,γ
LS Smes D+ f (t)

α = Γ (1 − α)

Z∞ 0

and α,γ
LS Smec D− f (t)

α = Γ (1 − α)

Z∞ 0

where γ ∈ C. It is to say that Eq. (2.571) and Eq. (2.572) are called the left-sided and right-sided tempered fractional derivatives involving the cosine function, and Eq.(2.573) and Eq. (2.574) are called the left-sided and right-sided tempered fractional derivatives involving the sine function, respectively.

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General Fractional Derivatives: Theory, Methods and Applications

2.13.11

Torres tempered fractional derivatives

Definition 2.68 The left-sided tempered fractional derivative within the kernel of the power-exponential function in the real line R, introduced by Torres in 2017, is defined as (see [291]): 

α,λ Ct D+ f



Zt

α (t) = λ f (t) + Γ (1 − α) α

f (t) − f (τ ) α+1

−∞

(t − τ )

e−λ(t−τ ) dτ ,

(2.575)

and the right-sided tempered fractional derivative within the kernel of the power-exponential function on the real line R as 

α,λ Ct D− f



α (t) = λ f (t) + Γ (1 − α) α

Z∞

f (t) − f (τ ) α+1

t

(τ − t)

e−λ(t−τ ) dτ ,

(2.576)

where 1 > α > 0 and λ ∈ C, which are called the left-sided and right-sided Torres tempered fractional derivatives (in honor of Torres), respectively. When the Torres idea can be extended to that in the interval [a, b], we have the following results. Definition 2.69 The left-sided tempered fractional derivative in the kernel of the power-exponential function on the interval [a, b] is defined as: 

α,λ Ct Da+ f



α (t) = λ f (t) + Γ (1 − α) α

Zt a

f (t) − f (τ ) (t − τ )

α+1

e−λ(t−τ ) dτ ,

(2.577)

and the right-sided tempered fractional derivative in the kernel of the powerexponential function on the interval [a, b] as 

α,λ Ct Db− f



α (t) = λ f (t) + Γ (1 − α) α

Zb

f (t) − f (τ ) α+1

t

(τ − t)

e−λ(t−τ ) dτ ,

(2.578)

where 1 > α > 0, −∞ < a < b < ∞ and λ ∈ C. Definition 2.70 The left-sided Torres tempered fractional derivatives involving the kernel of the power-cosine function are defined as α,γ
Ct D+ f (t)

α = λ f (t) + Γ (1 − α) α

Zt −∞

f (t) − f (τ ) (t − τ )

α+1

cos (−γ (t − τ )) dτ , (2.579)

Fractional Derivatives of Constant Order and Applications

117

the right-sided Torres tempered fractional derivative involving the kernel of the power-cosine function as 

α,λ Ct D− f



Z∞

α (t) = λ f (t) + Γ (1 − α) α

f (t) − f (τ ) α+1

t

(τ − t)

cos (−γ (τ − t)) dτ ,

(2.580) the left-sided Torres tempered fractional derivative involving the kernel of the power-sine function as α,γ
Ct D+ f (t)

Zt

α = λ f (t) + Γ (1 − α) α

f (t) − f (τ )

sin (−γ (t − τ )) dτ ,

α+1

−∞

(t − τ )

(2.581) and the right-sided Torres tempered fractional derivative involving the kernel of the power-sine function as 

α,λ Ct D− f



Z∞

α (t) = λ f (t) + Γ (1 − α) α

f (t) − f (τ ) α+1

t

(τ − t)

sin (−γ (τ − t)) dτ , (2.582)

where 1 > α > 0 and λ ∈ C. Similarly, the left-sided tempered fractional derivative within the kernel of the power-cosine function is defined as 

 α,λ α Ctc D+ f (t) = λ f (t) +

Zt

α Γ (1 − α)

f (t) − f (τ ) α+1

−∞

(t − τ )

cos (−γ (t − τ )) dτ ,

(2.583) the right-sided tempered fractional derivative within the kernel of the powercosine function as 

α,λ Ctc D− f



α (t) = λ f (t) + Γ (1 − α) α

Z∞

f (t) − f (τ ) α+1

t

(τ − t)

cos (−γ (τ − t)) dτ ,

(2.584) the left-sided tempered fractional derivative within the kernel of the powersine function as 

α,λ Cts D+ f



α (t) = λ f (t) + Γ (1 − α) α

Zt −∞

f (t) − f (τ ) (t − τ )

α+1

sin (−γ (t − τ )) dτ , (2.585)

118

General Fractional Derivatives: Theory, Methods and Applications

the right-sided tempered fractional derivative within the kernel of the powersine function as Z∞   α f (t) − f (τ ) α,λ α Cts D− f (t) = λ f (t) + α+1 sin (−γ (τ − t)) dτ , Γ (1 − α) (τ − t) t

(2.586) the left-sided tempered fractional derivative within the kernel of the powercosine function is defined as Zt   α f (t) − f (τ ) α,λ α Ctc Da+ f (t) = λ f (t) + α+1 cos (−γ (t − τ )) dτ , Γ (1 − α) (t − τ ) a

(2.587) the right-sided tempered fractional derivative within the kernel of the powercosine function as Zb   α f (t) − f (τ ) α,λ α Ctc Db− f (t) = λ f (t) + α+1 cos (−γ (τ − t)) dτ , Γ (1 − α) (τ − t) t

(2.588) the left-sided tempered fractional derivative within the kernel of the powersine function as Zt   α f (t) − f (τ ) α,λ α Cts Da+ f (t) = λ f (t) + α+1 sin (−γ (t − τ )) dτ , Γ (1 − α) (t − τ ) a

(2.589) and the right-sided tempered fractional derivative within the kernel of the power-sine function as 

α,λ Cts Db− f



α (t) = λ f (t) + Γ (1 − α) α

Zb

f (t) − f (τ ) α+1

t

(τ − t)

sin (−γ (τ − t)) dτ , (2.590)

respectively, where 1 > α > 0 and λ ∈ C. Definition 2.71 In this case, we define the left-sided tempered fractional derivative within the kernel of the power-exponential function in the real number R as Zt   α f (t) − f (τ ) −λ(t−τ ) α,λ dτ , (2.591) Cy D+ f (t) = α+1 e Γ (1 − α) (t − τ ) −∞

and the right-sided tempered fractional derivative within the kernel of the power-exponential function in the real number R as Z∞   α f (t) − f (τ ) −λ(t−τ ) α,λ dτ , (2.592) Cy D− f (t) = α+1 e Γ (1 − α) (τ − t) t

Fractional Derivatives of Constant Order and Applications

119

where 1 > α > 0 and λ ∈ C. In particular, we have the following results. Definition 2.72 The left-sided tempered fractional derivative within the kernel of the power-exponential function on the interval [a, b] is defined as 

 α,λ Cy Da+ f (t) =

α Γ (1 − α)

Zt

f (t) − f (τ ) α+1

a

(t − τ )

e−λ(t−τ ) dτ ,

(2.593)

and the right-sided tempered fractional derivative within the kernel of the power-exponential function on the interval [a, b] as 

 α,λ Cy Db− f (t) =

α Γ (1 − α)

Zb

f (t) − f (τ ) α+1

t

(τ − t)

e−λ(t−τ ) dτ ,

(2.594)

where 1 > α > 0 and λ ∈ C. Meanwhile, the left-sided tempered fractional derivative within the kernel of the power-cosine function is defined as 

α,λ Cyc Da+ f



α (t) = Γ (1 − α)

Zt a

f (t) − f (τ ) (t − τ )

α+1

cos (−γ (t − τ )) dτ ,

(2.595)

the left-sided tempered fractional derivative within the kernel of the powercosine function as 

α,λ Cyc Db− f



α (t) = Γ (1 − α)

Zb

f (t) − f (τ ) α+1

t

(τ − t)

cos (−γ (τ − t)) dτ ,

(2.596)

the left-sided tempered fractional derivative within the kernel of the powersine function as 

α,λ Cys Da+ f



α (t) = Γ (1 − α)

Zt

f (t) − f (τ ) α+1

a

(t − τ )

sin (−γ (t − τ )) dτ ,

(2.597)

and the left-sided tempered fractional derivative within the kernel of the power-sine function as 

α,λ Cys Db− f



α (t) = Γ (1 − α)

Zb

f (t) − f (τ ) α+1

t

(τ − t)

where 1 > α > 0, −∞ < a < b < ∞ and γ ∈ R.

sin (−γ (τ − t)) dτ ,

(2.598)

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General Fractional Derivatives: Theory, Methods and Applications

In particular, when λ = 0, we have the following relations:  
α,0 α Cy D+ f (t) = M D+ f (t) , and



α,0 Cy D− f



(t) =

α M D− f




(t) ,

(2.599)

(2.600)

where 1 > α > 0. Definition 2.73 The left-sided tempered fractional derivative within the kernel of the power-exponential function in the interval [a, b] is defined as 

α,λ Cy Da+ f



α (t) = Γ (1 − α)

Zt

f (t) − f (τ ) α+1

a

(t − τ )

e−λ(t−τ ) dτ ,

(2.601)

and the right-sided tempered fractional derivative within the kernel of the power-exponential function as 

α,λ Cy D− f



α (t) = Γ (1 − α)

Zb

f (t) − f (τ ) α+1

t

(t − τ )

e−λ(t−τ ) dτ ,

(2.602)

where 1 > α > 0 and λ ∈ C, respectively. In particular, when a = 0 and b = ∞, the left-sided tempered fractional derivative within the kernel of the power-exponential function is represented as Zt   α f (t) − f (τ ) −λ(t−τ ) α,λ dτ , (2.603) Cy D0+ f (t) := α+1 e Γ (1 − α) (t − τ ) 0

and the right-sided tempered fractional derivative within the kernel of the power-exponential function as 

α,λ Cy D− f



α (t) = Γ (1 − α)

Z∞

f (t) − f (τ ) α+1

t

(t − τ )

e−λ(t−τ ) dτ ,

(2.604)

where 1 > α > 0 and λ ∈ C. In fact, we also define the left-sided tempered fractional derivative within the kernel of the power-exponential function in sense of the Riesz type on the infinite interval [a, b] as 

α,λ Cs Da+ f



α (t) = Γ (1 − α)

Zt

f (t) − f (τ ) α+1

a

|t − τ |

e−λ(t−τ ) dτ ,

(2.605)

Fractional Derivatives of Constant Order and Applications

121

and the right-sided tempered fractional derivative within the kernel of the power-exponential function in the sense of the Riesz type on the infinite interval [a, b] as 

α,λ Cs D− f



α (t) = Γ (1 − α)

Zb

f (t) − f (τ ) α+1

t

|t − τ |

e−λ(t−τ ) dτ ,

(2.606)

which yields that 

α,λ Cs Da+ f



α (t) = Γ (1 − α)

Zt

f (t) − f (τ ) α+1

|t − τ |

−∞

e−λ(t−τ ) dτ ,

(2.607)

e−λ(t−τ ) dτ ,

(2.608)

and 

α,λ Cs D− f



α (t) = Γ (1 − α)

−∞ Z

f (t) − f (τ ) α+1

|t − τ |

t

respectively, where 1 > α > 0 and λ ∈ C. In a similar way, we also define the left-sided tempered fractional derivative within the kernel of the power-exponential function in the sense of the Riesz type on the infinite interval [a, b] by 

α,λ Ch Da+ f



α (t) = λ f (t) + Γ (1 − α) α

Zt

f (t) − f (τ ) α+1

a

|t − τ |

e−λ(t−τ ) dτ ,

(2.609)

and the right-sided tempered fractional derivative within the kernel of the power-exponential function in the sense of the Riesz type on the infinite interval [a, b] by 

α,λ Ct Db− f



α (t) = λ f (t) + Γ (1 − α) α

Zb t

f (t) − f (τ ) |t − τ |

α+1

e−λ(t−τ ) dτ ,

(2.610)

which deduce that the left-sided tempered fractional derivative within the kernel of the power-exponential function in the sense of the Riesz type on the real line R is defined as Zt   α f (t) − f (τ ) −λ(t−τ ) α,λ α dτ , (2.611) Ch D+ f (t) = λ f (t) + α+1 e Γ (1 − α) |t − τ | −∞

and the right-sided tempered fractional derivative within the kernel of the power-exponential function in the sense of the Riesz type on the real line R as Z∞   α f (t) − f (τ ) −λ(t−τ ) α,λ α dτ , (2.612) Ct D− f (t) = λ f (t) + α+1 e Γ (1 − α) |t − τ | t

where 1 > α > 0 and λ ∈ C.

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General Fractional Derivatives: Theory, Methods and Applications

2.14

Tempered fractional derivatives involving MittagLeffler function

In this section, we consider the tempered fractional derivatives within the weak singularity and the features of the Mittag-Leffler kernel (see [235]). Definition 2.74 For λ ∈ C, the left-sided tempered fractional derivative involving the kernel of the Mittag-Leffler function on the interval [a, b] is defined as   Zt Eα λ (t − τ )β   1 LS α,β,λ f (1) (τ ) dτ , (2.613) α Ev Da+ f (t) = Γ (1 − α) (t − τ ) a

and the right-sided tempered derivative involving the kernel of the MittagLeffler function on the interval [a, b] as   Zb Eα −λ (τ − t)β   1 LS α,β,λ f (t) = − f (1) (τ ) dτ , (2.614) α Ev Db− Γ (1 − α) (τ − t) t

where 1 > α > 0, 1 > β > 0 and λ ∈ C. Definition 2.75 For λ ∈ C, the left-sided tempered fractional derivative involving the kernel of the Mittag-Leffler function on the interval [a, b] is defined as   Zt Eα λ (t − τ )β   1 d LS α,β,λ f (τ ) dτ , (2.615) α Ev Da+ f (t) = Γ (1 − α) dt (t − τ ) a

and the right-sided tempered derivative involving the kernel of the MittagLeffler function on the interval [a, b] as   Zb Eα −λ (τ − t)β   1 d LS α,β,λ f (t) = − f (τ ) dτ , (2.616) α Ev Db− Γ (1 − α) dt (τ − t) t

where 1 > α > 0, 1 > β > 0 and λ ∈ C. Definition 2.76 For λ ∈ C, the left-sided tempered fractional derivative involving the kernel of the Mittag-Leffler function on the interval [a, b] is defined as 

LS α,α,λ Ev Da+ f



1 d (t) = Γ (1 − α) dt

Zt a

α

E (λ (t − τ ) ) f (τ ) dτ , α (t − τ )

(2.617)

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123

and the right-sided tempered derivative involving the kernel of the MittagLeffler function on the interval [a, b] as 

LS α,α,λ f Ev Db−



Zb

1 d (t) = − Γ (1 − α) dt

α

Eα (−λ (τ − t) ) f (τ ) dτ , α (τ − t)

(2.618)

t

where 1 > α > 0 and λ ∈ C. Similarly, the left-sided tempered fractional derivative involving the kernel of the Mittag-Leffler function is defined as 

LS α,α,λ f Ee D+



α (t) = Γ (1 − α)

Z∞

Eα (−λτ α ) (f (t) − f (t − τ )) dτ , τ 1+α

(2.619)

0

and the right-sided tempered fractional derivative involving the kernel of the Mittag-Leffler function as 

LS α,α,λ f Ee D−



α (t) = Γ (1 − α)

Z∞

Eα (−λτ α ) (f (t) − f (t + τ )) dτ , τ 1+α

(2.620)

0

where 1 > α > 0 and λ ∈ C.

2.15

Liouville-Weyl fractional calculus

In this section, we introduce the Liouville-Weyl fractional integral and Liouville-Weyl fractional derivative (in honor of Liouville and Weyl) (see [15, 16, 33]; also see [236]). Definition 2.77 Let α ∈ C, Re (α) > 0 and −∞ < x < b < ∞. The leftsided Liouville-Weyl fractional integral is defined as (see [3, 75]) LW α Cp I+ f




Zt

1 (t) = Γ (α)

−∞

1

1−α f

(t − τ )

(τ ) dτ .

(2.621)

Definition 2.78 Let α ∈ C, Re (α) > 0 and a < x < ∞. The right-sided Liouville-Weyl fractional integral is defined as (see [3, 75]) LW α Cp I− f




1 (t) = Γ (α)

Z∞ t

1

1−α f

(τ − t)

(τ ) dτ .

(2.622)

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General Fractional Derivatives: Theory, Methods and Applications

Definition 2.79 Let κ+1 > α > κ and −∞ < x < b. The left-sided LiouvilleWeyl fractional derivative is defined as (see [236]) LW α Cp D+ f




1 dκ Γ(κ−α) dtκ

(t) =

Rt

1 f (t−τ )α−κ+1

−∞

(τ ) dτ .

(2.623)

Definition 2.80 Let κ+1 > α > κ and a < x < ∞. The right-sided LiouvilleWeyl fractional derivative is defined as (see [236])


LW α Cp D− f

(t) =

1 dκ Γ(κ−α) dtκ

R∞ t

1 f (τ −t)α−κ+1

(τ ) dτ .

(2.624)

Here, the Liouville-Weyl fractional derivatives and the Liouville-Weyl fractional integrals are called the Liouville-Weyl fractional calculus. The properties of the Liouville-Weyl fractional derivatives and the Liouville-Weyl fractional integrals are represented as follows. Property 2.42 (see [236]) Let Re (α) > 0 and Re (β) > 0. Then we have     LW α+β LW α LW β I f (t) (2.625) I I f (t) = Cp + Cp + Cp + and



LW α LW β Cp I− Cp I− f



(t) =



LW α+β f Cp I−



(t) .

(2.626)

Property 2.43 (see [236]) Let Re (α) > 0. Then we have F



LW α Cp I+ f




f (ω) ei(sgnω)απ/2 (t) (ω) = , α = α (−iω) |ω|

(2.627)

F



LW α Cp I− f




f (ω) e−i(sgnω)απ/2 (t) (ω) = , α = α (iω) |ω|

(2.628)

F



LW α Cp D+ f

F



LW α Cp D− f




α α (t) (ω) = (−iω) f (ω) = |ω| e−i(sgnω)απ/2

(2.629)

α α (t) (ω) = (−iω) f (ω) = |ω| ei(sgnω)απ/2 .

(2.630)

and

2.16




Kilbas-Srivastava-Trujillo fractional calculus with respect to another function

The concepts of Liouville-Weyl fractional derivatives and Liouville-Weyl fractional integrals with respect to another function, introduced by Kilbas, Srivastava and Trujillo in 2004 (see [152]), which are called the Kilbas-SrivastavaTrujillo fractional calculus with respect to another function in honor of Kilbas, Srivastava and Trujillo, are given as follows.

Fractional Derivatives of Constant Order and Applications

125

Definition 2.81 Let α ∈ C, Re (α) > 0, −∞ < x < b < ∞ and h(1) (t) > 0. The left-sided Kilbas-Srivastava-Trujillo fractional integral with respect to another function is defined as (see [152]) LW α Cp I+,h f




Zt

1 (t) = Γ (α)

h(1) (τ )

1−α f

(h (t) − h (τ ))

−∞

(τ ) dτ ,

(2.631)

and the right-sided Kilbas-Srivastava-Trujillo fractional integral with respect to another function is defined as (see [152]) LW α Cp I−,h f




1 (t) = Γ (α)

Z∞ t

h(1) (τ )

1−α f

(τ ) dτ .

(h (τ ) − h (t))

(2.632)

Definition 2.82 Let α ∈ C, κ + 1 > Re (α) > κ , −∞ < x < b < ∞ and h(1) (t) > 0. The left-sided Kilbas-Srivastava-Trujillo fractional derivative with respect to another function is defined as (see [152]) 

LW α Cp D+,h f



(t) =

1 Γ(κ−α)



1 h(1) (t)

d dt

κ Rt −∞

h(1) (t) f (h(t)−h(τ ))α−κ+1

(τ ) dτ , .

(2.633) and the right-sided Kilbas-Srivastava-Trujillo fractional derivative with respect to another function is defined as (see [152]) 

LW α Cp D−,h f



(t) =

1 Γ(κ−α)



d − h(1)1 (t) dt

κ R∞ t

h(1) (τ ) f (h(τ )−h(t))α−κ+1

(τ ) dτ . (2.634)

Property 2.44 (see [152]) Let Re (α) > 0 , Re (β) > 0 and h(1) (t) > 0. Then we have (see [152])     LW α LW β LW α+β (2.635) Cp I+,h Cp I+,h f (t) = Cp I+,h f (t) and



LW α LW β Cp I−,h Cp I−,h f



(t) =



LW α+β Cp I−,h f



(t) .

(2.636)

Property 2.45 (see [152]) Let Re (α) > 0 , f (t) ∈ Lκ (−∞, ∞) and h(1) (t) > 0. Then we have (see [152])   LW α LW β (2.637) Cp D+,h Cp I+,h f (t) = f (t) and



LW α LW β Cp D−,h Cp I−,h f



(t) = f (t) .

(2.638)

126

General Fractional Derivatives: Theory, Methods and Applications

2.17

Sousa-de Oliveira fractional derivative with respect to another function

The conceptions of the Sousa-de Oliveira fractional derivatives with respect to another function in honor of Sousa and de Oliveira are presented as follows [292]. Definition 2.83 Let κ + 1 > α > κ, −∞ < x < b, f (t) ∈ ACκ (−∞, ∞) and h(1) (t) > 0. The left-sided Liouville fractional derivative with respect to another function is defined as (see [292]) 

L α Cp D+,h f



(t) =

1 Γ(κ−α)

Rt −∞

h(1) (τ ) (h(t)−h(τ ))α−κ+1



d 1 h(1) (τ ) dτ

κ

(κ)

fh (τ ) dτ . (2.639)

Definition 2.84 Let κ + 1 > α > κ, a < x < ∞, f (t) ∈ ACκ (−∞, ∞) and h(1) (t) > 0. The right-sided Liouville fractional derivative with respect to another function is defined as (see [292]) 

α L Cp D−,h f



(t) =

1 Γ(κ−α)

R∞ t

h(1) (τ ) (h(τ )−h(t))α−κ+1



d − h(1)1(τ ) dτ

κ

(κ)

fh (τ ) dτ . (2.640)

Definition 2.85 Let α ∈ C, 1 > Re (α) > 0, −∞ < x < b, f (t) ∈ AC1 (−∞, ∞) and h(1) (t) > 0. The left-sided Liouville fractional derivative with respect to another function is defined as 

α L Cp D+,h f



(t) =

1 Γ(1−α)

Rt −∞

h(1) (τ ) (h(t)−h(τ ))α



d 1 h(1) (τ ) dτ



(1)

fh (τ ) dτ .

(2.641)

Definition 2.86 Let 1 > Re (α) > 0 ,a < x < ∞, f (t) ∈ AC1 (−∞, ∞) and h(1) (t) > 0. The right-sided Liouville fractional derivative with respect to another function is defined as 

α L Cp D−,h f



(t) =

1 Γ(1−α)

R∞ t

h(1) (τ ) (h(τ )−h(t))α



 (1) d − h(1)1(τ ) dτ fh (τ ) dτ .

(2.642)

Property 2.46 Let Re (α) > 0, f (t) ∈ ACκ (−∞, ∞) and h(1) (t) > 0. Then we have   L α LW β (2.643) Cp D+,h Cp I+,h f (t) = f (t) and



L α LW β Cp D−,h Cp I−,h f



(t) = f (t) .

(2.644)

Fractional Derivatives of Constant Order and Applications

2.18

127

Liouville-Weyl fractional calculus with respect to another function in the Sense of Riesz, Feller and Richard types

The conceptions of the Liouville-Weyl fractional integrals, Liouville-Weyl fractional derivatives and Liouville fractional derivatives with respect to another function in the sense of Riesz, Feller and Richard types are given as follows.

2.18.1

Liouville-Weyl fractional integrals with respect to another function in the sense of Riesz, Feller and Richard types

Definition 2.87 Let 0 < α < 1, f ∈ L1 (−∞, ∞), a = −∞, b = ∞ and h(1) (t) > 0. The fractional integral with respect to another function in the sense of Liouville-Weyl and Riesz types is defined as   α α I+,h f + I−,h f (t) Hi α , (2.645) Rz I[−∞,∞],h f (t) = 2 cos (πα/2) the Liouville-Weyl fractional integral with respect to another function in the sense of Riemann-Liouville and Feller types with respect to another function as Fe α Rz I[−∞,∞],h f

α α (t) = H− (ϑ, α) I+,h f (t) + H+ (ϑ, α) I−,h f (t) ,

(2.646)

and the fractional Liouville-Weyl integral with respect to another function in the sense of Riemann-Liouville and Richard types as   α α I+,h f + I−,h f (t) Ri α . (2.647) Rz I[−∞,∞],h f (t) = 2 sin (πα/2)

2.18.2

Liouville-Weyl fractional derivatives with respect to another function in the sense of Riesz, Feller and Richard types

Definition 2.88 Let 0 < α < 1, f ∈ L1 (−∞, ∞) a = −∞ and b = ∞. The fractional derivative with respect to another function in the sense of LiouvilleWeyl and Riesz types is defined as   α α D+,h f + D−,h f (t) Hi α , (2.648) Rz D[−∞,∞],h f (t) = − 2 cos (πα/2)

128

General Fractional Derivatives: Theory, Methods and Applications

the fractional derivative with respect to another function in the sense of Liouville-Weyl and Feller types with respect to another function as   Fe α α α D f (t) = − H (ϑ, α) D f (t) + H (ϑ, α) D f (t) , + − Rz [−∞,∞],h +,h −,h (2.649) and the fractional derivative with respect to another function in the sense of Liouville-Weyl and Richard types as   α α D+,h f + D−,h f (t) Ri α . (2.650) Rz D[−∞,∞],h f (t) = − 2 sin (πα/2) Definition 2.89 Let 0 < α < 1, a = −∞, b = ∞ and f ∈ ACκ (−∞, ∞). The fractional derivative with respect to another function in the sense of Liouville and Riesz types is defined as   α α LCS D+,h f +LCS D−,h f (t) Ri α , (2.651) Rzv D[−∞,∞],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Liouville and Feller types as   Fe α α α D f (t) = − H (ϑ, α) D f (t) + H (ϑ, α) D f (t) , + − LCS Rzv [−∞,∞],h +,h −,h LCS (2.652) and the fractional derivative with respect to another function in the sense of Liouville and Richard types as   α α LCS D+,h f + LCS D−,h f (t) Ri α . (2.653) Rzv D[−∞,∞],h f (t) = − 2 sin (πα/2) Definition 2.90 Let 0 < α < 1, a = −∞, b = ∞ and f ∈ ACκ (−∞, ∞). The fractional derivative with respect to another function in the sense of Liouville and Riesz types is defined as   α α LC D+,h f +LC D−,h f (t) Riv α , (2.654) Rzv D[−∞,∞],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Liouville and Feller types as   F ev α α α Rzv D[−∞,∞],h f (t) = − H+ (ϑ, α)LC D+,h f (t) + H− (ϑ, α)LC D−,h f (t) , (2.655) and the fractional derivative with respect to another function in the sense of Liouville and Richard types as   α α LC D+,h f +LC D−,h f (t) Riv α . (2.656) Rzv D[−∞,∞],h f (t) = 2 sin (πα/2)

Fractional Derivatives of Constant Order and Applications

129

Definition 2.91 Let 0 < Re (α) < 1, f ∈ L1 (−∞, ∞), a = −∞, b = ∞ and h(1) (t) > 0. The fractional integral with respect to another function in the sense of Liouville-Weyl and Riesz types is defined as   α α I+,h f + I−,h f (t) Hi α , (2.657) Rz I[−∞,∞],h f (t) = 2 cos (πα/2) the Liouville-Weyl fractional integral with respect to another function in the sense of Riemann-Liouville and Feller types with respect to another function as Fe α Rz I[−∞,∞],h f

α α (t) = H− (ϑ, α) I+,h f (t) + H+ (ϑ, α) I−,h f (t) ,

(2.658)

and the fractional Liouville-Weyl integral with respect to another function in the sense of Riemann-Liouville and Richard types as   α α I+,h f + I−,h f (t) Ri α . (2.659) Rz I[−∞,∞],h f (t) = 2 sin (πα/2) Definition 2.92 Let 0 < Re (α) < 1, f ∈ L1 (−∞, ∞) a = −∞ and b = ∞. The fractional derivative with respect to another function in the sense of Liouville-Weyl and Riesz types is defined as   α α D+,h f + D−,h f (t) Hi α , (2.660) Rz D[−∞,∞],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Liouville-Weyl and Feller types with respect to another function as   Fe α α α D f (t) = − H (ϑ, α) D f (t) + H (ϑ, α) D f (t) , + − Rz [−∞,∞],h +,h −,h (2.661) and the fractional derivative with respect to another function in the sense of Liouville-Weyl and Richard types as   α α D+,h f + D−,h f (t) Ri α . (2.662) Rz D[−∞,∞],h f (t) = − 2 sin (πα/2) Definition 2.93 Let 0 < Re (α) < 1, a = −∞, b = ∞ and f ∈ ACκ (−∞, ∞). The fractional derivative with respect to another function in the sense of Liouville and Riesz types is defined as   α α LCS D+,h f +LCS D−,h f (t) Ri α , (2.663) Rzv D[−∞,∞],h f (t) = − 2 cos (πα/2)

130

General Fractional Derivatives: Theory, Methods and Applications

the fractional derivative with respect to another function in the sense of Liouville and Feller types as   Fe α α α Rzv D[−∞,∞],h f (t) = − H+ (ϑ, α)LCS D+,h f (t) + H− (ϑ, α) LCS D−,h f (t) , (2.664) and the fractional derivative with respect to another function in the sense of Liouville and Richard types as   α α LCS D+,h f +LCS D−,h f (t) Ri α . (2.665) Rzv D[−∞,∞],h f (t) = − 2 sin (πα/2) Definition 2.94 Let 0 < Re (α) < 1, a = −∞, b = ∞ andf ∈ ACκ (−∞, ∞). The fractional derivative with respect to another function in the sense of Liouville and Riesz types is defined as   α α D f + D f (t) LC +,h LC −,h Riv α , (2.666) Rzv D[−∞,∞],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Liouville and Feller types as   F ev α α α Rzv D[−∞,∞],h f (t) = − H+ (ϑ, α)LC D+,h f (t) + H− (ϑ, α)LC D−,h f (t) , (2.667) and the fractional derivative with respect to another function in the sense of Liouville and Richard types as   α α LC D+,h f +LC D−,h f (t) Riv α . (2.668) Rzv D[−∞,∞],h f (t) = 2 sin (πα/2) Definition 2.95 Let 0 < Re (α) < 1, f ∈ L1 (−∞, ∞), a = −∞, b = ∞ and h(1) (t) > 0. The Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Riesz type is defined as h i 1 α V α V α (2.669) T a Dt;[−∞,∞],h f (t) = 2cos(απ/2) M D+,h f (t) +M D−,h f (t) , the Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Feller type as α T a Dt;[−∞,∞],h f

α α (t) = H+ (ϑ, α) × VM D+,h f (t) + H− (ϑ, α) × VM D−,h f (t) , (2.670) and the Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Richard type as h i 1 α V α V α (2.671) T a Dt;[−∞,∞],h f (t) = − 2sin(απ/2) M D+,h f (t) + M D−,h f (t) .

Fractional Derivatives of Constant Order and Applications

131

Definition 2.96 Let 0 < α < 1, f ∈ L1 (−∞, ∞), a = −∞, b = ∞ and h(1) (t) > 0. The Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Riesz type is defined as h i 1 V α V α α (2.672) T a Dt;[−∞,∞],h f (t) = 2cos(απ/2) M D+,h f (t) + M D−,h f (t) , the Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Feller type as α T a Dt;[−∞,∞],h f

α α (t) = H+ (ϑ, α) ×VM D+,h f (t) + H− (ϑ, α) ×VM D−,h f (t) , (2.673) and the Tarasov type fractional derivative with respect to another function in the sense of Gr¨ unwald–Letnikov–Richard type as i h 1 α V α V α (2.674) T a Dt;[−∞,∞],h f (t) = − 2sin(απ/2) M D+,h f (t) +M D−,h f (t) .

2.19

Hilfer derivatives

The conceptions of the fractional derivatives [292], corned in 2000 by Hilfer, which have the Riemann-Liouville and the Caputo derivatives as specific cases [293, 294] and the related topics are presented as follows.

2.19.1

Hilfer derivatives

Definition 2.97 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided fractional derivative is defined as (see [292])    β(1−α) d (1−β)(1−α) α,β Da+ f (t) = Ia+ Ia+ f (t) , (2.675) dt (1−β)(1−α)

where Ia+ as

f (t) ∈ AC1 (a, b), and the right-sided fractional derivative    β(1−α) d (1−β)(1−α) α,β Db− f (t) = −Ib− I f (t) , (2.676) dt b−

(1−β)(1−α)

where Ib− f (t) ∈ AC1 (−∞, ∞), which are called the left-sided and right-sided Hilfer derivatives (in honor of Hilfer), respectively. Definition 2.98 Let κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞. The left-sided Hilfer fractional derivative is defined as   κ  (1−β)(κ−α) β(κ−α) d α,β,κ Da+ f (t) = Ia+ I f (t) , (2.677) dtκ a+

132

General Fractional Derivatives: Theory, Methods and Applications (1−β)(1−α)

where Ia+ derivative as

α,β,κ Db− f

f (t) ∈ ACκ (a, b), and the right-sided Hilfer fractional

  κ  (1−β)(κ−α) κ β(κ−α) d I f (t) , (t) = (−1) Ib− dtκ b−

(1−β)(1−α)

where Ib−

(2.678)

f (t) ∈ ACκ (a, b), respectively.

In this case, we set up the following relations: Property 2.47 Let 0 < α < 1, 0 ≤ β ≤ 1 and f ∈ L1 (a, b). Then we have α,β α,β,1 Da+ f (t) = Da+ f (t)

(2.679)

α,β α,β,1 Db− f (t) = Db− f (t) .

(2.680)

and Property 2.48 (1) Suppose that 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and (1−β)(1−α) Ia+ f (t) ∈ AC1 (a, b), then we have (see [293]) n o (1−β)(1−α) α,β L D0+ f (t) (s) = sα f (s) − sβ(1−α) I0+ f (0) . (2.681) (2) Suppose that κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b), (1−β)(1−α) −∞ < a < b < ∞ and I0+ f (t) ∈ ACκ (a, b), then we have n o (1−β)(κ−α) α,β,κ L D0+ f (t) (s) = sα f (s) − sβ(κ−α) I0+ f (0) . (2.682) Definition 2.99 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), a = −∞ and b = ∞. The left-sided fractional derivative in the sense of Hilfer type can be represented as    β(1−α) d (1−β)(1−α) α,β D+ f (t) = I+ f I+ (t) , (2.683) dt (1−β)(1−α)

f (t) ∈ AC1 (−∞, ∞), and the right-sided Hilfer fractional where I+ derivative in the sense of Hilfer type as    β(1−α) d (1−β)(1−α) α,β D− f (t) = −I− I− f (t) , (2.684) dt (1−β)(1−α)

where I−

f (t) ∈ AC1 (−∞, ∞).

Definition 2.100 Let κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (−∞, ∞), a = −∞ and b = ∞. The left-sided Hilfer fractional derivative can be represented as   κ  β(1−α) d (1−β)(κ−α) α,β,κ I f (t) , (2.685) D+ f (t) = I+ dtκ +

Fractional Derivatives of Constant Order and Applications (1−β)(κ−α)

where I+ tive as

f (t) ∈ ACκ (−∞, ∞), and the right-sided fractional deriva-

α,β,κ D− f (t) = (1−β)(κ−α)

where I−

133



κ

β(1−α)

(−1) I−

 dκ  (1−β)(κ−α)  I f (t) , dtκ −

(2.686)

f (t) ∈ ACκ (−∞, ∞).

Definition 2.101 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < (1−β)(1−α) (1−β)(1−α) b < ∞, Ia+ f (t) ∈ AC1 (a, b) and Ib− f (t) ∈ AC1 (a, b). The fractional derivative in the sense of Riesz and Hilfer types is defined as   α,β α,β Da+ f + Db− f (t) Hi α,β , (2.687) Rz D[a,b] f (t) = − 2 cos (πα/2) the fractional derivative in the sense of Feller and Hilfer types as F e α,β Rz D[a,b] f

α,β (t) = − sin((α+ϑ)π/2) × Da+ f (t) − sin(πϑ)

sin((α−ϑ)π/2) sin(πϑ)

α,β × Db− f (t) , (2.688) and the fractional derivative in the sense of Richard and Hilfer types as   α,β α,β Da+ f + Db− f (t) Ri α,β . (2.689) Rz D[a,b] f (t) = − 2 sin (πα/2)

Definition 2.102 Let 0 < Re (α) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided Hilfer fractional derivative is defined as    β(1−α) d (1−β)(1−α) α,β Da+ f (t) = Ia+ I f (t) , (2.690) dt a+ (1−β)(1−α)

where Ia+ derivative as

f (t) ∈ AC1 (a, b), and the right-sided Hilfer fractional

α,β Db− f (t) = (1−β)(1−α)

where Ib−



β(1−α)

−Ib−

 d  (1−β)(1−α)  Ib− f (t) , dt

(2.691)

f (t) ∈ AC1 (−∞, ∞), respectively.

Definition 2.103 Let κ < Re (α) < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞. The left-sided Hilfer fractional derivative is defined as   κ  β(κ−α) d (1−β)(κ−α) α,β,κ I , (2.692) Da+ f (t) = Ia+ dtκ a+ (1−β)(1−α)

where Ia+ derivative as

α,β,κ Db− f

  κ  (1−β)(κ−α) κ β(κ−α) d (t) = (−1) Ib− I f (t) , dtκ b−

(1−β)(1−α)

where Ib−

f (t) ∈ ACκ (a, b), and the right-sided Hilfer fractional

f (t) ∈ ACκ (a, b), respectively.

(2.693)

134

General Fractional Derivatives: Theory, Methods and Applications

Definition 2.104 Let κ < Re (α) < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b), a = −∞ and b = ∞. The left-sided Hilfer fractional derivative is defined as   κ  (1−β)(κ−α) β(κ−α) d α,β,κ I , (2.694) D+ f (t) = I+ dtκ + (1−β)(1−α)

where I+ derivative as

α,β,κ D− f

  κ  (1−β)(κ−α) κ β(κ−α) d (t) = (−1) I− I f (t) , dtκ −

(1−β)(1−α)

where I−

2.19.2

f (t) ∈ ACκ (a, b), and the right-sided Hilfer fractional (2.695)

f (t) ∈ ACκ (a, b), respectively.

Sousa-de Oliveira fractional derivatives with respect to another function in the sense of Hilfer type

The conceptions of the Hilfer fractional derivative with respect to another function, which are called the Sousa-de Oliveira fractional derivatives with respect to another function, are presented as follows [295]. Definition 2.105 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞ and h(1) (t) > 0. The left-sided Hilfer fractional derivative with respect to another function is defined as     1 d  (1−β)(1−α)  β(1−α) α,β Da+,h f (t) = Ia+,h Ia+,h f (t) , (2.696) h(1) (t) dt (1−β)(1−α)

where Ia+,h f (t) ∈ AC1 (a, b), and the right-sided Hilfer fractional derivative with respect to another function as     1 d  (1−β)(1−α)  β(1−α) α,β Db−,h f (t) = −Ib−,h Ib−,h f (t) , (2.697) h(1) (t) dt (1−β)(1−α)

where Ib−,h

f (t) ∈ AC1 (−∞, ∞), respectively.

Definition 2.106 Let 0 < Re (α) < 1, 0 ≤ Re (β) ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞ and h(1) (t) > 0. The left-sided Hilfer fractional derivative with respect to another function is defined as     1 d  (1−β)(1−α)  β(1−α) α,β Ia+,h f (t) , (2.698) Da+,h f (t) = Ia+,h h(1) (t) dt (1−β)(1−α)

where Ia+,h f (t) ∈ AC1 (a, b), and the right-sided Hilfer fractional derivative with respect to another function as     1 d  (1−β)(1−α)  β(1−α) α,β Db−,h f (t) = −Ib−,h I f (t) , (2.699) b−,h h(1) (t) dt (1−β)(1−α)

where Ib−,h

f (t) ∈ AC1 (−∞, ∞), respectively.

Fractional Derivatives of Constant Order and Applications

2.19.3

135

Riesz, Feller and Richard fractional derivatives with respect to another function via Hilfer fractional derivative

The conceptions of the Riesz, Feller and Richard fractional derivatives with respect to another function coupling with Hilfer fractional derivative are given as follows. Definition 2.107 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < (1−β)(1−α) (1−β)(1−α) b < ∞, Ia+,h f (t) ∈ AC1 (a, b) and Ib−,h f (t) ∈ AC1 (a, b). The fractional derivative in the sense of Riesz and Hilfer types with respect to another function is defined as   α,β α,β Da+,h f + Db−,h f (t) Hi α,β , (2.700) Rz D[a,b],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Feller and Hilfer types as   α,β α,β F e α,β D f (t) = − H (ϑ, α) D f (t) + H (ϑ, α) D f (t) , + − Rz [a,b],h a+,h b−,h (2.701) and the fractional derivative with respect to another function in the sense of Richard and Hilfer types as   α,β α,β Da+,h f + Db−,h f (t) Ri α,β . (2.702) Rz D[a,b],h f (t) = − 2 sin (πα/2) Definition 2.108 Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), a = −∞, b = (1−β)(1−α) (1−β)(1−α) ∞, I+,h f (t) ∈ AC1 (−∞, ∞) and I−,h f (t) ∈ AC1 (−∞, ∞). The fractional derivative in the sense of Riesz and Hilfer types with respect to another function is defined as   α,β α,β D+,h f + D−,h f (t) Hi α,β , (2.703) Rz D[−∞,∞],h f (t) = − 2 cos (πα/2) the fractional derivative with respect to another function in the sense of Feller and Hilfer types as   α,β α,β F e α,β D f (t) = − H (ϑ, α) D f (t) + H (ϑ, α) D f (t) , + − Rz [−∞,∞],h +,h −,h (2.704) and the fractional derivative with respect to another function in the sense of Richard and Hilfer types as   α,β α,β D+,h f + D−,h f (t) Ri α,β . (2.705) Rz D[−∞,∞],h f (t) = − 2 sin (πα/2)

136

General Fractional Derivatives: Theory, Methods and Applications

2.20 2.20.1

Applications Relaxation equations within fractional derivatives

In view of the anomalous relaxation behaviors in nature, one may utilize the fractional derivatives to propose the mathematical models. Now we give the examples and the open problems for the anomalous relaxation in complex phenomenon. Example 2.1 The mathematical model of the relaxation within the RiemannLiouville fractional derivative can be given as (see [152]; also see [301])
α D0+ f (t) + γf (t) = 0 (0 < α < 1) , (2.706) subjected to the initial condition
1−α I0+ f (0+) = h,

(2.707)

and Fourier-transformation solution is (for example, K. S. Cole and R. H. Cole introduced to the Cole-Cole model in 1941, see [302]) f (ω) =

h , α (iω) + γ

(2.708)

which reduces to (see [152]) f (t) = htα−1 Eα,α (−γtα ) ,

(2.709)

where h and γ are the constants and the Hille-Tamarkin function (also called the α−exponential function (see [152]), structured in 1930, can be written as (see [37]) dEα (γtα ) α−1 α eγt = t E (γt ) = . (2.710) α,α α dt Example 2.2 The mathematical model of the relaxation within the LiouvilleSonine fractional derivative can be written as (see [304])
α (2.711) LS D0+ f (t) + γf (t) = 0 (0 < α < 1) , subjected to the initial condition f (0+) = h,

(2.712)

and the Laplace-transformation solution can be represented in the form [152]: f (s) =

h , sα + γ

(2.713)

which reduces to [152] f (t) = htα−1 Eα,α (−γtα ) , where h and γ are the constants.

(2.714)

Fractional Derivatives of Constant Order and Applications

137

Example 2.3 The mathematical model of the relaxation within the LiouvilleGr¨ unwald-Letnikov fractional derivative can be written as α d D0+ f

(t) + γf (t) = hδ (t) (0 < α < 1) ,

(2.715)

with the Laplace-transformation solution represented as f (s) =

h , sα + γ

(2.716)

which reduces to [152] f (t) = htα−1 Eα,α (−γtα ) ,

(2.717)

where h and γ are the constants. Example 2.4 The mathematical model of the relaxation within the Hilfer fractional derivative can be written as [292] α,β D0+ f (t) + γf (t) = 0 (0 < α < 1) ,

(2.718)

subjected to the condition (1−α)(1−β)

I0+

= f (0) ,

(2.719)

and the Laplace transformation solution of the model is represented in the form sβ(1−α) f (0) sβ(1−α)−α f (0) f (s) = = (2.720) sα + γ 1 + γs−α with the solution f (t) = f (0) tα−β(1−α)−1 Eα,α−β(1−α) (−γtα ) , where f (0) and γ are the constants. There are some open problems for the mathematical model of the relaxation as follows. Example 2.5 The mathematical model of the relaxation within the Tarasov fractional derivative can be written as α T a Dt;[a,b] f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.721)

which reduces to α 2Γ (1 − α) cos (απ/2) where γ is the constant.

α d Da+ f


α (t) +d Db− f (t) + γf (t) = 0,

(2.722)

138

General Fractional Derivatives: Theory, Methods and Applications

Example 2.6 The mathematical model of the relaxation within the Tarasov type fractional derivative of the Gr¨ unwald–Letnikov–Feller type can be written as α (2.723) F V Dt;[a,b];ϑ f (t) + γf (t) = 0 (0 < α < 1) , which reduces to α T a Dt;[a,b] f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.724)

α α where F V Dt;[a,b];0 f (t) =T a Dt;[a,b] f (t), and the mathematical model of the relaxation within the Tarasov type fractional derivative of the Gr¨ unwald– Letnikov–Richard type can be represented in the form: α L Dt;[a,b] f

where

FV

(t) + γf (t) = 0 (0 < α < 1) ,

(2.725)

α α Dt;[a,b];1 f (t) =L Dt;[a,b] f (t) and γ is the constant.

Example 2.7 The mathematical model of the relaxation within the Erd´elyiKober type fractional derivative is represented in the form:
α Da+;σ,η f (t) + γf (t) = 0 (0 < α < 1) , (2.726) which yields that the mathematical model of the relaxation within the fractional derivative of Erd´elyi-Kober-Riesz type is given as α Rw DR f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.727)

the mathematical model of the relaxation within the fractional derivative of Erd´elyi-Kober-Feller type as α Rv IR f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.728)

and the mathematical model of the relaxation within the fractional derivative of Erd´elyi-Kober-Richard type as α Ro DR f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.729)

where γ is the constant. Example 2.8 The mathematical model of the relaxation within the Katugampola fractional derivative can be presented as
α (2.730) K Da+;σ f (t) + γf (t) = 0 (0 < α < 1) , where γ is the constant. Example 2.9 The mathematical model of the relaxation within the Hadamard type fractional derivative can be represented as
α (2.731) Hv Da+ f (t) + γf (t) = 0 (0 < α < 1) , where γ is the constant.

Fractional Derivatives of Constant Order and Applications

139

Example 2.10 The mathematical models of the relaxation within the tempered fractional derivative are presented as   RL α,λ (2.732) Cp Da+ f (t) + γf (t) = 0, 

RL α,λ Cp Da+;h f





LS α,λ Cp Da+ f





(t) + γf (t) = 0,

(2.733)

(t) + γf (t) = 0,

(2.734)


RL α,γ Cb Da+ f (t)

+ γf (t) = 0,

(2.735)


RL α,γ Cc Da+ f (t)

+ γf (t) = 0,

(2.736)

LS α,λ,γ V v Da+ f



(t) + γf (t) = 0,  α,λ,γ LS V cd Da+ f (t) + γf (t) = 0,   RL α,λ,γ Y V Da+ f (t) + γf (t) = 0,   α,λ Cs Da+ f (t) + γf (t) = 0,   α,λ Ch Da+ f (t) + γf (t) = 0



and



LS α,β,λ Ev Da+ f



(t) + γf (t) = 0,

(2.737) (2.738) (2.739) (2.740) (2.741) (2.742)

where γ is the constant and α (0 < α < 1). Example 2.11 The mathematical model of the relaxation within the Sousade Oliveira fractional derivative with respect to another function is represented as α,β Da+,h f (t) + γf (t) = 0 (0 < α < 1) , (2.743) where γ is the constant. Example 2.12 The mathematical model of the relaxation within the Riesz fractional derivative with respect to another function is written as Hi α,β Rz D[a,b],h f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.744)

the mathematical model of the relaxation within the Feller fractional derivative with respect to another function as F e α,β Rz D[a,b],h f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.745)

the mathematical model of the relaxation within the Richard fractional derivative with respect to another function as Ri α,β Rz D[a,b],h f

(t) + γf (t) = 0 (0 < α < 1) ,

(2.746)

140

General Fractional Derivatives: Theory, Methods and Applications

the mathematical model of the relaxation within the Riemann-Liouville fractional derivative with respect to another function as
α Da+,h f (t) + γf (t) = 0 (0 < α < 1) , (2.747) and the mathematical model of the relaxation within the Almeida fractional derivatives with respect to another function as
α (2.748) LS Da+,h f (t) + γf (t) = 0 (0 < α < 1) , where γ is the constant.

2.20.2

Rheological models within fractional derivative

The Nutting phenomenon, first proposed in 1921 by Nutting, was considered model to the elastic viscous deformation (see [339, 340]). In order to solve the above problems, the mathematical model for the deformation (see [338]), proposed by Blair in 1947, which is called the Scott Blair model in honor of the G.W. Scott Blair, was developed by many scientists, for example, Bagley and Torvik (see [139]), Mainardi and Spada (see [341]), Stiassnie (see [342]) and so on. Gerasimov reported the general law of the deformation based on the Liouville fractional derivative (see [269]) in 1948. The mathematical model via Liouville-Sonine fractional derivative was proposed by Smit and De Vries in 1970 and by Caputo and Mainardi in 1971. However, the Liouville and Sonine work of the study in the operator is not noticed in the results (see [36, 68, 270, 271]). We now present the proposed models for the law of deformation and some of the open problems of the mathematical models for the law of deformation as follows. Example 2.13 The Scott Blair model for the law of deformation via Riemann-Liouville (or Riemann) fractional derivative is represented in the form [338]:
α σ (t) = ξ D0+ ε (t) (0 < α < 1) , (2.749) which can be written as σ (t) = ξ

1 d Γ (1 − α) dt

Zt

ε (τ ) α dτ (0 < α < 1) , (τ − t)

(2.750)

0

where α (0 < α < 1) and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 2.14 The Gerasimov model for the law of deformation via Liouville fractional derivative can be written as (also see [269]) α σ (t) = ξL D−∞ ε (t) (0 < α < 1) ,

(2.751)

Fractional Derivatives of Constant Order and Applications

141

which becomes ξ σ (t) = Γ (1 − α)

Zt

ε(1) (τ ) α dτ (0 < α < 1) , (t − τ )

(2.752)

−∞

and the Gerasimov model for the law of deformation via Liouville-Weyl fractional derivative as (also see [269]; also see [150])
α σ (t) = ξ LW (2.753) Cp D− ε (t) (0 < α < 1) , which is ξ d σ (t) = Γ (1 − α) dt

Zt

1 α ε (τ ) dτ , (τ − t)

(2.754)

−∞

where α (0 < α < 1) and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 2.15 The mathematical model for the law of deformation via Liouville-Sonine fractional derivative is presented as (see [36, 68, 270, 271]) ξ σ (t) = Γ (1 − α)

Zt

ε(1) (τ ) α dτ , (t − τ )

(2.755)

0

where α (0 < α < 1) and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 2.16 The mathematical model for the law of deformation via Hilfer fractional derivative is represented as    β(1−α) d (1−β)(1−α) σ (t) = ξ I0+ I0+ ε (t) (0 < α < 1) , (2.756) dt the mathematical model for the law of deformation via Sousa-de Oliveira fractional derivative with respect to another function as    β(1−α) d (1−β)(1−α) σ (t) = ξ I0+,h I ε (t) (0 < α < 1) , (2.757) dt 0+,h and the mathematical model for the law of deformation via Hilfer fractional derivative with respect to another function in the sense of Liouville-Weyl type as    β(1−α) d (1−β)(1−α) σ (t) = ξ I+,h I+,h ε (t) (0 < α < 1) , (2.758) dt where α (0 < α < 1) and ξ are the material constants, h (t) is the basic function, σ (t) is the stress, and ε (t) is the strain.

142

General Fractional Derivatives: Theory, Methods and Applications

Example 2.17 The Colombaro-Garra-Giusti-Mainardi model for the law of deformation via Almeida fractional derivative with respect to another function in the sense of Liouville-Sonine type is presented as (see [343]) ξ σ (t) = Γ (1 − α)

Zt

h(1) (t) α (h (t) − h (τ ))



 1 (1) ε (τ ) dτ , h(1) (τ ) h

(2.759)

0

where α (0 < α < 1)and ξ are the material constants, h (t) is the basic function, σ (t) is the stress, and ε (t) is the strain. Example 2.18 The Garra-Mainardi model for the law of deformation via Riemann-Liouville fractional derivative with respect to another function is presented as (see [344]) d ξ σ (t) = (1) Γ (1 − α) h (t) dt

Zt

h(1) (t) α ε (τ ) dτ , (h (t) − h (τ ))

(2.760)

0

where α (0 < α < 1) and ξ are the material constants, h (t) is the basic function, σ (t) is the stress, and ε (t) is the strain. Example 2.19 The mathematical model for the law of deformation via Liouville fractional derivative with respect to another function is represented as ξ σ (t) = Γ (1 − α)

Zt

h(1) (t) α (h (t) − h (τ ))



 1 (1) ε (τ ) dτ , h(1) (τ ) h

(2.761)

−∞

and the mathematical model for the law of deformation via Kilbas-SrivastavaTrujillo fractional derivative with respect to another function as ξ d σ (t) = Γ (1 − α) h(1) (t) dt

Zt

h(1) (t) α ε (τ ) dτ , (h (t) − h (τ ))

(2.762)

−∞

where α (0 < α < 1) and ξ are the material constants, h (t) is the basic function, σ (t) is the stress, and ε (t) is the strain. Example 2.20 The mathematical model for the law of deformation via tempered fractional derivative in the sense of Riemann-Liouville type is represented as Zt ξ d ε (τ ) −λ(t−τ ) σ (t) = dτ , (2.763) αe Γ (1 − α) dt (t − τ ) 0

the mathematical model for the law of deformation via tempered fractional derivative in the sense of Liouville-Sonine type as ξ σ (t) = Γ (1 − α)

Zt 0

e−λ(t−τ ) (1) (τ ) dτ , αε (t − τ )

(2.764)

Fractional Derivatives of Constant Order and Applications

143

and the mathematical model for the law of deformation via tempered fractional derivative with respect to another function in the sense of Riemann-Liouville type as ξ σ (t) = Γ (1 − α)



1

d (1) h (t) dt

 Zt

e−λ(h(t)−h(τ )) h(1) (τ ) ε (τ ) dτ , α (h (t) − h (τ ))

(2.765)

−∞

where α (0 < α < 1) and ξ are the material constants, h (t) is the basic function, σ (t) is the stress, and ε (t) is the strain.

Chapter 3 General Fractional Derivatives of Constant Order and Applications

In this chapter, we present the general fractional derivatives and integrals of constant order involving the kernel of the special functions, such as the Mittag-Leffler function, Kohlrausch-Williams-Watts function, Wiman function, Prabhakar function, Gorenflo-Mainardi function, Miller-Ross function, Lorenzo-Hartley function, subcosine function and subsine function. The general fractional derivatives and the general fractional integrals with respect to another function are also presented. The general fractional derivatives are also considered to describe the relaxation and rheological models in complex phenomena.

3.1

General fractional derivatives involving the kernel of Mittag-Leffler function

In this section, the family of the general fractional derivatives involving the kernel of Mittag-Leffler function is introduced in detail.

3.1.1

General fractional derivatives involving the kernel of Mittag-Leffler function without power law

Definition 3.1 Let κ ≥ 0, ν ∈ C, λ ∈ C and Re (ν) > 0. The left-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Liouville-Sonine-Caputo type is defined as LSC κ,λ;ν M l Da+ f

Zt (t) =

Eν (−λ (t − τ )) f (κ) (τ ) dτ ,

(3.1)

a

145

146

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Liouville-Sonine-Caputo type as LSC κ,λ;ν M l Db− f

(t) := (−1)

Zb

κ

Eν (−λ (τ − t)) f (κ) (τ ) dτ .

(3.2)

t

Definition 3.2 The left-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Riemann-Liouville type is defined as Zt

dκ (t) = κ dt

RL κ,λ;ν M l Da+ f

Eν (−λ (t − τ )) f (τ ) dτ ,

(3.3)

a

and the right-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Riemann-Liouville type is defined as RL κ,λ;ν M l Db− f

dκ dtκ

κ

(t) = (−1)

Zb Eν (−λ (τ − t)) f (τ ) dτ .

(3.4)

t

In particular, when κ = 1, ν ∈ C, λ ∈ C and Re (ν) > 0, the left-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Liouville-Sonine type is defined as (see [299]) LSC 1,λ;ν M l Da+ f

Zt

Eν (−λ (t − τ )) f (1) (τ ) dτ ,

(t) =

(3.5)

a

and the right-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Liouville-Sonine type as LSC 1,λ;ν M l Db− f

Zb (t) = −

Eν (−λ (τ − t)) f (1) (τ ) dτ .

(3.6)

t

On the other hand, the left-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Riemann-Liouville type is defined as RL 1,λ;ν M l Da+ f

(t) =

d dt

Zt Eν (−λ (t − τ )) f (τ ) dτ ,

(3.7)

a

and the right-sided generalized derivative involving the kernel of Mittag-Leffler function in the sense of Riemann-Liouville type is defined as RL 1,λ;ν M l Db− f

d (t) = − dt

Zb Eν (−λ (τ − t)) f (τ ) dτ . t

(3.8)

General Fractional Derivatives of Constant Order and Applications

3.1.2

147

General fractional derivatives involving the kernel of Mittag-Leffler-Gauss function with power law 2

For J (α) = √ 1+α α

π (1−α)

and λ =

α 1−α ,

the left-sided general fractional derivative

involving the kernel of Mittag-Leffler-Gauss function with power law in the sense of Liouville-Sonine-Caputo type is defined as (see [299]) LSC α,κ,λ M le Da+ f Rt

(t)   2α f (κ) (τ ) dτ = J (β) Eα −λ (t − τ ) . a   t R 2 2α α Eα − 1−α = √ 1+α (t − τ ) f (κ) (τ ) dτ α

(3.9)

π (1−α) a

Definition 3.3 Let α ∈ R, κ = [α] + 1, κ < α < κ + 1 and λ ∈ C. The left-sided general fractional derivative involving the kernel of Mittag-LefflerGauss function with power law in the sense of Liouville-Sonine-Caputo type is defined as LSC α,κ,λ M lt Da+ f

Zt (t) =

  2α E2α −λ (t − τ ) f (κ) (τ ) dτ ,

(3.10)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler-Gauss function with power law in the sense of Liouville-Sonine-Caputo type as LSC α,κ,λ M lt Db− f

κ

Zb

(t) = (−1)

  2α E2α −λ (τ − t) f (κ) (τ ) dτ .

(3.11)

t

Definition 3.4 The left-sided general fractional derivative involving the kernel of Mittag-Leffler-Gauss function with power law in the sense of RiemannLiouville type is defined as α,κ,λ RL M lt Da+ f

dκ (t) = κ dt

Zt

  2α f (τ ) dτ , E2α −λ (t − τ )

(3.12)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler-Gauss function with power law in the sense of Riemann-Liouville type is defined as α,κ,λ RL M lt Db− f

dκ dtκ

Zb

  2α E2α −λ (τ − t) f (τ ) dτ .

(3.13)

  2α α,κ,λ (t) = RL D f (t) + E −λ (t − a) f (a) 2α M lt a+

(3.14)

κ

(t) = (−1)

t

For κ = 1 we have: LSC α,κ,λ M lt Da+ f

148

General Fractional Derivatives: Theory, Methods and Applications

and   2α α,κ,λ (t) = RL f (b) . M lt Db− f (t) + E2α −λ (b − t)

LSC α,κ,λ M lt Db− f

(3.15)

Definition 3.5 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional integral, as the inverse operator of the general fractional derivative involving the kernel of Mittag-Leffler-Gauss function with power law, is defined as RL α,κ,λ M lt Ia+ f

Zt (t) = a

1 (t − τ )

  2α −1 E −λ (t − τ ) f (τ ) dτ , 2α,κ−2α 2α+1−κ

(3.16)

and the right-sided general fractional integral as RL α,κ,λ f M lt Ib−

(t) = (−1)

κ

Zb t

1

  2α −1 E −λ (τ − t) f (τ ) dτ . 2α+1−κ 2α,κ−2α (τ − t) (3.17)

Property 3.1 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C. Then we have o n
−1 α,κ,λ κ−1 1 + λs−2α f (s) , (3.18) L RL M lt D0+ f (t) = s

L

n

LSC α,κ,λ M lt D0+ f

o

(t) = s−1

  κ X
−1 sκ f (s) − sκ−j f (j−1) (0) 1 + λs−2α j=1

(3.19) and

n

RL α,κ,λ f M lt I0+

L

Note that, when Dκ L and when Dκ





RL α,κ,λ f M lt I0+

n

α,κ,λ RL M lt D0+ f

RL α,κ,λ f M lt I0+

L

n

κ−1 X

(3.20)

 (t) |t=0 = 0, there is

o
−1 (t) = sκ−1 1 + λs−2α f (s) .

(3.21)

 (t) |t=0 6= 0, there is

α,κ,λ RL M lt D0+ f

−

o
(t) = s1−κ 1 + λs−2α f (s) .

o
−1 (t) = sκ−1 1 + λs−2α f (s)

i  h α,κ,λ f (0+) . sκ−µ−1 Dµ RL I M lt 0+

(3.22)

κ=0

In particular, when κ = 1, we have n o
−1 α,1,λ α,1,λ L RL D f (t) = 1 + λs−2α f (s) − RL M lt 0+ M lt I0+ f (0+) .

(3.23)

General Fractional Derivatives of Constant Order and Applications

149

Property 3.2 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,κ,λ RL α,κ,λ RL (3.24) M lt Da+ M lt Ia+ f (t) = f (t) , and



α,κ,λ RL α,κ,λ RL f M lt D+ M lt I+



(t) = f (t) .

(3.25)

Property 3.3 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ AC κ (a, b). Then we have   LCS α,κ,λ RL α,κ,λ D I f (t) = f (t) , (3.26) M lt a+ M lt a+ and

3.1.3



LCS α,κ,λ RL α,κ,λ f M lt Db− M lt Ib−



(t) = f (t) .

(3.27)

General fractional derivatives involving the kernel of Mittag-Leffler function with power law

The definitions and properties of the general fractional derivatives involving the kernel of Mittag-Leffler function with power law are given as follows. Definition 3.6 Let α ∈ R, κ = [α] + 1, κ < α < κ + 1, υ ∈ C, λ ∈ C and Re (υ) > 0. The left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type is defined as (for λ = ±1, see [116]) LSC α,κ,λ M l Da+ f

Zt (t) =

α

Eα (−λ (t − τ ) ) f (κ) (τ ) dτ ,

(3.28)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Liouville-Sonine-Caputo type as (for λ = ±1, see [116]) LSC α,κ,λ M l Db− f

κ

Zb

(t) = (−1)

α

Eα (−λ (τ − t) ) f (κ) (τ ) dτ .

(3.29)

t

The left-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as (for λ = ±1, see [116, 301] and for κ = 1, see [37]) RL α,κ,λ M l Da+ f

dκ (t) = κ dt

Zt

α

Eα (−λ (t − τ ) ) f (τ ) dτ , a

(3.30)

150

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as (for λ = ±1, see [116]) RL α,κ,λ M l Db− f

Zb

dκ (t) = (−1) dtκ κ

α

Eα (−λ (τ − t) ) f (τ ) dτ .

(3.31)

t

When κ = 1, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Riemann-Liouville type is defined as (see [301, 302]) RL α,λ M l Da+ f

(t) =

d dt

Rt

α

Eα (−λ (t − τ ) ) f (τ ) dτ ,

(3.32)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as RL α,λ M l Db− f

Rb

d (t) = − dt

α

Eα (−λ (τ − t) ) f (τ ) dτ .

(3.33)

t

This is also called the Hille-Tamarkin general fractional derivative (see [301, 302]). Here, for κ = 1 we have (see [302]) LSC α,1,λ M l Da+ f

α

(3.34)

α

(3.35)

α,1,λ (t) = RL M l Da+ f (t) + Eα (−λ (t − a) ) f (a)

and LSC α,κ,λ M l Db− f

α,κ,λ (t) = LSC M l Db− f (t) + Eα (−λ (b − t) ) f (b) .

In particular, when κ = 1 and a = 0, we have d dt

Zt

Zt

α

Eα (−λ (t − τ ) ) f (τ ) dτ = 0

α

Eα (−λ (t − τ ) )f (1) (τ )dτ −Eα (−λtα )f (0) ,

0

(3.36) such that, by using the expression Zt

α

Eα (−λτ ) f

(1)

Zt (t − τ ) dτ =

0

α

Eα (−λ (t − τ ) ) f (1) (τ ) dτ ,

(3.37)

0

there is the Gorenflo-Mainardi formulation (see [303]) d dt

Zt

α

Zt

Eα (−λ (t − τ ) ) f (τ ) dτ = 0

Eα (−λτ α ) f (1) (t − τ ) dτ +Eα (−λtα ) f (0) .

0

(3.38)

General Fractional Derivatives of Constant Order and Applications

151

In particular, when κ = 1, J (α) = =(α) 1−α , where = (α) is the normalization α , the constant with the conditions = (0) = 1 and = (1) = 1, and λ = 1−α left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type is defined as (see [105]) LSC α,λ M lA Da+ f

= (α) (t) = 1−α

Zt

  α α Eα − (t − τ ) f (1) (τ ) dτ , 1−α

(3.39)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Liouville-Sonine-Caputo type as   Zb = (α) α α LSC α,λ Eα − (τ − t) f (1) (τ ) dτ . (3.40) M lA Db− f (t) = 1−α 1−α t

The left-sided general fractional derivative involving the kernel of Prabhakar function with power law in the sense of Riemann-Liouville type is defined as (see [105]) α,κ,λ RL M lA Da+ f

d (t) = J (α) dt

Zt

α

Eα (−λ (t − τ ) ) f (τ ) dτ ,

(3.41)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as (see [105]) α,κ,λ RL M lA Db− f

d (t) = (−1) J (α) dt κ

Zb

α

Eα (−λ (τ − t) ) f (τ ) dτ .

(3.42)

t

In particular, when κ > 1, J (α) = =(α) 1−α , where = (α) is the normalization α , the constant with the conditions = (0) = 1 and = (1) = 1, and λ = 1−α left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type is defined as (see [105]) LSC α,κ,λ M lA Da+ f

= (α) (t) = 1−α

Zt

  α α Eα − (t − τ ) f (κ+1) (τ ) dτ 1−α

(3.43)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Liouville-Sonine-Caputo type as   Zb = (α) α α LSC α,κ,λ Eα − (τ − t) f (1+κ) (τ ) dτ . (3.44) M lA Db− f (t) = 1−α 1−α t

152

General Fractional Derivatives: Theory, Methods and Applications

The left-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as (see [105]) α,κ,λ RL M lA Da+ f

dκ+1 (t) = J (α) κ+1 dt

Zt

α

Eα (−λ (t − τ ) ) f (τ ) dτ ,

(3.45)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as (see [105]) α,κ,λ RL M lA Db− f

dκ+1 (t) = (−1) J (α) κ+1 dt κ

Zb

α

Eα (−λ (τ − t) ) f (τ ) dτ .

(3.46)

t

In fact, for α > 0, 1+κ > α > κ, κ = [α]+1, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral, as the inverse operator of the general fractional derivative involving the kernel of Mittag-Leffler function with power law, is defined as RL α,κ,λ M l Ia+ f

Zt (t) =

1

−1 α+1−κ Eα,κ−α

(t − τ )

a

α

(−λ (t − τ ) ) f (τ ) dτ ,

(3.47)

and the right-sided general fractional integral as RL α,κ,λ f M l Ib−

κ

Zb

(t) = (−1)

t

1

−1 α+1−κ Eα,κ−α

(τ − t)

α

(−λ (τ − t) ) f (τ ) dτ . (3.48)

Definition 3.7 For a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Riemann-Liouville type is defined as RL α,κ,λ f M l D+

Zt

dκ (t) = κ dt

α

Eα (−λ (t − τ ) ) f (τ ) dτ ,

(3.49)

−∞

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type is defined as RL α,κ,λ f M l D−

dκ (t) = (−1) dtκ κ

Z∞

α

Eα (−λ (τ − t) ) f (τ ) dτ . t

(3.50)

General Fractional Derivatives of Constant Order and Applications

153

Definition 3.8 For a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type is defined as LSC α,κ,λ f M l D+

Zt (t) =

α

Eα (−λ (t − τ ) ) f (κ) (τ ) dτ ,

(3.51)

−∞

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Liouville-Sonine-Caputo type as Z∞ κ α LSC α,κ,λ f (t) = (−1) Eα (−λ (τ − t) ) f (κ) (τ ) dτ . (3.52) M l D− t

Definition 3.9 For a = −∞ and b = ∞, the left-sided general fractional integral, as the inverse operator of the general fractional derivative involving the kernel of Mittag-Leffler function with power law, is defined as RL α,κ,λ f M l I+

Zt (t) = −∞

1

−1 α+1−κ Eα,κ−α

(t − τ )

α

(−λ (t − τ ) ) f (τ ) dτ ,

(3.53)

and the right-sided general fractional integral as RL α,κ,λ f M l I−

κ

−∞ Z

(t) = (−1)

t

1

−1 α+1−κ Eα,κ−α

(τ − t)

α

(−λ (τ − t) ) f (τ ) dτ . (3.54)

The properties of the general fractional derivative involving the kernel of Mittag-Leffler function with power law are given as follows. Property 3.4 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C. Then we have n o
−1 α,κ,λ κ−1 L RL 1 + λs−α f (s) , (3.55) M l D0+ f (t) = s   κ o X
−1 α,κ,λ −1 sκ f (s) − sκ−j f (j−1) (0) L LSC 1 + λs−α M l D0+ f (t) = s n

j=1

(3.56) and

n

o
α,κ,λ L RL I f (t) = s1−κ 1 + λs−α f (s) . M l 0+

Note that, when Dκ L

n



RL α,κ,λ f M l I0+

RL α,κ,λ M l D0+ f

(3.57)

 (t) |t=0 = 0, there is

o
−1 (t) = sκ−1 1 + λs−α f (s)

(3.58)

154

General Fractional Derivatives: Theory, Methods and Applications   α,κ,λ and when Dκ RL I f (t) |t=0 6= 0, there is M l 0+ L

n

RL α,κ,λ M l D0+ f

−

κ−1 X

o
−1 f (s) (t) = sκ−1 1 + λs−α

 h i α,κ,λ f (+0) . sκ−µ−1 Dµ RL M l I0+

(3.59)

κ=0

In particular, when κ = 1, we have n o
−1 α,1,λ α,1,λ L RL f (s) − RL D f (t) = 1 + λs−α M l 0+ M l I0+ f (+0) .

(3.60)

Property 3.5 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   RL α,κ,λ RL α,κ,λ (3.61) M l Da+ M l Ia+ f (t) = f (t) , 

RL α,κ,λ RL α,κ,λ f M l I+ M l D+



(t) = f (t) ,  RL α,κ,λ RL α,κ,λ f (t) = f (t) M l Db− M l Ib−

(3.62)

 and



RL α,κ,λ RL α,κ,λ f M l D− M l I−



(3.63)

(t) = f (t) .

(3.64)

Property 3.6 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ AC κ (a, b). Then we have   LCS α,κ,λ RL α,κ,λ I f (t) = f (t) , (3.65) D Ml a+ M l a+ 

LCS α,κ,λ RL α,κ,λ f M l Db− M l Ib−



(t) = f (t) .   LCS α,κ,λ RL α,κ,λ f (t) = f (t) M l D+ M l I+

and



LCS α,κ,λ RL α,κ,λ f M l D− M l I−



(t) = f (t) .

(3.66) (3.67) (3.68)

For more details of the general fractional derivatives involving the kernel of Mittag-Leffler function, see [297, 298, 299, 300, 301, 302, 303, 304, 305].

3.1.4

General fractional derivatives involving the kernel of Mittag-Leffler function with the negative power law

Definition 3.10 For α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral is defined as RL α,κ,λ M lr Ia+ f

Zt (t) = a

1 |t − τ |

−1 α+1−κ Eα,κ−α

α

(−λ |t − τ | ) f (τ ) dτ ,

(3.69)

General Fractional Derivatives of Constant Order and Applications

155

and the right-sided general fractional integral with the negative power law as RL α,κ,λ f M lr Ib−

Zb (t) = t

1

−1 α+1−κ Eα,κ−α

|τ − t|

α

(−λ |τ − t| ) f (τ ) dτ .

(3.70)

Definition 3.11 For α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of RiemannLiouville type is defined as α,κ,λ RL M lr Da+ f

dκ (t) = κ dt

Zt

α

Eα (−λ |t − τ | ) f (τ ) dτ ,

(3.71)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with the negative power law in the sense of Riemann-Liouville type is defined as α,κ,λ RL M lr Db− f

dκ (t) = (−1) dtκ κ

Zb

α

Eα (−λ |τ − t| ) f (τ ) dτ .

(3.72)

t

Definition 3.12 For α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of LiouvilleSonine-Caputo type is defined as LSC α,κ,λ M lr Da+ f

Zt (t) =

α

Eα (−λ |t − τ | ) f (κ) (τ ) dτ ,

(3.73)

a

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with the negative power law in the sense of Liouville-SonineCaputo type as LSC α,κ,λ M lr Db− f

κ

Zb

(t) = (−1)

α

Eα (−λ |τ − t| ) f (κ) (τ ) dτ .

(3.74)

t

In particular, for κ = 1 we have: α

LSC α,1,λ M lr Da+ f

α,1,λ (t) =RL M lr Da+ f (t) + Eα (−λ |t − a| ) f (a)

LSC α,1,λ M lr Db− f

α,1,λ (t) =RL M lr Db− f (t) + Eα (−λ |b − t| ) f (b) .

(3.75)

and α

(3.76)

156

General Fractional Derivatives: Theory, Methods and Applications

For α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral with the negative power law is defined as RL α,κ,λ f M lr I+

Zt

1

−1 α+1−κ Eα,κ−α

(t) =

|t − τ |

−∞

α

(−λ |t − τ | ) f (τ ) dτ ,

(3.77)

and the right-sided general fractional integral with power law as RL α,κ,λ f M lr I−

Z∞ (t) = t

1

−1 α+1−κ Eα,κ−α

|τ − t|

α

(−λ |τ − t| ) f (τ ) dτ .

(3.78)

Definition 3.13 Fr a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of Riemann-Liouville type is defined as α,κ,λ RL f M lr D+

Zt

dκ (t) = κ dt

α

Eα (−λ |t − τ | ) f (τ ) dτ ,

(3.79)

−∞

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with the negative power law in the sense of Riemann-Liouville type is defined as α,κ,λ RL f M lr D−

dκ (t) = (−1) dtκ κ

Z∞

α

Eα (−λ |τ − t| ) f (τ ) dτ .

(3.80)

t

Definition 3.14 For a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of Liouville-Sonine-Caputo type is defined as LSC α,κ,λ f M lr D+

Zt (t) =

α

Eα (−λ |t − τ | ) f (κ) (τ ) dτ ,

(3.81)

−∞

and the right-sided general fractional derivative involving the kernel of MittagLeffler function with the negative power law in the sense of Liouville-SonineCaputo type as LSC α,κ,λ f M lr D−

κ

Z∞

(t) = (−1)

t

α

Eα (−λ |τ − t| ) f (κ) (τ ) dτ .

(3.82)

General Fractional Derivatives of Constant Order and Applications

3.2

157

General fractional derivatives involving the kernel of Mittag-Leffler function in the sense of Riesz, Feller and Richard types

The general fractional calculus involving the kernel of Mittag-Leffler function in the sense of Riesz, Feller and Richard types is given as follows. In the above conditions, the general fractional integral in the sense of Riesz type is defined as Rze α,κ,λ M l I[a,b] f

= +

1 2 cos(πα/2)

Rb t

α,κ,λ α,κ,λ f +RL f )(t) (RL M l Ia+ M l Ib−

(t) = t R

2 cos(πα/2) 1 E −1 (t−τ )α+1−κ α,κ−α

a

1 E −1 (τ −t)α+1−κ α,κ−α

α

(−λ (t − τ ) ) f (τ ) dτ # α

(−λ (τ − t) ) f (τ ) dτ ,

the general fractional integral in the sense of Feller type as   Rzf α,κ,λ RL α,κ,λ RL α,κ,λ I f (t) = H (ϑ, α) I f + H (ϑ, α) I f (t) − − M l a+ M l b− M l [a,b] t R α −1 × (t−τ )1α+1−κ Eα,κ−α (−λ (t − τ ) ) f (τ ) dτ = sin((α−ϑ)π/2) sin(πϑ) a

+ sin((α+ϑ)π/2) × sin(πϑ)

Rb t

(3.83)

1 E −1 (τ −t)α+1−κ α,κ−α

(3.84)

α

(−λ (τ − t) ) f (τ ) dτ ,

and the general fractional integral in the sense of Richard type as Rzg α,κ,λ M l I[a,b] f

= +

(t) = t R

1 2 sin(πα/2)

Rb t

α,κ,λ α,κ,λ f +RL f )(t) (RL M l Ia+ M l Ib−

2 sin(πα/2)

1 E −1 (t−τ )α+1−κ α,κ−α

a

1 E −1 (τ −t)α+1−κ α,κ−α

α

(−λ (t − τ ) ) f (τ ) dτ #

(3.85)

α

(−λ (τ − t) ) f (τ ) dτ .

Definition 3.15 The general fractional derivatives in the sense of Riesz type are defined as Rzh α,κ,λ M l D[a,b] f

(t) = 

1 = − 2 cos(πα/2) κ dκ dtκ

+ (−1)

Rb t

α,κ,λ α,κ,λ f +RL f )(t) (RL M l Da+ M l Db−

2 cos(πα/2) κ

d dtκ

Rt a

α

Eν (−λ (t − τ ) ) f (τ ) dτ # α

Eα (−λ (τ − t) ) f (τ ) dτ ,

(3.86)

158

General Fractional Derivatives: Theory, Methods and Applications

and

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M l Db− (t) = − M l a+ 2 cos(πα/2) t R α 1 Eν (−λ (t − τ ) ) f (κ) (τ ) dτ = − 2 cos(πα/2) a # b κR α (κ) + (−1) Eα (−λ (τ − t) ) f (τ ) dτ ,

Rzh α,κ,λ M lr D[a,b] f

(3.87)

t

the general fractional derivative in the sense of Feller type as   RL α,κ,λ Rzj α,κ,λ RL α,κ,λ D f + H (ϑ, α) D f (t) f (t) = − H (ϑ, α) D − + a+ Ml Ml [a,b]   M l b− t R α dκ = − sin((α+ϑ)π/2) × dt Eν (−λ (t − τ ) ) f (τ ) dτ κ sin(πϑ) a ! b κ dκ R α sin((α−ϑ)π/2) × (−1) dtκ Eα (−λ (τ − t) ) f (τ ) dτ , − sin(πϑ) t

(3.88) and   LSC LSC α,κ,λ α,κ,λ (t) = − H+ (ϑ, α)M l Da+ f + H− (ϑ, α)M l Db− f (t) t  R α (κ) × E (−λ (t − τ ) ) f (τ ) dτ = − sin((α+ϑ)π/2) ν sin(πϑ) a ! b κR α sin((α−ϑ)π/2) (κ) − sin(πϑ) × (−1) Eα (−λ (τ − t) ) f (τ ) dτ , Rzj α,κ,λ M lr D[a,b] f

t

(3.89) and the general fractional derivative in the sense of Richard type as Rzl α,κ,λ M l D[a,b] f

=

(t) = 

1 − 2 sin(πα/2) κ

κ d dtκ

+ (−1) and

Rb

α,κ,λ α,κ,λ f +RL f )(t) (RL M l Da+ M l Db−

dκ dtκ

2 cos(πα/2)

Rt a

α

Eν (−λ (t − τ ) ) f (τ ) dτ #

(3.90)

α

Eα (−λ (τ − t) ) f (τ ) dτ ,

t

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M l Db− (t) = − M l a+ 2 cos(πα/2) t R α 1 = − 2 sin(πα/2) Eν (−λ (t − τ ) ) f (κ) (τ ) dτ a # b κR α (κ) + (−1) Eα (−λ (τ − t) ) f (τ ) dτ .

α,κ,λ Rzl M lr D[a,b] f

t

(3.91)

General Fractional Derivatives of Constant Order and Applications

159

In another way, the general fractional integral in the sense of Riesz type is defined as Rze α,κ,λ M lu I[a,b] f

= +

1 2 cos(πα/2)

Rb t

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr Ia+ M lr Ib−

(t) = t R a

2 cos(πα/2) α

1

|t−τ |α+1−κ

1 E −1 |t−τ |α+1−κ α,κ−α

−1 Eα,κ−α (−λ |t − τ | ) f (τ ) dτ # α

(−λ |t − τ | ) f (τ ) dτ ,

the general fractional integral in the sense of Feller type as   RL RL Rzf α,κ,λ α,κ,λ α,κ,λ f (t) M ly I[a,b] f (t) = H− (ϑ, α)M lr Ia+ f + H− (ϑ, α)M lr Ib− Rt α 1 E −1 (−λ |t − τ | ) f (τ ) dτ = sin((α−ϑ)π/2) × sin(πϑ) |t−τ |α+1−κ α,κ−α a

+ sin((α+ϑ)π/2) × sin(πϑ)

Rb t

(3.92)

1 E −1 |t−τ |α+1−κ α,κ−α

(3.93)

α

(−λ |t − τ | ) f (τ ) dτ ,

and the general fractional integral in the sense of Richard type as Rzg α,κ,λ M lh I[a,b] f

= +

(t) = t R

1 2 sin(πα/2)

Rb t

a

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr Ia+ M lr Ib−

2 sin(πα/2) α

1

|t−τ |α+1−κ

1 E −1 |t−τ |α+1−κ α,κ−α

−1 Eα,κ−α (−λ |t − τ | ) f (τ ) dτ #

(3.94)

α

(−λ |t − τ | ) f (τ ) dτ .

Definition 3.16 The general fractional derivatives in the sense of Riesz type are defined as Rzh α,κ,λ M lu D[a,b] f

=

1 − 2 cos(πα/2) κ dκ dtκ

+ (−1) and

(t) =  Rb

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr Da+ M lr Db−

dκ dtκ

2 cos(πα/2)

Rt a

α

Eν (−λ |t − τ | ) f (τ ) dτ #

(3.95)

α

Eα (−λ |t − τ | ) f (τ ) dτ ,

t

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M lr Db− (t) = − M lr a+ 2 cos(πα/2) t R α 1 = − 2 cos(πα/2) Eν (−λ (t − τ ) ) f (κ) (τ ) dτ a # b κR α (κ) + (−1) Eα (−λ (τ − t) ) f (τ ) dτ ,

Rzh α,κ,λ M lv D[a,b] f

t

(3.96)

160

General Fractional Derivatives: Theory, Methods and Applications

the general fractional derivative in the sense of Feller type as   RL RL α,κ,λ α,κ,λ Rzj α,κ,λ f (t) = − H (ϑ, α) D f + H (ϑ, α) D f (t) D + − a+ M lr M ly [a,b]   M lr b− t R κ α d = − sin((α+ϑ)π/2) × dt Eν (−λ |t − τ | ) f (τ ) dτ κ sin(πϑ) a ! b α κ dκ R sin((α−ϑ)π/2) − sin(πϑ) × (−1) dtκ Eα (−λ |t − τ | ) f (τ ) dτ , t

(3.97) and   LSC α,κ,λ α,κ,λ (t) = − H+ (ϑ, α)M lr Da+ f + H− (ϑ, α) LSC D f (t) M lr b− t  R α × Eν (−λ |t − τ | ) f (κ) (τ ) dτ = − sin((α+ϑ)π/2) sin(πϑ) a ! b κR α sin((α−ϑ)π/2) (κ) − sin(πϑ) × (−1) Eα (−λ |t − τ | ) f (τ ) dτ , Rzj α,κ,λ M lz D[a,b] f

t

(3.98) and the general fractional derivative in the sense of Richard type as α,κ,λ Rzl M lh D[a,b] f

=

(t) = 

1 − 2 sin(πα/2) κ dκ dtκ

+ (−1)

Rb

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr Da+ M lr Db−

dκ dtκ

2 cos(πα/2)

Rt a

α

Eν (−λ |t − τ | ) f (τ ) dτ #

(3.99)

α

Eα (−λ |t − τ | ) f (τ ) dτ ,

t

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M lr Db− (t) = − M lr a+ 2 cos(πα/2) t R α 1 Eν (−λ |t − τ | ) f (κ) (τ ) dτ = − 2 sin(πα/2) a # b κR α (κ) + (−1) Eα (−λ |t − τ | ) f (τ ) dτ .

α,κ,λ Rzl M lj D[a,b] f

(3.100)

t

When a = −∞ and b = ∞, the general fractional integral in the sense of Riesz type is defined as α,κ,λ f )(t) (RL I α,κ,λ f +RL M lr I− Rze α,κ,λ (t) = M lr + 2 cos(πα/2) M lr I[−∞,∞] f " Rt α 1 1 = 2 cos(πα/2) E −1 (−λ |t − τ | ) f |t−τ |α+1−κ α,κ−α −∞  −∞ R α −1 1 + E (−λ |t − τ | ) f (τ ) dτ , |t−τ |α+1−κ α,κ−α t

(τ ) dτ

(3.101)

General Fractional Derivatives of Constant Order and Applications

161

the general fractional integral in the sense of Feller type as   RL RL α,κ,λ α,κ,λ Rzf α,κ,λ f (t) = H (ϑ, α) I f + H (ϑ, α) I f (t) I − − + − M lr M lr M lr [−∞,∞] t R α 1 = sin((α−ϑ)π/2) × E −1 (−λ |t − τ | ) f (τ ) dτ sin(πϑ) |t−τ |α+1−κ α,κ−α −∞

+ sin((α+ϑ)π/2) sin(πϑ)

×

R∞ t

1 E −1 |t−τ |α+1−κ α,κ−α

α

(−λ |t − τ | ) f (τ ) dτ , (3.102)

and the general fractional integral in the sense of Richard type as α,κ,λ α,κ,λ f +RL f )(t) (RL Rzg α,κ,λ M lr I+ M lr I− M lr I[−∞,∞] f"(t) = 2 sin(πα/2) Rt α 1 1 E −1 (−λ |t − τ | ) f = 2 sin(πα/2) |t−τ |α+1−κ α,κ−α −∞  R∞ α −1 1 + |t−τ |α+1−κ Eα,κ−α (−λ |t − τ | ) f (τ ) dτ . t

(τ ) dτ

(3.103)

Definition 3.17 The general fractional derivative in the sense of Riesz type are defined as Rzh α,κ,λ M lr D[−∞,∞] f 1 = − 2 cos(πα/2) κ dκ dtκ

+ (−1) and

(t) = " κ

d dtκ

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr D+ M lr D−

2 cos(πα/2)

Rt

α

Eν (−λ |t − τ | ) f (τ ) dτ  R∞ α Eα (−λ |t − τ | ) f (τ ) dτ , −∞

(3.104)

t

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M lr D− (t) = − M lr + 2 cos(πα/2) " Rt α 1 = − 2 cos(πα/2) Eν (−λ |t − τ | ) f (κ) (τ ) dτ −∞  ∞ κ R α (κ) + (−1) Eα (−λ |t − τ | ) f (τ ) dτ ,

Rzh α,κ,λ M lr D[−∞,∞] f

(3.105)

t

the general fractional derivative in the sense of Feller type as   RL Rzj α,κ,λ α,κ,λ α,κ,λ f (t) f + H− (ϑ, α) RL M lr D− M l D[−∞,∞] f (t) = − H+ (ϑ, α)M lr D+ ! t R κ α d = − sin((α+ϑ)π/2) × dt Eν (−λ |t − τ | ) f (τ ) dτ κ sin(πϑ) −∞   ∞ κ dκ R α − sin((α−ϑ)π/2) × (−1) E (−λ |t − τ | ) f (τ ) dτ , α sin(πϑ) dtκ t

(3.106)

162

General Fractional Derivatives: Theory, Methods and Applications

and   LSC LSC α,κ,λ α,κ,λ (t) = − H+ (ϑ, α)M lr D+ f + H− (ϑ, α)M lr D− f (t) ! Rt α sin((α+ϑ)π/2) (κ) = − sin(πϑ) Eν (−λ |t − τ | ) f (τ ) dτ ×  −∞ ∞  κ R α sin((α−ϑ)π/2) (κ) − sin(πϑ) × (−1) Eα (−λ |t − τ | ) f (τ ) dτ , Rzj α,κ,λ M lr D[−∞,∞] f

t

(3.107) and the general fractional derivative in the sense of Richard type as α,κ,λ Rzl M lr D[−∞,∞] f 1 = − 2 sin(πα/2) κ dκ dtκ

+ (−1) and

(t) = " κ

d dtκ

α,κ,λ α,κ,λ f +RL f )(t) (RL M lr D+ M lr D−

2 cos(πα/2)

Rt

α

Eν (−λ |t − τ | ) f (τ ) dτ  R∞ α Eα (−λ |t − τ | ) f (τ ) dτ , −∞

(3.108)

t

α,κ,λ f )(t) (LSC Dα,κ,λ f +LSC M lr Db− = − M lr a+ 2 cos(πα/2) " Rt α 1 = − 2 sin(πα/2) Eν (−λ |t − τ | ) f (κ) (τ ) dτ −∞  ∞ κ R α + (−1) Eα (−λ |t − τ | ) f (κ) (τ ) dτ .

α,κ,λ Rzl M lr D[−∞,∞]

(3.109)

t

3.3

General fractional derivatives involving the kernel of Mittag-Leffler function with respect to another function

The definitions and properties of the general fractional derivatives involving the kernel of Mittag-Leffler function with respect to another function are presented as follows. Definition 3.18 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and h(α) (t) > 0. The left-sided fractional integral with respect to another function, as the inverse operator of the general fractional derivative involving the kernel of Mittag-Leffler function with power law, is defined as RL α,κ,λ M l Ia+,h f Rt

=

a

(t)

1 E −1 (h(t)−h(τ ))α+1−κ α,κ−α

α

(−λ (h (t) − h (τ )) ) f (τ ) h(1) (τ ) dτ ,

(3.110)

General Fractional Derivatives of Constant Order and Applications

163

and the right-sided fractional integral with respect to another function as RL α,κ,λ M l Ib−,h f (t) b κR 1 = (−1) E −1 (h(τ )−h(t))α+1−κ α,κ−α t

α

(−λ (h (τ ) − h (t)) ) f (τ ) h(1) (τ ) dτ , (3.111)

respectively. Definition 3.19 The left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Riemann-Liouville type with respect to another function is defined as κ Rt  α d 1 RL α,κ,λ Eα (−λ (h (t) − h (τ )) ) f (τ ) h(1) (τ ) dτ , M l Da+,h f (t) = h(1) (t) dt a

(3.112) and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Riemann-Liouville type with respect to another function as RL α,κ,λ M l Db−,h f

(t) = (−1)

κ



1

d (1) h (t) dt

κ Zb

α

Eα (−λ (h (τ )−h (t)) ) f (τ ) h(1) (τ ) dτ .

t

(3.113) Definition 3.20 In a similar way, the left-sided general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type with respect to another function is defined as  κ Zt 1 (1) α LSC α,κ,λ D f (t) = E (−λ (h (t) − h (τ )) ) f (τ ) (τ ) h(1) (τ ) dτ , α Ml a+,h h(1) (τ ) h a

(3.114) and the right-sided general fractional derivative involving the kernel of MittagLeffler function with power law in the sense of Liouville-Sonine-Caputo type with respect to another function as LSC α,κ,λ M l Db−,h f

(t)  Z∞ κ α = (−1) Eα (−λ (h (τ ) − h (t)) )

κ 1 (1) f (τ ) (τ ) h(1) (τ ) dτ , h(1) (τ ) h

t

(3.115) where



κ  κ (1) df (h(τ )) 1 1 f (τ ) (τ ) = , respectively. (1) (1) h dh(τ ) h (τ ) h (τ )

Property 3.7 Let α > 0,1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   RL α,κ,λ RL α,κ,λ (3.116) M l Da+,h M l Ia+,h f (t) = f (t) ,

164 and

General Fractional Derivatives: Theory, Methods and Applications 

RL α,κ,λ RL α,κ,λ M l Db−,h M l Ib−,h f



(t) = f (t) .

(3.117)

Property 3.8 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ AC κ (a, b). Then we have   LCS α,κ,λ RL α,κ,λ (3.118) M l Da+,h M l Ia+,h f (t) = f (t) , and



LCS α,κ,λ RL α,κ,λ M l Db−,h M l Ib−,h f



(t) = f (t) .

(3.119)

For more details of the general fractional derivative involving the kernel of Mittag-Leffler function, see [297, 298, 299, 301, 302, 304].

3.4

General fractional derivatives involving the kernel of Kohlrausch-Williams-Watts function

The general fractional derivatives involving the kernel of Kohlrausch-WilliamsWatts function are considered in details. Definition 3.21 Let α > 0, 1 + κ > α > κ, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of RiemannLiouville type is defined as α,κ,λ RL Kww Da+ f

dκ (t) = κ dt

Zt

α

e−λ(t−τ ) f (τ ) dτ ,

(3.120)

a

and the right-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of RiemannLiouville type as α,κ,λ RL Kww Db− f

dκ (t) = (−1) dtκ κ

Zb

α

e−λ(τ −t) f (τ ) dτ .

(3.121)

t

Definition 3.22 The left-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of Liouville-Sonine-Caputo type is defined as α,κ,λ LSC Kww Da+ f

Zt (t) = a

α

e−λ(t−τ ) f (κ) (τ ) dτ ,

(3.122)

General Fractional Derivatives of Constant Order and Applications

165

and the right-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of LiouvilleSonine-Caputo type as α,κ,λ LSC Kww Db− f

(t) = (−1)

κ

Zb

α

e−λ(τ −t) f (κ) (τ ) dτ .

(3.123)

t

Definition 3.23 For α > 0, 1 + κ > α > κ, κ = [α] + 1, λ ∈ C, a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of RiemannLiouville type is defined as α,κ,λ RL f Kww D+

Zt

dκ (t) = κ dt

α

e−λ(t−τ ) f (τ ) dτ ,

(3.124)

−∞

and the right-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of RiemannLiouville type as α,κ,λ RL f Kww D−

dκ (t) = (−1) dtκ κ

Z∞

α

e−λ(τ −t) f (τ ) dτ .

(3.125)

t

Definition 3.24 For α > 0, 1 + κ > α > κ, κ = [α] + 1, λ ∈ C, a = −∞ and b = ∞, the left-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of LiouvilleSonine-Caputo type is defined as α,κ,λ LSC f Kww D+

Zt (t) =

α

e−λ(t−τ ) f (κ) (τ ) dτ ,

(3.126)

−∞

and the right-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of LiouvilleSonine-Caputo type as α,κ,λ LSC f Kww D−

(t) = (−1)

κ

Z∞

α

e−λ(τ −t) f (κ) (τ ) dτ .

(3.127)

t

When κ = 1, the relationships α,λ −λ(t−a) (t) =RL f (a) Kww Da+ f (t) + e

α,λ LS Kww Db− f

α,λ −λ(b−t) (t) =RL f (b) Kww Db− f (t) + e

and are valid.

α

α,λ LS Kww Da+ f

α

(3.128) (3.129)

166

General Fractional Derivatives: Theory, Methods and Applications

In fact, the left-sided general fractional derivative involving the kernel of Kohlrausch-Williams-Watts function with power law in the sense of LiouvilleSonine type, introduced by Sun, Hao, Zhang and Baleanu in 2017, can be represented in the form (see [113]; also see [299]). LS α,λ Ks Da+ f

Rt α LS α,1,λ (t) = J (α)Kww Da+ f (t) = J (α) e−λ(t−τ ) f (1) (τ ) dτ , a

(3.130) where J (β) =

3.4.1

Γ(1+α) (1−α)1/α

and λ =

α 1−α .

Special cases: the kernel of Gauss function

As the special case, when κ ≥ 0 and λ ∈ C, the left-sided generalized derivative involving the kernel of Gauss function in the sense of Liouville-Sonine-Caputo type is defined as LSC κ,λ Gt Da+ f

Zt (t) =

2

e−λ(t−τ ) f (κ) (τ ) dτ ,

(3.131)

a

and the right-sided generalized derivative involving the kernel of Gauss function in the sense of Liouville-Sonine-Caputo type as LSC κ,λ Gt Db− f

κ

Zb

(t) = (−1)

2

e−λ(τ −t) f (κ) (τ ) dτ .

(3.132)

t

Definition 3.25 The left-sided generalized derivative involving the kernel of Gauss function in the sense of Riemann-Liouville type is defined as RL κ,λ Gt Da+ f

dκ (t) = κ dt

Zt

2

e−λ(t−τ ) f (τ ) dτ ,

(3.133)

a

and the right-sided generalized derivative involving the kernel of Gauss function in the sense of Riemann-Liouville type as RL κ,λ Gt Db− f

dκ (t) = (−1) dtκ κ

Zb

2

e−λ(τ −t) f (τ ) dτ .

(3.134)

t

When κ = 1, there are the relationships as follows: 2

RL κ,λ Gt Da+ f

κ,λ (t) =LSC Da+ f (t) − e−λ(t−a) f (a) , Gt

RL 1,λ Gt Db− f

1,λ (t) =LSC Db− f (t) − e−λ(b−t) f (b) . Gt

and

2

(3.135) (3.136)

General Fractional Derivatives of Constant Order and Applications

167

Definition 3.26 Let κ ≥ 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided generalized integral, as the inverse operator of the generalized derivative involving the kernel of Gauss function with power law, is defined as RL κ,λ Gt Ia+ f

Zt (t) =

1 (t − τ )

a

  2 −1 E −λ (t − τ ) f (τ ) dτ , 2,κ−2 3−κ

(3.137)

and the right-sided generalized integral as RL κ,λ Gt Ib− f

κ

Zb

(t) = (−1)

t

1

  2 −1 E −λ (τ − t) f (τ ) dτ . 3−κ 2,κ−2 (τ − t)

(3.138)

Property 3.9 Let κ ≥ 0, −∞ < a < b < ∞ and λ ∈ C. Then we have n o
−1 κ,λ L RL D f (t) = sκ−1 1 + λs−2 f (s) , (3.139) Gt 0+

L

n

LSC κ,λ Gt D0+ f

o

(t) = s−1

  κ X
−1 sκ f (s) − 1 + λs−2 sκ−j f (j−1) (0) j=1

(3.140) and L

n

RL κ,λ Gt I0+ f

Note that, when Dκ L and when Dκ

L

n

RL κ,λ Gt D0+ f





n


(t) = s1−κ 1 + λs−2 f (s) .

RL κ,λ Gt I0+ f

RL α,κ,λ M l D0+ f

RL κ,λ Gt I0+ f

o

(3.141)

 (t) |t=0 = 0, there is

o
−1 (t) = sκ−1 1 + λs−2 f (s)

(3.142)

 (t) |t=0 6= 0, there is

κ−1 o  h i X
−1 κ,λ (t) = sκ−1 1 + λs−2 f (s)− sκ−µ−1 Dµ RL . Gt I0+ f (+0) κ=0

(3.143) In particular, when κ = 1, we have o n
1,λ 1,λ −2 −1 L RL f (s) − RL Gt D0+ f (t) = 1 + λs Gt I0+ f (+0) .

(3.144)

Property 3.10 Let κ ≥ 0, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   RL κ,λ RL κ,λ (3.145) Gt Da+ Gt Ia+ f (t) = f (t) , and



RL κ,λ RL κ,λ Gt D+ Gt I+ f



(t) = f (t) .

(3.146)

168

General Fractional Derivatives: Theory, Methods and Applications

Property 3.11 Let κ ≥ 0, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ AC κ (a, b). Then we have   LCS κ,λ RL κ,λ (3.147) Gt Da+ Gt Ia+ f (t) = f (t) , and



LCS α,κ,λ RL α,κ,λ f Gt Db− Gt Ib−



(t) = f (t) .

(3.148)

In fact, when −∞ < a < b < ∞, λ ∈ C, f (t) ∈ AC (a, b), J (α) = 2 α √ 1+α , the left-sided generalized derivative involving the and λ = 1−α α π (1−α)

kernel of Gauss function in the sense of Liouville-Sonine-Caputo type can be expressed by (see [305]) LSC 1,λ Gtl Da+ f

Rt

Rt 2 2 (t) = J (α) e−λ(t−τ ) f (1) (τ ) dτ = √ 1+α α

π (1−α) a

a

α

2

e− 1−α (t−τ ) f (1) (τ ) dτ . (3.149)

3.4.2

Special cases: the kernel of Gaussian-like function

On one hand, when κ ≥ 0 and λ ∈ C, the left-sided general fractional derivative involving the kernel of Gaussian-like function in the sense of LiouvilleSonine-Caputo type is defined as LSC α,κ,λ Gtl Da+ f

Zt (t) =

2α

e−λ(t−τ ) f (κ) (τ ) dτ ,

(3.150)

a

and the right-sided general fractional derivative involving the kernel of Gaussian-like function in the sense of Liouville-Sonine-Caputo type as LSC α,κ,λ Gtl Db− f

κ

Zb

(t) = (−1)

2α

e−λ(τ −t) f (κ) (τ ) dτ .

(3.151)

t

On the other hand, the left-sided general fractional derivative involving the kernel of Gaussian-like function in the sense of Riemann-Liouville type can be expressed as Zt 2α dκ RL α,κ,λ e−λ(t−τ ) f (τ ) dτ , (3.152) Gtl Da+ f (t) = κ dt a

and the right-sided general fractional derivative involving the kernel of Gaussian-like function in the sense of Riemann-Liouville type as RL α,κ,λ Gtl Db− f

dκ (t) = (−1) dtκ κ

Zb t

2α

e−λ(τ −t) f (τ ) dτ .

(3.153)

General Fractional Derivatives of Constant Order and Applications

169

When κ = 1, the formulations 2α

RL α,1,λ Gtl Da+ f

α,1,λ −λ(t−a) (t) =LSC f (a) Gtl Da+ f (t) − e

RL α,1,λ Gtl Db− f

α,1,λ −λ(b−t) (t) =LSC f (b) Gtl Db− f (t) − e

and

2α

(3.154)

(3.155)

holds. As a special case, the left-sided general fractional derivative involving the kernel of Gaussian-like function in the sense of Liouville-Sonine type, introduced by Yang in 2017, is can be represented in the form ([299]) LSC α,λ Gtl Da+ f

Rt − α (t−τ )2α (1) α,1,λ 1−α f (τ ) dτ , (t) = J (α) LSC Gtl Da+ f (t) = J (α) e a

(3.156) 2

where J (α) = √ 1+α α

π (1−α)

3.5

and λ =

α 1−α .

General fractional derivatives in the Miller-Ross kernel

The definitions and properties of the general fractional derivatives in the Miller-Ross kernel and the general fractional integrals are presented as follows. Definition 3.27 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as 

α,λ RL M R Da+ f



(t) =

d dt



α,λ M R Ia+ f

 (t) =

d dt

Rt

α

Mα (−λ (t − τ ) ) f (τ ) dτ ,

a

(3.157) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as 

α,λ RL M R Db− f



d (t) = − dt

 Rb α α,λ d I f (t) = − dt Mα (−λ (τ − t) ) f (τ ) dτ , M R b−



t

(3.158) where the left-sided and right-sided general fractional integral operators within the Miller-Ross kernel are defined as α,λ M R Ia+ f

Zt

α

Mα (−λ (t − τ ) ) f (τ ) dτ

(t) = a

(3.159)

170

General Fractional Derivatives: Theory, Methods and Applications

and α,λ M R Ib− f

Zb

α

Mα (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.160)

t

respectively. Furthermore, when α ∈ C, Re (α) > 0, κ = [Re (α)]+1, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as 

α,λ RL M R Da+ f



(t) =

dκ dtκ



α,λ M R Ia+ f

 (t) =

dκ dtκ

Rt

α

Mα (−λ (t − τ ) ) f (τ ) dτ ,

a

(3.161) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as 

α,λ RL M R Db− f



d (t) = − dt


κ 

α,λ M R Ib− f

 b α κ dκ R Mα (−λ (τ − t) ) f (τ ) dτ. (t) = (−1) dt κ t

(3.162) On the other hand, when α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is defined as 



α,λ LS M R Da+ f


Rt α α,λ (t) =M R Ia+ f (1) (t) = Mα (−λ (t − τ ) ) f (1) (τ ) dτ , a

(3.163) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type as 

α,λ LS M R Db− f




Rb α α,λ (t) =M R Ib− −f (1) (t) = − Mα (−λ (τ − t) ) f (1) (τ ) dτ . t

(3.164) Furthermore, when α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type is defined as 

LSC α,λ M R Da+ f




Rt α α,λ (t) =M R Ia+ f (κ) (t) = Mα (−λ (t − τ ) ) f (κ) (τ ) dτ , a

(3.165) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type as 

LSC α,λ M R Db− f



b
κ κR α α,λ (t) = M R Ib− (−1) f (κ) (t) = (−1) Mα (−λ (τ − t) )f (κ) (τ ) dτ. t

(3.166) We now consider the inverse operator of the above general fractional derivatives within the Miller-Ross kernel as follows.

General Fractional Derivatives of Constant Order and Applications

171

Definition 3.28 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R M R Ia+ f



Zt (t) =

1 (t − τ )

a

α+2−κ E1,κ−α−1

(−λ (t − τ )) f (τ ) dτ ,

(3.167)

and the right-sided general fractional integral as 

α,λ R M R Ib− f

Zb



=

1

α+2−κ E1,κ−α−1

(t − τ )

t

(−λ (τ − t)) f (τ ) dτ .

(3.168)

In particular, when α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,λ R M R Ia+ f



Zt (t) = a

1

α+1 E1,−α

(t − τ )

(−λ (t − τ )) f (τ ) dτ ,

(3.169)

and the right-sided general fractional integral as 

α,λ R M R Ib− f



Zb = t

1

α+1 E1,−α

(τ − t)

(−λ (τ − t)) f (τ ) dτ .

(3.170)

As the direct results, the properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.12 If κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have  o n
−1 α,λ D f (t) = sκ−α−1 1 + λs−1 f (s) , (3.171) L RL M R 0+ L

n

α,λ RL M R D0+ f



o
−1 (t) = s−α 1 + λs−1 f (s) ,

(3.172)

o
−1 (t) = s−α−1 1 + λs−1 (sf (s) − f (0)) , (3.173)   κ n  o X
−1 α,λ −α−1 sκ f (s) − L LSC 1 + λs−1 sκ−j f (j−1) (0) , M R D0+ f (t) = s L

n

α,λ LS M R D0+ f



j=1

L

n

and L

α,λ R M R I0+ f

n



α,λ M R I0+ f

o

α+1−κ

(t) = s



−1

1 + λs




f (s)

o
(t) = sα+1 1 + λs−1 f (s) .

(3.174) (3.175) (3.176)

172

General Fractional Derivatives: Theory, Methods and Applications   α,λ Note that, when Dκ R I f (t) |t=0 = 0, there is M R 0+ L

and when Dκ

L

n

RL α,λ M R D0+ f



n

α,λ RL M R D0+ f

α,λ R M R I0+ f

o
−1 f (s) , (t) = sκ−γ 1 + λs−α

(3.177)

 (t) |t=0 6= 0, there is

κ−1 o i X κ−µ−1  µ hR α,λ
−1 (t) = sκ−γ 1 + λs−α f (s) − s D M R I0+ f (+0) . µ=0

(3.178)

In particular, when κ = 1, we have n o
−1 α,λ α,λ 1−γ L RL 1 + λs−α f (s) −R M R D0+ f (t) = s M R I0+ f (+0) , and when κ = 1 and D L



n

α,λ R M R I0+ f

α,λ RL M R D0+ f

(3.179)

 (t) |t=0 = 0, we have

o
−1 (t) = s1−γ 1 + λs−α f (s) .

(3.180)

Property 3.13 Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,λ R α,λ RL (3.181) M R Da+ M R Ia+ f (t) = f (t) , 

and

α,λ R α,λ RL M R Db− M R Ib− f



(t) = f (t) ,

(3.182)



α,λ R α,λ RL M R Da+ M R Ia+ f



(t) = f (t) ,

(3.183)



α,λ R α,λ RL M R Db− M R Ib− f



(t) = f (t) .

(3.184)

Property 3.14 If −∞ < a < b < ∞, λ ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,λ R α,λ LS M R Da+ M R Ia+ f (t) = f (t) , 

and

α,λ R α,λ LS M R Db− M R Ib− f

C and f (t)

∈

(3.185)



(t) = f (t) ,

(3.186)

(t) = f (t) ,

(3.187)

(t) = f (t) .

(3.188)



α,λ LSC α,λ R M R Da+ M R Ia+ f





α,λ LSC α,λ R M R Db− M R Ib− f



General Fractional Derivatives of Constant Order and Applications

173

Definition 3.29 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as     Rt α α,λ α,λ d d RL I f (t) = Mα (−λ (t − τ ) ) f (τ ) dτ , D f (t) = M R + MR + dt dt −∞

(3.189) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as     R∞ α α,λ α,λ d d RL Mα (−λ (τ − t) ) f (τ ) dτ , M R D− f (t) = − dt M R I− f (t) = − dt t

(3.190) where the left-sided and right-sided general fractional integral operators within the Miller-Ross kernel are defined as α,λ M R I+ f

Zt

α

Mα (−λ (t − τ ) ) f (τ ) dτ

(t) =

(3.191)

−∞

and α,λ M R I− f (t) =

Z∞

α

Mα (−λ (τ − t) ) f (τ ) dτ ,

(3.192)

t

respectively. Furthermore, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, a = −∞, b = ∞ and λ ∈ C, the left-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as     Rt α α,λ α,λ dκ dκ RL I f (t) = dt Mα (−λ (t − τ ) ) f (τ ) dτ , D f (t) = κ MR + dtκ M R + −∞

(3.193) and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as    ∞
 κ dκ R α α,λ α,λ d κ RL Mα (−λ (τ − t) )f (τ ) dτ. M R I− f (t) = (−1) dtκ M R D− f (t) = − dt t

(3.194) Furthermore, when α ∈ C, κ+1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C, the left-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type is defined as  
Rt α α,λ LSC α,λ f (κ) (t) = Mα (−λ (t − τ ) ) f (κ) (τ ) dτ M R D+ f (t) =M R I+ −∞

(3.195)

174

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type as   ∞
κ (κ) κR α α,λ LSC α,λ D f (t) = I (−1) f (t) = (−1) Mα (−λ (τ − t) ) f (κ) (τ ) dτ. M R − − MR t

(3.196) The inverse operators of the above fractional derivatives within the MillerRoss kernel are considered as follows. Definition 3.30 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R M R I+ f



Zt (t) = −∞

1 (t − τ )

α+2−κ E1,κ−α−1

(−λ (t − τ )) f (τ ) dτ

(3.197)

and the right-sided general fractional integral as 

α,λ R M R I− f



Z∞ =

1

α+2−κ E1,κ−α−1

t

(t − τ )

(−λ (τ − t)) f (τ ) dτ .

(3.198)

Definition 3.31 Let κ + 1 > α > κ, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R M R I+ f



Zt (t) = −∞

1 (t − τ )

α+2−κ E1,κ−α−1

(−λ (t − τ )) f (τ ) dτ

(3.199)

and the right-sided general fractional integral as 

R α,γ,λ f A I−



−∞ Z

= t

1 (t − τ )

α+2−κ E1,κ−α−1

(−λ (τ − t)) f (τ ) dτ .

(3.200)

We now give the properties of the general fractional derivatives and the general fractional integrals as follows. Property 3.15 Let κ + 1 > α > κ, a = −∞, b = ∞, λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,λ R α,λ RL (3.201) M R D+ M R I+ f (t) = f (t) ,

and



α,λ R α,λ RL M R D− M R I− f



(t) = f (t) ,

(3.202)



α,λ R α,λ RL M R D+ M R I+ f



(t) = f (t) ,

(3.203)



α,λ R α,λ RL M R D− M R I− f



(t) = f (t) .

(3.204)

General Fractional Derivatives of Constant Order and Applications

175

Property 3.16 If a = −∞, b = ∞, λ ∈ C and f (t) ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,λ R α,λ LS (3.205) M R D+ M R I+ f (t) = f (t) , 

and

3.6

α,λ R α,λ LS M R D− M R I− f



(t) = f (t) ,

(3.206)

(t) = f (t) ,

(3.207)

(t) = f (t) .

(3.208)



α,λ LSC α,λ R M R D+ M R I+ f





α,λ LSC α,λ R M R D− M R I− f



Hilfer type general fractional derivatives in the Miller-Ross kernel

The definitions of the Hilfer type general fractional derivatives containing the kernel of the Miller-Ross function are given as follows. Definition 3.32 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative containing the kernel of the Miller-Ross function is defined as α,β,λ Hi M R Da+ f



Rt

  α α,β f (τ ) dτ Mα (−λ (t − τ ) ) Da+ a    Rt β(1−α) d (1−β)(1−α) α = Mα (−λ (t − τ ) ) Ia+ I f (τ ) dτ , a+ dτ 

(t) =

(3.209)

a

and the right-sided Hilfer type general fractional derivative containing the kernel of the Miller-Ross function as   Rb α α,β (t) = − Mα (−λ (τ − t) ) Db− f (τ ) dτ t    Rb (1−β)(1−α) β(1−α) d α f (τ ) dτ . = − Mα (−λ (τ − t) ) Ib− dτ Ib−



α,β,λ Hi f M R Db−



(3.210)

t

Definition 3.33 Let κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative containing the kernel of the Miller-Ross function is defined as 

a



Rt

  α α,β,κ Mα (−λ (t − τ ) ) Da+ f (τ ) dτ a    Rt β(κ−α) dκ (1−β)(κ−α) α = Mα (−λ (t − τ ) ) Ia+ I f (τ ) dτ , a+ dτ κ Hir α,β,λ M R Da+ f

(t) =

(3.211)

176

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Hilfer type general fractional derivative containing the kernel of the Miller-Ross function as Rb

  α α,β,κ Mα (−λ (τ − t) ) Db− f (τ ) dτ t    b β(κ−α) dκ (1−β)(κ−α) κR α = (−1) Mα (−λ (τ − t) ) Ib− f (τ ) dτ . dτ κ Ib−



Hir α,β,λ f M R Db−



κ

(t) = (−1)

(3.212)

t

Definition 3.34 Let κ + 1 > α > κ, β = 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Liouville-Sonine-Caputo type general fractional derivative containing the kernel of the Miller-Ross function is defined as 

Hir α,1,λ M R Da+ f



(t) =

Rt

α

Mα (−λ (t − τ ) )

α,κ LSC Da+ f


(τ ) dτ

a

=

Rt

(3.213)

α

κ−α (κ) f Ia+

Mα (−λ (t − τ ) )




(τ ) dτ ,

a

and the right-sided Liouville-Sonine-Caputo type general fractional derivative containing the kernel of the Miller-Ross function as 

Hir α,1,λ M R Db− f κ

= (−1)

Rb t



κ

(t) = (−1)

Rb

α

Mα (−λ (τ − t) )

t

α,κ LSC Db− f


(τ ) dτ (3.214)

α


κ−α (κ) Mα (−λ (τ − t) ) Ib− f (τ ) (τ ) dτ .

Definition 3.35 Let κ + 1 > α > κ, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative containing the kernel of the Miller-Ross function is defined as 

Hir α,0,λ M R Da+ f



(t) =

Rt


α α,κ Mα (−λ (t − τ ) ) Da+ f (τ ) dτ

a

=

Rt

(3.215) dκ dτ κ

α

Mα (−λ (t − τ ) )

a

κ−α Ia+ f





(τ ) dτ

and the right-sided Riemann-Liouville type general fractional derivative containing the kernel of the Miller-Ross function as 

Hir α,0,λ M R Db− f κ

= (−1)

Rb t



κ

(t) = (−1)

Rb t

α

Mα (−λ (τ − t) )


α α,κ Mα (−λ (τ − t) ) Db− f (τ ) dτ (3.216) dκ dτ κ



κ−α Ib− f (τ ) dτ .

General Fractional Derivatives of Constant Order and Applications

3.7

177

General fractional derivatives in the one-parametric Lorenzo-Hartley kernel

The definitions and properties of the general fractional derivatives in the oneparametric Lorenzo-Hartley kernel and the general fractional integrals are given as follows. Definition 3.36 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and γ ∈ C. The left-sided Riemann-Liouville type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as
RL α,γ A Da+ f (t)

=

d dt

α,γ A Ia+ f


(t) =

d dt

Rt

α

Fα (−γ (t − τ ) ) f (τ ) dτ ,

(3.217)

a

and the right-sided Riemann-Liouville type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as
RL α,γ A Db− f (t)

d = − dt


Rb α d (t) = − dt Fα (−γ (τ − t) ) f (τ ) dτ ,

α,γ A Ib− f

t

(3.218) where the left-sided and right-sided general fractional integral operators involving the one-parametric Lorenzo-Hartley kernel are defined as α,γ A Ia+ f

Zt

α

(3.219)

α

(3.220)

Fα (−γ (t − τ ) ) f (τ ) dτ

(t) = a

and α,γ A Ib− f

Zb

Fα (−γ (τ − t) ) f (τ ) dτ ,

(t) = t

respectively. Definition 3.37 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and γ ∈ C. The left-sided Riemann-Liouville type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as
RL α,γ A Da+ f (t)

=

dκ dtκ

Rt

α

Fα (−γ (t − τ ) ) f (τ ) dτ ,

(3.221)

a

and the right-sided Riemann-Liouville type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as
RL α,γ A Db− f (t)

κ dκ dtκ

= (−1)

Rb t

α

Fα (−γ (τ − t) ) f (τ ) dτ .

(3.222)

178

General Fractional Derivatives: Theory, Methods and Applications

Definition 3.38 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and γ ∈ C. The left-sided Liouville-Sonine type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as
LS α,γ A Da+ f (t)


Rt α α,γ =A Ia+ f (1) (t) = Fα (−γ (t − τ ) ) f (1) (τ ) dτ ,

(3.223)

a

and the right-sided Liouville-Sonine type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as 

LS α,λ A Db− f




Rb α α,γ (t) =A Ib− −f (1) (t) = − Fα (− (τ − t) ) f (1) (τ ) dτ . t

(3.224) Definition 3.39 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and γ ∈ C. The left-sided Liouville-Sonine-Caputo type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as
LSC α,γ Da+ f (t) A

Rt

=

α

Fα (−γ (t − τ ) ) f (κ) (τ ) dτ ,

(3.225)

a

and the right-sided Liouville-Sonine-Caputo type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as 

LSC α,λ Db− f A



(t) = (−1)

κ

Rb

α

Fα (−γ (τ − t) ) f (κ) (τ ) dτ .

(3.226)

t

In this case, we give the following inverse operators: Definition 3.40 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and γ ∈ C. The left-sided general fractional integral in the complex parameters is defined as
R α,γ A Ia+ f (t)

Zt

κ−α−1

(t − τ )

=

α

−1 Eα,κ−α (−γ (t − τ ) ) f (τ ) dτ (t > a)

a

(3.227) and the right-sided general fractional integral in the complex parameters as
R α,γ A Ib− f

κ

Zb

κ−α−1

(τ − t)

= (−1)

α

−1 Eα,κ−α (−γ (τ − t) ) f (τ ) dτ (t < b) .

t

(3.228) Definition 3.41 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and γ ∈ C. The left-sided general fractional integral in the complex parameters is defined as Zt −1 α
Eα,1−α (−γ (t − τ ) ) R α,γ f (τ ) dτ (t > a) (3.229) α A Ia+ f (t) = (t − τ ) a

General Fractional Derivatives of Constant Order and Applications

179

and the right-sided general fractional integral in the complex parameters as
R α,γ A Ib− f

Zb =−

α

−1 Eα,1−α (−γ (τ − t) ) f (τ ) dτ (t < b) . α (τ − t)

(3.230)

t

The properties of the general fractional derivatives involving the oneparametric Lorenzo-Hartley kernel and the general fractional integrals are presented as follows. Property 3.17 If κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have L


RL α,γ A D0+ f (t)



= sκ−α 1 + γs−α


−1

f (s) ,

(3.231)


−1 = s1−α 1 + γs−α f (s) , (3.232) 
−1 α,γ
−α L LS 1 + γs−α (sf (s) − f (0)) , (3.233) A D0+ f (t) = s   κ X  LSC α,γ

−1 sκ f (s) − L A D0+ f (t) = s−α 1 + γs−α sκ−j f (j−1) (0) , L


RL α,γ A D0+ f (t)



j=1




R α,γ A I0+ f (t)


= sα−κ 1 + γs−α f (s)

(3.234) (3.235)



α,γ
A I0+ f (t)


= sα−1 1 + γs−α f (s) ,

(3.236)

L and L where


−1 L {Aα,γ (−t)} = s−α 1 + γs−α .
α,γ Note that, when Dκ R A I0+ f (t) |t=0 = 0, there is
−1 α,γ D0+ f (t) = sκ−α 1 + γs−α f (s) ,
R α,γ A I0+ f (t) |t=0 6= 0, there is L

and when Dκ L

RL A

RL A

(3.237)

(3.238)

κ−1 X
−1  α,γ 
α,γ D0+ f (t) = sκ−α 1 + γs−α f (s)− sκ−µ−1 Dµ R A I0+ f (+0) . µ=0

(3.239) In particular, when κ = 1, we have
−1 α,γ α,γ D0+ f (t) = s1−α 1 + γs−α f (s) −R A I0+ f (+0) ,
α,γ and when κ = 1 and D R A I0+ f (t) |t=0 = 0, we have L

RL A

L

RL A


−1 α,γ D0+ f (t) = s1−α 1 + γs−α f (s) .

(3.240)

(3.241)

180

General Fractional Derivatives: Theory, Methods and Applications

Property 3.18 Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have
RL α,γ R α,γ (3.242) A Da+ A Ia+ f (t) = f (t) ,
RL α,γ R α,γ (3.243) A Db− A Ib− f (t) = f (t) ,
RL α,γ R α,γ (3.244) A Da+ A Ia+ f (t) = f (t) , and
RL α,γ R α,γ A Db− A Ib− f (t)

= f (t) .

(3.245)

Property 3.19 If −∞ < a < b < ∞, γ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have
LS α,γ R α,γ (3.246) A Da+ A Ia+ f (t) = f (t) ,
LS α,γ R α,γ (3.247) A Db− A Ib− f (t) = f (t) ,
α,γ α,γ LSC Da+ R (3.248) A A Ia+ f (t) = f (t) , and
LSC α,γ R α,γ Db− A Ib− f (t) A

3.8

= f (t) .

(3.249)

Hilfer type general fractional derivatives involving the one-parametric Lorenzo-Hartley kernel

We now start with the definitions of the Hilfer type general fractional derivatives involving the one-parametric Lorenzo-Hartley kernel. Definition 3.42 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞ and γ ∈ C. The left-sided Hilfer type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as 



Rt

  α α,β Fα (−γ (t − τ ) ) Da+ f (τ ) dτ a    Rt β(1−α) d (1−β)(1−α) α = Fα (−γ (t − τ ) ) Ia+ f (τ ) dτ , dτ Ia+ Hi α,β,γ A Da+ f

(t) =

(3.250)

a

and the right-sided Hilfer type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel as   Rb α α,β (t) = − Aα,γ (−γ (τ − t) ) Db− f (τ ) dτ t    Rb β(1−α) d (1−β)(1−α) α = − Aα,γ (−γ (τ − t) ) Ib− I f (τ ) dτ . b− dτ 

Hi α,β,γ A Db− f

t



(3.251)

General Fractional Derivatives of Constant Order and Applications

181

Definition 3.43 Let κ+1 > α > κ, κ = [α]+1, 0 ≤ β ≤ 1, −∞ < a < b < ∞ and γ ∈ C. The left-sided Hilfer type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as     Rt α α,β,κ Hir α,β,γ F (−γ (t − τ ) ) D f (τ ) dτ D f (t) = α a+ a+ A a (3.252)    Rt β(κ−α) dκ (1−β)(κ−α) α = Fα (−γ (t − τ ) ) Ia+ I f (τ ) dτ a+ dτ κ a

and the right-sided Hilfer type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel as 



Rb

  α α,β,κ Fα (−γ (τ − t) ) Da+ f (τ ) dτ t     b β(κ−α) dκ (1−β)(κ−α) κR α = (−1) Fα (−γ (τ − t) ) Ib− I f (τ ) dτ . κ b− dτ Hir α,β,γ A Db− f

κ

(t) = (−1)

t

(3.253)

3.9

General fractional derivatives in the subcosine kernel via Mittag-Leffler function

The definitions and properties of the general fractional derivatives within the subcosine kernel via Mittag-Leffler function and the general fractional integrals are given as follows. Definition 3.44 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as     Rt α α,λ d d RL α,λ I f (t) = dt LCosα (−λ (t − τ ) ) f (τ ) dτ , D f (t) = Scs a+ dt Scs a+ a

(3.254) and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

RL α,λ Scs Db− f



d (t) = − dt



α,λ Scs Ib− f

 Rb α d (t) = − dt LCosα (−λ (τ − t) ) f (τ ) dτ , t

(3.255) where the left-sided and right-sided general fractional integral operators within the subcosine kernel are defined as α,λ Scs Ia+ f

Zt

α

LCosα (−λ (t − τ ) ) f (τ ) dτ

(t) = a

(3.256)

182

General Fractional Derivatives: Theory, Methods and Applications

and α,λ Scs Ib− f

Zb

α

LCosα (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.257)

t

respectively. Definition 3.45 Let α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as     Rt α α,λ dκ dκ RL α,λ LCosα (−λ (t − τ ) ) f (τ ) dτ , Scs Da+ f (t) = dtκ Scs Ia+ f (t) = dtκ a

(3.258) and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as  κ     d α,λ RL α,λ Scs Ib− f (t) Scs Db− f (t) = − dt b κ Z α κ d LCosα (−λ (τ − t) ) f (τ ) dτ . (3.259) = (−1) dtκ t

Definition 3.46 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as  
Rt α α,λ α,λ LS (1) (t) = LCosα (−λ (t − τ ) ) f (1) (τ ) dτ , Scs Da+ f (t) =Scs Ia+ f a

(3.260) and the right-sided Liouville-Sonine type general fractional derivative within the subcosine kernel via Mittag-Leffler function as  
Rb α α,λ α,λ LS (1) (t) = − LCosα (−λ (τ − t) ) f (1) (τ ) dτ . Scs Db− f (t) =Scs Ib− −f t

(3.261) Definition 3.47 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. The left-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as  
Rt α α,λ LSC α,λ (κ) (t) = LCosα (−λ (t − τ ) ) f (κ) (τ ) dτ , Scs Da+ f (t) =Scs Ia+ f a

(3.262) and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 



LSC α,λ Scs Db− f

  Rb α,λ (t) = ScsIb− (−1)κ f (κ) (t) = (−1)κ LCosα (−λ (τ − t)α )f (κ) (τ ) dτ. t

(3.263)

General Fractional Derivatives of Constant Order and Applications

183

The inverse operators of the above fractional derivatives within the subcosine kernel via Mittag-Leffler function are presented as follows. Definition 3.48 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. As the inverse operator, the left-sided general fractional integral is defined as 

α,λ R Scs Ia+ f



Zt

κ−2

(t − τ )

(t) =

  2α −1 E2α,κ−1 −λ (t − τ ) f (τ ) dτ

(3.264)

a

and the right-sided general fractional integral as 

α,λ R Scs Ib− f



1 = Γ (κ)

Zb

κ−2

(τ − t)

  2α −1 E2α,κ−1 −λ (τ − t) f (τ ) dτ .

(3.265)

t

In particular, when α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,λ R Scs Ia+ f



Zt (t) =

  1 2α −1 E2α,0 −λ (t − τ ) f (τ ) dτ t−τ

(3.266)

a

and the right-sided general fractional integral as 

α,λ R Scs Ib− f



=

1 Γ (κ)

Zb

  1 2α −1 E2α,0 −λ (τ − t) f (τ ) dτ . τ −t

(3.267)

t

Here, the properties of the general fractional derivatives and the general fractional integrals are presented as follows. Property 3.20 If κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have n  o
−1 α,λ L RL D f (t) = sκ−1 1 + λ2 s−2α f (s) , (3.268) Scs 0+ L

L

L

n

n

α,λ LS Scs D0+ f

RL α,λ Scs D0+ f





o
−1 (t) = 1 + λ2 s−2α f (s) ,

o
−1 (t) = s−1 1 + λ2 s−2α (sf (s) − f (0)) ,

(3.269)

(3.270)

  κ o X
−1 LSC α,λ −1 sκ f (s) − 1 + λ2 s−2α sκ−j f (j−1) (0) , Scs D0+ f (t) = s

n



j=1

(3.271)

184

General Fractional Derivatives: Theory, Methods and Applications

L and

n

α,λ R Scs I0+ f

n



o
(t) = 1 + λ2 s−2α f (s) ,

(3.272)

o
(t) = s1−κ 1 + λ2 s−2α f (s) .   α,λ Note that, when Dκ R Scs I0+ f (t) |t=0 = 0, there is n o
−1 α,λ f (s) , L RL f (t) = sκ−1 1 + λ2 s−2α D Scs 0+   α,λ and when Dκ R Scs I0+ f (t) |t=0 6= 0, there is L

α,λ R Scs I0+ f



(3.273)

(3.274)

κ−1 n o  h i X
−1 α,λ α,λ κ−1 L RL 1 + λ2 s−2α f (s)− sκ−µ−1 Dµ R Scs D0+ f (t) = s Scs I0+ f (+0) . µ=0

(3.275) In particular, when κ = 1, we have o n
α,λ α,λ 2 −2α −1 f (s) −R L RL Scs D0+ f (t) = 1 + λ s Scs I0+ f (+0) ,   α,λ I f (t) |t=0 = 0, we have and when κ = 1 and D R Scs 0+ n o
α,λ 2 −2α −1 L RL f (s) . Scs D0+ f (t) = 1 + λ s

(3.276)

(3.277)

Property 3.21 Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,λ RL α,λ R (3.278) Scs Da+ Scs Ia+ f (t) = f (t) ,   α,λ RL α,λ R I f (t) = f (t) , (3.279) D Scs b− Scs b−   α,λ RL α,λ R (3.280) Scs Da+ Scs Ia+ f (t) = f (t) , and



α,λ RL α,λ R Scs Db− Scs Ib− f



(t) = f (t) .

Property 3.22 If −∞ < a < b < ∞, λ ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,λ R α,λ LS D I f (t) = f (t) , Scs a+ Scs a+   α,λ R α,λ LS Scs Db− Scs Ib− f (t) = f (t) ,   α,λ LSC α,λ R Scs Da+ Scs Ia+ f (t) = f (t) , and



α,λ LSC α,λ R Scs Db− Scs Ib− f



(t) = f (t) .

(3.281) C and f (t)

∈

(3.282) (3.283) (3.284) (3.285)

General Fractional Derivatives of Constant Order and Applications

185

Definition 3.49 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

RL α,λ Scs D+ f



(t) =

d dt



 (t) =

α,λ Scs I+ f

d dt

Rt

α

LCosα (−λ (t − τ ) ) f (τ ) dτ ,

−∞

(3.286) and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

RL α,λ Scs D− f



d (t) = − dt



 R∞ α α,λ d I f (t) = − dt LCosα (−λ (τ − t) ) f (τ ) dτ , Scs − t

(3.287) where the left-sided and right-sided general fractional integral operators within the subcosine kernel via Mittag-Leffler function are defined as α,λ Scs I+ f

Zt

α

LCosα (−λ (t − τ ) ) f (τ ) dτ

(t) =

(3.288)

−∞

and α,λ Scs I− f

Z∞

α

LCosα (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.289)

t

respectively. More generally, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, a = −∞, b = ∞ and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

RL α,λ Scs D+ f



(t) =

dκ dtκ

Rt

α

LCosα (−λ (t − τ ) ) f (τ ) dτ ,

(3.290)

−∞

and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

RL α,λ Scs D− f



κ dκ dtκ

(t) = (−1)

R∞

α

LCosα (−λ (τ − t) ) f (τ ) dτ .

(3.291)

t

On the other hand, when α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C, the left-sided Liouville-Sonine type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

α,λ LS Scs D+ f




Rt α α,λ f (1) (t) = LCosα (−λ (t − τ ) ) f (1) (τ ) dτ , (t) =Scs I+ −∞

(3.292)

186

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Liouville-Sonine type general fractional derivative within the subcosine kernel via Mittag-Leffler function as   −∞
R α α,λ α,λ LS D f (t) =Scs I− LCosα (−λ (τ − t) ) f (1) (τ ) dτ . −f (1) (t) = − Scs − t

(3.293) More generally, when α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as  
Rt α α,λ LSC α,λ (κ) D f (t) = I f (t) = LCosα (−λ (t − τ ) ) f (κ) (τ ) dτ , Scs + + Scs −∞

(3.294) and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function as     κ (κ) α,λ LSC α,λ D f (t) = I (−1) f (t) Scs − Scs − κ

Z∞

= (−1)

α

LCosα (−λ (τ − t) ) f (κ) (τ ) dτ .

(3.295)

t

The inverse operators of the above general fractional derivatives within the subcosine kernel via Mittag-Leffler function can be presented as follows. Definition 3.50 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R Scs I+ f



Zt

κ−2

(t − τ )

(t) =

  2α −1 E2α,κ−1 −λ (t − τ ) f (τ ) dτ ,

(3.296)

−∞

and the right-sided general fractional integral as 

α,λ R Scs I− f



1 = Γ (κ)

Z∞

κ−2

(τ − t)

  2α −1 E2α,κ−1 −λ (τ − t) f (τ ) dτ .

(3.297)

t

In particular, when α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,λ R Scs I+ f



Zt (t) =

  1 2α −1 E2α,0 −λ (t − τ ) f (τ ) dτ , t−τ

(3.298)

−∞

and the right-sided general fractional integral as 

α,λ R Scs I− f



1 = Γ (κ)

Z∞ t

  1 2α −1 E2α,0 −λ (τ − t) f (τ ) dτ . τ −t

(3.299)

General Fractional Derivatives of Constant Order and Applications

187

The properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.23 Let κ + 1 > α > κ, a = −∞, b = ∞, λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,λ RL α,λ R (3.300) Scs D+ Scs I+ f (t) = f (t) , 

and

α,λ RL α,λ R Scs D− Scs I− f



(t) = f (t) ,

(3.301)



α,λ RL α,λ R Scs D+ Scs I+ f



(t) = f (t) ,

(3.302)



α,λ RL α,λ R Scs D− Scs I− f



(t) = f (t) .

(3.303)

Property 3.24 If a = −∞, b = ∞, λ ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,λ R α,λ LS Scs D+ Scs I+ f (t) = f (t) ,   and



3.10

α,λ R α,λ LS Scs D− Scs I− f

C and f (t)



(t) = f (t) ,  α,λ LSC α,λ R Scs D+ Scs I+ f (t) = f (t) , α,λ LSC α,λ R Scs D− Scs I− f



(t) = f (t) .

∈

(3.304) (3.305) (3.306) (3.307)

Hilfer type general fractional derivatives within the subcosine kernel via Mittag-Leffler function

The definitions and properties of the Hilfer type general fractional derivatives within the subcosine kernel via Mittag-Leffler function are given as follows. Definition 3.51 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

a



Rt

  α α,β LCosα (−λ (t − τ ) ) Da+ f (τ ) dτ a    Rt (1−β)(1−α) β(1−α) d α I f (τ ) dτ , = LCosα (−λ (t − τ ) ) Ia+ a+ dτ α,β,λ Hi Scs Da+ f

(t) =

(3.308)

188

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Hilfer type general fractional derivative within the subcosine kernel via Mittag-Leffler function as   Rb α α,β f (τ ) dτ (t) = − LCosα (−λ (τ − t) ) Db− t    Rb β(1−α) d (1−β)(1−α) α = − LCosα (−λ (τ − t) ) Ib− f (τ ) dτ . dτ Ib−



α,β,λ Hi f Scs Db−



(3.309)

t

Generally, when κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 



Rt

  α α,β,κ LCosα (−λ (t − τ ) ) Da+ f (τ ) dτ a    Rt (1−β)(κ−α) β(κ−α) dκ α f (τ ) dτ , I = LCosα (−λ (t − τ ) ) Ia+ κ a+ dτ Hir α,β,λ Scs Da+ f

(t) =

(3.310)

a

and the right-sided Hilfer type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 



Rb

  α α,β,κ LCosα (−λ (τ − t) ) Db− f (τ ) dτ t    b (1−β)(κ−α) β(κ−α) dκ κR α f (τ ) dτ . = (−1) LCosα (−λ (τ − t) ) Ib− dτ κ Ib− Hir α,β,λ f Scs Db−

(t) = (−1)

κ

t

(3.311) Definition 3.52 Let κ + 1 > α > κ, β = 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

Hir α,1,λ Scs Da+ f



(t) =

Rt


α κ−α (κ) LCosα (−λ (t − τ ) ) Ia+ f (τ ) dτ ,

(3.312)

a

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

Hir α,1,λ Scs Db− f



κ

(t) = (−1)

Rb t


α κ−α (κ) LCosα (−λ (τ − t) ) Ib− f (τ ) (τ ) dτ . (3.313)

In particular, when 1 > α > 0, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

α,0,λ Hi Scs Da+ f



(t) =

Rt a

α

LCosα (−λ (t − τ ) )

d dτ

1−α Ia+ f





(τ ) dτ ,

(3.314)

General Fractional Derivatives of Constant Order and Applications

189

and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

α,0,λ Hi Scs Db− f



Rb α (t) = − LCosα (−λ (τ − t) ) t

1−α Ib− f

d dτ





(τ ) dτ .

(3.315)

Furthermore, when κ+1 > α > κ, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function is defined as 

Hir α,0,λ Scs Da+ f



(t) =

Rt

α

LCosα (−λ (t − τ ) )

a

dκ dτ κ



κ−α Ia+ f (τ ) dτ ,

(3.316)

and the right-sided Riemann-Liouville type general fractional derivative within the subcosine kernel via Mittag-Leffler function as 

Hir α,0,λ Scs Db− f



κ

(t) = (−1)

Rb

α

LCosα (−λ (τ − t) )

t

dκ dτ κ



κ−α Ib− f (τ ) dτ . (3.317)

3.11

General fractional derivatives in the subsine kernel via Mittag-Leffler function

The definitions and properties of the general fractional derivatives within the subsine kernel via Mittag-Leffler function and the general fractional integrals are given as follows. Definition 3.53 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as 

RL α,λ Sls Da+ f



(t) =

d dt



α,λ Sls Ia+ f

 (t) =

d dt

Rt

α

LSinα (−λ (t − τ ) ) f (τ ) dτ ,

a

(3.318) and the right-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function as 

RL α,λ Sls Db− f



d (t) = − dt



α,λ Sls Ib− f

 Rb α d (t) = − dt LSinα (−λ (τ − t) ) f (τ ) dτ , t

(3.319)

190

General Fractional Derivatives: Theory, Methods and Applications

where the left-sided and right-sided general fractional integral operators within the subsine kernel are defined as α,λ Sls Ia+ f

Zt

α

(3.320)

α

(3.321)

LSinα (−λ (t − τ ) ) f (τ ) dτ

(t) = a

and α,λ Sls Ib− f

Zb

LSinα (−λ (τ − t) ) f (τ ) dτ ,

(t) = t

respectively. Furthermore, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as     Rt α α,λ dκ dκ RL α,λ D f (t) = f (t) = dt I LSinα (−λ (t − τ ) ) f (τ ) dτ , κ Sls a+ dtκ Sls a+ a

(3.322) and the right-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function as  κ     d α,λ RL α,λ Sls Ib− f (t) Sls Db− f (t) = − dt b κ Z κ d α = (−1) LSinα (−λ (τ − t) ) f (τ ) dτ . (3.323) dtκ t

Definition 3.54 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as  
Rt α α,λ LS α,λ (1) (t) = LSinα (−λ (t − τ ) ) f (1) (τ ) dτ , Sls Da+ f (t) =Sls Ia+ f a

(3.324) and the right-sided Liouville-Sonine type general fractional derivative within the subsine kernel via Mittag-Leffler function as  
Rb α α,λ LS α,λ D f (t) =Sls Ib− −f (1) (t) = − LSinα (−λ (τ − t) ) f (1) (τ ) dτ . Sls b− t

(3.325) More generally, when α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as  
Rt α α,λ (κ) LSC α,λ (t) = LSinα (−λ (t − τ ) ) f (κ) (τ ) dτ , D f (t) = I f Sls a+ a+ Sls a

(3.326)

General Fractional Derivatives of Constant Order and Applications

191

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function as     κ α,λ LSC α,λ (−1) f (κ) (t) Sls Db− f (t) =Sls Ib− Zb

κ

= (−1)

α

LSinα (−λ (τ − t) ) f (κ) (τ ) dτ .

(3.327)

t

The inverse operators of the above fractional derivatives within the subsine kernel via Mittag-Leffler function are considered as follows. Definition 3.55 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R Sls Ia+ f



Zt

1 (t) = λ

κ−α−2

(t − τ )

  2α −1 E2α,κ−α−1 −λ2 (t − τ ) f (τ ) dτ

a

(3.328) and the right-sided general fractional integral as 

α,λ R Sls Ib− f



1 = λ

Zb

κ−α−2

(τ − t)

  2α −1 E2α,κ−α−1 −λ2 (τ − t) f (τ ) dτ .

t

(3.329) In particular, when α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,λ R Sls Ia+ f



1 (t) = λ

Zt a

1

−1 α+1 E2α,−α

(t − τ )



2α

−λ2 (t − τ )



f (τ ) dτ

(3.330)

and the right-sided general fractional integral as 

α,λ R Sls Ib− f



1 = λ

Zb t

1

  2α −1 2 E −λ (τ − t) f (τ ) dτ . α+1 2α,−α (τ − t)

(3.331)

The properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.25 If κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have n  o
−1 α,λ L RL D f (t) = λsκ−(1+α) 1 + λ2 s−2α f (s) , (3.332) Sls 0+ L

n

RL α,λ Sls D0+ f



o
−1 (t) = λs1−(1+α) 1 + λ2 s−2α f (s) ,

(3.333)

192

General Fractional Derivatives: Theory, Methods and Applications n  o
−1 α,λ L LS (sf (s) − f (0)) , (3.334) D f (t) = λs−(1+α) 1 + λ2 s−2α Sls 0+   κ n  o X
−1 α,λ −(1+α) sκf (s) − L LSC 1 + λ2 s−2α sκ−j f (j−1) (0), Sls D0+ f (t) = λs j=1

L

n

α,λ R Sls I0+ f



o
(t) = λ−1 sα 1 + λ2 s−2α f (s) (0 < α < 1)

(3.335) (3.336)

and n

o
(t) = λ−1 sα+1−κ 1 + λ2 s−2α f (s) (κ < α < κ + 1) . (3.337)   α,λ Note that, when Dκ R Sls I0+ f (t) |t=0 = 0, there is L

α,λ R Sls I0+ f

L

n

RL α,λ Sls D0+ f

and when Dκ L





o
−1 (t) = λsκ−(1+α) 1 + λ2 s−2α f (s) ,

α,λ R Sls I0+ f

n

RL α,λ Sls D0+ f

(3.338)

 (t) |t=0 6= 0, there is

o
−1 (t) = λsκ−(1+α) 1 + λ2 s−2α f (s) −

κ−1 X

 i h α,λ sκ−µ−1 Dµ R I f (+0) . Sls 0+

(3.339)

µ=0

In particular, when κ = 1, we have o n
−1 α,λ α,λ D f (t) = λs−α 1 + λ2 s−2α f (s) −R L RL Sls 0+ Sls I0+ f (+0) , and when κ = 1 and D L



n

α,λ R Sls I0+ f

RL α,λ Sls D0+ f

(3.340)

 (t) |t=0 = 0, we have

o
−1 (t) = λs−α 1 + λ2 s−2α f (s) .

(3.341)

Property 3.26 Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,λ RL α,λ R (3.342) Sls Da+ Sls Ia+ f (t) = f (t) , 

and

α,λ RL α,λ R Sls Db− Sls Ib− f



(t) = f (t) ,

(3.343)



α,λ RL α,λ R Sls Da+ Sls Ia+ f



(t) = f (t) ,

(3.344)



α,λ RL α,λ R Sls Db− Sls Ib− f



(t) = f (t) .

(3.345)

General Fractional Derivatives of Constant Order and Applications

193

Property 3.27 If −∞ < a < b < ∞, λ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,λ LS α,λ R D I f (t) = f (t) , (3.346) Sls a+ Sls a+ 

and

α,λ LS α,λ R Sls Db− Sls Ib− f



(t) = f (t) ,   α,λ LSC α,λ R Sls Da+ Sls Ia+ f (t) = f (t) ,

(3.347)



(3.349)

α,λ LSC α,λ R Sls Db− Sls Ib− f



(3.348)

(t) = f (t) .

Definition 3.56 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as 

RL α,λ Sls D+ f



(t) =

d dt



α,λ Sls I+ f

 (t) =

d dt

Rt

α

LSinα (−λ (t − τ ) ) f (τ ) dτ ,

−∞

(3.350) and the right-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function as 

RL α,λ Sls D− f



d (t) = − dt



 R∞ α α,λ d I f (t) = − dt LSinα (−λ (τ − t) ) f (τ ) dτ , Sls − t

(3.351) where the left-sided and right-sided general fractional integral operators within the subsine kernel via Mittag-Leffler function are defined as α,λ Sls I+ f

Zt

α

LSinα (−λ (t − τ ) ) f (τ ) dτ

(t) =

(3.352)

−∞

and α,λ Sls I− f

Z∞

α

LSinα (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.353)

t

respectively. Generally, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, a = −∞, b = ∞ and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as 

RL α,λ Sls D+ f



(t) =

dκ dtκ



α,λ Sls I+ f

 (t) =

dκ dtκ

Rt

α

LSinα (−λ (t − τ ) ) f (τ ) dτ ,

−∞

(3.354)

194

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function as    ∞
 α κ dκ R α,λ d κ RL α,λ Sls I− f (t) = (−1) dtκ LSinα (−λ (τ − t) )f (τ ) dτ . Sls D− f (t) = − dt t

(3.355) Definition 3.57 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as  
Rt α α,λ LS α,λ D f (t) =Sls I+ f (1) (t) = LSinα (−λ (t − τ ) ) f (1) (τ ) dτ Sls + −∞

(3.356) and the right-sided Liouville-Sonine type general fractional derivative within the subsine kernel via Mittag-Leffler function as   −∞
R α α,λ (1) LS α,λ −f (t) = − f (t) = I LSinα (−λ (τ − t) ) f (1) (τ ) dτ . D Sls − − Sls t

(3.357) Furthermore, when α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as   Rt α LSC α,λ LSinα (−λ (t − τ ) ) f (κ) (τ ) dτ (3.358) Sls D+ f (t) = −∞

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function as   ∞ κ R α LSC α,λ LSinα (−λ (τ − t) ) f (κ) (τ ) dτ . (3.359) Sls D− f (t) = (−1) t

The inverse operators of the above general fractional derivatives within the subsine kernel via Mittag-Leffler function are presented as follows. Definition 3.58 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,λ R Sls I+ f



Zt

1 (t) = λ

κ−α−2

(t − τ )

  2α −1 E2α,κ−α−1 −λ2 (t − τ ) f (τ ) dτ

−∞

(3.360) and the right-sided general fractional integral as 

α,λ R Sls I− f



=

1 λ

Z∞

κ−α−2

(τ − t)

  2α −1 E2α,κ−α−1 −λ2 (τ − t) f (τ ) dτ .

t

(3.361)

General Fractional Derivatives of Constant Order and Applications

195

In particular, when α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,λ R Sls I+ f



(t) =

Zt

1 λ

1

−∞

(t − τ )

−1 α+1 E2α,−α

  2α −λ2 (t − τ ) f (τ ) dτ

(3.362)

and the right-sided general fractional integral as 

α,λ R Sls I− f



1 = λ

Z∞ t

1

  2α −1 2 E −λ (τ − t) f (τ ) dτ . 2α,−α α+1 (τ − t)

(3.363)

In this case, the properties of the general fractional derivatives and fractional integrals are given as follows. Property 3.28 Let κ + 1 > α > κ, a = −∞, b = ∞, λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,λ RL α,λ R (3.364) Sls D+ Sls I+ f (t) = f (t) ,

and



α,λ RL α,λ R Sls D− Sls I− f



(t) = f (t) ,

(3.365)



α,λ RL α,λ R Sls D+ Sls I+ f



(t) = f (t) ,

(3.366)



α,λ RL α,λ R Sls D− Sls I− f



(t) = f (t) .

(3.367)

Property 3.29 If a = −∞, b = ∞, λ ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,λ LS α,λ R I f (t) = f (t) , D Sls + Sls + 

α,λ LS α,λ R Sls D− Sls I− f



(t) = f (t) ,  α,λ LSC α,λ R Sls D+ Sls I+ f (t) = f (t) ,

 and



α,λ LSC α,λ R Sls D− Sls I− f

3.12



(t) = f (t) .

C and f (t)

∈

(3.368) (3.369) (3.370) (3.371)

Hilfer type general fractional derivatives within the subsine kernel via Mittag-Leffler function

The definitions and properties of the Hilfer type general fractional derivatives within the subsine kernel via Mittag-Leffler function are given as follows.

196

General Fractional Derivatives: Theory, Methods and Applications

Definition 3.59 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as     Rt α α,β α,β,λ Hi LSinα (−λ (t − τ ) ) Da+ f (τ ) dτ Sls Da+ f (t) = a (3.372)    Rt β(1−α) d (1−β)(1−α) α = LSinα (−λ (t − τ ) ) Ia+ I f (τ ) dτ , a+ dτ a

and the right-sided Hilfer type general fractional derivative within the subsine kernel via Mittag-Leffler function as     Rb α α,β,λ α,β Hi f (t) = − LSinα (−λ (τ − t) ) Db− f (τ ) dτ Sls Db− t (3.373)    Rb (1−β)(1−α) β(1−α) d α = − LSinα (−λ (τ − t) ) Ib− I f (τ ) dτ . b− dτ t

Generally, when κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as     Rt α α,β,κ Hir α,β,λ LSinα (−λ (t − τ ) ) Da+ f (τ ) dτ Sls Da+ f (t) = a (3.374)    Rt (1−β)(κ−α) β(κ−α) dκ α f (τ ) dτ , = LSinα (−λ (t − τ ) ) Ia+ I κ a+ dτ a

and the right-sided Hilfer type general fractional derivative within the subsine kernel via Mittag-Leffler function as     b κR α α,β,κ Hir α,β,λ f (t) = (−1) LSinα (−λ (τ − t) ) Db− f (τ ) dτ Sls Db− t    b β(κ−α) dκ (1−β)(κ−α) κR α = (−1) LSinα (−λ (τ − t) ) Ib− I f (τ ) dτ . κ b− dτ t

(3.375) On one hand, when κ + 1 > α > κ, β = 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as  
Rt α α,κ Hir α,1,λ LSinα (−λ (t − τ ) ) LSC Da+ f (τ ) dτ Sls Da+ f (t) = a (3.376)
Rt α κ−α (κ) = LSinα (−λ (t − τ ) ) Ia+ f (τ ) dτ , a

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the subsine kernel via Mittag-Leffler function as   b
κR α α,κ Hir α,1,λ LSinα (−λ (τ − t) ) LSC Db− f (τ ) dτ Sls Db− f (t) = (−1) t

κ

= (−1)

Rb t


α κ−α (κ) LSinα (−λ (τ − t) ) Ib− f (τ ) (τ ) dτ . (3.377)

General Fractional Derivatives of Constant Order and Applications

197

On the other hand, when κ + 1 > α > κ, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function is defined as 

Hir α,0,λ Sls Da+ f



(t) =

Rt


α α,κ LSinα (−λ (t − τ ) ) Da+ f (τ ) dτ

a

=

Rt

(3.378) dκ dτ κ

α

LSinα (−λ (t − τ ) )

a

κ−α Ia+ f







(τ ) dτ ,

and the right-sided Riemann-Liouville type general fractional derivative within the subsine kernel via Mittag-Leffler function as 

Hir α,0,λ Sls Db− f κ

= (−1)

Rb



(t) = (−1)

κ

Rb t


α α,κ LSinα (−λ (τ − t) ) Db− f (τ ) dτ (3.379) dκ dτ κ

α

LSinα (−λ (τ − t) )

t

3.13



κ−α Ib− f (τ ) dτ .

General fractional derivatives in the two-parametric Lorenzo-Hartley kernel

The definitions and properties of the general fractional derivatives in the twoparametric Lorenzo-Hartley kernel and the general fractional integrals are given as follows. Definition 3.60 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as 

RL α,υ,λ HL Da+ f



(t) =

d dt



α,υ,λ HL Ia+ f

 (t) =

d dt

Rt

α

Rα,υ (−λ (t − τ ) ) f (τ ) dτ ,

a

(3.380) and the right-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as 

RL α,υ,λ HL Db− f



d (t) = − dt



 Rb α α,υ,λ d I f (t) = − dt Rα,υ (−λ (τ − t) ) f (τ ) dτ , HL b− t

(3.381) where the left-sided and right-sided general fractional integral operators within the two-parametric Lorenzo-Hartley kernel are defined as α,υ,λ HL Ia+ f

Zt

α

Rα,υ (−λ (t − τ ) ) f (τ ) dτ

(t) = a

(3.382)

198

General Fractional Derivatives: Theory, Methods and Applications

and α,υ,λ f HL Ib−

Zb

α

Rα,υ (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.383)

t

respectively. More generally, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞ and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as     Rt α α,υ,λ dκ dκ RL α,υ,λ I f (t) = Rα,υ (−λ (t − τ ) ) f (τ ) dτ , D f (t) = κ κ HL a+ HL a+ dt dt a

(3.384) and the right-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as    b
 κ dκ R α α,υ,λ d κ RL α,υ,λ f (t) = (−1) dt κ Rα,υ (−λ (τ − t) )f (τ ) dτ . HL Ib− HL Db− f (t) = − dt t

(3.385) Definition 3.61 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as  
Rt α α,υ,λ α,υ,λ LS f (1) (t) = Rα,υ (−λ (t − τ ) ) f (1) (τ ) dτ , HL Da+ f (t) =HL Ia+ a

(3.386) and the right-sided Liouville-Sonine type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as  
Rb α α,υ,λ α,υ,λ LS −f (1) (t) = − Rα,υ (−λ (τ − t) ) f (1) (τ ) dτ . HL Db− f (t) =HL Ib− t

(3.387) More generally, for α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as  
Rt α α,υ,λ LSC α,υ,λ (κ) D f (t) = I f (t) = Rα,υ (−λ (t − τ ) ) f (κ) (τ ) dτ , HL a+ a+ HL a

(3.388) and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as   LSC α,υ,λ HL Db− f (t) b
κ κR α α,υ,λ =HL Ib− (−1) f (κ) (t) = (−1) Rα,υ (−λ (τ − t) ) f (κ) (τ ) dτ . t

(3.389)

General Fractional Derivatives of Constant Order and Applications

199

The inverse operators of the above general fractional derivatives within the two-parametric Lorenzo-Hartley kernel are shown as follows. Definition 3.62 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,υ,λ R HL Ia+ f



Zt

κ−(α−υ)−1

(t − τ )

(t) =

α

−1 Eα,κ−(α−υ) (−γ (t − τ ) ) f (τ ) dτ

a

(3.390) and the right-sided general fractional integral as 

α,υ,λ R f M R Ib−



Zb

κ−(α−υ)−1

(τ − t)

=

α

−1 Eα,κ−(α−υ) (−γ (τ − t) ) f (τ ) dτ .

t

(3.391) In particular, when α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,υ,λ R HL Ia+ f



Zt (t) = a

1 (t − τ )

−1 α−υ Eα,1−(α−υ)

α

(−γ (t − τ ) ) f (τ ) dτ

(3.392)

and the right-sided general fractional integral as 

α,υ,λ R f M R Ib−



Zb = t

1

−1 α−υ Eα,κ−(α−υ)

(τ − t)

α

(−γ (τ − t) ) f (τ ) dτ .

(3.393)

The properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.30 If κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have n  o
−1 α,υ,λ κ−(α−υ) L RL 1 + λs−α f (s) , (3.394) HL D0+ f (t) = s  o n
−1 α,υ,λ κ−(α−υ) L RL 1 + λs−α f (s) , (3.395) HL D0+ f (t) = s n  o
−1 α,υ,λ −(α−υ) L LS 1 − λs−α (sf (s) − f (0)) , (3.396) HL D0+ f (t) = s   κ n  o X
−1 α,υ,λ −(α−υ) L LSC 1 − λs−α sκ f (s)− sκ−j f (j−1) (0), HL D0+ f (t) = s j=1

and L

o
(t) = s(α−υ)−κ 1 − λs−α f (s)

(3.397) (3.398)

o
(t) = s(α−υ)−1 1 − λs−α f (s) .

(3.399)

n



n



L

α,υ,λ R f HL I0+

α,υ,λ R f HL I0+

200

General Fractional Derivatives: Theory, Methods and Applications   α,υ,λ Note that, when Dκ R I f (t) |t=0 = 0, there is HL 0+ L

n

o
−1 f (s) , (t) = sκ−(α−υ) 1 + λs−α

RL α,υ,λ HL D0+ f



and when Dκ L

α,υ,λ R f HL I0+

n

 (t) |t=0 6= 0, there is

o
−1 (t) = sκ−(α−υ) 1 + λs−α f (s)

RL α,υ,λ HL D0+ f

−

(3.400)

κ−1 X

 h i α,υ,λ sκ−µ−1 Dµ R f (+0) . HL I0+

(3.401)

µ=0

In particular, when κ = 1, we have n o
−1 α,υ,λ α,λ 1−(α−υ) L RL 1 + λs−α f (s) −R HL D0+ f (t) = s M R I0+ f (+0) , (3.402) and when κ = 1 and D L



n

RL α,υ,λ HL D0+ f

α,υ,λ R f HL I0+

 (t) |t=0 = 0, we have

o
−1 (t) = s1−(α−υ) 1 + λs−α f (s) .

(3.403)

Property 3.31 Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,λ RL α,λ R (3.404) HL Da+ HL Ia+ f (t) = f (t) , 

and

α,λ RL α,λ R HL Db− HL Ib− f



(t) = f (t) ,

(3.405)



α,λ RL α,λ R HL Da+ HL Ia+ f



(t) = f (t) ,

(3.406)



α,λ RL α,λ R HL Db− HL Ib− f



(t) = f (t) .

(3.407)

Property 3.32 If −∞ < a < b < ∞, λ ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,λ R α,λ LS HL Da+ HL Ia+ f (t) = f (t) ,   and



α,λ R α,λ LS HL Db− HL Ib− f



(t) = f (t) ,  α,λ LSC α,λ R HL Da+ HL Ia+ f (t) = f (t) ,

α,λ LSC α,λ R HL Db− HL Ib− f



(t) = f (t) .

C and f (t)

∈

(3.408) (3.409) (3.410) (3.411)

General Fractional Derivatives of Constant Order and Applications

201

Definition 3.63 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as 

RL α,υ,λ f HL D+



(t) =

d dt



α,υ,λ f HL I+

 (t) =

d dt

Rt

α

Rα,υ (−λ (t − τ ) ) f (τ ) dτ ,

−∞

(3.412) and the right-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as     R∞ α α,υ,λ d d RL α,υ,λ I f (t) = − dt Rα,υ (−λ (τ − t) ) f (τ ) dτ , D f (t) = − HL − HL − dt t

(3.413) where the left-sided and right-sided general fractional integral operators within the two-parametric Lorenzo-Hartley kernel are defined as α,υ,λ f HL I+

Zt

α

Rα,υ (−λ (t − τ ) ) f (τ ) dτ

(t) =

(3.414)

−∞

and α,υ,λ f HL I−

Z∞

α

Rα,υ (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.415)

t

respectively. For α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, a = −∞, b = ∞ and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as 

RL α,υ,λ f HL D+



(t) =

dκ dtκ

Rt

α

Rα,υ (−λ (t − τ ) ) f (τ ) dτ ,

(3.416)

−∞

and the right-sided Riemann-Liouville type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as   ∞ κ dκ R α RL α,υ,λ D f (t) = (−1) dt Rα,υ (−λ (τ − t) ) f (τ ) dτ . (3.417) κ HL − t

Definition 3.64 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as 

α,υ,λ LS f HL D+




Rt α α,υ,λ f (1) (t) = Rα,υ (−λ (t − τ ) ) f (1) (τ ) dτ , (t) =HL I+ −∞

(3.418)

202

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Liouville-Sonine type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as   −∞
R α α,υ,λ α,υ,λ LS f (t) = HL I− −f (1) (t) = − Rα,υ (−λ (τ − t) ) f (1) (τ ) dτ . HL D− t

(3.419) Definition 3.65 For α ∈ C, κ+1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the two-parametric Lorenzo-Hartley kernel is defined as  
Rt α α,υ,λ LSC α,υ,λ D f (t) =HL I+ f (κ) (t) = Rα,υ (−λ (t − τ ) ) f (κ) (τ ) dτ , + HL −∞

(3.420) and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the two-parametric Lorenzo-Hartley kernel as  
κ α,υ,λ LSC α,υ,λ D f (t) =HL I− (−1) f (κ) (t) − HL ∞ (3.421) κ R α = (−1) Rα,υ (−λ (τ − t) ) f (κ) (τ ) dτ . t

The inverse operators of the above general fractional derivatives within the two-parametric Lorenzo-Hartley kernel can be considered as follows. Definition 3.66 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞ and λ ∈ C. The left-sided general fractional integral is defined as 

α,υ,λ R f HL I+



Zt

κ−(α−υ)−1

(t − τ )

(t) =

α

−1 Eα,κ−(α−υ) (−γ (t − τ ) ) f (τ ) dτ ,

−∞

(3.422) and the right-sided general fractional integral as   Z∞ κ−(α−υ)−1 −1 α α,υ,λ R I f = (τ − t) Eα,κ−(α−υ) (−γ (τ − t) ) f (τ ) dτ . MR − t

(3.423) In particular, when α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞ and λ ∈ C, the left-sided general fractional integral is defined as 

α,υ,λ R f HL I+



Zt (t) = −∞

1

−1 α−υ Eα,1−(α−υ)

(t − τ )

α

(−γ (t − τ ) ) f (τ ) dτ , (3.424)

and the right-sided general fractional integral as   Z∞ 1 α α,υ,λ −1 R f = M R I− α−υ Eα,κ−(α−υ) (−γ (τ − t) ) f (τ ) dτ . (τ − t) t

(3.425)

General Fractional Derivatives of Constant Order and Applications

203

The properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.33 Let κ + 1 > α > κ, a = −∞, b = ∞, λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,υ,λ RL α,υ,λ R D I f (t) = f (t) , (3.426) HL + HL +

and



α,υ,λ RL α,υ,λ R f HL D− HL I−



(t) = f (t) ,

(3.427)



α,υ,λ RL α,υ,λ R f HL D+ HL I+



(t) = f (t) ,

(3.428)



α,υ,λ RL α,υ,λ R f HL D− HL I−



(t) = f (t) .

(3.429)

Property 3.34 If a = −∞, b = ∞, λ ∈ C and f (t) ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,υ,λ α,υ,λ R LS f (t) = f (t) , (3.430) HL I+ HL D+   and

3.14



α,υ,λ α,υ,λ R LS f HL I− HL D−



(t) = f (t) ,  α,υ,λ LSC α,υ,λ R f (t) = f (t) , HL I+ HL D+ α,υ,λ LSC α,υ,λ R f HL I− HL D−



(3.431) (3.432)

(t) = f (t) .

(3.433)

General fractional derivatives in the GorenfloMainardi kernel via Wiman Function

The definitions and properties of the general fractional derivatives in the Gorenflo-Mainardi kernel via Wiman function and general fractional integrals are given as follows. Definition 3.67 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type is defined as 

α,γ,λ RL GM Da+ f



(t) =

d dt



 α,γ,λ I f (t) = GM a+

d dt

Rt

α

Gα,γ (−λ (t − τ ) ) f (τ ) dτ ,

a

(3.434)

204

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type as     Rb α α,γ,λ α,γ,λ d d RL f (t) = − dt Gα,γ (−λ (τ − t) ) f (τ ) dτ , GM Db− f (t) = − dt GM Ib− t

(3.435) where the left-sided and right-sided general fractional integral operators containing the kernel of the Gorenflo-Mainardi function via Wiman function are defined as Zt α α,γ,λ Gα,γ (−λ (t − τ ) ) f (τ ) dτ (3.436) GM Ia+ f (t) = a

and α,γ,λ f GM Ib−

Zb

α

Gα,γ (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.437)

t

respectively. For α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type is defined as   Rt α α,γ,λ dκ RL (3.438) Gα,γ (−λ (t − τ ) ) f (τ ) dτ , GM Da+ f (t) = dtκ a

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type as   b κ dκ R α α,γ,λ RL (3.439) Gα,γ (−λ (τ − t) ) f (τ ) dτ . GM Db− f (t) = (−1) dtκ t

Definition 3.68 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine type is defined as  
Rt α α,γ,λ α,γ,λ LS f (1) (t) = Gα,γ (−λ (t − τ ) ) f (1) (τ ) dτ , D f (t) =GM Ia+ GM a+ a

(3.440) and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine type as  
Rb α α,γ,λ α,γ,λ LS −f (1) (t) = − Gα,γ (−λ (τ − t) ) f (1) (τ ) dτ . GM Db− f (t) =GM Ib− t

(3.441)

General Fractional Derivatives of Constant Order and Applications

205

More generally, for α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of Liouville-Sonine-Caputo type is defined as 

LSC α,γ,λ GM Da+ f



(t) =

Rt

α

Gα,γ (−λ (t − τ ) ) f (κ) (τ ) dτ ,

(3.442)

a

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine-Caputo type as 

LSC α,γ,λ GM Db− f



κ

(t) = (−1)

Rb

α

Gα,γ (−λ (τ − t) ) f (κ) (τ ) dτ .

(3.443)

t

Definition 3.69 For α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided general fractional integral is defined as 

α,γ,λ R GM Ia+ f



Zt

κ−γ−1

(t − τ )

(t) =

α

−1 Eα,κ−γ (−λ (t − τ ) ) f (τ ) dτ ,

(3.444)

a

and the right-sided general fractional integral as 

α,γ,λ R f GM Ib−



κ

Zb

κ−γ−1

(τ − t)

= (−1)

α

−1 Eα,κ−γ (−λ (τ − t) ) f (τ ) dτ . (3.445)

t

In particular, for α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided general fractional integral is defined as 

α,γ,λ R GM Ia+ f



Zt (t − τ )

(t) =

−γ

α

(3.446)

α

(3.447)

−1 Eα,1−γ (−λ (t − τ ) ) f (τ ) dτ ,

a

and the right-sided general fractional integral as 

α,γ,λ R f GM Ib−



Zb =−

−γ

(τ − t)

−1 Eα,1−γ (−λ (τ − t) ) f (τ ) dτ .

t

The properties of the general fractional derivatives and the general fractional integrals are given as follows. Property 3.35 If κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ C, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have n  o
−1 α,γ,λ L RL D f (t) = sκ−γ 1 + λs−α f (s) , (3.448) GM 0+

206

General Fractional Derivatives: Theory, Methods and Applications n  o
−1 α,γ,λ L RL f (s) , (3.449) D f (t) = s1−γ 1 + λs−α GM 0+ n  o
−1 α,γ,λ −γ L LS 1 + λs−α (sf (s) − f (0)) , (3.450) GM D0+ f (t) = s   κ n  o X
−1 α,γ,λ −γ sκ f (s) − L LSC 1 + λs−α sκ−j f (j−1) (0) , GM D0+ f (t) = s j=1

L

n

L

n

and

α,γ,λ R GM I0+ f

α,γ,λ GM I0+ f

Note that, when Dκ L and when Dκ







(t) = s

f (s)

(3.451) (3.452)



o
(t) = s−γ 1 + λs−α f (s) .

(3.453)

o

α,γ,λ R GM I0+ f

n

α,γ,λ RL GM D0+ f

α,γ,λ R GM I0+ f

γ−κ

−α

1 + λs




 (t) |t=0 = 0, there is

o
−1 (t) = sκ−γ 1 + λs−α f (s) .

(3.454)

 (t) |t=0 6= 0, there is

κ−1 n o  h i X
−1 α,γ,λ α,γ,λ κ−γ L RL 1 + λs−α f (s)− sκ−µ−1 Dµ R GM D0+ f (t) = s GM I0+ f (+0) . µ=0

(3.455) In particular, for κ = 1 we have n o
−1 α,γ,λ α,γ,λ L RL D f (t) = s1−γ 1 + λs−α f (s) −R GM 0+ GM I0+ f (+0) , and, for κ = 1 and D L



n

α,γ,λ R GM I0+ f

α,γ,λ RL GM D0+ f

(3.456)

 (t) |t=0 = 0 we have

o
−1 (t) = s1−γ 1 + λs−α f (s) .

(3.457)

For the details of the general fractional derivatives and the general fractional integrals, see [301, 302]. Property 3.36 Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ C, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,γ,λ R α,γ,λ RL (3.458) GM Da+ GM Ia+ f (t) = f (t) , 

and

α,γ,λ R α,γ,λ RL f GM Db− GM Ib−



(t) = f (t) ,

(3.459)



α,γ,λ R α,γ,λ RL GM Da+ GM Ia+ f



(t) = f (t) ,

(3.460)



α,γ,λ R α,γ,λ RL f GM Db− GM Ib−



(t) = f (t) .

(3.461)

General Fractional Derivatives of Constant Order and Applications

207

Property 3.37 If −∞ < a < b < ∞, γ ∈ C, λ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,γ,λ R α,γ,λ LS (3.462) GM Da+ GM Ia+ f (t) = f (t) , 

and

α,γ,λ R α,γ,λ LS f GM Db− GM Ib−



(t) = f (t) ,

(3.463)



α,γ,λ LSC α,γ,λ R GM Da+ GM Ia+ f



(t) = f (t) ,

(3.464)



α,γ,λ LSC α,γ,λ R f GM Db− GM Ib−



(t) = f (t) .

(3.465)

Definition 3.70 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, γ ∈ C and λ ∈ C. The left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type is defined as 

α,γ,λ RL f GM D+



(t) =

d dt



α,γ,λ f GM I+

 (t) =

d dt

Rt

α

Gα,γ (−λ (t − τ ) ) f (τ ) dτ ,

−∞

(3.466) and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type as     R∞ α α,γ,λ α,γ,λ d d RL D f (t) = − I f (t) = − Gα,γ (−λ (τ − t) ) f (τ ) dτ , GM − GM − dt dt t

(3.467) where the left-sided and right-sided general fractional integral operators containing the kernel of the Gorenflo-Mainardi function via Wiman function are defined as Zt α α,γ,λ f (t) = Gα,γ (−λ (t − τ ) ) f (τ ) dτ (3.468) GM I+ −∞

and α,γ,λ f GM I−

Z∞

α

Gα,γ (−λ (τ − t) ) f (τ ) dτ ,

(t) =

(3.469)

t

respectively. On the other hand, when α ∈ C, Re (α) > 0, κ = [Re (α)] + 1,a = −∞, b = ∞, γ ∈ C and λ ∈ C, the left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of Riemann-Liouville type is defined as 

α,γ,λ RL f GM D+



(t) =

dκ dtκ

Rt −∞

α

Gα,γ (−λ (t − τ ) ) f (τ ) dτ ,

(3.470)

208

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type as   ∞ α κ dκ R α,γ,λ RL Gα,γ (−λ (τ − t) ) f (τ ) dτ . f (t) = (−1) dt (3.471) κ GM D− t

Definition 3.71 For α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, γ ∈ C and λ ∈ C, the left-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine type is defined as   Rt α α,γ,λ LS f (t) = Gα,γ (−λ (t − τ ) ) f (1) (τ ) dτ , (3.472) GM D+ −∞

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine type as   R∞ α α,γ,λ LS f (t) = − Gα,γ (−λ (τ − t) ) f (1) (τ ) dτ . (3.473) GM D− t

For α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞, γ ∈ C and λ ∈ C, the left-sided general fractional derivative containing the kernel of the GorenfloMainardi function via Wiman function in the sense of Liouville-Sonine-Caputo type is defined as   Rt α LSC α,γ,λ f (t) = D Gα,γ (−λ (t − τ ) ) f (κ) (τ ) dτ , (3.474) + GM −∞

and the right-sided general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine-Caputo type as   ∞ κ R α LSC α,γ,λ D f (t) = (−1) Gα,γ (−λ (τ − t) ) f (κ) (τ ) dτ . (3.475) − GM t

Their inverse operators are presented as follows. Definition 3.72 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞, γ ∈ C and λ ∈ C. The left-sided general fractional integral is defined as 

α,γ,λ R f GM I+



Zt

κ−γ−1

(t − τ )

(t) =

α

−1 Eα,κ−γ (−λ (t − τ ) ) f (τ ) dτ ,

(3.476)

−∞

and the right-sided general fractional integral as 

α,γ,λ R f GM I−



κ

= (−1)

−∞ Z κ−γ−1 −1 α (τ − t) Eα,κ−γ (−λ (τ − t) ) f (τ ) dτ . t

(3.477)

General Fractional Derivatives of Constant Order and Applications

209

In particular, for α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, γ ∈ C and λ ∈ C, the left-sided general fractional integral is defined as 

α,γ,λ R f GM I+



Zt

−γ

(t − τ )

(t) =

α

−1 Eα,1−γ (−λ (t − τ ) ) f (τ ) dτ

(3.478)

−∞

and the right-sided general fractional integral as 

α,γ,λ R f GM I−



Z∞

−γ

(τ − t)

=−

α

−1 Eα,1−γ (−λ (τ − t) ) f (τ ) dτ .

(3.479)

t

The properties of the general fractional derivatives and the general fractional integrals can be given as follows. Property 3.38 Let κ + 1 > α > κ, a = −∞, b = ∞, γ ∈ C, λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,γ,λ α,γ,λ R RL f (t) = f (t) , (3.480) GM I+ GM D+

and



α,γ,λ R α,γ,λ RL f GM D− GM I−



(t) = f (t) ,

(3.481)



α,γ,λ R α,γ,λ RL f GM D+ GM I+



(t) = f (t) ,

(3.482)



α,γ,λ R α,γ,λ RL f GM D− GM I−



(t) = f (t) .

(3.483)

Property 3.39 If a = −∞, b = ∞, γ ∈ C, λ ∈ C and f (t) ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,γ,λ R α,γ,λ LS f (t) = f (t) , (3.484) GM D+ GM I+ 

and

α,γ,λ R α,γ,λ LS f GM D− GM I−



(t) = f (t) ,

(3.485)

(t) = f (t) ,

(3.486)

(t) = f (t) .

(3.487)



α,γ,λ LSC α,γ,λ R f GM D+ GM I+





α,γ,λ LSC α,γ,λ R f GM D− GM I−



210

3.15

General Fractional Derivatives: Theory, Methods and Applications

Hilfer type general fractional derivatives involving the kernel of the Gorenflo-Mainardi function

The definitions and properties of the Hilfer type general fractional derivatives involving the kernel of the Gorenflo-Mainardi function via Wiman function can be given as follows. Definition 3.73 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as     Rt α α,β,γ,λ α,β Hi f (t) = Gα,γ (−λ (t − τ ) ) Da+ f (τ ) dτ GM Da+ a (3.488)    Rt (1−β)(1−α) β(1−α) d α f (τ ) dτ , I = Gα,γ (−λ (t − τ ) ) Ia+ a+ dτ a

and the right-sided Hilfer type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as     Rb α α,β,γ,λ α,β Hi f (t) = − Gα,γ (−λ (τ − t) ) Db− f (τ ) dτ GM Db− t (3.489)    Rb β(1−α) d (1−β)(1−α) α = − Gα,γ (−λ (τ − t) ) Ib− I f (τ ) dτ . b− dτ t

On the other hand, for κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as      Rt β(κ−α) dκ (1−β)(κ−α) α Hir α,β,γ,λ D f (t) = G (−λ (t − τ ) ) I f (τ ) dτ , I κ α,γ a+ a+ GM a+ dτ a

(3.490) and the right-sided Hilfer type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as   Hir α,β,γ,λ f (t) GM Db−    b β(κ−α) dκ (1−β)(κ−α) κR α = (−1) Gα,γ (−λ (τ − t) ) Ib− I f (τ ) dτ . κ b− dτ t

(3.491) For 1 > α > 0, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as  

Rt α α,0,γ,λ 1−α d Hi (3.492) D f (t) = Gα,γ (−λ (t − τ ) ) dτ Ia+ f (τ ) dτ , GM a+ a

General Fractional Derivatives of Constant Order and Applications

211

and the right-sided Riemann-Liouville type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as  

Rb α α,0,γ,λ 1−α d Hi D f (t) = − Gα,γ (−λ (τ − t) ) dτ Ib− f (τ ) dτ . (3.493) GM b− t

More generally, for κ + 1 > α > κ, β = 0, −∞ < a < b < ∞, γ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as  

Rt α κ−α dκ Hir α,0,γ,λ Ia+ f (τ ) dτ , (3.494) f (t) = Gα,γ (−λ (t − τ ) ) dτ κ GM Da+ a

and the right-sided Riemann-Liouville type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as   b

κR α κ−α dκ Hir α,0,γ,λ D f (t) = (−1) Gα,γ (−λ (τ − t) ) dτ Ib− f (τ ) dτ . κ GM b− t

(3.495)

3.16

General fractional derivatives in the generalized Prabhakar kernel

The definitions and properties of the general fractional derivatives within the generalized Prabhakar kernel via Prabhakar function and the general fractional integrals, which are proposed by Kilbas, Saigo and Saxena (see [97]; also see [101]), are presented as follows. Definition 3.74 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined by     α,υ,ϕ,λ d RL α,υ,ϕ,λ D f (t) = I f (t) , (3.496) GP a+ dt GP a+ and the right-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function by     α,υ,ϕ,λ d RL α,υ,ϕ,λ D f (t) = − I f (t) , (3.497) GP GP b− b− dt where the left-sided and right-sided general fractional integral operators within the Prabhakar kernel via Prabhakar function are defined as α,υ,ϕ,λ f GP Ia+

Zt (t − τ )

(t) = a

υ−1

α

ϕ Eα,υ (−λ (t − τ ) ) f (τ ) dτ

(3.498)

212

General Fractional Derivatives: Theory, Methods and Applications

and α,υ,ϕ,λ f GP Ib−

Zb

υ−1

(τ − t)

(t) =

α

ϕ Eα,υ (−λ (τ − t) ) f (τ ) dτ ,

(3.499)

t

respectively. More generally, for α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as (see [97]; also see [101])   Rt υ−1 ϕ α dκ RL α,υ,ϕ,λ D f (t) = (t − τ ) Eα,υ (−λ (t − τ ) ) f (τ ) dτ , (3.500) κ GP a+ dt a

and the right-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as   b α υ−1 ϕ κ dκ R RL α,υ,ϕ,λ (τ − t) Eα,υ (−λ (τ − t) ) f (τ ) dτ , f (t) = (−1) dt κ GP Db− t

(3.501) Definition 3.75 Let α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided Liouville-Sonine type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as   Rt α υ−1 ϕ α,υ,ϕ,λ LS f (t) = (t − τ ) Eα,υ (−λ (t − τ ) ) f (1) (τ ) dτ , (3.502) GP Da+ a

and the right-sided Liouville-Sonine type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as   Rb υ−1 ϕ α α,υ,ϕ,λ LS f (t) = − (τ − t) Eα,υ (−λ (τ − t) ) f (1) (τ ) dτ . (3.503) GP Db− t

Generally, for α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as (see [101]) 

LSC α,ϕ,υ,λ f GP Da+



(t) =

Rt

υ−1

(t − τ )

α

ϕ Eα,υ (−λ (t − τ ) ) f (κ) (τ ) dτ ,

(3.504)

a

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as   b κR υ−1 ϕ α LSC α,ϕ,υ,λ f (t) = (−1) (τ − t) Eα,υ (−λ (τ − t) ) f (κ) (τ ) dτ . GP Db− t

(3.505)

General Fractional Derivatives of Constant Order and Applications

213

The inverse operators of the above fractional derivatives within the generalized Prabhakar kernel via Prabhakar function are presented as follows. Definition 3.76 Let α ∈ C, κ + 1 > Re (α) > κ, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C. As the inverse operator, the left-sided general fractional integral via Prabhakar function is defined by (see [97]; also see [101]) 

α,ϕ,υ,λ R f GP Ia+



Zt (t − τ )

(t) =

κ−υ−1

−ϕ Eα,κ−υ (−λ (t − τ ) ) f (τ ) dτ α

(3.506)

a

and the right-sided general fractional integral via Prabhakar function by 

α,ϕ,υ,λ R f GP Ib−



Zb

κ−υ−1

(τ − t)

=

−ϕ (−λ (τ − t) ) f (τ ) dτ . Eα,κ−υ α

(3.507)

t

In particular, for α ∈ C, 1 > Re (α) > 0, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided general fractional integral via Prabhakar function is defined as 

α,ϕ,υ,λ R f GP Ia+



Zt (t) =

1 α −ϕ (−λ (t − τ ) ) f (τ ) dτ υE (t − τ ) α,κ−υ

(3.508)

a

and the right-sided general fractional integral via Prabhakar function as 

α,ϕ,υ,λ R f GP Ib−



Zb =

1 α −ϕ (−λ (τ − t) ) f (τ ) dτ . υE (τ − t) α,κ−υ

(3.509)

t

The properties of the general fractional derivatives and the general fractional integrals based on the Prabhakar function are given as follows. Property 3.40 If κ + 1 > α > κ, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞), then we have (see [101])  o n
−ϕ α,υ,ϕ,λ f (t) = sκ−υ 1 + λs−α f (s) , (3.510) L RL D GP 0+ L n

n

RL α,υ,ϕ,λ f GP D0+



o
−ϕ (t) = s1−υ 1 + λs−α f (s) ,

(3.511) o
−ϕ α,υ,ϕ,λ L LS f (t) = s−υ 1 + λs−α (sf (s) − f (0)) , (3.512) GP D0+   κ  o n X
−ϕ α,υ,ϕ,λ sκ f (s) − sκ−j f (j−1) (0) , L LSC f (t) = s−υ 1 + λs−α GP D0+ 

j=1

(3.513)

214

General Fractional Derivatives: Theory, Methods and Applications n  o
ϕ α,υ,ϕ,λ L R (3.514) I f (t) = sυ−κ 1 + λs−α f (s) , GP 0+

and L

n

α,υ,ϕ,λ R f GP I0+

Note that, when Dκ L and when Dκ L

n



RL α,υ,ϕ,λ f GP D0+





o
ϕ (t) = sυ−1 1 + λs−α f (s) .

α,υ,ϕ,λ R f GP I0+

n

RL α,υ,ϕ,λ f GP D0+

α,υ,ϕ,λ R f GP I0+

(3.515)

 (t) |t=0 = 0, there is

o
−1 (t) = sκ−1 1 + λs−α f (s) .

(3.516)

 (t) |t=0 6= 0, there is

κ−1 i o X κ−µ−1  µ hRL α,κ,λ
−1 s D M l I0+ f (+0) . f (s)− (t) = sκ−1 1 + λs−α µ=0

(3.517)

In particular, for κ = 1 we have o n
−1 α,υ,ϕ,λ α,υ,ϕ,λ f (t) = 1 + λs−α f (s) −R f (+0) . L RL GP D0+ GP I0+ and for κ = 1 and D L



n

α,υ,ϕ,λ R f GP I0+

RL α,υ,ϕ,λ f GP D0+

(3.518)

 (t) |t=0 = 0 we have

o
−1 (t) = 1 + λs−α f (s) .

(3.519)

Property 3.41 Let κ + 1 > α > κ, −∞ < a < b < ∞ , ϕ ∈ C, υ ∈ C, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have (see [97]; also see [101])   α,υ,ϕ,λ RL α,υ,ϕ,λ R f (t) = f (t) , (3.520) GP Ia+ GP Da+ 

and

α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP Db− GP Ib−



(t) = f (t) ,

(3.521)



α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP Ia+ GP Da+



(t) = f (t) ,

(3.522)



α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP Db− GP Ib−



(t) = f (t) .

(3.523)

Property 3.42 If −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C, λ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,υ,ϕ,λ R α,υ,ϕ,λ LS f (t) = f (t) , (3.524) D I GP a+ GP a+   and



α,υ,ϕ,λ R α,υ,ϕ,λ LS f GP Db− GP Ib−



(t) = f (t) ,  α,υ,ϕ,λ LSC α,υ,ϕ,λ R f (t) = f (t) , GP Da+ GP Ia+ α,υ,ϕ,λ LSC α,υ,ϕ,λ R f GP Db− GP Ib−



(t) = f (t) .

(3.525) (3.526) (3.527)

General Fractional Derivatives of Constant Order and Applications

215

Definition 3.77 Let α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as     α,υ,ϕ,λ d RL α,υ,ϕ,λ f (t) f (t) = dt GP I+ GP D+ (3.528) Rt υ−1 ϕ α d = dt (t − τ ) Eα,υ (−λ (t − τ ) ) f (τ ) dτ , −∞

and the right-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as     α,υ,ϕ,λ d RL α,υ,ϕ,λ f (t) = − dt f (t) GP I− GP D− (3.529) R∞ α υ−1 ϕ d = − dt (τ − t) Eα,υ (−λ (τ − t) ) f (τ ) dτ , t

where the left-sided and right-sided general fractional integral operators within the Prabhakar kernel via Prabhakar function are defined as α,υ,ϕ,λ f GP I+

Zt

υ−1

(t − τ )

(t) =

α

ϕ Eα,υ (−λ (t − τ ) ) f (τ ) dτ

(3.530)

−∞

and α,υ,ϕ,λ f GP I−

Z∞

υ−1

(τ − t)

(t) =

α

ϕ Eα,υ (−λ (τ − t) ) f (τ ) dτ ,

(3.531)

t

respectively. More generally, for α ∈ C, Re (α) > 0, κ = [Re (α)] + 1, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as 

RL α,υ,ϕ,λ f GP D+



(t) =

dκ dtκ

Rt

υ−1

(t − τ )

−∞

α

ϕ Eα,υ (−λ (t − τ ) ) f (τ ) dτ ,

(3.532) and the right-sided Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as 

RL α,υ,ϕ,λ f GP D−



κ dκ dtκ

(t) = (−1)

R∞

υ−1

(τ − t)

α

ϕ Eα,υ (−λ (τ − t) ) f (τ ) dτ .

t

(3.533)

216

General Fractional Derivatives: Theory, Methods and Applications

Definition 3.78 For α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Liouville-Sonine type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as  
α,υ,ϕ,λ α,υ,ϕ,λ LS D f (t) =GP I+ f (1) (t) GP + (3.534) Rt υ−1 ϕ α (t − τ ) Eα,υ (−λ (t − τ ) ) f (1) (τ ) dτ , = −∞

and the right-sided Liouville-Sonine type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as  
α,υ,ϕ,λ α,υ,ϕ,λ LS f (t) =GP I− −f (1) (t) GP D− −∞ (3.535) R α υ−1 ϕ =− (τ − t) Eα,υ (−λ (τ − t) ) f (1) (τ ) dτ . t

More generally, when α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as  
α,υ,ϕ,λ LSC α,υ,ϕ,λ f (t) =GP I+ f (κ) (t) GP D+ (3.536) Rt υ−1 ϕ α = (t − τ ) Eα,υ (−λ (t − τ ) ) f (κ) (τ ) dτ , −∞

and the right-sided Liouville-Sonine-Caputo type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as  
κ α,υ,ϕ,λ LSC α,υ,ϕ,λ D f (t) =GP I− (−1) f (κ) (t) − GP −∞ (3.537) α κ R υ−1 ϕ = (−1) (τ − t) Eα,υ (−λ (τ − t) ) f (κ) (τ ) dτ . t

The inverse operators of the above general fractional derivatives within the generalized Prabhakar kernel via Prabhakar function can be considered as follows. Definition 3.79 Let α ∈ C, κ + 1 > Re (α) > κ, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided general fractional integral via Prabhakar function is defined as 

α,ϕ,υ,λ R f GP I+



Zt

κ−υ−1

(t − τ )

(t) =

−ϕ Eα,κ−υ (−λ (t − τ ) ) f (τ ) dτ , (3.538) α

−∞

and the right-sided general fractional integral via Prabhakar function as 

α,ϕ,υ,λ R f GP I−



Z∞

κ−υ−1

(τ − t)

= t

−ϕ Eα,κ−υ (−λ (τ − t) ) f (τ ) dτ . α

(3.539)

General Fractional Derivatives of Constant Order and Applications

217

In particular, when α ∈ C, 1 > Re (α) > 0, a = −∞, b = ∞, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided general fractional integral via Prabhakar function is defined as 

α,ϕ,υ,λ R f GP I+



Zt (t) =

1 α −ϕ (−λ (t − τ ) ) f (τ ) dτ , υE (t − τ ) α,κ−υ

(3.540)

−∞

and the right-sided general fractional integral via Prabhakar function as 

α,ϕ,υ,λ R f GP I−



Z∞ =

1 α −ϕ (−λ (τ − t) ) f (τ ) dτ . υE (τ − t) α,κ−υ

(3.541)

t

The properties of the general fractional derivatives and the general fractional integrals via Prabhakar function are presented as follows. Property 3.43 Let κ + 1 > α > κ, a = −∞, b = ∞ , ϕ ∈ C, υ ∈ C and λ ∈ C and f (t) ∈ Ll (−∞, ∞) (1 ≤ l < ∞). Then we have   α,υ,ϕ,λ RL α,υ,ϕ,λ R f (t) = f (t) , (3.542) GP I+ GP D+ 

and

α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP D− GP I−



(t) = f (t) ,

(3.543)



α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP D+ GP I+



(t) = f (t) ,

(3.544)



α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP I− GP D−



(t) = f (t) .

(3.545)

Property 3.44 If a = −∞,b = ∞ , ϕ ∈ C, υ ∈ C, λ ∈ C and f (t) ∈ ACκ (−∞, ∞) (1 ≤ κ < ∞), then we have   α,υ,ϕ,λ R α,υ,ϕ,λ LS f (t) = f (t) , (3.546) GP D+ GP I+   and



α,υ,ϕ,λ R α,υ,ϕ,λ LS f GP D− GP I−



(t) = f (t) ,  α,υ,ϕ,λ LSC α,υ,ϕ,λ R f (t) = f (t) , GP D+ GP I+ α,υ,ϕ,λ LSC α,υ,ϕ,λ R f GP D− GP I−



(t) = f (t) .

(3.547) (3.548) (3.549)

218

3.17

General Fractional Derivatives: Theory, Methods and Applications

General fractional derivatives involving the generalized Prabhakar kernel with respect to another function

The definitions and properties of the general fractional derivatives and general fractional integrals involving the generalized Prabhakar kernel via Prabhakar function with respect to another function are presented as follows. Definition 3.80 Let 1 > α > 0, −∞ < a < b < ∞, ϕ ∈ C, υ ∈ C, h(1) (t) > 0, λ ∈ C and h(1) (t) > 0. The left-sided Riemann-Liouville type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function is defined as      α,υ,ϕ,λ d 1 RL α,υ,ϕ,λ f (t) GP Ia+,h GP Da+,h f (t) = h(1) (t) dt   Rt υ−1 ϕ α d = h(1)1 (t) dt (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) f (τ ) h(1) (τ ) dτ , a

(3.550) and the right-sided Riemann-Liouville type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function as      α,υ,ϕ,λ d RL α,υ,ϕ,λ f (t) = − h(1)1 (t) dt f (t) GP Ib−,h GP Db−,h  Rb  α υ−1 ϕ d (h (τ ) − h (t)) Eα,υ (−λ (h (τ ) − h (t)) ) h(1) (τ ) f (τ ) dτ , = − h(1)1 (t) dt t

(3.551) where the left-sided and right-sided general fractional integral operators involving the generalized Prabhakar kernel via Prabhakar function with respect to another function are represented as α,υ,ϕ,λ f GP Ia+,h

Zt υ−1 ϕ α (t) = (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) h(1) (τ ) f (τ ) dτ , a

(3.552) and α,υ,ϕ,λ f GP Ib−,h

Zb υ−1 ϕ α (t) = (h (τ )−h (t)) Eα,υ (−λ (h (τ ) − h (t)) ) h(1) (τ ) f (τ ) dτ , t

(3.553) respectively. Definition 3.81 For κ + 1 > α > κ, −∞ < a < b < ∞, ϕ ∈ C, h(1) (t) > 0, υ ∈ C and λ ∈ C, the left-sided Liouville-Sonine-Caputo type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar

General Fractional Derivatives of Constant Order and Applications

219

function with respect to another function is defined as    κ  (κ) α,ϕ,υ,λ 1 d LSC α,ϕ,υ,λ D f (t) = I f (t) GP (1) GP a+,h a+,h h h (t) dt κ  Rt (κ) υ−1 ϕ α d fh (τ ) dτ , = (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) h(1)1(τ ) dτ a

(3.554) and the right-sided Liouville-Sonine-Caputo type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function as κ     (κ) α,ϕ,υ,λ d 1 LSC α,ϕ,υ,λ f (t) D f (t) = I − GP GP h b−,h b−,h h(1) (t) dt κ  Rb (κ) α υ−1 ϕ d fh (τ ) dτ . = (h (τ ) − h (t)) Eα,υ (−λ (h (τ ) − h (t)) ) − h(1)1(τ ) dτ t

(3.555) Definition 3.82 Let κ + 1 > α > κ, −∞ < a < b < ∞, ϕ ∈ C, h(1) (t) > 0, υ ∈ C and λ ∈ C. The left-sided general fractional integral via Prabhakar function with respect to another function is defined as   α,ϕ,υ,λ R I f (t) GP a+,h Rt κ−υ−1 −ϕ α = (h (t) − h (τ )) Eα,κ−υ (−λ (h (t) − h (τ )) ) h(1) (τ ) f (τ ) dτ , a

(3.556) and the right-sided general fractional integral via Prabhakar function with respect to another function as   α,ϕ,υ,λ R I f GP b−,h Rb κ−υ−1 −ϕ α = (h (τ ) − h (t)) Eα,κ−υ (−λ (h (τ ) − h (t)) ) h(1) (τ ) f (τ ) dτ . t

(3.557) In particular, when 1 > α > 0, −∞ < a < b < ∞, ϕ ∈ C, h(1) (t) > 0, υ ∈ C and λ ∈ C, the left-sided general fractional integral via Prabhakar function with respect to another function is defined as   α,ϕ,υ,λ R I f (t) GP a+,h (3.558) Rt α −ϕ 1 (1) = (h(t)−h(τ (τ ) f (τ ) dτ , ))υ Eα,1−υ (−λ (h (t) − h (τ )) ) h a

and the right-sided general fractional integral via Prabhakar function with respect to another function as   α,ϕ,υ,λ R f GP Ib−,h (3.559) Rb α −ϕ 1 (1) (τ ) f (τ ) dτ . = (h(τ )−h(t)) υ Eα,1−υ (−λ (h (τ ) − h (t)) ) h t

220

General Fractional Derivatives: Theory, Methods and Applications

The properties of the general fractional derivatives and general fractional integrals with respect to another function are presented as follows. Property 3.45 Let κ + 1 > α > κ, −∞ < a < b < ∞, ϕ ∈ C, ,h(1) (t) > 0, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then we have   α,υ,ϕ,λ RL α,υ,ϕ,λ R D I f (t) = f (t) , GP a+,h GP a+,h   α,υ,ϕ,λ RL α,υ,ϕ,λ R f (t) = f (t) , GP Db−,h GP Ib−,h   α,υ,ϕ,λ RL α,υ,ϕ,λ R D I f (t) = f (t) , GP a+,h GP a+,h and



α,υ,ϕ,λ RL α,υ,ϕ,λ R f GP Db−,h GP Ib−,h



(t) = f (t) .

υ ∈ C (3.560) (3.561) (3.562) (3.563)

Property 3.46 If −∞ < a < b < ∞ , ϕ ∈ C, υ ∈ C , h(1) (t) > 0, λ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then we have   α,υ,ϕ,λ α,υ,ϕ,λ R LS I f (t) = f (t) , (3.564) D GP a+,h GP a+,h   α,υ,ϕ,λ R α,υ,ϕ,λ LS f (t) = f (t) , (3.565) GP Db−,h GP Ib−,h   α,υ,ϕ,λ LSC α,υ,ϕ,λ R f (t) = f (t) , (3.566) GP Da+,h GP Ia+,h and



3.18

α,υ,ϕ,λ LSC α,υ,ϕ,λ R f GP Db−,h GP Ib−,h



(t) = f (t) .

(3.567)

Hilfer type general fractional derivatives within the Prabhakar kernel

The definitions and properties of the Hilfer type general fractional derivatives within the generalized Prabhakar kernel via Prabhakar function are given as follows. Definition 3.83 Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as (see [101])     Rt υ−1 ϕ α α,β,υ,ϕ,λ α,β Hi f (t) = (t − τ ) Eα,υ (−λ (t − τ ) ) Da+ f (τ ) dτ GP Da+ a    Rt (1−β)(1−α) β(1−α) d υ−1 ϕ α I f (τ ) dτ , = (t − τ ) Eα,υ (−λ (t − τ ) ) Ia+ a+ dτ a

(3.568)

General Fractional Derivatives of Constant Order and Applications

221

and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as   Rb υ−1 ϕ α α,β Eα,υ (−λ (τ − t) ) Db− f (τ ) dτ (t) = − (τ − t) t    Rb β(1−α) d (1−β)(1−α) υ−1 ϕ α = − (τ − t) Eα,υ (−λ (τ − t) ) Ib− f (τ ) dτ . dτ Ib−





α,β,υ,ϕ,λ Hi f GP Db−

t

(3.569) Generally, when κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as   Hir α,β,υ,ϕ,λ f (t) GP Da+    Rt β(κ−α) dκ (1−β)(κ−α) α υ−1 ϕ = (t − τ ) f (τ ) dτ , Eα,υ (−λ (t − τ ) ) Ia+ I κ a+ dτ a

(3.570) and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as   Hir α,β,υ,ϕ,λ D f (t) GP b−    b β(κ−α) dκ (1−β)(κ−α) α κR υ−1 ϕ = (−1) (τ − t) Eα,υ (−λ (τ − t) ) Ib− I f (τ ) dτ . κ b− dτ t

(3.571) In particular, for 1 > α > 0, β = 1, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as (see [101]) 

α,1,υ,ϕ,λ Hi f GP Da+



(t) =

Rt

υ−1

(t − τ )


α 1−α (1) ϕ (−λ (t − τ ) ) Ia+ f (τ ) dτ , Eα,υ

a

(3.572) and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as 

α,1,υ,ϕ,λ Hi f GP Db−




Rb υ−1 ϕ α 1−α (1) (t) = − (τ − t) Eα,υ (−λ (τ − t) ) Ib− f (τ ) (τ ) dτ . t

(3.573) Furthermore, for κ + 1 > α > κ, β = 1, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as 

Hir α,1,υ,ϕ,λ f GP Da+



(t) =

Rt

υ−1

(t − τ )



α κ−α ϕ Eα,υ (−λ (t − τ ) ) Ia+ f (κ) (τ ) dτ ,

a

(3.574)

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General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as     b β(κ−α) (κ) κR υ−1 ϕ α Hir α,1,υ,ϕ,λ D f (t) = (−1) (τ − t) E (−λ (τ − t) ) I f (τ ) dτ. α,υ GP b− b− t

(3.575) In particular, for 1 > α > 0, β = 0, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as  

Rt υ−1 ϕ α α,0,υ,ϕ,λ 1−α d Hi D f (t) = (t − τ ) Eα,υ (−λ (t − τ ) ) dτ Ia+ f (τ ) dτ , a+ GP a

(3.576) and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as  

Rb α υ−1 ϕ α,0,υ,ϕ,λ 1−α d Hi f (t) = − (τ − t) Ib− f (τ ) dτ . Eα,υ (−λ (τ − t) ) dτ GP Db− t

(3.577) Furthermore, when κ + 1 > α > κ, β = 0, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as     Rt υ−1 ϕ α α,0,κ Hir α,0,υ,ϕ,λ D f (t) = (t − τ ) E (−λ (t − τ ) ) D f (τ ) dτ α,υ a+ a+ GP a

=

Rt

(t − τ )

υ−1

ϕ Eα,υ

α

(−λ (t − τ ) )

a

dκ dτ κ



κ−α f (τ ) dτ , Ia+ (3.578)

and the right-sided Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as     b α κR υ−1 ϕ α,0,κ Hir α,0,υ,ϕ,λ f (τ ) dτ f (t) = (−1) (τ − t) Eα,υ (−λ (τ − t) ) Db− GP Db− t

κ

= (−1)

Rb t

υ−1

(τ − t)

ϕ Eα,υ

α

(−λ (τ − t) )

dκ dτ κ

κ−α Ib− f (τ )





dτ . (3.579)

3.19

Hilfer type general fractional derivatives within the Prabhakar kernel with respect to another function

The definitions and properties of the Hilfer type general fractional derivatives involving the generalized Prabhakar kernel via Prabhakar function with respect to another function are given as follows.

General Fractional Derivatives of Constant Order and Applications

223

Definition 3.84 Let 1 > α > 0, 0 ≤ β ≤ 1, − ∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C. The left-sided Hilfer type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function is defined as   α,β,υ,ϕ,λ Hi f (t) GP Da+,h   Rt υ−1 ϕ α α,β = (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) Da+,h f (τ ) h(1) (τ ) dτ , a

(3.580) and the right-sided Hilfer type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function as   α,β,υ,ϕ,λ Hi D f (t) GP b−,h   b R α υ−1 ϕ α,β = − (h (τ ) − h (t)) f (τ ) h(1) (τ ) dτ . Eα,υ (−λ (h (τ ) − h (t)) ) Db−,h t

(3.581) More generally, when κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ C, ϕ ∈ C, υ ∈ C and λ ∈ C, the left-sided Hilfer type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function is defined as   Hir α,β,υ,ϕ,λ f (t) GP Da+,h   Rt υ−1 ϕ α α,β,κ = (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) Da+,h f (τ ) h(1) (τ ) dτ , a

(3.582) and the right-sided Hilfer type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function as   Hir α,β,υ,ϕ,λ D f (t) GP b−,h   b α υ−1 ϕ κR α,β,κ Eα,υ (−λ (h (τ ) − h (t)) ) Db−,h f (τ ) h(1) (τ ) dτ . = (−1) (h (τ )−h (t)) t

(3.583)

3.20 3.20.1

Applications Relaxation models within general fractional derivatives

In this section, we present the general fractional derivatives to model the relaxation models in complex phenomenon. To begin with, we illustrate the open

224

General Fractional Derivatives: Theory, Methods and Applications

problems on the relaxation models within the general fractional derivatives involving the special functions. Example 3.1 The mathematical model of the relaxation involving the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be suggested as follows α,1,λ RL M lt D0+ f

(t) + γf (t) = 0

subjected to the initial condition f (0+) = h,

(3.584)

where λ, γ and h are the constants. Example 3.2 The mathematical model of the relaxation involving the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be illustrated as LSC α,1,λ M le D0+ f

(t) + γf (t) = 0

subjected to the initial condition f (0+) = h,

(3.585)

where λ, γ and h are the constants. Example 3.3 The mathematical model of the relaxation involving the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be shown as follows: LSC α,1,λ M l D0+ f

(t) + γf (t) = 0

subjected to the initial condition f (0+) = h,

(3.586)

where λ, γ and h are the constants. Example 3.4 The mathematical model of the relaxation involving the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be given as follows: LSC 1,λ;ν M l D0+ f

(t) + γf (t) = 0,

(3.587)

subjected to the initial condition f (0+) = h, where λ, γ and h are the constants.

(3.588)

General Fractional Derivatives of Constant Order and Applications

225

Example 3.5 The mathematical model of the relaxation within the general fractional derivative in the sense of Richard type can be shown as follows: Rzh α,1,λ M l D[a,b] f

(t) + γf (t) = 0

(3.589)

where λ and γ are the constants. Example 3.6 The mathematical model of the relaxation within the general fractional derivative in the sense of Feller type can be written as Rzj α,1,λ M lz D[a,b] f

(t) + γf (t) = 0

(3.590)

where λ and γ are the constants. Example 3.7 The mathematical model of the relaxation within the general fractional derivative in the sense of Richard type can be presented as α,λ Rzl M lj D[a,b] f

(t) + γf (t) = 0

(3.591)

where λ and γ are the constants. Example 3.8 The mathematical model of the relaxation based on the general fractional derivative involving the kernel of Mittag-Leffler function in the sense of Riemann-Liouville type with respect to another function can be represented as RL α,λ (3.592) M l D0+,h f (t) + γf (t) = 0 where λ and γ are the constants, h(1) (t) > 0. Example 3.9 The mathematical model of the relaxation based on the general fractional derivative involving the kernel of Mittag-Leffler function with power law in the sense of Liouville-Sonine-Caputo type with respect to another function can be written as LSC α,λ M l D0+,h f

(t) + γf (t) = 0

(3.593)

where λ and γ are the constants and h(1) (t) > 0. Example 3.10 The mathematical model of the relaxation based on the general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of Riemann-Liouville type can be represented as   α,γ,λ RL f (t) + µf (t) = 0, (3.594) D GM 0+ where λ , µ and γ are the constants.

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General Fractional Derivatives: Theory, Methods and Applications

Example 3.11 The mathematical model of the relaxation based on the general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of Liouville-Sonine type can be represented as   α,γ,λ LS (3.595) GM D0+ f (t) + µf (t) = 0, where λ , µ and γ are the constants. Example 3.12 The mathematical model of the relaxation based on the Hilfer type general fractional derivative containing the kernel of the GorenfloMainardi function via Wiman function can be given as   α,β,γ,λ Hi f (t) + µf (t) = 0, (3.596) GM D0+ where λ , µ and γ are the constants and 0 ≤ β ≤ 1. Example 3.13 The mathematical model of the relaxation based on the general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type can be represented as   α,λ RL D f (t) + µf (t) = 0, (3.597) M R 0+ where λ and µ are the constants. Example 3.14 The mathematical model of the relaxation based on the general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type can be written as   α,λ LS D f (t) + µf (t) = 0, (3.598) M R 0+ where λ and µ are the constants. Example 3.15 The mathematical model of the relaxation based on the Hilfer type general fractional derivative containing the kernel of the Miller-Ross function can be given as   α,β,λ Hi (3.599) M R D0+ f (t) + µf (t) = 0, where λ and µ are the constants and 0 ≤ β ≤ 1. Example 3.16 The mathematical model of the relaxation based on the Riemann-Liouville type general fractional derivative involving the twoparametric Lorenzo-Hartley kernel can be represented as   RL α,υ,λ (3.600) HL D0+ f (t) + µf (t) = 0, where λ and µ are the constants and υ > 0.

General Fractional Derivatives of Constant Order and Applications

227

Example 3.17 The mathematical model of the relaxation based on the Liouville-Sonine type general fractional derivative involving the twoparametric Lorenzo-Hartley kernel can be written as   α,υ,λ LS (3.601) HL D0+ f (t) + µf (t) = 0, where γ and µ are the constants and υ > 0. Example 3.18 The mathematical model of the relaxation based on the Riemann-Liouville type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel can be represented as
RL α,γ (3.602) A D0+ f (t) + µf (t) = 0, where γ and µ are the constants. Example 3.19 The mathematical model of the relaxation based on the Liouville-Sonine type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel can be written as
LS α,γ (3.603) A D0+ f (t) + µf (t) = 0, where γ and µ are the constants. Example 3.20 The mathematical model of the relaxation based on the Hilfer type general fractional derivative involving the one-parametric LorenzoHartley kernel can be given as   Hi α,β,γ D f (t) + µf (t) = 0, (3.604) A 0+ where γ , 0 ≤ β ≤ 1 and µ are the constants. Note that there are many mathematical models for the relaxation involving the special functions, such as the Davidson-Cole model (see [306]), Havriliak-Negami model (see [307]), Jurlewicz-Trzmiel-Weron model (see [308]), Oliveira-Mainardi-Vaz model (see [309]) and so on.

3.20.2

Rheological models within general fractional derivatives

In this section, we present the mathematical models of the rheological materials based on the general fractional derivatives involving the special functions. The open problems for the rheological models in general fractional derivatives are discussed in detail. To begin with the ideas, we present the mathematical models for the rheological materials as follows.

228

General Fractional Derivatives: Theory, Methods and Applications

Example 3.21 The mathematical model for the law of deformation based on the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function is represented in the form α,1,λ RL σ (t) = ξM lt D0+ ε (t) (0 < α < 1) ,

(3.605)

which can be written as Zt

d σ (t) = ξ dt

  2α E2α −λ (t − τ ) ε (τ ) dτ (0 < α < 1) ,

(3.606)

0

where α, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.22 The mathematical model for the law of deformation based on the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be written as LSC α,1,λ σ (t) = ξM l D0+ ε (t) (0 < α < 1) ,

(3.607)

which can be represented in the form Zt σ (t) = ξ

α

Eα (−λ (t − τ ) ) ε(1) (τ ) dτ (0 < α < 1) ,

(3.608)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. In particular, when λ = 1, the mathematical model for the law of deformation based on the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function can be expressed as (see [301, 302]) LSC α,1,1 σ (t) = ξM l D0+ ε (t) (0 < α < 1) ,

(3.609)

which is expressed as Zt σ (t) = ξ

α

Eα (− (t − τ ) ) ε(1) (τ ) dτ (0 < α < 1) ,

(3.610)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.23 The mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative within the kernel of Mittag-Leffler function can be given as RL α,λ σ (t) = ξM l D0+ ε (t) (0 < α < 1) ,

(3.611)

General Fractional Derivatives of Constant Order and Applications

229

which yields that d σ (t) = ξ dt

Zt

α

Eα (−λ (t − τ ) ) ε (τ ) dτ ,

(3.612)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. In particular, when λ = 1, the mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative within the kernel of Mittag-Leffler function can be given as (see [302]) RL α,1 σ (t) = ξM l D0+ ε (t) (0 < α < 1) ,

which becomes d σ (t) = ξ dt

Zt

α

Eα (− (t − τ ) ) ε (τ ) dτ ,

(3.613)

(3.614)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.24 The mathematical model for the law of deformation based on the general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of Riemann-Liouville type can be written as α,1,λ RL σ (t) = ξM (3.615) lr D0+ ε (t) (0 < α < 1) , which can be expressed as σ (t) = ξ

d dt

Zt

α

Eα (−λ |t − τ | ) ε (τ ) dτ ,

(3.616)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.25 The mathematical model for the law of deformation based on the general fractional derivative involving the kernel of Mittag-Leffler function with the negative power law in the sense of Liouville-Sonine-Caputo type can be written as LSC α,1,λ σ (t) = ξM (3.617) lr Da+ ε (t) (0 < α < 1) , which implies that Zt σ (t) = ξ

α

Eα (−λ |t − τ | ) ε(1) (τ ) dτ ,

(3.618)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

230

General Fractional Derivatives: Theory, Methods and Applications

Example 3.26 The mathematical model for the law of deformation based on the Liouville-Sonine-Caputo type general fractional derivative within the kernel of Mittag-Leffler function with respect to another function can be expressed as LSC α,1,λ σ (t) = ξM (3.619) l D0+,h ε (t) (0 < α < 1) , which becomes  Zt α σ (t) = ξ Eα (−λ (h (t) − h (τ )) )

 1 (1) ε (τ ) (τ ) h(1) (τ ) dτ (0 < α < 1), h(1) (τ ) h

0

(3.620) where α (0 < α < 1), λ and ξ are the material constants, h(1) (t) > 0, σ (t) is the stress, and ε (t) is the strain. Example 3.27 The mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative within the kernel of Mittag-Leffler function with respect to another function can be written as RL α,1,λ σ (t) = ξM l D0+,h ε (t) (0 < α < 1) ,

(3.621)

which can be repeated as  σ (t) = ξ

d (1) h (t) dt 1

 Zt

α

Eα (−λ (h (t) − h (τ )) ) f (τ ) h(1) (τ ) dτ ,

(3.622)

0

where α (0 < α < 1), λ and ξ are the material constants, h(1) (t) > 0, σ (t) is the stress, and ε (t) is the strain. Example 3.28 The mathematical model for the law of deformation based on the general fractional derivative containing the kernel of the GorenfloMainardi function via Wiman function in the sense of Riemann-Liouville type can be illustrated as   α,γ,λ σ (t) = ξ RL (3.623) GM D0+ ε (t) (0 < α < 1) , which is rewritten as σ (t) = ξ

d dt

Zt

α

Gα,γ (−λ (t − τ ) ) ε (τ ) dτ ,

(3.624)

0

where α (0 < α < 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. In particular, when λ = 1, the mathematical model for the law of deformation based on the general fractional derivative containing the kernel of

General Fractional Derivatives of Constant Order and Applications

231

the Gorenflo-Mainardi function via Wiman function in the sense of RiemannLiouville type can be illustrated as   α,γ,λ σ (t) = ξ RL (3.625) GM D0+ ε (t) (0 < α < 1) , which is rewritten as d σ (t) = ξ dt

Zt

α

Gα,γ (− (t − τ ) ) ε (τ ) dτ ,

(3.626)

0

where α (0 < α < 1), γ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.29 The mathematical model for the law of deformation based on the general fractional derivative containing the kernel of the GorenfloMainardi function via Wiman function in the sense of Liouville-Sonine type can be given as Zt σ (t) = ξ

α

Gα,γ (−λ (t − τ ) ) ε(1) (τ ) dτ ,

(3.627)

0

where α (0 < α < 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. In particular, when λ = 1, the mathematical model for the law of deformation based on the general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function in the sense of LiouvilleSonine type can be given as (see [302]) Zt σ (t) = ξ

α

Gα,γ (− (t − τ ) ) ε(1) (τ ) dτ ,

(3.628)

0

where α (0 < α < 1), γ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.30 The mathematical model for the law of deformation based on the Hilfer type general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function can be illustrated as Zt σ (t) = ξ

   β(1−α) d (1−β)(1−α) α Gα,γ (−λ (t − τ ) ) Ia+ Ia+ ε (τ ) dτ , (3.629) dτ

0

where α (0 < α < 1), β (0 ≤ α ≤ 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

232

General Fractional Derivatives: Theory, Methods and Applications

Example 3.31 The mathematical model for the law of deformation based on the general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type can be illustrated as d σ (t) = ξ dt

Zt

α

Mα (−λ (t − τ ) ) ε (τ ) dτ ,

(3.630)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.32 The mathematical model for the law of deformation based on the general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type can be considered as Zt σ (t) = ξ

α

Mα (−λ (t − τ ) ) ε(1) (τ ) dτ ,

(3.631)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.33 The mathematical model for the law of deformation based on the Hilfer type general fractional derivative containing the kernel of the MillerRoss function can be illustrated as Zt

α



Mα (−λ (t − τ ) )

σ (t) = ξ

β(1−α) Ia+

 d  (1−β)(1−α)  ε (τ ) dτ , I dτ a+

(3.632)

0

where α (0 < α < 1), β (0 ≤ α ≤ 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.34 The mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel can be illustrated as d σ (t) = ξ dt

Zt

α

Fα (−λ (t − τ ) ) ε (τ ) dτ ,

(3.633)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.35 The mathematical model for the law of deformation based on the Liouville-Sonine type general fractional derivative involving the oneparametric Lorenzo-Hartley kernel can be considered as Zt σ (t) = ξ 0

α

Fα (−λ (t − τ ) ) ε(1) (τ ) dτ ,

(3.634)

General Fractional Derivatives of Constant Order and Applications

233

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.36 The mathematical model for the law of deformation based on the Hilfer type general fractional derivative involving the one-parametric Lorenzo-Hartley kernel can be illustrated as Zt



α

Fα (−λ (t − τ ) )

σ (t) = ξ

β(1−α) Ia+

 d  (1−β)(1−α)  I ε (τ ) dτ , dτ a+

(3.635)

0

where α (0 < α < 1), β (0 ≤ α ≤ 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.37 The mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function can be illustrated as d σ (t) = ξ dt

Zt (t − τ )

υ−1

α

ϕ Eα,υ (−λ (t − τ ) ) ε (τ ) dτ ,

(3.636)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.38 The mathematical model for the law of deformation based on the Liouville-Sonine type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function can be considered as Zt (t − τ )

σ (t) = ξ

υ−1

α

ϕ Eα,υ (−λ (t − τ ) ) ε(1) (τ ) dτ ,

(3.637)

0

where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.39 The mathematical model for the law of deformation based on the Hilfer type general fractional derivative within the generalized Prabhakar kernel via Prabhakar function can be illustrated as Zt (t − τ )

σ (t) = ξ

υ−1

ϕ Eα,υ

α

(−λ (t − τ ) )



β(1−α) Ia+

 d  (1−β)(1−α)  I ε (τ ) dτ , dτ a+

0

(3.638) where α (0 < α < 1), β (0 ≤ α ≤ 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

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General Fractional Derivatives: Theory, Methods and Applications

Example 3.40 The mathematical model for the law of deformation based on the Riemann-Liouville type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function can be illustrated as σ (t) =  Zt 1 d υ−1 ϕ α (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) ) ε (τ ) h(1) (τ ) dτ, ξ h(1) (t) dt 0

(3.639) where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.41 The mathematical model for the law of deformation based on the Liouville-Sonine type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function can be illustrated as  Zt υ−1 ϕ α σ (t) = ξ (h (t) − h (τ )) Eα,υ (−λ (h (t) − h (τ )) )

1 d h(1) (τ ) dτ



(1)

εh (τ ) dτ ,

0

(3.640) where α (0 < α < 1), λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 3.42 The mathematical model for the law of deformation based on the Hilfer type general fractional derivative involving the generalized Prabhakar kernel via Prabhakar function with respect to another function can be illustrated as Zt

υ−1

(h (t) − h (τ ))

σ (t) = ξ

  α α,β ϕ Eα,υ (−λ (h (t) − h (τ )) ) Da+,h ε (τ ) dτ ,

a

(3.641) where α (0 < α < 1), β (0 ≤ α ≤ 1), γ, λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

Chapter 4 Fractional Derivatives of Variable Order and Applications

In this chapter, we investigate the variable-order fractional calculus and the variable-order general fractional calculus involving the kernel of the special functions, such as the Mittag-Leffler function, weak power-law function, Lorenzo-Hartley function, Wiman function, Miller-Ross function and Prabhakar function. The Hilfer type fractional derivatives of variable order, the Riesz type fractional derivatives of variable order, the Feller type fractional derivatives of variable order and the Richard type fractional derivatives of variable order are also considered. The general fractional derivatives and general fractional integrals of the variable order involving the special functions are also discussed. The mathematical models for the relaxation and rheological models are addressed in detail.

4.1

Fractional derivatives of variable order involving the kernel of the singular power function

In this section, we introduce the fractional integrals and fractional derivatives of variable order involving the kernel of the singular power function. Definition 4.1 Let α (t) > 0 and −∞ < a < b < ∞. The left-sided RiemannLiouville type fractional integral of variable order is defined as (see [95]) 

α(·) RL Ia+ f



Zt (t) = a

1 f (τ ) dτ , Γ (α (τ )) (t − τ )1−α(τ )

(4.1)

and the right-sided Riemann-Liouville type fractional integral of variable order as Zb   1 f (τ ) α(·) dτ . (4.2) RL Ib− f (t) = Γ (α (τ )) (τ − t)1−α(τ ) t

235

236

General Fractional Derivatives: Theory, Methods and Applications

Definition 4.2 Let κ < α (t) < 1 + κ and −∞ < a < b < ∞. The left-sided Riemann-Liouville type fractional derivative of variable order is defined as 

α(·) RL Da+ f




d κ dt

(t) =



κ−α(·) f RL Ia+



(t) =

dκ dtκ

Rt a

f (τ ) 1 Γ(κ−α(τ )) (t−τ )α(τ )−κ+1 dτ ,

(4.3) and the right-sided Riemann-Liouville type fractional derivative of variable order as 

α(·) RL Db− f



d (t) = − dt


κ 

α(·) RL Ib− f



d (t) = − dt


κ Rb t

f (τ ) 1 Γ(κ−α(τ )) (τ −t)α(τ )−κ+1 dτ .

(4.4) Definition 4.3 Let κ < α (t) < 1 + κ and −∞ < a < b < ∞. The leftsided Riemann-Liouville type fractional derivative of variable order is defined as (see [310]) 

CGS α(·) RL Da+ f



1 dκ (t) = Γ (κ − α (τ )) dtκ

Zt

f (τ ) (t − τ )

a

α(τ )−κ+1

dτ ,

(4.5)

and the right-sided Riemann-Liouville type fractional derivative of variable order as 

CGS α(·) RL Db− f



1 (t) = Γ (κ − α (τ ))

 κ Zb d f (τ ) dτ . − α(τ )−κ+1 dt (τ − t)

(4.6)

t

In particular, when 0 < α (t) < 1 and −∞ < a < b < ∞, the left-sided Riemann-Liouville type fractional derivative of variable order is defined as (see [310]) 

α(·) RL Da+ f



(t) =

d dt



1−α(·) f RL Ia+



(t) =

d dt

Rt a

f (τ ) 1 Γ(1−α(τ )) (t−τ )α(τ ) dτ ,

(4.7)

and the right-sided Riemann-Liouville type fractional derivative of variable order as 

α(·) RL Db− f



  Rb α(·) d d (t) = − dt Ib− f (t) = − dt t

f (τ ) 1 Γ(1−α(τ )) (τ −t)α(τ ) dτ .

(4.8)

Definition 4.4 Let α (t) ≥ 0 and κ = [α (t)] + 1. The left-sided LiouvilleSonine-Caputo type fractional derivative of variable order is defined as (see [95]) α(·) LSC D+a f

α(·)

(t) = I+a

 (κ)  Rt f (t) = a

f (κ) (τ ) 1 Γ(κ−α(τ )) (t−τ )α(τ )−κ+1 dτ ,

(4.9)

Fractional Derivatives of Variable Order and Applications

237

and the right-sided Liouville-Sonine-Caputo type fractional derivative of variable order as α(·) LSC Db− f

b  (κ)  κR f (x) = (−1)

α(·)

(x) = Ib−

t

f (κ) (τ ) 1 Γ(κ−α(τ )) (τ −t)α(τ )−κ+1 dτ .

(4.10) In particular, when 0 < α (t) < 1, the left-sided Liouville-Sonine fractional derivative of variable order as (see [95]) α(·) LS D+a f

α(·)

(t) = I+a

 (1)  Rt f (t) = a

f (1) (τ ) 1 Γ(1−α(τ )) (t−τ )α(τ ) dτ ,

(4.11)

the right-sided Liouville-Sonine fractional derivative of variable order as α(·) LS Db− f

α(·)

(t) = Ib−

 (1)  Rb f (t) = − t

f (1) (τ ) 1 Γ(1−α(τ )) (τ −t)α(τ ) dτ .

(4.12)

More generally, when 0 < α (τ, t) < 1, the left-sided Riemann-Liouville type fractional integral of variable order is defined as (see [311]; also see [312]) 

HL α(·) RLq Ia+ f



Zt (t) = a

1 f (τ ) dτ , Γ (α (τ, t)) (t − τ )1−α(τ,t)

(4.13)

the right-sided Riemann-Liouville type fractional integral of variable order as (see [312]) 

HL α(·) RLq Ib− f



Zb (t) = t

1 f (τ ) dτ , Γ (α (τ, t)) (τ − t)1−α(τ,t)

(4.14)

the left-sided Riemann-Liouville type fractional derivative of variable order is defined as (see [312]) 

α(·) RLq Da+ f



(t) =

d dt



1−α(·) f RLq Ia+



(t) =

d dt

Rt a

f (τ ) 1 Γ(1−α(τ,t)) (t−τ )α(τ,t) dτ ,

(4.15) the right-sided Riemann-Liouville type fractional derivative of variable order as     Rb α(·) α(·) f (τ ) d d 1 D f (t) = − I f (t) = − dt RLq b− dt RLq b− Γ(1−α(τ,t)) (τ −t)α(τ,t) dτ , t

(4.16) the left-sided Riemann-Liouville type fractional derivative of variable order is defined as (see [313]) 

α(·) RLq Da+ f



1 d (t) = Γ (1 − α (τ, t)) dt

Zt

f (τ ) α(τ,t)

a

(t − τ )

dτ ,

(4.17)

238

General Fractional Derivatives: Theory, Methods and Applications

the right-sided Riemann-Liouville type fractional derivative of variable order as (see [313]) 

α(·) RLq Db− f



1 (t) = Γ (1 − α (τ, t))



d − dt

 Zb

f (τ ) α(τ,t)

(τ − t)

t

dτ ,

(4.18)

the left-sided Liouville-Sonine fractional derivative of variable order as (see [312]) α(·) LSq D+a f

α(·)

(t) =RLq I+a



 Rt f (1) (t) = a

f (1) (τ ) 1 Γ(1−α(τ,t)) (t−τ )α(τ,t) dτ ,

(4.19)

and the right-sided Liouville-Sonine fractional derivative of variable order as (see [312]) α(·) LSq Db− f

α(·)

(t) =RLq Ib−

 (1)  Rb f (t) = − t

f (1) (τ ) 1 Γ(1−α(τ,t)) (τ −t)α(τ,t) dτ .

(4.20)

In particular, when α (τ, t) = α (t), the left-sided Riemann-Liouville type fractional integral of variable order is defined as (see [93]) 

α(·) RLqs Ia+ f



(t) =

Rt a

f (τ ) 1 Γ(α(t)) (t−τ )1−α(t) dτ

=

1 Γ(α(t))

Rt a

f (τ ) dτ , (t−τ )1−α(t)

(4.21)

the right-sided Riemann-Liouville type fractional integral of variable order as 

α(·) RLqs Ib− f



(t) =

Rb t

f (τ ) 1 Γ(α(t)) (τ −t)1−α(t) dτ

=

1 Γ(α(t))

Rb t

f (τ ) dτ , (τ −t)1−α(t)

(4.22)

the left-sided Riemann-Liouville type fractional derivative of variable order is defined as (see [93]) 

α(·) SR RLqs Da+ f



1 d (t) = Γ (1 − α (t)) dt

Zt

f (τ ) α(t)

(t − τ )

a

dτ ,

(4.23)

the right-sided Riemann-Liouville type fractional derivative of variable order as Zb   d f (τ ) 1 α(·) dτ , (4.24) RLqs Db− f (t) = − α(t) Γ (1 − α (t)) dt (t − τ ) t

the left-sided Riemann-Liouville type fractional derivative of variable order is defined as       Rt f (τ ) α(·) 1−α(·) d d 1 f (t) = dt Γ(1−α(t)) (t−τ )α(t) dτ , RLqs Da+ f (t) = dt RLqs Ia+ a

(4.25)

Fractional Derivatives of Variable Order and Applications

239

the right-sided Riemann-Liouville type fractional derivative of variable order as !     b R α(·) α(·) f (τ ) d d 1 dτ , RLqs Db− f (t) = − dt RLqs Ib− f (t) = − dt Γ(1−α(t)) (τ −t)α(t) t

(4.26) the left-sided Liouville-Sonine fractional derivative of variable order as (see [314]; with the complement function, see [95]) α(·) LSqs D+a f

α(·)

(t) =RLqs I+a

 (1)  f (t) =

1 Γ(1−α(t))

Rt a

f (1) (τ ) dτ , (t−τ )α(t)

(4.27)

and the right-sided Liouville-Sonine fractional derivative of variable order as α(·) LSqs Db− f

α(·)

(t) =RLqs Ib−

 (1)  Rb 1 f (t) = − Γ(1−α(t)) t

f (1) (τ ) dτ . (τ −t)α(t)

(4.28)

For more information of the theory and applications of the variable-order fractional calculus within the kernel of the singular power function, see [315, 316, 317, 318, 319, 320, 321, 322, 323].

4.2

Riesz, Feller and Richard types fractional derivatives of variable order

In this section, we introduce the Riesz type fractional derivatives of variable order (see [313]), proposed by Zayernouri and Karniadakis in 2015, and give the Feller and Richard types fractional derivatives of variable order based on the Riemann-Liouville fractional derivatives. Meanwhile, we present the definitions of the Riesz, Feller and Richard types fractional derivatives of variable order based on the Liouville-Sonine fractional derivatives. Definition 4.5 The Riesz type fractional derivatives of variable order based on the Riemann-Liouville fractional derivative of variable order is defined as (see [313])        α(·) α(·) α(·) 1 Ri RLqs Da+ f (t) + RLqs Db− f (t) ZK D[a,b] f (t) = − 2 cos(πα(t)/2) × Rt f (τ ) 1 1 d = − 2 cos(πα(t)/2) × Γ(1−α(t)) dτ dt (t−τ )α(t) a

1 + 2 cos(πα(t)/2)

×

1 d Γ(1−α(t)) dt

Rb t

f (τ ) dτ , (τ −t)α(t)

(4.29) where 

 α(·) RLqs Da+ f (t) =

1 d Γ (1 − α (t)) dt

Zt

f (τ ) α(t)

a

(t − τ )

dτ

(4.30)

240

General Fractional Derivatives: Theory, Methods and Applications

and 

α(·) RLqs Db− f



1 d (t) = − Γ (1 − α (t)) dt

Zb

f (τ ) α(t)

t

(τ − t)

dτ .

(4.31)

More generally, the Riesz type fractional derivatives of variable order based on the Riemann-Liouville fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Ri D f (t) = − × D f (t) + D f (t) RLq b− RLq a+ GZK [a,b] 2 cos(πα(t)/2) t R f (τ ) 1 d 1 = − 2 cos(πα(t)/2) × dt Γ(1−α(τ,t)) (t−τ )α(τ,t) dτ a

1 + 2 cos(πα(t)/2)

×

Rb

d dt

t

f (τ ) 1 Γ(1−α(τ,t)) (τ −t)α(τ,t) dτ ,

(4.32) where 

α(·) RLq Da+ f



d (t) = dt

Zt a

1 f (τ ) dτ , Γ (1 − α (τ, t)) (t − τ )α(τ,t)

(4.33)

and 

α(·) RLq Db− f



d (t) = − dt

Zb t

1 f (τ ) dτ . Γ (1 − α (τ, t)) (τ − t)α(τ,t)

(4.34)

In particular, when α (τ, t) = α (t), the Riesz type fractional derivatives of variable order based on the Riemann-Liouville type fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Ri D f (t) = − f (t) + D f (t) × D RL b− RL a+ SZK [a,b] 2 cos(πα(t)/2) t R f (τ ) 1 d 1 = − 2 cos(πα(t)/2) × dt Γ(1−α(τ )) (t−τ )α(τ ) dτ a

1 + 2 cos(πα(t)/2)

×

d dt

Rb t

f (τ ) 1 Γ(1−α(τ )) (τ −t)α(τ ) dτ ,

(4.35) where 

α(·) RL Da+ f



d (t) = dt

Zt a

1 f (τ ) dτ , Γ (1 − α (τ )) (t − τ )α(τ )

(4.36)

and 

α(·) RL Db− f



d (t) = − dt

Zb t

1 f (τ ) dτ . Γ (1 − α (τ )) (τ − t)α(τ )

(4.37)

Fractional Derivatives of Variable Order and Applications

241

Definition 4.6 The Riesz type fractional derivatives of variable order based on the Liouville-Sonine fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Ri D f (t) = − × D f (t) + D f (t) LSqs LSqs a+ LSqs [a,b] b− 2 cos(πα(t)/2) t R (1) f (τ ) 1 1 = − 2 cos(πα(t)/2) × Γ(1−α(t)) dτ (t−τ )α(t) a

1 + 2 cos(πα(t)/2)

×

Rb

1 Γ(1−α(t))

t

f (1) (τ ) dτ , (τ −t)α(t)

(4.38) where 

α(·) LSqs Da+ f



1 (t) = Γ (1 − α (t))

Zt a

f (1) (τ ) (t − τ )

α(t)

dτ

(4.39)

dτ .

(4.40)

and 

α(·) LSqs Db− f



1 (t) = − Γ (1 − α (t))

Zb

f (1) (τ ) α(t)

t

(τ − t)

More generally, the Riesz type fractional derivatives of variable order based on the Liouville-Sonine fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Ri LSq Da+ f (t) + LSq Db− f (t) GLSq D[a,b] f (t) = − 2 cos(πα(t)/2) × Rt f (1) (τ ) 1 1 = − 2 cos(πα(t)/2) dτ × Γ(1−α(τ,t)) (t−τ )α(τ,t) a

1 + 2 cos(πα(t)/2) ×

Rb

(1)

f (τ ) 1 Γ(1−α(τ,t)) (τ −t)α(τ,t) dτ ,

t

(4.41) where 

α(·) LSq Da+ f



Zt (t) = a

f (1) (τ ) 1 dτ , Γ (1 − α (τ, t)) (t − τ )α(τ,t)

(4.42)

and 

α(·) RLq Db− f



Zb (t) = − t

1 f (1) (τ ) dτ . Γ (1 − α (τ, t)) (τ − t)α(τ,t)

(4.43)

In particular, when α (τ, t) = α (t), the Riesz type fractional derivatives of variable order based on the Liouville-Sonine fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Ri D f (t) = − × D f (t) + D f (t) GLC GLC a+ GLC [a,b] b− 2 cos(πα(t)/2) t R (1) f (τ ) 1 1 = − 2 cos(πα(t)/2) × Γ(1−α(τ )) (t−τ )α(τ ) dτ a

1 + 2 cos(πα(t)/2)

×

Rb t

f (1) (τ ) 1 Γ(1−α(τ )) (τ −t)α(τ ) dτ ,

(4.44)

242

General Fractional Derivatives: Theory, Methods and Applications

where 

α(·) GLC Da+ f



Zt (t) = a

1 f (1) (τ ) dτ , Γ (1 − α (τ )) (t − τ )α(τ )

(4.45)

and 

α(·) GLC Db− f



Zb (t) = − t

1 f (1) (τ ) dτ . Γ (1 − α (τ )) (τ − t)α(τ )

(4.46)

Definition 4.7 The Feller type fractional derivatives of variable order based on the Riemann-Liouville fractional derivative of variable order is defined as   α(·) Fe ZK D[a,b] f (t)       α(·) α(·) = − H+ (ϑ, α (t)) RLqs Da+ f (t) + H− (ϑ, α (t)) RLqs Db− f (t) Rt f (τ ) 1 d = − sin((α(t)+ϑ)π/2) dτ × Γ(1−α(t)) sin(πϑ) dt (t−τ )α(t) a

+ sin((α(t)−ϑ)π/2) sin(πϑ)

×

1 d Γ(1−α(t)) dt

Rb t

f (τ ) dτ , (τ −t)α(t)

(4.47) where 

α(·) RLqs Da+ f



1 d (t) = Γ (1 − α (t)) dt

Zt

f (τ ) α(t)

(t − τ )

a

dτ

(4.48)

dτ .

(4.49)

and 

α(·) RLqs Db− f



1 d (t) = − Γ (1 − α (t)) dt

Zb

f (τ ) α(t)

t

(τ − t)

Definition 4.8 The Feller type fractional derivatives of variable order based on the Liouville-Sonine fractional derivative of variable order is defined as   α(·) Fe ZKLC D[a,b] f (t)       α(·) α(·) = − H+ (ϑ, α (t)) RLqs Da+ f (t) + H− (ϑ, α (t)) RLqs Db− f (t) Rt f (1) (τ ) 1 = − sin((α(t)+ϑ)π/2) × Γ(1−α(t)) dτ sin(πϑ) (t−τ )α(t) a

+ sin((α(t)−ϑ)π/2) sin(πϑ)

×

1 Γ(1−α(t))

Rb t

f (1) (τ ) dτ , (τ −t)α(t)

(4.50) where 

α(·) LSqs Da+ f



1 (t) = Γ (1 − α (t))

Zt a

f (1) (τ ) (t − τ )

α(t)

dτ

(4.51)

Fractional Derivatives of Variable Order and Applications

243

and 

α(·) LSqs Db− f



1 (t) = − Γ (1 − α (t))

Zb

f (1) (τ ) α(t)

(τ − t)

t

dτ .

(4.52)

Definition 4.9 The Richard type fractional derivatives of variable order based on the Riemann-Liouville fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Rc RLqs Da+ f (t) + RLqs Db− f (t) ZK D[a,b] f (t) = − 2 cos(πα(t)/2) × Rt f (τ ) 1 d 1 × Γ(1−α(t)) = − 2 cos(πα(t)/2) dτ dt (t−τ )α(t) a

1 × + 2 cos(πα(t)/2)

1 d Γ(1−α(t)) dt

Rb t

f (τ ) dτ , (τ −t)α(t)

(4.53) where 

α(·) RLqs Da+ f



Zt

1 d (t) = Γ (1 − α (t)) dt

f (τ ) α(t)

(t − τ )

a

dτ

(4.54)

dτ .

(4.55)

and 

α(·) RLqs Db− f



1 d (t) = − Γ (1 − α (t)) dt

Zb

f (τ ) α(t)

t

(τ − t)

Definition 4.10 The Richard type fractional derivatives of variable order based on the Liouville-Sonine fractional derivative of variable order is defined as        α(·) α(·) α(·) 1 Rc RLqs Da+ f (t) + RLqs Db− f (t) ZKLC D[a,b] f (t) = − 2 cos(πα(t)/2) × Rt f (1) (τ ) 1 1 = − 2 cos(πα(t)/2) × Γ(1−α(t)) dτ (t−τ )α(t) a

1 + 2 cos(πα(t)/2) ×

Rb

1 Γ(1−α(t))

t

f (1) (τ ) dτ , (τ −t)α(t)

(4.56) where 

α(·) LSqs Da+ f



1 (t) = Γ (1 − α (t))

Zt a

f (1) (τ ) (t − τ )

α(t)

dτ

(4.57)

dτ .

(4.58)

and 

α(·) LSqs Db− f



1 (t) = − Γ (1 − α (t))

Zb

f (1) (τ ) α(t)

t

(τ − t)

244

General Fractional Derivatives: Theory, Methods and Applications

4.3

Hilfer type fractional derivatives of variable order involving the kernel of the singular power function

In this section, we introduce the Hilfer type fractional derivatives of variable order involving the kernel of the singular power function. Definition 4.11 Let κ < α (τ, t) < 1 + κ, κ = [α (τ, t)] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer fractional derivative is defined as   κ  HL β(κ−α(·)) d Hi α(·),β,κ HL (1−β)(κ−α(·)) I f (t) = f (t) , (4.59) D I Hq a+ Hq a+ dtκ Hq a+ and the right-sided variable-order Hilfer fractional derivative as   κ  κ HL β(κ−α(·)) d HL (1−β)(κ−α(·)) Hi α(·),β,κ I f f (t) = (−1) Hq Ib− (t) , Hq Db− dtκ Hq b− (4.60) t Z   1 f (τ ) HL β(κ−α(·)) f (t) = dτ , (4.61) Hq Ia+ Γ (β (κ − α (τ, t))) (t − τ )1−β(κ−α(τ,t)) a



HL β(κ−α(·)) f Hq Ib−



Zb (t) = t

Zt   HL (1−β)(κ−α(·)) f (t) = Hq Ia+ a

f (τ ) 1 dτ , 1−β(κ−α(τ,t)) Γ (β (κ − α (τ, t))) (τ − t)

(4.62)

1 f (τ ) dτ Γ ((1 − β) (κ − α (τ, t))) (t − τ )1−(1−β)(κ−α(τ,t)) (4.63)

and 

HL (1−β)(κ−α(·)) f Hq Ib−



Zb (t) = t

1 f (τ ) dτ . Γ ((1 − β) (κ − α (τ, t))) (τ − t)1−(1−β)(κ−α(τ,t)) (4.64)

Definition 4.12 Let κ < α (t) < 1 + κ, κ = [α (t)] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer fractional derivative is defined as   κ  Hi α(·),β,κ HL β(κ−α(·)) d HL (1−β)(κ−α(·)) D f (t) = I I f (t) , (4.65) H H a+ a+ dtκ H a+ and the right-sided variable-order Hilfer fractional derivative as   κ  κ HL β(κ−α(·)) d Hi α(·),β,κ HL (1−β)(κ−α(·)) f (t) = (−1) H Ib− I f (t) , H Db− dtκ H b− (4.66)

Fractional Derivatives of Variable Order and Applications 

HL β(κ−α(·)) f H Ia+



Zt (t) = a



HL β(κ−α(·)) f H Ib−



Zb (t) = t



HL (1−β)(κ−α(·)) H Ia+



245

1 f (τ ) dτ , Γ (β (κ − α (τ ))) (t − τ )1−β(κ−α(τ ))

(4.67)

1 f (τ ) dτ , Γ (β (κ − α (τ ))) (τ − t)1−β(κ−α(τ ))

(4.68)

Zt

f (τ ) 1 dτ Γ ((1 − β) (κ − α (τ ))) (t − τ )1−(1−β)(κ−α(τ ))

f (t) = a

(4.69) and 

HL (1−β)(κ−α(·)) f H Ib−



Zb (t) = t

f (τ ) 1 dτ . Γ ((1 − β) (κ − α (τ ))) (τ − t)1−(1−β)(κ−α(τ )) (4.70)

Definition 4.13 Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer type fractional derivative is defined as    β(1−α(·)) d α(·),β (1−β)(1−α(·)) HL Hi HL I D f (t) = I f (t) , (4.71) RLqs a+ Sqs a+ dt RLqs a+ and the right-sided variable-order Hilfer type fractional derivative as    α(·),β β(1−α(·)) d (1−β)(1−α(·)) HL Hi HL f (t) = −RLqs Ib− I f (t) , Sqs Db− dt RLqs b−

(4.72)

where 





β(1−α(·)) HL f RLqs Ia+

β(1−α(·)) HL f RLqs Ib−

(1−β)(1−α(·)) HL RLqs Ia+







(t) =

(t) =

f (t) =

1 Γ(β(1−α(t)))

1 Γ(β(1−α(t)))

Rt a

Rb t

1 Γ((1−β)(1−α(t)))

f (τ ) dτ , (t−τ )1−β(1−α(t))

(4.73)

f (τ ) dτ , (τ −t)1−β(1−α(t))

(4.74)

Rt a

f (τ ) dτ , (t−τ )1−(1−β)(1−α(t))

(4.75)

f (τ ) dτ . (τ −t)1−(1−β)(1−α(t))

(4.76)

and 

(1−β)(1−α(·)) HL f RLqs Ib−



(t) =

1 Γ((1−β)(1−α(t)))

Rb t

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General Fractional Derivatives: Theory, Methods and Applications

Definition 4.14 Let κ < α (t) < 1 + κ, κ = [α (t)] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer fractional derivative is defined as   κ  α(·),β,κ HL (1−β)(κ−α(·)) Hi HL β(κ−α(·)) d I f (t) , (4.77) f (t) = Hqs Ia+ Hqs Da+ dtκ Hqs a+ and the right-sided variable-order Hilfer fractional derivative as   κ  α(·),β,κ κ HL β(κ−α(·)) d Hi HL (1−β)(κ−α(·)) I f (t) = (−1) Hqs Ib− f (t) , Hqs Db− dtκ Hqs b− (4.78)   t R β(κ−α(·)) f (τ ) 1 HL (4.79) dτ , f (t) = Γ(β(κ−α(t))) Hqs Ia+ (t−τ )1−β(κ−α(t)) a

 

HL (1−β)(κ−α(·)) Hqs Ia+

and 

4.4

HL β(κ−α(·)) f Hqs Ib−



HL (1−β)(κ−α(·)) f Hqs Ib−



(t) =

f (t) =



(t) =

1 Γ(β(κ−α(t)))

Rb t

1 Γ((1−β)(κ−α(t)))

1 Γ((1−β)(κ−α(t)))

f (τ ) dτ , (τ −t)1−β(κ−α(t))

Rt a

Rb t

(4.80)

f (τ ) dτ , (t−τ )1−(1−β)(κ−α(t))

(4.81)

f (τ ) dτ . (τ −t)1−(1−β)(κ−α(t))

(4.82)

Tempered fractional derivatives of variable order involving the kernel of the weak power-law function

In this section, the variable-order general fractional derivatives and the variable-order tempered fractional integrals involving the kernel of the weak power-law function are given as follows. Definition 4.15 let κ < α (t) < κ + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order Riemann-Liouville type tempered fractional derivative is defined as 

RL α(·),λ f Cp Da+



dκ (t) = κ dt

Zt a

1 f (τ ) e−λ(t−τ ) dτ , (4.83) Γ (κ − α (τ )) (t − τ )α(τ )−κ+1

and the right-sided variable-order Riemann-Liouville type tempered fractional derivative as 

RL α(·),λ f Cp Db−



(t) = (−1)

κ

dκ dtκ

Zb t

1 f (τ ) e−λ(τ −t) dτ . Γ (κ − α (τ )) (τ − t)α(τ )−κ+1 (4.84)

Fractional Derivatives of Variable Order and Applications

247

Definition 4.16 Let α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order Liouville-Sonine-Caputo type tempered fractional derivative is defined as 

LSC α(·),λ f Cp Da+



Zt (t) = a

1 e−λ(t−τ ) f (κ) (τ ) dτ , Γ (κ − α (τ )) (t − τ )α(τ )−κ+1

(4.85)

and the right-sided variable-order Liouville-Sonine-Caputo type tempered fractional derivative as 

LSC α(·),λ f Cp Db−



(t) = (−1)

κ

Zb t

e−λ(τ −t) 1 f (κ) (τ ) dτ , Γ (κ − α (τ )) (τ − t)α(τ )−κ+1 (4.86)

respectively. Definition 4.17 Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral is defined as 

RL α(·),λ f Cp Ia+



Zt (t) = a

1 (t − τ )

1−α(τ )

α(τ )−κ

E1,α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(4.87)

and the right-sided variable-order tempered fractional integral as 

RL α(·),λ f Cp Ib−



Zb (t) =

1−α(τ )

t

4.5

1 (τ − t)

α(τ )−κ

E1,α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.88)

General fractional derivatives of variable order involving the kernel of Mittag-Leffler function

In this section, we present the general fractional derivatives and general fractional integrals of variable order involving the kernel of Mittag-Leffler function. Definition 4.18 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Liouville-Sonine-Caputo type variable-order general fractional derivative involving the kernel of Mittag-Leffler function is defined by LSC α(·),κ,λ f M lt Da+

Zt (t) = a

  2α(τ ) E2α(τ ) −λ (t − τ ) f (κ) (τ ) dτ ,

(4.89)

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General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative involving the kernel of Mittag-Leffler function by LSC α(·),κ,λ f M lt Db−

κ

Zb

(t) = (−1)

  2α(τ ) E2α(τ ) −λ (τ − t) f (κ) (τ ) dτ .

(4.90)

t

Definition 4.19 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative involving the kernel of Mittag-Leffler function is defined by α(·),κ,λ RL f M lt Da+

dκ (t) = κ dt

Zt

  2α(τ ) E2α(τ ) −λ (t − τ ) f (τ ) dτ ,

(4.91)

a

and the right-sided Riemann-Liouville type variable-order general fractional derivative involving the kernel of Mittag-Leffler function by α(·),κ,λ RL f M lt Db−

dκ (t) = (−1) dtκ κ

Zb

  2α(τ ) E2α(τ ) −λ (τ − t) f (τ ) dτ .

(4.92)

t

Definition 4.20 Let α (t) > 0, 1 + κ > α (t) > κ, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order fractional integral is defined as Zt (t) =

1

  2α(τ ) −1 E −λ (t − τ ) f (τ ) dτ , 2α(τ )+1−κ 2α(τ ),κ−2α(τ ) (t − τ ) a (4.93) and the right-sided variable-order fractional integral as RL α(·),κ,λ f M lt Ia+

RL α(·),κ,λ f M lt Ib−

(t) = (−1)

κ

Zb t

1

(τ − t)

  2α(τ ) −1 −λ (τ − t) f (τ ) dτ . E 2α(τ ),κ−2α(τ ) 2α(τ )+1−κ (4.94)

In fact, the general fractional derivatives and general fractional integrals of variable order involving the kernel of Mittag-Leffler function with the normalization function was discussed in [110]. We now consider the general fractional derivatives and general fractional integrals of variable order via Mittag-Leffler function as follows. Definition 4.21 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Liouville-Sonine-Caputo type variable-order general fractional derivative with the Mittag-Leffler function is defined by LSC α(·),κ,λ f M l Da+

Zt (t) = a

  α(τ ) Eα(τ ) −λ (t − τ ) f (κ) (τ ) dτ ,

(4.95)

Fractional Derivatives of Variable Order and Applications

249

and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative with the Mittag-Leffler function by LSC α(·),κ,λ f M l Db−

Zb

κ

(t) = (−1)

  α(τ ) Eα(τ ) −λ (τ − t) f (κ) (τ ) dτ .

(4.96)

t

Definition 4.22 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative with the Mittag-Leffler function is defined as RL α(·),κ,λ f M l Da+

dκ (t) = κ dt

Zt

  α(τ ) f (τ ) dτ , Eα(τ ) −λ (t − τ )

(4.97)

a

and the right-sided Riemann-Liouville type variable-order general fractional derivative with the Mittag-Leffler function as RL α(·),κ,λ f M l Db−

dκ (t) = (−1) dtκ κ

Zb

  α(τ ) Eα(τ ) −λ (τ − t) f (τ ) dτ .

(4.98)

t

Definition 4.23 Let α (t) > 0, 1 + κ > α (t) > κ, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order fractional integral is defined as Zt

1

  α(τ ) −1 E −λ (t − τ ) f (τ ) dτ , α(τ ),κ−α(τ ) α(τ )+1−κ (t − τ ) a (4.99) and the right-sided variable-order fractional integral as RL α(·),κ,λ f M l Ia+

RL α(·),κ,λ f M l Ib−

(t) =

(t) = (−1)κ

Zb t

1 (τ − t)

α(τ )+1−κ

  −1 α(τ ) Eα(τ f (τ ) dτ . ),κ−α(τ ) −λ (τ − t) (4.100)

We have to claim that the Liouville-Sonine type general fractional derivative of variable order involving the kernel of Mittag-Leffler function with the normalization function was discussed in [110]. Meanwhile, the RiemannLiouville type variable-order general fractional derivative with the MittagLeffler function is the special case of the Sousa and de Oliveira results (see [324]).

250

General Fractional Derivatives: Theory, Methods and Applications

4.6

General fractional derivatives of variable order involving the kernel of Wiman function

In this section, we present the variable-order general fractional calculus containing the kernel of the Gorenflo-Mainardi function via Wiman function. Definition 4.24 Let α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as     α(·),γ,λ α(·),γ,λ dκ RL f (t) , (4.101) I f (t) = D GM a+ dtκ GM a+ and the right-sided Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as   
 α(·),γ,λ α(·),γ,λ d κ RL f (t) = − dt f (t) , (4.102) GM Ib− GM Db− where the left-sided and right-sided variable-order general fractional integral operators containing the kernel of the Gorenflo-Mainardi function via Wiman function can be written as α(·),γ,λ f GM Ia+

Zt

  α(τ ) Gα(τ ),γ −λ (t − τ ) f (τ ) dτ

(4.103)

  α(τ ) f (τ ) dτ , Gα(τ ),γ −λ (τ − t)

(4.104)

(t) = a

and α(·),γ,λ f GM Ib−

Zb (t) = t

respectively. Definition 4.25 Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined by     Rt α(·),γ,λ α(τ ) LS (4.105) f (t) = G −λ (t − τ ) f (1) (τ ) dτ , D α(τ ),γ a+ GM a

and the right-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function by     Rb α(·),γ,λ α(τ ) LS (4.106) D f (t) = − G −λ (τ − t) f (1) (τ ) dτ . α(τ ),γ GM b− t

Fractional Derivatives of Variable Order and Applications

251

For κ + 1 > α (t) > κ, −∞ < a < b < ∞, γ ∈ R and λ ∈ R, the leftsided Liouville-Sonine-Caputo type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined by 

LSC α(·),γ,λ f GM Da+



(t) =

Rt

  α(τ ) Gα(τ ),γ −λ (t − τ ) f (κ) (τ ) dτ ,

(4.107)

a

and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function by 

LSC α(·),γ,λ f GM Db−



κ

(t) = (−1)

Rb

  α(τ ) Gα(τ ),γ −λ (τ − t) f (κ) (τ ) dτ .

t

(4.108) Definition 4.26 Let κ + 1 > α (t) > κ, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order general fractional integral is defined as 

α(·),γ,λ R f GM Ia+



Zt

κ−γ−1

(t − τ )

(t) =

  α(τ ) −1 f (τ ) dτ Eα(τ ),κ−γ −λ (t − τ )

a

(4.109) and the right-sided variable-order general fractional integral as 

α(·),γ,λ R f GM Ib−



κ

Zb

κ−γ−1

(τ − t)

= (−1)

  α(τ ) −1 −λ (τ − t) f (τ ) dτ . Eα(τ ),κ−γ

t

(4.110) More generally, we have the following definitions of the variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function and the variable-order general fractional integrals as follows. Definition 4.27 Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined as     α(·),γ,λ α(·),γ,λ d RL f (t) = dt f (t) , (4.111) GM q Ia+ GM q Da+ and the right-sided Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function as   
 α(·),γ,λ α(·),γ,λ d RL D f (t) = − I f (t) , (4.112) GM q GM q b− b− dt

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General Fractional Derivatives: Theory, Methods and Applications

where the left-sided and right-sided variable-order general fractional integral operators containing the kernel of the Gorenflo-Mainardi function via Wiman function can be written as α(·),γ,λ

GM q Ia+

Zt f (t) =

  α(t,τ ) Gα(t,τ ),γ −λ (t − τ ) f (τ ) dτ

(4.113)

  α(t,τ ) Gα(t,τ ),γ −λ (τ − t) f (τ ) dτ ,

(4.114)

a

and α(·),γ,λ

GM q Ib−

Zb f (t) = t

respectively. Definition 4.28 Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined by  
α(·),γ,λ α(·),γ,λ LS D f (t) =GM q Ia+ f (1) (t) , (4.115) GM q a+ and the right-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function by  
α(·),γ,λ α(·),γ,λ LS D f (t) =GM q Ib− −f (1) (t) . (4.116) GM q b− Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order general fractional integral is defined as 

α(·),γ,λ R f GM q Ia+



Zt

−γ

(t − τ )

(t) =

  α(t,τ ) −1 −λ (t − τ ) f (τ ) dτ , Eα(t,τ ),1−γ

a

(4.117) and the right-sided variable-order general fractional integral as 

α(·),γ,λ R f GM q Ib−



Zb =−

−γ

(τ − t)

  α(t,τ ) −1 Eα(t,τ −λ (τ − t) f (τ ) dτ . ),1−γ

t

(4.118) Definition 4.29 Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is defined by     Rt α(·),γ,λ α(t) LS (4.119) −λ (t − τ ) f (1) (τ ) dτ , D f (t) = G α(t),γ GM qs a+ a

Fractional Derivatives of Variable Order and Applications

253

and the right-sided Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function by     Rb α(·),γ,λ α(t) LS (4.120) D f (t) = − G −λ (τ − t) f (1) (τ ) dτ . α(t),γ GM qs b− t

Definition 4.30 Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order general fractional integral is defined as 

α(·),γ,λ R f GM qs Ia+



Zt

−γ

(t − τ )

(t) =

  α(t) −1 Eα(t),1−γ −λ (t − τ ) f (τ ) dτ ,

a

(4.121) and the right-sided variable-order general fractional integral as 

α(·),γ,λ R f GM qs Ib−



Zb =−

−γ

(τ − t)

  α(t) −1 Eα(t),1−γ −λ (τ − t) f (τ ) dτ .

t

(4.122)

4.7

General fractional derivatives of variable order involving the one-parametric Lorenzo-Hartley function

In this section, we present the variable-order general fractional calculus involving the one-parametric Lorenzo-Hartley function. Definition 4.31 Let 1 > α (t) > 0, −∞ < a < b < ∞ and γ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as     α(·),γ d RL α(·),γ f (t) = dt f (t) , (4.123) A Ia+ A Da+ and the right-sided Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as   
 α(·),γ d RL α(·),γ I f (t) , (4.124) D f (t) = − A A b− b− dt where the left-sided and right-sided variable-order general fractional integral operators involving the one-parametric Lorenzo-Hartley kernel can be written as Zt   α(·),γ α(τ ) f (t) = Fα(τ ) −γ (t − τ ) f (τ ) dτ (4.125) A Ia+ a

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General Fractional Derivatives: Theory, Methods and Applications

and α(·),γ f A Ib−

Zb (t) =

  α(τ ) Fα(τ ) −γ (τ − t) f (τ ) dτ ,

(4.126)

t

respectively. More generally, when α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞ and γ ∈ C, the left-sided Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as 

RL α(·),γ f A Da+



(t) =

dκ dtκ

Rt

  α(τ ) Fα(τ ) −γ (t − τ ) f (τ ) dτ ,

(4.127)

a

and the right-sided Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as 

RL α(·),γ f A Db−



d (t) = − dt


κ Rb

  α(τ ) Fα(τ ) −γ (τ − t) f (τ ) dτ .

(4.128)

t

Definition 4.32 Let 1 > α (t) > 0, −∞ < a < b < ∞ and γ ∈ C. The left-sided Liouville-Sonine type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as 

LS α(·),γ f A Da+



(t) =

Rt

  α(τ ) Fα(τ ) −γ (t − τ ) f (1) (τ ) dτ ,

(4.129)

a

and the right-sided Liouville-Sonine type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as 

LS α(·),λ f A Db−



  Rb α(τ ) (t) = − Fα(τ ) −γ (τ − t) f (1) (τ ) dτ .

(4.130)

t

More generally, when κ + 1 > α (t) > κ, −∞ < a < b < ∞ and γ ∈ C, the left-sided Liouville-Sonine-Caputo type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is defined as 

LSC α(·),γ Da+ f A



(t) =

Rt

  α(τ ) Fα(τ ) −γ (t − τ ) f (κ) (τ ) dτ ,

(4.131)

a

and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel as 

LSC α(·),λ Db− f A



(t) = (−1)

κ

Rb t

  α(τ ) Fα(τ ) −γ (τ − t) f (κ) (τ ) dτ .

(4.132)

Fractional Derivatives of Variable Order and Applications

255

Definition 4.33 Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and γ ∈ C. The left-sided variable-order general fractional integral in the real parameters is defined as 

R α(·),γ f A Ia+



Zt

κ−α(τ )−1

(t − τ )

(t) =

  α(τ ) −1 Eα(τ f (τ ) dτ , ),κ−α(τ ) −γ (t − τ )

a

(4.133) and the right-sided variable-order general fractional integral in the real parameters as 

R α(·),γ f A Ib−



κ

Zb

κ−α(τ )−1

(τ − t)

= (−1)

  α(τ ) −1 Eα(τ f (τ ) dτ . ),κ−α(τ ) −γ (τ − t)

t

(4.134)

4.8

General fractional derivatives of variable order involving the Miller-Ross function

In this section, we present the variable-order general fractional calculus involving the Miller-Ross function. Definition 4.34 Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The leftsided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as     α(·),λ α(·),λ d RL f (t) = D f (t) , (4.135) I M R a+ a+ MR dt and the right-sided general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as   
 α(·),λ α(·),λ d RL f (t) = − dt f (t) , (4.136) M R Ib− M R Db− where the left-sided and right-sided variable-order general fractional integral operators within the Miller-Ross kernel are defined as α(·),λ f M R Ia+

Zt (t) =

  α(τ ) Mα(τ ) −λ (t − τ ) f (τ ) dτ

(4.137)

  α(τ ) Mα(τ ) −λ (τ − t) f (τ ) dτ ,

(4.138)

a

and α(·),λ f M R Ib−

Zb (t) = t

respectively.

256

General Fractional Derivatives: Theory, Methods and Applications

For α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R, the left-sided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is defined as     Rt α(·),λ α(τ ) dκ RL (4.139) D f (t) = M −λ (t − τ ) f (τ ) dτ , κ α(τ ) M R a+ dt a

and the right-sided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type as 

α(·),λ RL f M R Db−



d (t) = − dt


κ Rb

  α(τ ) Mα(τ ) −λ (τ − t) f (τ ) dτ .

(4.140)

t

Definition 4.35 Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The leftsided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is defined as     Rt α(·),λ α(τ ) LS (4.141) f (t) = Mα(τ ) −λ (t − τ ) f (1) (τ ) dτ , M R Da+ a

and the right-sided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type as 

α(·),λ LS f M R Db−



  Rb α(τ ) (t) = − Mα(τ ) −λ (τ − t) f (1) (τ ) dτ .

(4.142)

t

For κ+1 > α (t) > κ, −∞ < a < b < ∞ and λ ∈ R, the left-sided variableorder general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type is defined as     Rt α(τ ) LSC α(·),λ (4.143) f (t) = −λ (t − τ ) f (κ) (τ ) dτ , D M α(τ ) a+ MR a

and the right-sided variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine-Caputo type as 

LSC α(·),λ f M R Db−



(t) = (−1)

κ

Rb

  α(τ ) Mα(τ ) −λ (τ − t) f (κ) (τ ) dτ .

(4.144)

t

Definition 4.36 Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order general fractional integral is defined as 

α(·),λ R f M R Ia+



Zt (t) = a

1 (t − τ )

α(τ )+1

E1,−α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(4.145)

and the right-sided variable-order general fractional integral as 

α(·),γ,λ R f M R Ib−



Zb =

1 α(τ )+1

t

(τ − t)

E1,−α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.146)

Fractional Derivatives of Variable Order and Applications

257

More generally, when κ + 1 > α > κ, −∞ < a < b < ∞and λ ∈ R, the left-sided variable-order general fractional integral is defined as 

α(·),λ R f M R Ia+



Zt (t) =

1 α(τ )+2−κ

a

(t − τ )

E1,κ−α(τ )−1 (−λ (t − τ )) f (τ ) dτ , (4.147)

and the right-sided variable-order general fractional integral as 

α(·),γ,λ R f M R Ib−



Zb = t

1 (t − τ )

α(τ )+2−κ

E1,κ−α(τ )−1 (−λ (τ − t)) f (τ ) dτ . (4.148)

4.9

General fractional derivatives of variable order involving the Prabhakar function

In this section, we present the variable-order general fractional calculus involving the Prabhakar function. Definition 4.37 Let 1 > α (t) > 0, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as     α(·),υ,ϕ,λ d RL α(·),υ,ϕ,λ f (t) = dt f (t) , (4.149) GP Ia+ GP Da+ and the right-sided Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as     Rb υ−1 ϕ α(τ ) d RL α(·),υ,ϕ,λ f (t) = − dt (τ − t) Eα(τ ),υ −λ (τ − t) f (τ ) dτ , GP Db− t

(4.150) where the left-sided and right-sided variable-order general fractional integral operators within the Prabhakar kernel via Prabhakar function are represented in the forms α(·),υ,ϕ,λ f GP Ia+

Zt

υ−1

  α(τ ) ϕ f (τ ) dτ Eα(τ ),υ −λ (t − τ )

υ−1

  α(τ ) ϕ f (τ ) dτ , (4.152) Eα(τ −λ (τ − t) ),υ

(t − τ )

(t) =

(4.151)

a

and α(·),υ,ϕ,λ f GP Ib−

Zb

(τ − t)

(t) = t

respectively.

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General Fractional Derivatives: Theory, Methods and Applications

Definition 4.38 For α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R, the left-sided Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as 



RL α(·),υ,ϕ,λ f GP Da+

Rt

dκ dtκ

(t) =

(t − τ )

υ−1

a

  α(τ ) ϕ −λ (t − τ ) f (τ ) dτ , Eα(τ ),υ

(4.153) and the right-sided Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as 

RL α(·),υ,ϕ,λ f GP Db−



d (t) = − dt


κ Rb

υ−1

(τ − t)

t

  α(τ ) ϕ Eα(τ f (τ ) dτ . ),υ −λ (τ − t) (4.154)

Definition 4.39 Forκ + 1 > α (t) > κ, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R, the left-sided Liouville-Sonine-Caputo type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is defined as 

LSC α(·),ϕ,υ,λ f GP Da+



(t) =

Rt

(t − τ )

υ−1

a

  α(τ ) ϕ Eα(τ f (κ) (τ ) dτ , ),υ −λ (t − τ )

(4.155) and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function as 

LSC α(·),ϕ,υ,λ f GP Db−



(t) = (−1)

b κR

υ−1

(τ − t)

t

  α(τ ) ϕ Eα(τ −λ (τ − t) f (κ) (τ ) dτ . ),υ (4.156)

Definition 4.40 For κ + 1 > α (t) > κ, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R, the left-sided variable-order general fractional integral via Prabhakar function is defined as 

α(·),ϕ,υ,λ R f GP Ia+



Zt (t − τ )

(t) =

κ−υ−1

  α(τ ) −ϕ f (τ ) dτ Eα(τ ),κ−υ −λ (t − τ )

a

(4.157) and the right-sided variable-order general fractional integral via Prabhakar function as 

α(·),ϕ,υ,λ R f GP Ib−



Zb

κ−υ−1

(τ − t)

=

  α(τ ) −ϕ Eα(τ −λ (τ − t) f (τ ) dτ . ),κ−υ

t

(4.158)

Fractional Derivatives of Variable Order and Applications

4.10

259

Applications

4.10.1

Relaxation models within variable-order fractional derivatives

Now, we give the open problems on the relaxation models within the variableorder fractional derivatives. Example 4.1 The mathematical model of the relaxation involving the Riemann-Liouville type fractional derivative of variable order can be given as   α(·) (4.159) RL D0+ f (t) + γf (t) = 0, which deduces that d dt

Zt 0

f (τ ) 1 dτ + γf (t) = 0, Γ (1 − α (τ )) (τ − t)α(τ )

(4.160)

where γ is the relaxation constant. Example 4.2 The mathematical model of the relaxation involving the Liouville-Sonine type fractional derivative of variable order can be given as α(·) LS D0+ f

(t) + γf (t) = 0,

(4.161)

which becomes Zt 0

1 f (1) (τ ) dτ + γf (t) = 0, Γ (1 − α (τ )) (t − τ )α(τ )

(4.162)

where γ is the relaxation constant. Example 4.3 The mathematical model of the relaxation based on the Hilfer type fractional derivative of variable order involving the kernel of the singular power function is written as follows 

HL β(1−α(·)) RLq I0+

 d HL (1−β)(1−α(·))  f (t) + γf (t) = 0, I dt RLq 0+

(4.163)

where γ is the relaxation constant and β is the constant. Example 4.4 The mathematical model of the relaxation based on the Hilfer type fractional derivative of variable order involving the kernel of the singular power function is written as follows

260

General Fractional Derivatives: Theory, Methods and Applications



HL β(1−α(·)) RLq I0+

 d HL (1−β)(1−α(·))  I f (t) + γf (t) = 0, dt RLq 0+

(4.164)

where γ is the relaxation constant and β is the constant. Example 4.5 The mathematical model of the relaxation based on the variable-order Riemann-Liouville type tempered fractional derivative is presented as follows Zt

d dt

0

1 f (τ ) e−λ(t−τ ) dτ + γf (t) = 0, Γ (1 − α (τ )) (t − τ )α(τ )

(4.165)

where γ is the relaxation constant and λ is the constant. Example 4.6 The mathematical model of the relaxation based on the variable-order Liouville-Sonine type tempered fractional derivative is represented in the form: Zt 0

f (1) (τ ) −λ(t−τ ) 1 e dτ + γf (t) = 0, Γ (1 − α (τ )) (t − τ )α(τ )

(4.166)

where γ is the relaxation constant and λ is the constant. Example 4.7 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative with the Mittag-Leffler function is given as Zt

  α(τ ) Eα(τ ) −λ (t − τ ) f (1) (τ ) dτ + γf (t) = 0,

(4.167)

0

where γ is the relaxation constant and λ is the constant. Example 4.8 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative with the Mittag-Leffler function is given as d dt

Zt

  α(τ ) Eα(τ ) −λ (t − τ ) f (τ ) dτ + γf (t) = 0,

(4.168)

0

where γ is the relaxation constant and λ is the constant. Example 4.9 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is given as

Fractional Derivatives of Variable Order and Applications Zt

  α(τ ) Gα(τ ),γ −λ (t − τ ) f (1) (τ ) dτ + γf (t) = 0,

261

(4.169)

0

where γ is the relaxation constant and λ is the constant. Example 4.10 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is expressed as d dt

Zt

  α(τ ) Gα(τ ),γ −λ (t − τ ) f (τ ) dτ + γf (t) = 0,

(4.170)

0

where γ is the relaxation constant and λ is the constant. Example 4.11 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is given as d dt

Zt

  α(τ ) f (τ ) dτ + γf (t) = 0, Fα(τ ) −γ (t − τ )

(4.171)

0

where γ is the relaxation constant and λ is the constant. Example 4.12 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is expressed as Zt

  α(τ ) Fα(τ ) −γ (t − τ ) f (1) (τ ) dτ + γf (t) = 0,

(4.172)

0

where γ is the relaxation constant and λ is the constant. Example 4.13 The mathematical model of the relaxation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is given as d dt

Zt

  α(τ ) Mα(τ ) −λ (t − τ ) f (τ ) dτ + γf (t) = 0,

0

where γ is the relaxation constant and λ is the constant.

(4.173)

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General Fractional Derivatives: Theory, Methods and Applications

Example 4.14 The mathematical model of the relaxation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is expressed as Zt

  α(τ ) Mα(τ ) −λ (t − τ ) f (1) (τ ) dτ + γf (t) = 0,

(4.174)

0

where γ is the relaxation constant and λ is the constant. Example 4.15 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is given as

d dt

Zt

υ−1

(t − τ )

  α(τ ) ϕ Eα(τ −λ (t − τ ) f (τ ) dτ + γf (t) = 0, ),υ

(4.175)

0

where γ is the relaxation constant and both ϕ and υ are the constants. Example 4.16 The mathematical model of the relaxation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is expressed as Zt (t − τ )

υ−1

  α(τ ) ϕ Eα(τ −λ (t − τ ) f (1) (τ ) dτ + γf (t) = 0, ),υ

(4.176)

0

where γ is the relaxation constant and both ϕ and υ are the constants.

4.10.2

Rheological models within variable-order fractional derivatives

In this section, we present the mathematical models of the rheological materials based on the variable-order fractional derivatives. The open problems for the rheological models in variable-order fractional derivatives are presented in detail. To start with the ideas, we illustrate the mathematical models for the rheological materials as follows. Example 4.17 The mathematical model for the law of deformation based on the Riemann-Liouville type fractional derivative of variable order is represented in the form (see [325])   α(·) (4.177) σ (t) = ξ RL D0+ ε (t) ,

Fractional Derivatives of Variable Order and Applications

263

which can be written as d σ (t) = ξ dt

Zt 0

1 ε (τ ) dτ , Γ (1 − α (τ )) (τ − t)α(τ )

(4.178)

where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 4.18 The mathematical model for the law of deformation based on the Liouville-Sonine type fractional derivative of variable order is represented in the form (see [326])   α(·) σ (t) = ξ LS D0+ ε (t) , (4.179) which leads to

Zt σ (t) = ξ 0

ε(1) (τ ) 1 dτ , Γ (1 − α (τ )) (τ − t)α(τ )

(4.180)

where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 4.19 The mathematical model for the law of deformation based on the Hilfer type fractional derivative of variable order involving the kernel of the singular power function is written as follows    HL (1−β)(1−α(·)) HL β(1−α(·)) d I ε (t) , σ (t) = ξ RLq I0+ (4.181) dt RLq 0+ where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and β is the constant. Example 4.20 The mathematical model for the law of deformation based on the Hilfer type fractional derivative of variable order involving the kernel of the singular power function is written as follows    β(1−α(·)) d HL (1−β)(1−α(·)) I ε (t) , (4.182) σ (t) = ξ HL I RLq 0+ dt RLq 0+ where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and β is the constant. Example 4.21 The mathematical model for the law of deformation based on the variable-order Riemann-Liouville type tempered fractional derivative is presented as follows d σ (t) = ξ dt

Zt 0

1 ε (τ ) e−λ(t−τ ) dτ , Γ (1 − α (τ )) (t − τ )α(τ )

(4.183)

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant.

264

General Fractional Derivatives: Theory, Methods and Applications

Example 4.22 The mathematical model for the law of deformation based on the variable-order Liouville-Sonine type tempered fractional derivative is represented in the form: Zt

1 ε(1) (τ ) −λ(t−τ ) e dτ , Γ (1 − α (τ )) (t − τ )α(τ )

σ (t) = ξ 0

(4.184)

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.23 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative with the Mittag-Leffler function is given as Zt σ (t) = ξ

  α(τ ) Eα(τ ) −λ (t − τ ) ε(1) (τ ) dτ ,

(4.185)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.24 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative with the Mittag-Leffler function is given as d σ (t) = ξ dt

Zt

  α(τ ) Eα(τ ) −λ (t − τ ) ε (τ ) dτ ,

(4.186)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.25 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is given as Zt σ (t) = ξ

  α(τ ) Gα(τ ),γ −λ (t − τ ) ε(1) (τ ) dτ ,

(4.187)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.26 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative containing the kernel of the Gorenflo-Mainardi function via Wiman function is expressed as

Fractional Derivatives of Variable Order and Applications

d σ (t) = ξ dt

Zt

  α(τ ) Gα(τ ),γ −λ (t − τ ) ε (τ ) dτ ,

265

(4.188)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.27 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is given as

σ (t) = ξ

d dt

Zt

  α(τ ) ε (τ ) dτ , Fα(τ ) −γ (t − τ )

(4.189)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.28 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative involving the one-parametric Lorenzo-Hartley kernel is expressed as Zt σ (t) = ξ

  α(τ ) Fα(τ ) −γ (t − τ ) ε(1) (τ ) dτ ,

(4.190)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.29 The mathematical model for the law of deformation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Riemann-Liouville type is given as d σ (t) = ξ dt

Zt

  α(τ ) Mα(τ ) −λ (t − τ ) ε (τ ) dτ ,

(4.191)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant. Example 4.30 The mathematical model for the law of deformation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is expressed as Zt σ (t) = ξ

  α(τ ) Mα(τ ) −λ (t − τ ) ε(1) (τ ) dτ ,

(4.192)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain and λ is the constant.

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General Fractional Derivatives: Theory, Methods and Applications

Example 4.31 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative within the generalized Prabhakar kernel via Prabhakar function is given as

d σ (t) = ξ dt

Zt (t − τ )

υ−1

  α(τ ) ϕ ε (τ ) dτ , Eα(τ ),υ −λ (t − τ )

(4.193)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain, λ is the constant, and both ϕ and υ are the constants. Example 4.32 The mathematical model for the law of deformation based on the variable-order general fractional derivative within the Miller-Ross kernel in the sense of Liouville-Sonine type is expressed as Zt

υ−1

(t − τ )

σ (t) = ξ

  α(τ ) ϕ Eα(τ ε(1) (τ ) dτ , ),υ −λ (t − τ )

(4.194)

0

where ξ is the material constant, σ (t) is the stress, ε (t) is the strain, λ is the constant, and both ϕ and υ are the constants.

Chapter 5 Fractional Derivatives of Variable Order with Respect to Another Function and Applications

In this chapter, we introduce the variable-order fractional calculus with respect to another function, first proposed by Yang and Machado in 2017 [114], and the variable-order general fractional calculus involving the Mittag-Leffler function with respect to another function, proposed by Sousa and Oliveira in 2018 for the first time [324]. The variable-order general fractional derivatives containing the kernel of the special functions, such as the Mittag-Leffler function and weak power-law function are presented. The definitions of the variable-order fractional derivatives and the variable-order general fractional integrals with respect to another function are introduced. The Hilfer type fractional derivatives of variable order with respect to another function and the Riesz type fractional derivatives of variable order with respect to another function are also proposed. The mathematical models for the relaxation and rheological models are discussed in detail.

5.1

Fractional derivatives of variable order with respect to another function

In this section, we recall the fractional integrals and fractional derivatives of variable order with respect to another function. Definition 5.1 Let α (t) > 0, −∞ < a < b < ∞ and h(1) (t) > 0. The leftsided variable order Riemann-Liouville type fractional integral with respect to another function is defined as 

α(·) RL Ia+,h f



Zt (t) = a

h(1) (τ ) f (τ ) dτ , Γ (α (τ )) (h (t) − h (τ ))1−α(τ )

(5.1)

267

268

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided variable-order Riemann-Liouville type fractional integral with respect to another function as 

α(·) RL Ib−,h f



Zb (t) = t

h(1) (τ ) f (τ ) dτ . Γ (α (τ )) (h (τ ) − h (t))1−α(τ )

(5.2)

Definition 5.2 Let κ < α (t) < 1 + κ, −∞ < a < b < ∞ and h(1) (t) > 0. The left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as (see [114]) 

CGS α(·) RL Da+,h f



1 (t) = Γ (κ − α (τ ))



1

d (1) h (t) dt

κ Z t

h(1) (τ ) f (τ ) α(τ )−κ+1

(h (t) − h (τ ))

a

dτ ,

(5.3) and the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as 

CGS α(·) RL Db−,h f



 κ Zb 1 1 d h(1) (τ ) f (τ ) (t) = dτ . − (1) α(τ )−κ+1 Γ (κ − α (τ )) h (t) dt (h (τ ) − h (t)) t

(5.4) In particular, when 0 < α (t) < 1 , −∞ < a < b < ∞ and h(1) (t) > 0, the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as (see [114]; also see [324]) 

α(·) RL Da+,h f



(t) =



1 h(1) (t)

d dt

 Rt a

h(1) (τ ) f (τ ) Γ(1−α(τ )) (h(t)−h(τ ))α(τ ) dτ ,

(5.5)

and the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as (see [114]) 

α(·) RL Db−,h f



  Rb d (t) = − h(1)1 (t) dt t

f (τ ) h(1) (τ ) Γ(1−α(τ )) (h(τ )−h(t))α(τ ) dτ .

(5.6)

Definition 5.3 Let α (t) ≥ 0, κ = [α (t)] + 1 and h(1) (t) > 0. The leftsided Liouville-Sonine-Caputo type fractional derivative of variable order with respect to another function is defined as α(·)

LSC Da+,h f (t) =

Rt a

h(1) (τ ) 1 Γ(κ−α(τ )) (h(t)−h(τ ))α(τ )−κ+1



1

d h(1) (t) dt

κ

 (κ) fh (τ ) dτ ,

(5.7) and the right-sided Liouville-Sonine-Caputo type fractional derivative of variable order with respect to another function as α(·) LSC Db−,h f

(t) = (−1)κ

Rb t

h(1) (τ ) 1 Γ(κ−α(τ )) (h(τ )−h(t))α(τ )−κ+1



d − h(1)1(τ ) dτ

κ

(κ)

fh

 (τ ) dτ . (5.8)

Fractional Derivatives of Variable Order with Respect to Another Functions 269 In particular, when 0 < α (t) < 1 and h(1) (t) > 0, the left-sided LiouvilleSonine fractional derivative of variable order with respect to another function as (see [114]; also see [324]) α(·) LS Da+,h f

Rt

(t) =

a

h(1) (τ ) 1 Γ(1−α(τ )) (h(t)−h(τ ))α(τ )



1 h(1) (τ )

d dτ



 (1) fh (τ ) dτ ,

(5.9)

the right-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LS Db−,h f

(t) =

Rb t

h(1) (τ ) 1 Γ(1−α(τ )) (h(τ )−h(t))α(τ )



  (1) d − h(1)1(τ ) dτ fh (τ ) dτ .

(5.10) More generally, when 0 < α (τ, t) < 1 and h(1) (t) > 0, the left-sided Riemann-Liouville type fractional integral of variable order with respect to another function is defined as (see [114]) 

HL α(·) RLq Ia+,h f



Zt (t) = a

h(1) (τ ) f (τ ) dτ , Γ (α (τ, t)) (h (t) − h (τ ))1−α(τ,t)

(5.11)

the right-sided Riemann-Liouville type fractional integral of variable order with respect to another function as (see [114]) 

HL α(·) RLq Ib−,h f



Zb (t) = t

h(1) (τ ) f (τ ) dτ , Γ (α (τ, t)) (h (τ ) − h (t))1−α(τ,t)

(5.12)

the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as (see [114]) 

α(·) RLq Da+,h f



(t) =



1 h(1) (t)

d dt

 Rt a

f (τ ) h(1) (τ ) Γ(1−α(τ,t)) (h(t)−h(τ ))α(τ,t) dτ ,

(5.13)

the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as 

α(·) RLq Db−,h f



  Rb d (t) = − h(1)1 (t) dt t

h(1) (τ ) f (τ ) Γ(1−α(τ,t)) (h(τ )−h(t))α(τ,t) dτ ,

(5.14)

the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as 

α(·) RLq Da+,h f



1 (t) = Γ (1 − α (τ, t))



1

d h(1) (t) dt

 Zt

h(1) (τ ) f (τ ) α(τ,t)

a

(h (t) − h (τ ))

dτ ,

(5.15)

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General Fractional Derivatives: Theory, Methods and Applications

the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as 

α(·) RLq Db−,h f



1 (t) = Γ (1 − α (τ, t))



1

d − (1) h (t) dt

 Zb

h(1) (τ ) f (τ ) α(τ,t)

(h (τ ) − h (t))

t

dτ ,

(5.16) the left-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LSq Da+,h f

(x) =

Rt a

f (1) (t) h(1) (τ ) Γ(1−α(τ,t)) (h(t)−h(τ ))α(τ,t) dt,

(5.17)

and the right-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LSq Db−,h f

(t) = −

Rb t

h(1) (τ ) f (1) (t) Γ(1−α(τ,t)) (h(τ )−h(t))α(τ,t) dt.

(5.18)

In particular, when α (τ, t) = α (t), the left-sided Riemann-Liouville type fractional integral of variable order with respect to another function is defined as   Rt α(·) h(1) (τ )f (τ ) 1 (5.19) dτ , RLqs Ia+,h f (t) = Γ(α(t)) (h(t)−h(τ ))1−α(t) a

the right-sided Riemann-Liouville type fractional integral of variable order with respect to another function as   Rb α(·) h(1) (τ )f (τ ) 1 (5.20) dτ , RLqs Ib−,h f (t) = Γ(α(t)) (h(τ )−h(t))1−α(t) t

the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as 

α(·) SR RLqs Da+,h f



1 (t) = Γ (1 − α (t))



d (1) h (t) dt 1

 Zt

h(1) (τ ) f (τ ) α(t)

(h (t) − h (τ ))

a

dτ ,

(5.21) the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as 

α(·) RLqs Db−,h f



1 (t) = − Γ (1 − α (t))

 −

1

d (1) h (t) dt

 Zb

h(1) (τ ) f (τ ) α(t)

t

(h (τ ) − h (t))

dτ ,

(5.22) the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as      Rt h(1) (τ )f (τ ) α(·) 1 d 1 (5.23) dτ , D f (t) = RLqs a+,h Γ(1−α(t)) h(1) (t) dt (h(t)−h(τ ))α(t) a

Fractional Derivatives of Variable Order with Respect to Another Functions 271 the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as !     Rb h(1) (τ )f (τ ) α(·) d 1 1 dτ , (5.24) RLqs Db−,h f (t) = − h(1) (t) dt Γ(1−α(t)) (h(τ )−h(t))α(t) t

the left-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LSqs Da+ f

(t) =

Rt

1 Γ(1−α(t))

a

h(1) (τ ) (h(t)−h(τ ))α(t)



1 h(1) (τ )

d dτ



 (1) fh (τ ) dτ ,

(5.25) and the right-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LSqs Db−,h f

(t) =

Rb

1 Γ(1−α(t))

t

h(1) (τ ) (h(τ )−h(t))α(t)



  (1) d − h(1)1(τ ) dτ fh (τ ) dτ .

(5.26) More generally, when κ < α (t) < κ + 1, the left-sided Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as     κ  Rt α(·) h(1) (τ )f (τ ) 1 d 1 dτ , RLqs Da+,h f (t) = h(1) (t) dt Γ(κ−α(t)) (h(t)−h(τ ))α(t)−κ+1 a

(5.27) the right-sided Riemann-Liouville type fractional derivative of variable order with respect to another function as !    κ Rb α(·) h(1) (τ )f (τ ) 1 d 1 dτ , RLqs Db−,h f (t) = − h(1) (t) dt Γ(κ−α(t)) (h(τ )−h(t))α(t)−κ+1 t

(5.28) the left-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·)

LSqs Da+,h f (t) =

1 Γ(κ−α(t))

Rt a

h(1) (τ ) (h(t)−h(τ ))α(t)−κ+1



1 d h(1) (τ ) dτ

κ

 (κ) fh (τ ) dτ ,

(5.29) and the right-sided Liouville-Sonine fractional derivative of variable order with respect to another function as α(·) LSqs Db−,h f

(t) =

1 Γ(1−α(t))

Rb t

h(1) (τ ) (h(τ )−h(t))α(t)

 κ  (κ) d − h(1)1(τ ) dτ fh (τ ) dτ . (5.30)

272

General Fractional Derivatives: Theory, Methods and Applications

5.2

Riesz type fractional derivatives of variable order with respect to another function

In this section, we extend the Riesz type fractional derivatives of variable order (see [313]), proposed by Zayernouri and Karniadakis in 2015, and give the Riesz type fractional derivatives of variable order with respect to another function. Definition 5.4 The Riesz type fractional derivative of variable order based on the Riemann-Liouville fractional derivative of variable order with respect to another function is defined as   α(·) Ri D f (t) ZK [a,b],h      α(·) α(·) 1 × RLqs Da+,h f (t) + RLqs Db−,h f (t) = − 2 cos(πα(t)/2)   Rt (5.31) h(1) (τ )f (τ ) 1 1 d 1 = − 2 cos(πα(t)/2) × Γ(1−α(t)) dτ (1) h (t) dt (h(t)−h(τ ))α(t) a   Rb h(1) (τ )f (τ ) 1 1 1 d + 2 cos(πα(t)/2) dτ , × Γ(1−α(t)) h(1) (t) dt (h(τ )−h(t))α(t) t

where 

α(·) RLqs Da+,h f



1 (t) = Γ (1 − α (t))



1

d h(1) (t) dt

 Zt

h(1) (τ ) f (τ ) α(t)

a

(h (t) − h (τ ))

dτ (5.32)

and 

 α(·) RLqs Db−,h f (t) =

1 Γ (1 − α (t))

 −

1

d h(1) (t) dt

 Zb

h(1) (τ ) f (τ ) α(t)

t

(h (τ ) − h (t))

dτ . (5.33)

More generally, the Riesz type fractional derivative of variable order with respect to another function based on the Riemann-Liouville fractional derivative of variable order with respect to another function is defined as   α(·) Ri D f (t) GZK [a,b],h      α(·) α(·) 1 = − 2 cos(πα(t)/2) × RLq Da+,h f (t) + RLq Db−,h f (t)   Rt (5.34) h(1) (τ ) f (τ ) 1 d = − 2 cos(πα(t)/2) × h(1)1 (t) dt Γ(1−α(τ,t)) (h(t)−h(τ ))α(τ,t) dτ a   Rb h(1) (τ ) f (τ ) 1 d + 2 cos(πα(t)/2) × h(1)1 (t) dt Γ(1−α(τ,t)) (h(τ )−h(t))α(τ,t) dτ , t

Fractional Derivatives of Variable Order with Respect to Another Functions 273 where 

α(·) RLq Da+,h f



 (t) =

d (1) h (t) dt 1

 Zt

h(1) (τ ) f (τ ) dτ , Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)

a

(5.35) and 

α(·) RLq Db−,h f



 (t) =

d − (1) h (t) dt 1

 Zb t

h(1) (τ ) f (τ ) dτ . Γ (1 − α (τ, t)) (h (τ ) − h (t))α(τ,t)

(5.36) In particular, when α (τ, t) = α (t), the Riesz type fractional derivative of variable order with respect to another function based on the Riemann-Liouville type fractional derivative of variable order with respect to another function is defined as   α(·) Ri SZK D[a,b],h f (t)      α(·) α(·) 1 = − 2 cos(πα(t)/2) × RL Da+,h f (t) + RL Db−,h f (t)  Rt (1)  (5.37) h (τ ) f (τ ) 1 d = − 2 cos(πα(t)/2) × h(1)1 (t) dt Γ(1−α(τ )) (h(t)−h(τ ))α(τ ) dτ a   Rb (1) f (τ ) h (τ ) d 1 1 + 2 cos(πα(t)/2) × h(1) (t) dt Γ(1−α(τ )) (h(τ )−h(t))α(τ ) dτ , t

where 

α(·) RL Da+,h f



 (t) =

1

d (1) h (t) dt

 Zt a

f (τ ) h(1) (τ ) dτ (5.38) Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )

and 

α(·) RL Db−,h f



 (t) = −

1

d h(1) (t) dt

 Zb t

f (τ ) h(1) (τ ) dτ . Γ (1 − α (τ )) (h (τ ) − h (t))α(τ ) (5.39)

Definition 5.5 The Riesz type fractional derivative of variable order with respect to another function based on the Liouville-Sonine fractional derivative of variable order with respect to another function is defined as   α(·) Ri LSqs D[a,b],h f (t)      α(·) α(·) 1 = − 2 cos(πα(t)/2) × LSqs Da+,h f (t) + LSqs Db−,h f (t)   Rt (1) (5.40) h(1) (τ ) 1 1 d 1 = − 2 cos(πα(t)/2) × Γ(1−α(t)) fh (τ ) dτ (1) α(t) h (τ ) dτ (h(t)−h(τ )) a   Rb (1) (1) (τ ) 1 1 1 d + 2 cos(πα(t)/2) × Γ(1−α(t)) (h(τ h)−h(t)) fh (τ ) dτ , α(t) h(1) (τ ) dτ t

274

General Fractional Derivatives: Theory, Methods and Applications

where 

α(·) LSqs Da+,h f



1 (t) = Γ (1 − α (t))

Zt

h(1) (τ )

 α(t)

a

(h (t) − h (τ ))

d 1 (1) h (τ ) dτ



(1)

fh (τ ) dτ (5.41)

and 

α(·) LSqs Db− f



1 (t) = Γ (1 − α (t))

Zb

h(1) (τ )

 α(t)

t

(h (τ ) − h (t))

1 d − (1) h (τ ) dτ



(1)

fh (τ ) dτ . (5.42)

More generally, the Riesz type fractional derivative of variable order with respect to another function based on the Liouville-Sonine fractional derivative of variable order with respect to another function is defined as   α(·) Ri GLSq D[a,b],h f (t)      α(·) α(·) 1 × LSq Da+,h f (t) + LSq Db−,h f (t) = − 2 cos(πα(t)/2)   Rt h(1) (τ ) (1) 1 1 d 1 = − 2 cos(πα(t)/2) × Γ(1−α(τ,t)) fh (τ ) dτ α(τ,t) (1) (τ ) dτ h (h(t)−h(τ )) a   Rb h(1) (τ ) (1) 1 1 1 d + 2 cos(πα(t)/2) × Γ(1−α(τ,t)) (h(τ )−h(t)) fh (τ ) dτ , α(τ,t) (1) h (τ ) dτ t

(5.43) where 

α(·) LSq Da+,h f



Zt (t) = a

h(1) (τ ) 1 Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)



1

d h(1) (τ ) dτ



(1)

fh (τ ) dτ (5.44)

and 

α(·) RLq Db−,h f



Zb (t) =

h(1) (τ ) 1 Γ (1 − α (τ, t)) (h (τ ) − h (t))α(τ,t)

t

 −

1 d h(1) (τ ) dτ



(1)

fh (τ ) dτ . (5.45)

In particular, when α (τ, t) = α (t), the Riesz type fractional derivative of variable order with respect to another function based on the Liouville-Sonine fractional derivative of variable order with respect to another function is defined as   α(·) Ri GLC D[a,b],h f (t)      α(·) α(·) 1 × GLC Da+,h f (t) + GLC Db−,h f (t) = − 2 cos(πα(t)/2)   Rt (1) (5.46) h(1) (τ ) 1 1 d 1 = − 2 cos(πα(t)/2) × Γ(1−α(τ fh (τ ) dτ (1) α(τ ) )) (h(t)−h(τ )) h (τ ) dτ a   Rb (1) h(1) (τ ) 1 1 1 d + 2 cos(πα(t)/2) × Γ(1−α(τ fh (τ ) dτ , )) (h(τ )−h(t))α(τ ) h(1) (τ ) dτ t

Fractional Derivatives of Variable Order with Respect to Another Functions 275 where 

α(·) GLC Da+ f



  Zt 1 h(1) (τ ) 1 d (1) (t) = fh (τ ) dτ Γ (1 − α (τ )) (h (t) − h (τ ))α(τ ) h(1) (τ ) dτ a

(5.47) and 

α(·) GLC Db− f



Zb (t) = t

h(1) (τ ) 1 Γ (1 − α (τ )) (h (τ ) − h (t))α(τ )

 −

1

d h(1) (τ ) dτ



(1)

fh (τ ) dτ . (5.48)

5.3

Hilfer type fractional derivatives of variable order with respect to another function

In this section, we introduce the Hilfer type fractional derivatives of variable order with respect to another function. Definition 5.6 Let 0 < α (τ, t) < 1, 0 ≤ β ≤ 1, h(1) (t) > 0,f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer type fractional derivative with respect to another function is defined as     1 d HL (1−β)(1−α(·))  Hi α(·),β HL β(1−α(·)) f (t) , RLq Ia+,h Sq Da+,h f (t) = RLq Ia+,h h(1) (t) dt (5.49) and the right-sided variable-order Hilfer type fractional derivative with respect to another function as     1 d HL (1−β)(1−α(·))  Hi α(·),β HL β(1−α(·)) D f (t) = I − I f (t) , Sq b−,h RLq b−,h RLq b−,h h(1) (t) dt (5.50) where 

HL β(1−α(·)) f RLq Ia+,h



Zt (t) = a

f (τ ) h(1) (τ ) dτ , Γ (β (1 − α (τ, t))) (h (t) − h (τ ))1−β(1−α(τ,t)) (5.51)



HL β(1−α(·)) f RLq Ib−,h



Zb (t) = t

(1)

h (τ ) f (τ ) dτ , Γ (β (1 − α (τ, t))) (h (τ ) − h (t))1−β(1−α(τ,t)) (5.52)

276 

General Fractional Derivatives: Theory, Methods and Applications

HL (1−β)(1−α(·)) RLq Ia+,h

Zt  h(1) (τ ) f (τ ) f (t) = dτ Γ ((1 − β) (1 − α (τ, t))) (h (t) − h (τ ))1−(1−β)(1−α(τ,t)) a

(5.53)

and 

HL (1−β)(1−α(·)) f RLq Ib−,h

Zb  f (τ ) h(1) (τ ) (t) = dτ . Γ ((1 − β) (1 − α (τ, t))) (h (τ ) − h (t))1−(1−β)(1−α(τ,t)) t

(5.54)

Definition 5.7 Let 0 < α (t) < 1, 0 ≤ β ≤ 1, h(1) (t) > 0, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer type fractional derivative with respect to another function is defined as     d HL (1−β)(1−α(·))  1 Hi α(·),β HL β(1−α(·)) I f (t) , D f (t) = I RL a+,h S RL a+,h a+,h h(1) (t) dt (5.55) and the right-sided variable-order Hilfer type fractional derivative with respect to another function as     1 d HL (1−β)(1−α(·))  Hi α(·),β HL β(1−α(·)) I f (t) , D f (t) = I − RL b−,h S RL b−,h b−,h h(1) (t) dt (5.56) where 

HL β(1−α(·)) f RL Ia+,h



Zt (t) = a

h(1) (τ ) f (τ ) dτ , Γ (β (1 − α (τ ))) (h (t) − h (τ ))1−β(1−α(τ )) (5.57)



HL β(1−α(·)) f RL Ib−,h



Zb (t) = t

(1)

h (τ ) f (τ ) dτ , Γ (β (1 − α (τ ))) (h (τ ) − h (t))1−β(1−α(τ )) (5.58)



HL (1−β)(1−α(·)) RL Ia+,h



Zt f (t) = a

(1)

h (τ ) f (τ ) dτ Γ ((1 − β) (1 − α (τ ))) (h (t) − h (τ ))1−(1−β)(1−α(τ )) (5.59)

and 

HL (1−β)(1−α(·)) f RL Ib−,h



Zb (t) = t

f (τ ) h(1) (τ ) dτ . Γ ((1 − β) (1 − α (τ ))) (h (τ ) − h (t))1−(1−β)(1−α(τ )) (5.60)

Definition 5.8 Let 0 < α (t) < 1, 0 ≤ β ≤ 1, h(1) (t) > 0, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer type fractional derivative with respect to another function is defined as

Fractional Derivatives of Variable Order with Respect to Another Functions 277     1 d HL (1−β)(1−α(·))  α(·),β β(1−α(·)) Hi HL f (t) , Sqs Da+,h f (t) = RLqs Ia+,h RLqs Ia+,h h(1) (t) dt (5.61) and the right-sided variable-order Hilfer type fractional derivative with respect to another function as     d HL (1−β)(1−α(·))  1 α(·),β β(1−α(·)) Hi HL I f (t) , D f (t) = I − RLqs b−,h Sqs b−,h RLqs b−,h h(1) (t) dt (5.62) where 

β(1−α(·)) HL f RLqs Ia+,h



β(1−α(·)) HL f RLqs Ib−,h



(t) =

1 Γ(β(1−α(t)))

Rt a

h(1) (τ )f (τ ) dτ , (h(t)−h(τ ))1−β(1−α(t))

(5.63)

h(1) (τ )f (τ ) dτ , (h(τ )−h(t))1−β(1−α(t))

(5.64)

and 



(1−β)(1−α(·)) HL RLqs Ia+,h



(t) =

1 Γ(β(1−α(t)))

f (t) =

Rb t

1 Γ((1−β)(1−α(t)))

Rt a

h(1) (τ )f (τ ) dτ , (h(t)−h(τ ))1−(1−β)(1−α(t))

(5.65) and 

(1−β)(1−α(·)) HL f RLqs Ib−,h



(t) =

1 Γ((1−β)(1−α(t)))

Rb t

h(1) (τ )f (τ ) dτ . (h(τ )−h(t))1−(1−β)(1−α(t))

(5.66)

5.4

Tempered fractional derivatives of variable order with respect to another function

In this section, we propose the variable-order general fractional derivatives and the variable-order tempered fractional integrals with respect to another function. Definition 5.9 Let κ < α (t) < κ + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is defined as 

RL α(·),λ Cp Da+,h f



(t) =



1 h(1) (t)

d dt

κ Rt a

e−λ(h(t)−h(τ )) h(1) (τ ) 1 Γ(κ−α(τ )) (h(t)−h(τ ))α(τ )−κ+1 f

(τ ) dτ , (5.67)

278

General Fractional Derivatives: Theory, Methods and Applications

and the right-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function as  κ Rb   h(1) (τ )e−λ(h(τ )−h(t)) d 1 1 RL α(·),λ D f (t) = − (1) Cp b−,h Γ(κ−α(τ )) (h(τ )−h(t))α(τ )−κ+1 f (τ ) dτ . h (t) dt t

(5.68) Definition 5.10 Let α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞and λ ∈ R. The left-sided variable-order Liouville-Sonine-Caputo type tempered fractional derivative with respect to another function is defined as  κ    Rt (κ) h(1) (τ )e−λ(h(t)−h(τ )) 1 1 d LSC α(·),λ fh (τ ) dτ , Cp Da+,h f (t) = Γ(κ−α(τ )) (h(t)−h(τ ))α(τ )−κ+1 h(1) (τ ) dτ a

(5.69) and the right-sided variable-order Liouville-Sonine-Caputo type tempered fractional derivative with respect to another function as 

LSC α(·),λ Cp Db−,h f



(t) =

Rb t

h(1) (τ )e−λ(h(τ )−h(t)) 1 Γ(κ−α(τ )) (h(τ )−h(t))α(τ )−κ+1



d − h(1)1(τ ) dτ

κ

(κ)

fh

 (τ ) dτ , (5.70)

respectively. Definition 5.11 Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral with respect to another function is defined as   Rt α(τ )−κ 1 RL α(·),λ E (−λ (h (t) − h (τ ))) f (τ ) dτ , Cp Ia+,h f (t) = (h(t)−h(τ ))1−α(τ ) 1,α(τ ) a

(5.71) and the right-sided variable-order tempered fractional integral with respect to another function as   Rb α(τ )−κ 1 RL α(·),λ E (−λ (h (τ ) − h (t))) f (τ ) dτ . Cp Ib−,h f (t) = (h(τ )−h(t))1−α(τ ) 1,α(τ ) t

(5.72) Definition 5.12 Let κ < α (τ, t) < κ + 1, −∞ < a < b < ∞, h(1) (t) > 0 and λ ∈ R. The left-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is defined as    κ Rt α(·),λ e−λ(h(t)−h(τ )) h(1) (τ ) 1 1 d RL D f (t) = (1) Cpq a+,h Γ(κ−α(τ,t)) (h(t)−h(τ ))α(τ,t)−κ+1 f (τ ) dτ , h (t) dt a

(5.73) and the right-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function as    κ Rb α(·),λ h(1) (τ )e−λ(h(τ )−h(t)) 1 d 1 RL Cpq Db−,h f (t) = − h(1) (t) dt Γ(κ−α(τ,t)) (h(τ )−h(t))α(τ,t)−κ+1 f (τ ) dτ . t

(5.74)

Fractional Derivatives of Variable Order with Respect to Another Functions 279 Definition 5.13 Let κ < α (t) < κ + 1, −∞ < a < b < ∞, h(1) (t) > 0 and λ ∈ R. The left-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is defined as  κ     Rt e−λ(h(t)−h(τ )) h(1) (τ ) α(·),λ 1 1 d RL D f (t) = f (τ ) dτ , Cpqs a+,h Γ(κ−α(t)) h(1) (t) dt (h(t)−h(τ ))α(t)−κ+1 a

(5.75) and the right-sided variable-order Riemann-Liouville type tempered fractional derivative with respect to another function as ! κ    b (1) R −λ(h(τ )−h(t)) α(·),λ h (τ )e 1 d 1 RL f (τ ) dτ . Cpqs Db−,h f (t) = − h(1) (t) dt Γ(κ−α(t)) (h(τ )−h(t))α(t)−κ+1 t

(5.76)

5.5

General fractional derivatives of variable order via Mittag-Leffler function with respect to another function

In this section, we present the general fractional derivatives and general fractional integrals of variable order via Mittag-Leffler function with respect to another function, which was proposed in [324]. Definition 5.14 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Liouville-Sonine-Caputo type variable-order general fractional derivative via Mittag-Leffler function with respect to another function is defined as LSC α(·),κ,λ f M l Da+,h

(t) =

Rt a

    κ (κ) 1 d Eα(τ ) −λ (h (t) − h (τ ))α(τ ) f (τ ) dτ , (1) h h (τ ) dτ (5.77)

and the right-sided Liouville-Sonine-Caputo type variable-order general fractional derivative via Mittag-Leffler function with respect to another function as LSC α(·),κ,λ f M l Db−,h

(t) =

Rb t

    κ (κ) d Eα(τ ) −λ (h (τ ) − h (t))α(τ ) − h(1)1(τ ) dτ fh (τ ) dτ . (5.78)

Definition 5.15 Let α (t) > 0, κ = [α (t)] + 1, κ < α (t) < κ + 1 and λ ∈ R. The left-sided Riemann-Liouville type variable-order general fractional derivative via Mittag-Leffler function with respect to another function is defined by  κ Rt   α(τ ) 1 d RL α(·),κ,λ D f (t) = E −λ (h (t) − h (τ )) f (τ ) dτ , α(τ ) (1) M l a+,h h (t) dt a

(5.79)

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General Fractional Derivatives: Theory, Methods and Applications

and the right-sided Riemann-Liouville type variable-order general fractional derivative via Mittag-Leffler function with respect to another function by RL α(·),κ,λ f M l Db−,h

κ Rb    α(τ ) d Eα(τ ) −λ (h (τ ) − h (t)) f (τ ) dτ . (t) = − h(1)1 (t) dt t

(5.80) Definition 5.16 Let α (t) > 0, 1 + κ > α (t) > κ, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order fractional integral via Mittag-Leffler function with respect to another function is defined as RL α(·),κ,λ f M l Ia+,h

(t) =

Rt a

1 E −1 (h(t)−h(τ ))α(τ )+1−κ α(τ ),κ−α(τ )



 −λ (h (t) − h (τ ))α(τ ) f (τ ) dτ , (5.81)

and the right-sided variable-order fractional integral via Mittag-Leffler function with respect to another function as RL α(·),κ,λ f M l Ib−,h

(t) =

Rb t

1 E −1 (h(τ )−h(t))α(τ )+1−κ α(τ ),κ−α(τ )



 −λ (h (τ ) − h (t))α(τ ) f (τ ) dτ . (5.82)

5.6

Applications

In this section, we discuss the mathematical models for the relaxation and the law of deformation based on the variable-order fractional derivatives with respect to another function.

5.6.1

Relaxation models based on variable-order fractional derivatives with respect to another function

Now, the variable-order fractional derivative and variable-order general fractional derivatives with respect to another function are to model the relaxation models in complex phenomenon. To start with the ideas, we present the open problems on the relaxation models based on the variable-order fractional derivatives with respect to another function. Example 5.1 The mathematical model of the relaxation involving the Riemann-Liouville type fractional derivative of variable order with respect to another function can be given as   α(·) (5.83) RL D0+,h f (t) + γf (t) = 0,

Fractional Derivatives of Variable Order with Respect to Another Functions 281 which becomes 

1

d (1) h (t) dt

 Zt 0

h(1) (τ ) f (τ ) dτ + γf (t) = 0, Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )

(5.84)

where γ is the relaxation constant. Example 5.2 The mathematical model of the relaxation involving the Liouville-Sonine type fractional derivative of variable order with respect to another function can be given as α(·) LS Da+,h f

(t) + γf (t) = 0,

(5.85)

which implies that Zt 0

h(1) (τ ) 1 Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )



1 d h(1) (τ ) dτ



 (1) fh (τ ) dτ + γf (t) = 0, (5.86)

where γ is the relaxation constant. Example 5.3 The mathematical model of the relaxation involving the Riemann-Liouville type fractional derivative of variable order with respect to another function can be given as 

1

d h(1) (t) dt

 Zt 0

h(1) (τ ) f (τ ) dτ + γf (t) = 0, Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)

(5.87)

where γ is the relaxation constant. Example 5.4 The mathematical model of the relaxation based on the Hilfer type fractional derivative of variable order with respect to another function is written as follows: 

HL β(1−α(·)) RLq I0+,h



1

d h(1) (t) dt



HL (1−β)(1−α(·)) f RLq I0+,h



(t) + γf (t) = 0, (5.88)

where γ is the relaxation constant and β is the constant. Example 5.5 The mathematical model of the relaxation based on the Hilfer type fractional derivative of variable order with respect to another function is written as follows: 

HL β(1−α(·)) RLq I0+,h



1

d h(1) (t) dt



HL (1−β)(1−α(·)) f RLq I0+,h



(t) + γf (t) = 0, (5.89)

where γ is the relaxation constant and β is the constant.

282

General Fractional Derivatives: Theory, Methods and Applications

Example 5.6 The mathematical model of the relaxation based on the variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is presented as follows:



1

d h(1) (t) dt

 Zt 0

h(1) (τ ) e−λ(h(t)−h(τ )) f (τ ) dτ + γf (t) = 0, (5.90) Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )

where γ is the relaxation constant and λ is the constant. Example 5.7 The mathematical model of the relaxation based on the variable-order Liouville-Sonine type tempered fractional derivative with respect to another function is represented in the form: Zt 0

h(1) (τ ) e−λ(h(t)−h(τ )) Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )



1 d h(1) (τ ) dτ



 (1) fh (τ ) dτ + γf (t) = 0, (5.91)

where γ is the relaxation constant and λ is the constant. Example 5.8 The mathematical model of the relaxation based on the variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is suggested as follows:



1

d (1) h (t) dt

 Zt 0

e−λ(h(t)−h(τ )) h(1) (τ ) f (τ ) dτ + γf (t) = 0, Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t) (5.92)

where γ is the relaxation constant and λ is the constant. Example 5.9 The mathematical model of the relaxation based on the variable-order Liouville-Sonine type tempered fractional derivative with respect to another function is represented in the form: Zt 0

h(1) (τ ) e−λ(h(t)−h(τ )) Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)



1 d h(1) (τ ) dτ



 (1) fh (τ ) dτ +γf (t) = 0, (5.93)

where γ is the relaxation constant and λ is the constant. Example 5.10 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is given as

Fractional Derivatives of Variable Order with Respect to Another Functions 283

Zt   α(τ ) Eα(τ ) −λ (h (t) − h (τ ))

  1 d (1) f (τ ) h(1) (τ ) dτ +γf (t) = 0, h(1) (τ ) dτ h

0

(5.94) where γ is the relaxation constant and λ is the constant. Example 5.11 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is presented as follows:



d (1) h (t) dt 1

 Zt

  α(τ ) h(1) (τ ) f (τ ) dτ + γf (t) = 0, Eα(τ ) −λ (h (t) − h (τ ))

0

(5.95) where γ is the relaxation constant and λ is the constant. Example 5.12 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is suggested as follows: Zt   α(t,τ ) Eα(t,τ ) −λ (h (t)−h (τ ))

  1 d (1) f (τ ) h(1) (τ )dτ +γf (t) = 0, h(1) (τ ) dτ h

0

(5.96) where γ is the relaxation constant and λ is the constant. Example 5.13 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is reported as follows:



1

d (1) h (t) dt

 Zt

  α(t,τ ) Eα(t,τ ) −λ (h (t) − h (τ )) h(1) (τ ) f (τ ) dτ + γf (t) = 0,

0

(5.97) where γ is the relaxation constant and λ is the constant. Example 5.14 The mathematical model of the relaxation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is proposed as

284

Zt

General Fractional Derivatives: Theory, Methods and Applications

   Eα(t) −λ (h (t) − h (τ ))α(t)

1

d h(1) (τ ) dτ



 (1) fh (τ ) h(1) (τ ) dτ + γf (t) = 0,

0

(5.98)

where γ is the relaxation constant and λ is the constant. Example 5.15 The mathematical model of the relaxation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is written as follows:



d (1) h (t) dt 1

 Zt

  α(t) h(1) (τ ) f (τ ) dτ + γf (t) = 0, Eα(t) −λ (h (t) − h (τ ))

0

(5.99) where γ is the relaxation constant and λ is the constant.

5.6.2

Rheological models in variable-order fractional derivatives with respect to another function

Here, we address the mathematical models of the rheological materials based on the variable-order fractional derivatives with respect to another function. The open problems for the rheological models in variable-order fractional derivatives with respect to another function are presented as follows. Example 5.16 The mathematical model for the law of deformation involving the Riemann-Liouville type fractional derivative of variable order with respect to another function can be given as   α(·) (5.100) σ (t) = ξ RL D0+,h f (t) , which becomes  σ (t) = ξ

1

d (1) h (t) dt

 Zt 0

h(1) (τ ) f (τ ) dτ , Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )

(5.101)

where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 5.17 The mathematical model for the law of deformation involving the Liouville-Sonine type fractional derivative of variable order with respect to another function can be represented as α(·)

σ (t) = ξLS Da+,h f (t) ,

(5.102)

Fractional Derivatives of Variable Order with Respect to Another Functions 285 which can be written as Zt σ (t) = ξ 0

1 h(1) (τ ) Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )



1 d (1) h (τ ) dτ



(1) fh

 (τ ) dτ ,

(5.103) where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 5.18 The mathematical model for the law of deformation involving the Riemann-Liouville type fractional derivative of variable order with respect to another function can be represented as  σ (t) = ξ

d (1) h (t) dt 1

 Zt 0

f (τ ) h(1) (τ ) dτ , Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)

(5.104)

where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 5.19 The mathematical model for the law of deformation involving the Liouville-Sonine type fractional derivative of variable order with respect to another function can be given as Zt σ (t) = ξ 0

1 h(1) (τ ) Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)



1 d (1) h (τ ) dτ



(1) fh

 (τ ) dτ ,

(5.105) where ξ is the material constant, σ (t) is the stress, and ε (t) is the strain. Example 5.20 The mathematical model for the law of deformation based on the Hilfer type fractional derivative of variable order with respect to another function is written as follows  σ (t) = ξ

HL β(1−α(·)) RLq I0+,h



1

d h(1) (t) dt



HL (1−β)(1−α(·)) f RLq I0+,h



(t) ,

(5.106)

where β and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.21 The mathematical model for the law of deformation based on the Hilfer type fractional derivative of variable order with respect to another function can be expressed as  σ (t) = ξ

HL β(1−α(·)) RLq I0+,h



d (1) h (t) dt 1



HL (1−β)(1−α(·)) f RLq I0+,h



(t) ,

(5.107)

where β and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

286

General Fractional Derivatives: Theory, Methods and Applications

Example 5.22 The mathematical model for the law of deformation based on the variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is represented in the form

 σ (t) = ξ

1

d h(1) (t) dt

 Zt 0

h(1) (τ ) e−λ(h(t)−h(τ )) f (τ ) dτ , Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )

(5.108)

where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.23 The mathematical model for the law of deformation based on the variable-order Liouville-Sonine type tempered fractional derivative with respect to another function is represented as Zt σ (t) = ξ 0

e−λ(h(t)−h(τ )) h(1) (τ ) Γ (1 − α (τ )) (h (t) − h (τ ))α(τ )



1 d h(1) (τ ) dτ



(1) fh

 (τ ) dτ ,

(5.109) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.24 The mathematical model for the law of deformation based on the variable-order Riemann-Liouville type tempered fractional derivative with respect to another function is represented as

 σ (t) = ξ

1

d (1) h (t) dt

 Zt 0

h(1) (τ ) e−λ(h(t)−h(τ )) f (τ ) dτ , (5.110) Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)

where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.25 The mathematical model for the law of deformation based on the variable-order Liouville-Sonine type tempered fractional derivative with respect to another function is suggested as Zt σ (t) = ξ 0

h(1) (τ ) e−λ(h(t)−h(τ )) Γ (1 − α (τ, t)) (h (t) − h (τ ))α(τ,t)



1 d (1) h (τ ) dτ



(1) fh

 (τ ) dτ ,

(5.111) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

Fractional Derivatives of Variable Order with Respect to Another Functions 287 Example 5.26 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is given as Zt

   α(τ ) Eα(τ ) −λ (h (t) − h (τ ))

σ (t) = ξ

d 1 h(1) (τ ) dτ



 (1) fh (τ ) h(1) (τ ) dτ ,

0

(5.112) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.27 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is presented as

 σ (t) = ξ

1

d h(1) (t) dt

 Zt

  α(τ ) h(1) (τ ) f (τ ) dτ , Eα(τ ) −λ (h (t) − h (τ ))

0

(5.113) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.28 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is represented in the form Zt σ (t) = ξ

   Eα(t,τ ) −λ (h (t) − h (τ ))α(t,τ )

1

d h(1) (τ ) dτ



 (1) fh (τ ) h(1) (τ ) dτ ,

0

(5.114)

where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.29 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function takes the form

 σ (t) = ξ

d h(1) (t) dt 1

 Zt

  α(t,τ ) Eα(t,τ ) −λ (h (t) − h (τ )) h(1) (τ ) f (τ ) dτ ,

0

(5.115) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

288

General Fractional Derivatives: Theory, Methods and Applications

Example 5.30 The mathematical model for the law of deformation based on the Liouville-Sonine type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function can be expressed as Zt σ (t) = ξ

   α(t) Eα(t) −λ (h (t) − h (τ ))

1 d h(1) (τ ) dτ



(1) fh

 (τ ) h(1) (τ ) dτ ,

0

(5.116) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain. Example 5.31 The mathematical model for the law of deformation based on the Riemann-Liouville type variable-order general fractional derivative via the Mittag-Leffler function with respect to another function is expressed as

 σ (t) = ξ

d (1) h (t) dt 1

 Zt

  α(t) Eα(t) −λ (h (t) − h (τ )) h(1) (τ ) f (τ ) dτ ,

0

(5.117) where λ and ξ are the material constants, σ (t) is the stress, and ε (t) is the strain.

Appendix A Laplace Transforms of the Functions

Laplace gave the integral transform, which is the well-known Laplace transform of the function defined as (see [10, 152, 236]) Z∞ L [f (t)] := f (s) =

e−st f (t) dt

(A.1)

0

and the corresponding inverse Laplace transform is as follows (see [10, 152, 236]): r+iT Z 1 −1 lim est f (s) ds, (A.2) f (t) = L [f (s)] = 2πi T →∞ r−iT

where r ∈ R and s = r + iT . The relation between Laplace and Fourier transforms is given as follows (see [152]): f (s) = f (iω) , (A.3) where ω = is. For information on the Laplace transforms of the functions, see [236, 327].

The functions f (t) + g (t) tf (t) tκ f (t) f (1) (t) f (κ) (t)

Laplace transforms f (s) + g (s) −f (1) (s) κ (−1) f (κ) (s) sf (s) − f (0) κ P κ s f (s) − sκ−j f (j−1) (0) j=1

Rt

f (t) dt

0

Rt

f (s) s

f (at) (λ > 0)

f (s/λ) /λ

f (τ ) g (t − τ ) dτ

f (s) g (s)

0

289

290

General Fractional Derivatives: Theory, Methods and Applications

The functions The exponential decay e−λt The unit step function ϑ (τ ) The unit impulse function δ (τ ) t-ν /Γ (1 − ν) tν /Γ (1+ν) t-1 t t1/κ /Γ (1 + 1/κ) tν e−λt /Γ (1+ν) The sine function sin (λt) The cosine function cos (λt) The hyperbolic sine sinh (λt) The hyperbolic cosine cosh (λt)

Laplace transforms of the functions
−1 s−1 1 + λs−1 s−1 1 sν s−ν s s−1 s−(1+1/κ) −(ν+1) (s + λ)

tν sinh (−λt) /Γ (1+ν)

h i −(ν+1) (s + λ) + (s − λ) h i −(ν+1) −(ν+1) 1 (s + λ) − (s − λ) 2h i −(ν+1) −(ν+1) 1 + (s − iλ) 2 h(s + iλ) i −(ν+1) −(ν+1) 1 (s + iλ) − (s − iλ) 2

tν cosh (−λt) /Γ (1+ν) tν sin (−λt) /Γ (1+ν) tν cos (λt) /Γ (1+ν)

The special functions Eν (λtν ) υ−1 t Eν,υ (λtν ) µ+υ−1 t Eν,µ+υ (λtν ) tυ−1 E1,υ (λt) ϕ tυ−1 Eν,υ (λtν ) ϕ tµ+υ−1 Eν,µ+υ (λtν ) ϕ+$ tυ−1 Eν,υ (λtν ) ϕ+$ tµ+υ−1 Eν,µ+υ (λtν ) υ−1 ϕ t E1,υ (λt) ϕ Eν,1 (λtν ) ϕ E1,1 (λt)

1 2

λ s2 +λ2 s s2 +λ2 λ s2 −λ2 s s2 −λ2 −(ν+1)

Laplace transforms −1 s−1 (1 − λs−ν ) −1 s−υ (1 − λs−ν ) −(µ+υ) −ν −1 s (1 − λs )
−1 −υ s 1 − λs−1 −ϕ s−υ (1 − ℘s−ν ) −ϕ s−µ+υ (1 − λs−ν ) −(ϕ+$) s−υ (1 − λs−ν ) −(ϕ+$) s−(µ+υ) (1 − λs−ν )
−ϕ s−υ 1 − λs−1 −ϕ s−1 (1 − ℘s−ν )
−ϕ s−1 1 − ℘s−1

Appendix B Fourier Transforms of the Functions

Fourier gave the integral transform, which is the well-known Fourier transform of the function defined as (see [12, 236]) Z∞ F [f (t)] = f (ω) =

e−iωt f (t) dt

(B.1)

−∞

and the corresponding inverse Fourier transform is as follows (see [236]): f (t) = F

−1

Z∞ [f (ω)] =

eiωt f (ω) dω.

(B.2)

−∞

The relation between the Fourier and Laplace transforms is given as follows (see [236]): f (ω) = f (−is) , (B.3) where ω = −is. For information on the Fourier transforms of the functions, see [152, 236, 328].

291

292

General Fractional Derivatives: Theory, Methods and Applications The functions f (t) + g (t) tf (t) tκ f (t) f (1) (t) f (κ) (t) f (t − λ) Rt f (t) dt

Fourier transforms f (ω) + g (ω) −if (1) (ω) κ (−i) f (κ) (ω) iωf (ω) κ (iω) f (ω) −iλω e f (s) f (ω) iω

−∞ −iλt

Rt

e f (t) f (at) (λ > 0)

f (ω − λ) f (ω/λ) /λ

f (τ ) g (t − τ ) dτ

f (ω) g (ω)

−∞

The exponential decay e−iλt 1 The unit impulse function δ (τ ) t-ν /Γ (1 − ν) tν /Γ (1+ν) t t1/κ /Γ (1 + 1/κ) tν e−λt /Γ (1+ν)

The special functions Eν (λtν ) tυ−1 Eν,υ (λtν ) tµ+υ−1 Eν,µ+υ (λtν ) tυ−1 E1,υ (λt) ϕ tυ−1 Eν,υ (λtν ) ϕ tµ+υ−1 Eν,µ+υ (λtν ) ϕ+$ tυ−1 Eν,υ (λtν ) ϕ+$ tµ+υ−1 Eν,µ+υ (λtν ) ϕ tυ−1 E1,υ (λt) ϕ Eν,1 (λtν ) ϕ E1,1 (λt)

2πδ (ω − λ) 2πδτ (ω) 1 ν (−iω) −ν (−iω) −1 (−iω) −(1+1/κ) (iω) −(ν+1) (iω + λ)

Fourier transforms  −1 −1 −ν (iω) 1 − λ (iω)  −1 −υ −ν (iω) 1 − λ (iω)  −1 −(µ+υ) −ν (iω) 1 − λ (iω)  −1 −υ −1 (iω) 1 − λ (iω)  −ϕ −υ −ν (iω) 1 − ℘ (iω)  −ϕ −µ+υ −ν (iω) 1 − λ (iω)  −(ϕ+$) −υ −ν (iω) 1 − λ (iω)  −(ϕ+$) −(µ+υ) −ν (iω) 1 − λ (iω)  −ϕ −υ −1 (iω) 1 − λ (iω)  −ϕ −1 −ν (iω) 1 − ℘ (iω)  −ϕ −1 −1 (iω) 1 − ℘ (iω)

Appendix C Mellin Transforms of the Functions

Mellin gave the integral transform, which is the well-known Mellin transform of the function, defined as (see [236, 255, 329]) Z∞ M [f (t)] := f ($) =

t$−1 f (t) dt,

(C.1)

0

and the corresponding inverse Mellin transform is as follows (see [255, 236, 329]): r+iT Z 1 −1 f (t) = M [f ($)] = lim t−$ f ($) d$, (C.2) 2πi T →∞ r−iT

where r ∈ R and $ = r + iT . For information on the Mellin transforms of the functions, see [236, 255].

293

294

General Fractional Derivatives: Theory, Methods and Applications

The functions f (t) + g (t) tf (t) Rt

f (t) t

Mellin transforms f ($) + g ($) f ($+1) f ($ − 1)

f (t) dt

−$−1 f ($)

f (t) dt

$−1 f ($)

0

R∞ t

Rt 0

f (τ ) g

t τ


dτ τ

ν

f ($) g ($) −1

(t/a) (t > a) tν (t > a) −1 (t > a) e−λt Eν (−λt)

(ν + $) a$ −1 − (ν + $) aν+$ −1 $ $ a Γ ($) λ−$ Γ($)Γ(1−$) −$ Γ(1−ν$) λ

Eν,υ (−λt)

Γ($)Γ(1−$) −$ Γ(υ−ν$) λ Γ($)Γ(1−$) −$ Γ(µ+υ−ν$) λ Γ($)Γ(1−$) −$ Γ(υ−$) λ Γ($)Γ(ϕ−$) −$ Γ(ϕ)Γ(υ−ν$) λ Γ($)Γ(ϕ−$) −$ Γ(ϕ)Γ(µ+υ−ν$) λ Γ($)Γ(ϕ+$−$) −$ Γ(ϕ+$)Γ(υ−ν$) λ Γ($)Γ(ϕ+$−$) −$ Γ(ϕ)Γ(µ+υ−ν$) λ Γ($)Γ(ϕ−$) −$ Γ(ϕ)Γ(υ−$) λ Γ($)Γ(ϕ−$) −$ Γ(ϕ)Γ(1−ν$) λ Γ($)Γ(ϕ−$) −$ Γ(ϕ)Γ(1−$) λ

Eν,µ+υ (−λt) E1,υ (−λt) ϕ (−λt) Eν,υ ϕ Eν,µ+υ (−λt) ϕ+$ Eν,υ (−λt) ϕ+$ Eν,µ+υ (−λt) ϕ E1,υ (−λt) ϕ Eν,1 (−λt) ϕ E1,1 (−λt)

Appendix D The Special Functions and Their Expansions

D.1

The one-parametric special functions via MittagLeffler functions

We now present the definitions and properties of the subcosine, subcosine, hyperbolic subcosine and hyperbolic subcosine functions via Mittag-Leffler functions and their Laplace transforms. √ Let κ ∈ N, i = −1, λ ∈ C, α ∈ C, Re (λ) ≥ 0 and Re (α) ≥ 0. The Mittag-Leffler function with one-parameter constant λ is defined as [152] Eα (λtα ) = such that = (α) =

∞ X

λκ tκα , Γ (κα + 1) κ=0

∞ X

1 , Γ (κα + 1) κ=0

(D.1)

(D.2)

which, for α = 1, leads to ∞ X

1 = e = = (1) . Γ (κ + 1) κ=0

(D.3)

The subsine function, denoted by LSinα (tα ), is defined as [330, 331] LSinα (tα ) =

∞ X

κ

(−1) t(2κ+1)α , Γ ((2κ + 1) α + 1) κ=0

(D.4)

which implies that LSinα (λtα ) =

∞ κ X (−1) λ2κ+1 t(2κ+1)α , Γ ((2κ + 1) α + 1) κ=0

(D.5)

the subcosine function, denoted by LCosα (tα ), as [330, 331] ∞ κ X (−1) t2κα LCosα (t ) = , Γ (2κα + 1) κ=0 α

(D.6) 295

296

General Fractional Derivatives: Theory, Methods and Applications

which implies that LCosα (λtα ) =

∞ κ X (−1) λ2κ t2κα , Γ (2κα + 1) κ=0

(D.7)

the hyperbolic subsine function, denoted by LSinhα (tα ), as LSinhα (tα ) =

∞ X

t(2κ+1)α , Γ ((2κ + 1) α + 1) κ=0

(D.8)

which implies that LSinhα (λtα ) =

∞ X

λ2κ+1 t(2κ+1)α , Γ ((2κ + 1) α + 1) κ=0

(D.9)

and the hyperbolic subcosine function, denoted by LCoshα (tα ), as LCoshα (tα ) =

∞ X

t2κα , Γ (2κα + 1) κ=0

(D.10)

which implies that LCoshα (λtα ) =

∞ X

λ2κ t2κα . Γ (2κα + 1) κ=0

(D.11)

In fact, the Mittag-Leffler function with one-parameter constant can be written as [330, 331] Eα (λtα ) = LCoshα (λtα ) + LSinhα (λtα ) ,

(D.12)

which leads to [330, 331] Eα (iλtα ) = LCosα (λtα ) + iLSinα (λtα ) ,

(D.13)

and α

Eα ((iλt) ) =

∞ P κ=0

(−1)κα (λt)2κα Γ(2κα+1)

+ iα

∞ P κ=0

(−1)κα (λt)(2κ+1)α Γ((2κ+1)α+1) ,

(D.14)

which from Eq. (D.1) leads to α

where

Eα ((iλt) ) = RLCosα (λtα ) + iα RLSinα (λtα ) ,

(D.15)

∞ κα (2κ+1)α X (−1) (λt) RLSinα ((λt) ) := Γ ((2κ + 1) α + 1) κ=0

(D.16)

α

The Special Functions and Their Expansions and α

RLCosα ((λt) ) :=

∞ κα 2κα X (−1) (λt) . Γ (2κα + 1) κ=0

With the use of the formulation   tκα 1 L = κα , Γ (κα + 1) s

297

(D.17)

(D.18)

we have [332] −1

L {Eα (−λtα )} = s−1 (1 + λs−α )

(|λs−α | < 1)

(D.19)

such that L {LSinα (λtα )} = λs−(1+α) 1 + λ2 s−2α L {LCosα (λtα )} = s−1 1 + λ2 s−2α


−1


−1

(|λs−α | < 1) ,

(|λs−α | < 1) ,


−1 1 − λ2 s−2α (|λs−α | < 1) ,

L {LSinhα (λtα )} = λs−(1+α)

(D.20) (D.21) (D.22)

and L {LCoshα (λtα )} = s−1 1 − λ2 s−2α


−1

(|λs−α | < 1) .

(D.23)

Therefore, there are LSinα (λtα ) =

Eα (iλtα ) − Eα (−iλtα ) , 2i

Eα (iλtα ) + Eα (−iλtα ) , 2 Eα (λtα ) − Eα (−λtα ) LSinhα (λtα ) = 2

LCosα (λtα ) =

and LCoshα (λtα ) =

Eα (λtα ) + Eα (−λtα ) . 2

(D.24) (D.25) (D.26)

(D.27)

There is [330, 331] Eα (itα ) = LCosα (tα ) + iLSinα (tα ) .

(D.28)

In this case, there are [333] LSinα (tα ) =

Eα (itα ) − Eα (−itα ) , 2i

(D.29)

LCosα (tα ) =

Eα (itα ) + Eα (−itα ) , 2

(D.30)

298

General Fractional Derivatives: Theory, Methods and Applications LSinhα (tα ) =

Eα (tα ) − Eα (−tα ) 2

(D.31)

LCoshα (tα ) =

Eα (tα ) + Eα (−tα ) . 2

(D.32)

and

Thus, we have Eα (−itα ) = LCosα (tα ) − iLSinα (tα )

(D.33)

Eα (−iλtα ) = LCosα (λtα ) − iLSinα (λtα )

(D.34)

Eα (−itα ) = Re (Eα (−itα )) + iIm (Eα (−itα )) ,

(D.35)

Re (Eα (−itα )) = Cosα (−tα )

(D.36)

Im (Eα (−itα )) = LSinα (−tα ) .

(D.37)

and such that where and In a similar way, we have Eα (itα ) = LCosα (tα ) + iLSinα (tα )

(D.38)

Eα (iλtα ) = LCosα (λtα ) + iLSinα (λtα )

(D.39)

Eα (itα ) = Re (Eα (itα )) + iIm (Eα (itα )) ,

(D.40)

Re (Eα (itα )) = Cosα (tα )

(D.41)

Im (Eα (itα )) = LSinα (tα ) .

(D.42)

and such that where and In this case, we find that LCosα (−λtα ) = LCosα (λtα )

(D.43)

LSinα (−λtα ) = −LSinα (λtα ) .

(D.44)

and We obtain the following results: ϑ1 (α) :=

ϑ2 (α) :=

∞ X

1 , Γ (2κα + 1) κ=0

∞ X

t(2κ+1)α , Γ ((2κ + 1) α + 1) κ=0

(D.45)

(D.46)

The Special Functions and Their Expansions ϑ3 (α) := and ϑ4 (α) :=

D.2

∞ X

κ

(−1) Γ (2κα + 1) κ=0

∞ X

299 (D.47)

κ

(−1) . Γ ((2κ + 1) α + 1) κ=0

(D.48)

The one-parametric special functions via Plotnikov functions

The Plotnikov subsine function, suggested by Plotnikov in 1979, is defined as [334] ∞ κ X (−1) t(2−α)κ+1 P Sinα (tα ) = , (D.49) Γ ((2 − α) κ + 2) κ=0 which implies that P Sinα (λtα ) =

∞ κ X (−1) λ2κ+1 t(2−α)κ+1 , Γ ((2 − α) κ + 2) κ=0

(D.50)

Plotnikov subcosine function, denoted by P Cosα (tα ), as [334] ∞ κ X (−1) t(2−α)κ P Cosα (t ) = , Γ ((2 − α) κ + 1) κ=0 α

(D.51)

which implies that P Cosα (λtα ) =

∞ κ X (−1) λ2κ t(2−α)κ , Γ ((2 − α) κ + 1) κ=0

(D.52)

the hyperbolic subsine function, denoted by P Sinhα (tα ), as P Sinhα (tα ) =

∞ X

t(2−α)κ+1 , Γ ((2 − α) κ + 2) κ=0

(D.53)

which implies that P Sinhα (λtα ) =

∞ X λ2κ+1 t(2−α)κ+1 , Γ ((2 − α) κ + 2) κ=0

(D.54)

the hyperbolic subcosine function, denoted by P Coshα (tα ), as α

P Coshα (t ) =

∞ X

t(2−α)κ , Γ ((2 − α) κ + 1) κ=0

(D.55)

300

General Fractional Derivatives: Theory, Methods and Applications

which implies that P Coshα (λtα ) =

∞ X

λ2κ t(2−α)κ . Γ ((2 − α) κ + 1) κ=0

Finding the Laplace transforms of the above formulations, we have

−1 2 2−α λ s