General Fractional Derivatives with Applications in Viscoelasticity 0128172088, 9780128172087

Table of contents :
Contents
Preface
1 Special functions
1.1 Euler gamma and beta functions
1.1.1 Euler gamma function
1.1.2 Euler beta function
1.2 Laplace transform and properties
1.3 Mittag-Leffler function
1.4 Miller-Ross function
1.5 Rabotnov function
1.6 One-parameter Lorenzo-Hartley function
1.7 Prabhakar function
1.8 Wiman function
1.9 The two-parameter Lorenzo-Hartley function
1.10 Two-parameter Gorenflo-Mainardi function
1.11 Euler-type gamma and beta functions with respect to another function
1.12 Mittag-Leffler-type function with respect to another function
1.13 Miller-Ross-type function with respect to function
1.14 Rabotnov-type function with respect to another function
1.15 Lorenzo-Hartley-type function with respect to another function
1.16 Prabhakar-type function with respect to another function
1.17 Wiman-type function with respect to another function
1.18 Two-parameter Lorenzo-Hartley function with respect to another function
1.19 Gorenflo-Mainardi-type function with respect to another function
2 Fractional derivatives with singular kernels
2.1 The space of the functions
2.1.1 The set of Lebesgue measurable functions
2.1.2 The weighted space with the power weight
2.1.3 The space of absolutely continuous functions
2.1.4 The Kolmogorov-Fomin condition
2.1.5 The Samko-Kilbas-Marichev condition
2.2 Riemann-Liouville fractional calculus
2.2.1 Riemann-Liouville fractional integrals
2.2.2 Riemann-Liouville fractional derivatives
2.3 Osler fractional calculus
2.4 Liouville-Weyl fractional calculus
2.4.1 Liouville-Weyl fractional integrals
2.4.2 Liouville-Weyl fractional derivatives
2.5 Samko-Kilbas-Marichev fractional calculus
2.5.1 Samko-Kilbas-Marichev fractional integrals
2.5.2 Samko-Kilbas-Marichev fractional derivatives
2.6 Liouville-Sonine-Caputo fractional derivatives
2.6.1 History of Liouville-Sonine-Caputo fractional derivatives
2.7 Liouville fractional derivatives
2.8 Almeida fractional derivatives with respect to another function
2.9 Liouville-type fractional derivative with respect to another function
2.10 Liouville-Grünwald-Letnikov fractional derivatives
2.10.1 History of the Liouville-Grünwald-Letnikov fractional derivatives
2.10.2 Concepts of Liouville-Grünwald-Letnikov fractional derivatives
2.10.3 Liouville-Grünwald-Letnikov fractional derivatives on a bounded domain
2.11 Kilbas-Srivastava-Trujillo fractional difference derivatives
2.12 Riesz fractional calculus
2.12.1 Riesz fractional calculus
2.12.2 Riesz-type fractional calculus
2.12.3 Liouville-Sonine-Caputo-Riesz-type fractional derivatives
2.13 Feller fractional calculus
2.13.1 Feller fractional calculus
2.13.2 Feller-type fractional calculus
2.13.3 Liouville-Sonine-Caputo-Feller-type fractional derivatives
2.14 Herrmann fractional calculus
2.14.1 Herrmann fractional calculus
2.14.2 Herrmann-type fractional calculus
2.14.3 Liouville-Sonine-Caputo-Herrmann-type fractional derivatives
2.15 Samko-Kilbas-Marichev symmetric fractional difference derivative
2.16 Grünwald-Letnikov-Herrmann-type symmetric fractional difference derivative
2.17 Grünwald-Letnikov-Feller-type symmetric fractional difference derivative
2.18 Samko-Kilbas-Marichev symmetric fractional difference derivative on a bounded domain
2.19 Grünwald-Letnikov-Herrmann-type symmetric fractional difference derivative on a bounded domain
2.20 Grünwald-Letnikov-Feller-type symmetric fractional difference derivative on a bounded domain
2.21 Erdelyi-Kober-type calculus
2.21.1 Erdelyi-Kober-type fractional integrals
2.21.2 Erdelyi-Kober-type fractional derivatives
2.21.3 Erdelyi-Kober-Riesz-type fractional difference derivative on the real line
2.21.4 Erdelyi-Kober-Herrmann-type fractional difference derivative on the real line
2.21.5 Erdelyi-Kober-Feller-type fractional difference derivative on the real line
2.21.6 Erdelyi-Kober-Riesz-type fractional difference derivative on a bounded domain
2.21.7 Erdelyi-Kober-Herrmann-type fractional difference derivative on a bounded domain
2.21.8 Erdelyi-Kober-Feller-type fractional difference derivative on a bounded domain
2.22 Hadamard fractional calculus
2.22.1 Hadamard fractional calculus
2.22.2 Hadamard-Riesz-type fractional difference derivative on a bounded domain
2.22.3 Hadamard-Herrmann-type fractional difference derivative on a bounded domain
2.22.4 Hadamard-Feller-type fractional difference derivative on a bounded domain
2.22.5 Hadamard fractional calculus on the real line
2.22.6 Hadamard-type fractional calculus
2.22.7 Hadamard-type fractional calculus involving the exponential function
2.22.8 Hadamard-type fractional calculus involving the exponential function on the real line
2.22.9 Extended Hadamard-type fractional derivatives
2.23 Marchaud fractional derivatives
2.23.1 Definition of Marchaud fractional derivatives
2.23.2 Definition of Marchaud-type fractional derivatives
2.23.3 Marchaud-type fractional derivatives with respect to another function
2.24 Riemann-Liouville-type tempered fractional calculus
2.24.1 Riemann-Liouville-type tempered fractional derivatives
2.24.2 Riemann-Liouville-type tempered fractional integrals
2.25 Liouville-Weyl-type tempered fractional calculus
2.25.1 Liouville-Weyl-type tempered fractional derivatives on the real line
2.25.2 Liouville-Weyl-type tempered fractional integrals on the real line
2.25.3 Liouville-Sonine-Caputo-type tempered fractional derivatives
2.25.4 Liouville-Sonine-Caputo-type tempered fractional derivatives
2.25.5 Liouville-Weyl-Riesz type tempered fractional calculus
2.25.6 Riemann-Liouville-Riesz-type fractional calculus
2.25.7 Liouville-Sonine-Caputo-Riesz-type tempered fractional derivatives
2.25.8 Liouville-Weyl-Feller tempered fractional calculus
2.25.9 Riemann-Liouville-Feller-type tempered fractional calculus
2.25.10 Liouville-Sonine-Caputo-Feller-type tempered fractional derivatives
2.25.11 Liouville-Weyl-Herrmann tempered fractional calculus
2.25.12 Riemann-Liouville-Herrmann-type tempered fractional calculus
2.25.13 Liouville-Sonine-Caputo-Herrmann-type tempered fractional derivatives
2.26 Riemann-Liouville-type tempered fractional calculus with respect to another function
2.26.1 Riemann-Liouville-type tempered fractional integrals with respect to another function
2.26.2 Riemann-Liouville-type tempered fractional integrals with respect to another function
2.26.3 Riemann-Liouville-type tempered fractional derivatives with respect to another function on the real line
2.26.4 Liouville-type tempered fractional integrals with respect to another function on the real line
2.26.5 Liouville-Sonine-Caputo-type tempered fractional derivatives with respect to another function
2.26.6 Liouville-type tempered fractional derivatives with respect to another function
2.27 Hilfer derivatives
2.27.1 Liouville-Weyl-Hilfer-type derivatives
2.28 Mixed fractional derivatives
2.28.1 Riesz-Hilfer-type fractional derivative
2.28.2 Feller-Hilfer-type fractional derivative
2.28.3 Herrmann-Hilfer-type fractional derivative
2.28.4 Riesz-Hilfer fractional derivative
2.28.5 Feller-Hilfer fractional derivative
2.28.6 Herrmann-Hilfer fractional derivative
2.28.7 Sousa-de Oliveira fractional derivative with respect to another function
2.28.8 Liouville-Weyl-Sousa-de Oliveira-type fractional derivatives
2.28.9 Hilfer-Riesz type fractional derivative with respect to another function
2.28.10 Hilfer-Feller-type fractional derivative with respect to another function
2.28.11 Hilfer-Herrmann-type fractional derivative with respect to another function
2.28.12 Hilfer-Riesz fractional derivative with respect to another function
2.28.13 Hilfer-Feller type fractional derivative with respect to another function
2.28.14 Hilfer-Herrmann-type fractional derivative with respect to another function
3 Fractional derivatives with nonsingular kernels
3.1 History of fractional derivatives with nonsingular kernels
3.2 Sonine general fractional calculus with nonsingular kernels
3.2.1 Sonine general fractional integrals with nonsingular kernel
3.2.2 Sonine general fractional derivatives with nonsingular kernel
3.2.3 Sonine-Choudhary-type general fractional derivative with Sonine nonsingular kernel
3.2.4 Sonine-Wick-Choudhary-type general fractional calculus with Sonine nonsingular kernel
3.2.5 Sonine-Wick-Choudhary-type general fractional derivatives with Sonine nonsingular kernel
3.2.6 Sonine-Choudhary-type general fractional derivative with Sonine nonsingular kernel
3.2.7 Hilfer-Sonine-Choudhary-type general fractional derivative with nonsingular kernel
3.2.8 Sonine general fractional calculus with respect to another function
3.3 General fractional derivatives with Mittag-Leffler nonsingular kernel
3.3.1 Hille-Tamarkin general fractional derivative
3.3.2 Hille-Tamarkin general fractional integrals
3.3.3 Liouville-Weyl-Hille-Tamarkin-type general fractional calculus
3.3.4 Hilfer-Hille-Tamarkin-type general fractional derivative with nonsingular kernel
3.3.5 General fractional derivatives with respect to another function via Mittag-Leffler nonsingular kernel
3.3.6 Hille-Tamarkin general fractional integrals with respect to another function
3.3.7 Liouville-Weyl-Hille-Tamarkin type general fractional calculus with respect to another function
3.3.8 Hilfer-Hille-Tamarkin-type general fractional derivative with respect to another function
3.4 General fractional derivatives with Wiman nonsingular kernel
3.4.1 General fractional derivatives via Wiman function
3.4.2 Prabhakar general fractional integrals on the real line
3.4.3 Hilfer type general fractional derivative with Wiman nonsingular kernel
3.4.4 General fractional derivatives with respect to another function via Wiman function
3.5 General fractional derivatives with Prabhakar nonsingular kernel
3.5.1 Kilbas-Saigo-Saxena-type general fractional derivative with Prabhakar nonsingular kernel
3.5.2 Garra-Gorenflo-Polito-Tomovski-type general fractional derivative with Prabhakar nonsingular kernel
3.5.3 Prabhakar-type general fractional integrals
3.5.4 Kilbas-Saigo-Saxena type general fractional derivative with Prabhakar nonsingular kernel on the real line
3.5.5 Garra-Gorenflo-Polito-Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line
3.5.6 Prabhakar-type general fractional integrals on the real line
3.5.7 Hilfer-type general fractional derivative with Prabhakar nonsingular kernel
3.5.8 General fractional derivatives with respect to another function via Prabhakar nonsingular kernel
3.5.9 Prabhakar-type general fractional integrals with respect to another function
3.5.10 Kilbas-Saigo-Saxena-type general fractional derivative on the real line
3.5.11 Garra-Gorenflo-Polito-Tomovski-type general fractional derivative with respect to another function on the real line
3.5.12 Prabhakar-type general fractional integrals with respect to another function on the real line
3.5.13 Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel
3.6 General fractional derivatives with Gorenflo-Mainardi nonsingular kernel
3.6.1 Prabhakar-type general fractional integrals
3.6.2 Hilfer-type general fractional derivatives with Gorenflo-Mainardi nonsingular kernel
3.6.3 Riemann-Liouville-Hilfer general fractional derivatives with Gorenflo-Mainardi nonsingular kernel
3.7 General fractional derivatives with Miller-Ross nonsingular kernel
3.7.1 Riemann-Liouville-type general fractional derivative with Miller-Ross nonsingular kernel
3.7.2 Liouville-Sonine-Caputo-type general fractional derivative with Miller-Ross nonsingular kernel
3.7.3 Hilfer-type general fractional derivatives with Miller-Ross nonsingular kernel
3.8 General fractional derivatives with one-parameter Lorenzo-Hartley nonsingular kernel
3.8.1 Riemann-Liouville-type general fractional derivative with one-parameter Lorenzo-Hartley nonsingular kernel
3.8.2 Liouville-Sonine-Caputo-type general fractional derivatives with one-parameter Lorenzo-Hartley nonsingular kernel
3.8.3 Hilfer-type general fractional derivatives with one-parameter Lorenzo-Hartley nonsingular kernel
3.9 General fractional derivatives with two-parameter Lorenzo-Hartley nonsingular kernel
3.9.1 Riemann-Liouville-type general fractional derivative with two-parameter Lorenzo-Hartley nonsingular kernel
3.9.2 Liouville-Sonine-Caputo-type general fractional derivatives with two-parameter Lorenzo-Hartley nonsingular kernel
3.9.3 Hilfer-type general fractional derivatives with two-parameter Lorenzo-Hartley nonsingular kernel
4 Variable-order fractional derivatives with singular kernels
4.1 Riemann-Liouville-type variable-order fractional calculus with singular kernel
4.1.1 History of the variable-order fractional calculus with singular kernel
4.1.2 Riemann-Liouville-type variable-order fractional integrals
4.1.3 Riemann-Liouville-type variable-order fractional derivatives
4.2 Variable-order Hilfer-type fractional derivatives with singular kernel
4.3 Liouville-Weyl-type variable-order fractional calculus
4.3.1 Liouville-Weyl-type variable-order fractional integrals
4.3.2 Liouville-Weyl-type variable-order fractional derivatives
4.4 Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel
4.4.1 Riesz-type variable-order fractional derivative with singular kernel
4.4.2 Feller-type variable-order fractional derivative with singular kernel
4.4.3 Herrmann-type variable-order fractional derivative with singular kernel
4.5 Variable-order tempered fractional derivatives with weakly singular kernel
5 Variable-order general fractional derivatives with nonsingular kernels
5.1 Riemann-Liouville-type variable-order general fractional derivatives with Mittag-Leffler-Gauss-like nonsingular kernel
5.1.1 Liouville-Sonine-Caputo-type variable-order general fractional derivatives with Mittag-Leffler nonsingular kernel
5.2 Hilfer-type variable-order fractional derivatives with Mittag-Leffler nonsingular kernel
5.3 Variable-order general fractional derivatives with Gorenflo-Mainardi nonsingular kernel
5.4 Variable-order Hilfer-type fractional derivatives with Gorenflo-Mainardi nonsingular kernel
5.5 Variable-order general fractional derivatives with one-parameter Lorenzo-Hartley nonsingular kernel
5.6 Variable-order Hilfer-type fractional derivatives with Gorenflo-Mainardi nonsingular kernel
5.7 Variable-order general fractional derivative with Miller-Ross nonsingular kernel
5.8 Variable-order Hilfer-type fractional derivatives with Miller-Ross nonsingular kernel
5.9 Variable-order general fractional derivative with Prabhakar nonsingular kernel
5.10 Variable-order Hilfer-type fractional derivatives with Prabhakar nonsingular kernel
6 General derivatives
6.1 Classical derivatives
6.2 Derivatives with respect to another function
6.3 General derivatives with respect to power-law function
6.4 General derivatives with respect to exponential function
6.5 General derivatives with respect to logarithmic function
6.6 Other general derivatives
6.6.1 General derivative with respect to negative power-law function
6.6.2 General derivative with respect to logarithmic function with parameter
6.6.3 General derivative with respect to exponential function with parameter
7 Applications of fractional-order viscoelastic models
7.1 Mathematical models with classical derivatives
7.2 Mathematical models with general derivatives
7.3 Mathematical models with fractional derivatives
7.4 Mathematical models with fractional derivatives with nonsingular kernels
7.4.1 Mathematical models with fractional derivatives with Mittag-Leffler nonsingular kernel
7.4.2 Mathematical models with fractional derivatives with Wiman nonsingular kernel
7.4.3 Mathematical models with fractional derivatives with Prabhakar nonsingular kernel
7.5 Mathematical models with fractional derivatives with respect to another function
References
Index

Citation preview

General Fractional Derivatives With Applications in Viscoelasticity

General Fractional Derivatives With Applications in Viscoelasticity Xiao-Jun Yang Feng Gao Yang Ju

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-817208-7 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Candice Janco Editorial Project Manager: Sara Valentino Production Project Manager: Joy Christel Neumarin Honest Thangiah Designer: Miles Hitchen Typeset by VTeX

Contents

Preface

ix

1

1 3 5 6 12 16 19 23 47 53 56

2

Special functions 1.1 Euler gamma and beta functions 1.2 Laplace transform and properties 1.3 Mittag-Leffler function 1.4 Miller–Ross function 1.5 Rabotnov function 1.6 One-parameter Lorenzo–Hartley function 1.7 Prabhakar function 1.8 Wiman function 1.9 The two-parameter Lorenzo–Hartley function 1.10 Two-parameter Gorenflo–Mainardi function 1.11 Euler-type gamma and beta functions with respect to another function 1.12 Mittag-Leffler-type function with respect to another function 1.13 Miller–Ross-type function with respect to function 1.14 Rabotnov-type function with respect to another function 1.15 Lorenzo–Hartley-type function with respect to another function 1.16 Prabhakar-type function with respect to another function 1.17 Wiman-type function with respect to another function 1.18 Two-parameter Lorenzo–Hartley function with respect to another function 1.19 Gorenflo–Mainardi-type function with respect to another function Fractional derivatives with singular kernels 2.1 The space of the functions 2.2 Riemann–Liouville fractional calculus 2.3 Osler fractional calculus 2.4 Liouville–Weyl fractional calculus 2.5 Samko–Kilbas–Marichev fractional calculus 2.6 Liouville–Sonine–Caputo fractional derivatives 2.7 Liouville fractional derivatives 2.8 Almeida fractional derivatives with respect to another function 2.9 Liouville-type fractional derivative with respect to another function 2.10 Liouville–Grünwald–Letnikov fractional derivatives

59 61 64 66 68 70 85 89 91 95 98 100 107 111 113 115 120 120 122 123

vi

Contents

2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 3

4

Kilbas–Srivastava–Trujillo fractional difference derivatives Riesz fractional calculus Feller fractional calculus Herrmann fractional calculus Samko–Kilbas–Marichev symmetric fractional difference derivative Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative Grünwald–Letnikov–Feller-type symmetric fractional difference derivative Samko–Kilbas–Marichev symmetric fractional difference derivative on a bounded domain Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative on a bounded domain Grünwald–Letnikov–Feller-type symmetric fractional difference derivative on a bounded domain Erdelyi–Kober-type calculus Hadamard fractional calculus Marchaud fractional derivatives Riemann–Liouville-type tempered fractional calculus Liouville–Weyl-type tempered fractional calculus Riemann–Liouville-type tempered fractional calculus with respect to another function Hilfer derivatives Mixed fractional derivatives

Fractional derivatives with nonsingular kernels 3.1 History of fractional derivatives with nonsingular kernels 3.2 Sonine general fractional calculus with nonsingular kernels 3.3 General fractional derivatives with Mittag-Leffler nonsingular kernel 3.4 General fractional derivatives with Wiman nonsingular kernel 3.5 General fractional derivatives with Prabhakar nonsingular kernel 3.6 General fractional derivatives with Gorenflo–Mainardi nonsingular kernel 3.7 General fractional derivatives with Miller–Ross nonsingular kernel 3.8 General fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 3.9 General fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel Variable-order fractional derivatives with singular kernels 4.1 Riemann–Liouville-type variable-order fractional calculus with singular kernel

127 128 132 137 143 144 144 147 147 148 150 160 171 173 175 191 195 198 209 211 224 237 252 263 285 293 298 303 311 311

Contents

4.2 4.3 4.4 4.5

5

vii

Variable-order Hilfer-type fractional derivatives with singular kernel Liouville–Weyl-type variable-order fractional calculus Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel Variable-order tempered fractional derivatives with weakly singular kernel

Variable-order general fractional derivatives with nonsingular kernels 5.1 Riemann–Liouville-type variable-order general fractional derivatives with Mittag-Leffler–Gauss-like nonsingular kernel 5.2 Hilfer-type variable-order fractional derivatives with Mittag-Leffler nonsingular kernel 5.3 Variable-order general fractional derivatives with Gorenflo–Mainardi nonsingular kernel 5.4 Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel 5.5 Variable-order general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 5.6 Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel 5.7 Variable-order general fractional derivative with Miller–Ross nonsingular kernel 5.8 Variable-order Hilfer-type fractional derivatives with Miller–Ross nonsingular kernel 5.9 Variable-order general fractional derivative with Prabhakar nonsingular kernel 5.10 Variable-order Hilfer-type fractional derivatives with Prabhakar nonsingular kernel

318 323 330 340

349 350 357 359 364 365 370 371 376 377 382

6

General derivatives 6.1 Classical derivatives 6.2 Derivatives with respect to another function 6.3 General derivatives with respect to power-law function 6.4 General derivatives with respect to exponential function 6.5 General derivatives with respect to logarithmic function 6.6 Other general derivatives

385 385 386 390 392 394 395

7

Applications of fractional-order viscoelastic models 7.1 Mathematical models with classical derivatives 7.2 Mathematical models with general derivatives 7.3 Mathematical models with fractional derivatives 7.4 Mathematical models with fractional derivatives with nonsingular kernels

399 399 400 409 411

viii

Contents

7.5

Mathematical models with fractional derivatives with respect to another function

417

References

429

Index

439

Preface

The main purpose of this monograph is to provide an introduction to the newlyestablished fractional-order calculus operators involving singular and nonsingular kernels with applications to fractional-order viscoelastic models from the general fractional-order calculus operators’ view-point. Another aim is to present anomalous relaxation and rheological models in the light of nature complexity. The topics are important and interesting for scientists and engineers in the fields of mathematics, physics, chemistry, and elasticity. Due to the above-mentioned avenues of their potential applications in wide-spread real-world phenomena in the fields of physical and engineering sciences, we systematically illustrate the different calculi and the viscoelastic models with different derivatives. More specifically, we have clearly illustrated the special functions, fractional derivatives with singular, weakly-singular, and nonsingular kernels, variableorder fractional derivatives with singular, weakly-singular, and nonsingular kernels, and general derivatives. Moreover, we have investigated the viscoelastic models, e.g., dashpot, Maxwell-like, Kelvin–Voigt-like, Burgers-like, and Zener-like elements. The monograph is divided into seven chapters. Chapter 1 introduces the Euler gamma and beta functions, and the families of Mittag-Leffler, Miller–Ross, Rabotnov, Lorenzo–Hartley, Kilbas–Saigo–Saxena, Gorenflo–Mainardi, Wiman, an Prabhakar functions, including the subsine, subcosine, hyperbolic subsine, and hyperbolic subcosine functions. Chapter 2 investigates the functional spaces as well as the fractional derivative and integral operators with singular kernels, for example, power function, and the function related to the power-law. More specifically, we have introduced the Riemann–Liouville, Osler, Liouville–Weyl, Samko–Kilbas–Marichev fractional calculi, Liouville–Sonine–Caputo and Liouville fractional derivatives, Almeida and Liouville-type fractional derivatives with respect to another function, Liouville– Grünwald–Letnikov fractional derivatives, Kilbas–Srivastava–Trujillo fractional difference derivatives, Riesz fractional calculus, Liouville–Sonine–Caputo–Riesz-type fractional derivatives, Feller fractional calculus, Liouville–Sonine–Caputo–Feller type-fractional derivatives, Herrmann fractional calculus, Liouville–Sonine–Caputo– Herrmann-type fractional derivatives, Samko–Kilbas–Marichev symmetric fractional difference derivative, Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative, Grünwald–Letnikov–Feller-type symmetric fractional difference derivative, Erdelyi–Kober-type calculus, Hadamard fractional calculus, Hadamardtype fractional calculus involving the exponential function, Marchaud fractional derivatives Marchaud-type fractional derivatives with respect to another function, Riemann–Liouville-type tempered fractional calculus, Liouville–Weyl-type tempered fractional calculus, Liouville–Sonine–Caputo-type tempered fractional derivatives,

x

Preface

Liouville–Weyl–Riesz-type tempered fractional calculus, Riemann–Liouville–Riesztype fractional calculus, Liouville–Sonine–Caputo–Riesz-type tempered fractional derivatives, Liouville–Weyl–Feller tempered fractional calculus, Riemann–Liouville– Feller-type tempered type fractional calculus, Liouville–Sonine–Caputo–Feller-type tempered fractional derivatives, Liouville–Weyl–Herrmann tempered fractional calculus, Riemann–Liouville–Herrmann-type tempered type fractional calculus, Liouville– Sonine–Caputo–Herrmann-type tempered fractional derivatives, Riemann–Liouvilletype tempered fractional calculus with respect to another function, Hilfer derivative, Liouville–Weyl–Hilfer-type derivative, Riesz–Hilfer-type fractional derivative, Feller–Hilfer-type fractional derivative, Herrmann–Hilfer-type fractional derivative, Sousa–de Oliveira fractional derivatives with respect to another function, Liouville– Weyl–Sousa–de Oliveira-type fractional derivatives, Hilfer–Riesz-type fractional derivatives with respect to another function, Hilfer–Feller-type fractional derivatives with respect to another function, and Hilfer–Herrmann-type fractional derivatives with respect to another function. Chapter 3 presents the history of the fractional derivatives with nonsingular kernels, and general fractional derivatives with nonsingular kernels. We introduce the family of the general fractional derivatives with Sonine nonsingular kernel, general fractional derivatives with Mittag-Leffler nonsingular kernel, general fractional derivatives with Wiman nonsingular kernel, general fractional derivatives with Prabhakar nonsingular kernel, general fractional derivatives with Gorenflo–Mainardi nonsingular kernel, general fractional derivatives with Miller–Ross nonsingular kernel, general fractional derivatives with one-parametric Lorenzo–Hartley nonsingular kernel, and general fractional derivatives with two-parametric Lorenzo–Hartley nonsingular kernel, and their families of Hilfer-type general fractional derivatives. Chapter 4 discusses the concepts of the Riemann–Liouville-type variable-order fractional integrals with singular kernel, Riemann–Liouville-type variable-order fractional derivatives with singular kernel, variable-order Hilfer-type fractional derivatives with singular kernel, Liouville–Weyl-type variable-order fractional integrals with singular kernel, Liouville–Weyl-type variable-order fractional derivatives with singular kernel, Riesz-type variable-order fractional derivatives with singular kernel, Fellertype variable-order fractional derivatives with singular kernel, and Herrmann-type variable-order fractional derivatives with singular kernel. Chapter 5 illustrates the variable-order fractional derivatives with nonsingular kernels, which are called the variable-order general fractional derivatives. We introduce the variable-order general fractional derivatives with Mittag-Leffler–Gauss-like nonsingular kernel, variable-order general fractional derivatives with Mittag-Leffler nonsingular kernel, variable-order general fractional derivatives with Wiman nonsingular kernel, variable-order general fractional derivatives with Prabhakar nonsingular kernel, variable-order general fractional derivatives with one-parametric Lorenzo– Hartely nonsingular kernel, and variable-order general fractional derivatives with Miller–Ross nonsingular kernel. We have also proposed the Hilfer-type variableorder general fractional derivatives with the nonsingular kernels, e.g., Mittag-Leffler, Wiman, Prabhakar, Lorenzo–Hartely, and Miller–Ross.

Preface

xi

Chapter 6 addresses the general derivatives and integrals with respect to another function based on the Newton–Leibniz derivatives and integrals. Chapter 7 presents the dashpot, Maxwell-like, Kelvin–Voigt-like, Burgers-like, and Zener-like elements with the different derivatives. Professor Xiao-Jun Yang would like to express grateful thanks to Professor Wolfgang Sprößig, Professor H. M. Srivastava, Professor Simeon Oka, Professor Igor Emri, Professor Martin Bohner, Professor Irene Maria Sabadini, Professor Thiab Taha, Professor Zdzislaw Jackiewicz, Professor Rosa Maria Spitaleri, Professor BoMing Yu, Professor André Keller, Professor Martin Ostoja-Starzewski, Professor Imre Miklós Szilágyi, Professor Minvydas Ragulskis, Ms Karin Uhlemann, Professor Sung Yell Song, Professor Chin-Hong Park, Professor Dumitru Mihalache, Professor Vukman Bakic, Professor Delfim F. M. Torres, Professor J. A. Tenreiro Machado, Professor Dumitru Baleanu, Professor Carlo Cattani, Professor Ayman S. Abdel-Khalik, and Professor Syed Tauseef Mohyud-Din. Authors express their special thanks to Professor He-Ping Xie, Professor Guo-Qing Zhou, Professor Fu-Bao Zhou, Professor Hong-Wen Jing, and Professor Zhan-Guo Ma, 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), the State Key Research Development Program of the People’s Republic of China (Grant No. 2016YFC0600705), National Natural Science Foundation of China (Grant Nos. 51727807 and 51674251), and the Innovation Team Project of the Ten-Thousand Talents Program sponsored by the Ministry of Science and Technology of China (Grant No. 2016RA4067). Finally, I also wish to express my special thanks to Elsevier staff, especially, Sara Valentino, Michael Lutz, Glyn Jones, J. Scott Bentley, and Indhumathi Mani, for their cooperation in the production process of this book. Xiao-Jun Yang Feng Gao Yang Ju

Special functions

1

Contents 1.1

Euler gamma and beta functions 1.1.1 1.1.2

Euler gamma function Euler beta function

3

3 4

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Laplace transform and properties 5 Mittag-Leffler function 6 Miller–Ross function 12 Rabotnov function 16 One-parameter Lorenzo–Hartley function 19 Prabhakar function 23 Wiman function 47 The two-parameter Lorenzo–Hartley function 53 Two-parameter Gorenflo–Mainardi function 56 Euler-type gamma and beta functions with respect to another function Mittag-Leffler-type function with respect to another function 61 Miller–Ross-type function with respect to function 64 Rabotnov-type function with respect to another function 66 Lorenzo–Hartley-type function with respect to another function 68 Prabhakar-type function with respect to another function 70 Wiman-type function with respect to another function 85 Two-parameter Lorenzo–Hartley function with respect to another function 89 1.19 Gorenflo–Mainardi-type function with respect to another function 91

59

In this chapter, we introduce the Euler gamma and beta functions, Mittag-Leffler function, subsine function of Mittag-Leffler type, subcosine function of Mittag-Leffler type, hyperbolic subsine function of Mittag-Leffler type, hyperbolic subcosine function of Mittag-Leffler type, Miller–Ross function, subsine function of Miller–Ross type, subcosine function of Miller–Ross type, hyperbolic subsine function of Miller– Ross type, hyperbolic subcosine function of Miller–Ross type, Rabotnov function, subsine function of Rabotnov type, subcosine function of Rabotnov type, hyperbolic subsine function of Rabotnov type, hyperbolic subcosine function of Rabotnov type, Lorenzo–Hartley function, subsine function of Lorenzo–Hartley type, subcosine function of Lorenzo–Hartley type, hyperbolic subsine function of Lorenzo–Hartley type, hyperbolic subcosine function of Lorenzo–Hartley type, Prabhakar type function, subsine function of Prabhakar type, subcosine function of Prabhakar type, hyperbolic subsine function of Prabhakar type, hyperbolic subcosine function of Prabhakar type, Kilbas–Saigo–Saxena function, subsine function of Kilbas–Saigo–Saxena General Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00006-6 Copyright © 2020 Elsevier Inc. All rights reserved.

2

General Fractional Derivatives With Applications in Viscoelasticity

type, subcosine function of Kilbas–Saigo–Saxena type, hyperbolic subsine function of Kilbas–Saigo–Saxena type, hyperbolic subcosine function of Kilbas–Saigo–Saxena type, Wiman function, subsine function of Wiman type, subcosine function of Wiman type, hyperbolic subsine function of Wiman type, hyperbolic subcosine function of Wiman type, Gorenflo–Mainardi function, subsine function of Gorenflo–Mainardi type, subcosine function of Gorenflo–Mainardi type, hyperbolic subsine function of Gorenflo–Mainardi type, hyperbolic subcosine function of Gorenflo–Mainardi type, Euler-type gamma and beta functions with respect to another function, Mittag-Leffler type function with respect to another function, subsine function of Mittag-Leffler type with respect to another function, subcosine function of Mittag-Leffler type with respect to another function, hyperbolic subsine function of Mittag-Leffler type with respect to another function, hyperbolic subcosine function of Mittag-Leffler type with respect to another function, Miller–Ross type function with respect to another function, subsine function of Miller–Ross type with respect to another function, subcosine function of Miller–Ross type with respect to another function, hyperbolic subsine function of Miller–Ross type with respect to another function, hyperbolic subcosine function of Miller–Ross type with respect to another function, Rabotnov type function with respect to another function, subsine function of Rabotnov type with respect to another function, subcosine function of Rabotnov type with respect to another function, hyperbolic subsine function of Rabotnov type with respect to another function, hyperbolic subcosine function of Rabotnov type with respect to another function, Lorenzo–Hartley type function with respect to another function, subsine function of Lorenzo–Hartley type with respect to another function, subcosine function of Lorenzo–Hartley type with respect to another function, hyperbolic subsine function of Lorenzo–Hartley type with respect to another function, hyperbolic subcosine function of Lorenzo–Hartley type with respect to another function, Prabhakar type function with respect to another function, subsine function of Prabhakar type with respect to another function, subcosine function of Prabhakar type with respect to another function, hyperbolic subsine function of Prabhakar type with respect to another function, hyperbolic subcosine function of Prabhakar type with respect to another function, Kilbas–Saigo–Saxena type function with respect to another function, subsine function of Kilbas–Saigo–Saxena type with respect to another function, subcosine function of Kilbas–Saigo–Saxena type with respect to another function, hyperbolic subsine function of Kilbas–Saigo–Saxena type with respect to another function, hyperbolic subcosine function of Kilbas–Saigo–Saxena type with respect to another function, Wiman type function with respect to another function, subsine function of Wiman type with respect to another function, subcosine function of Wiman type with respect to another function, hyperbolic subsine function of Wiman type with respect to another function, hyperbolic subcosine function of Wiman type with respect to another function, Gorenflo–Mainardi type function with respect to another function, subsine function of Gorenflo–Mainardi type with respect to another function, subcosine function of Gorenflo–Mainardi type with respect to another function, hyperbolic subsine function of Gorenflo–Mainardi type with respect to another function, and hyperbolic subcosine function of Gorenflo–Mainardi type with respect to another function.

Special functions

1.1

3

Euler gamma and beta functions

1.1.1 Euler gamma function In 1729, Euler [1] showed that ∞  (x) =

e−t t x−1 dt,

Re (x) > 0,

(1.1)

0

which is called the Euler gamma function and was introduced in 1809 by Legendre as [2]:   1 22x−1 ,  (2x) = √  (2x)  x + 2 π

2x = −1, −2, ....

The properties of the Euler gamma function are given as follows [3–5]: (1) For Re (x) > −1, the recurrence equation can be given as follows [1,6]:  (x + 1) =  (x) x. (2) For x ∈ C, the Euler’s reflection formula is given as follows [7,8]:  (1 − x)  (x) =

π . sin (πx)

(3) For x ∈ C one has [9,10] ∞  1 x  −x = xe−γ x  1 + e n,  (x + 1) n n=1

where γ is Euler’s constant. (4) For x, p ∈ C Re (x) > 0 and Re (p) > 0 one has [11] ∞

e−pt t x−1 dt =

 (p) . px

0

(5) For x, p ∈ C, ∞

Re (x) > 0 and Re (p) > 0, one has [11–13]: 

 1 dt =  +1 , p

e

−t p

e

−t p x−1

0

∞ 0

t

  1 x dt =  , p p

4

General Fractional Derivatives With Applications in Viscoelasticity

∞ e

−pt 2 x−1

t

x p− 2  x  dt =  . 2 2

0

(6) For p, q ∈ C, Re (q) > 0 and Re (p) > 0, one has [13–15] ∞ e

−pt q

dt = p

− q1



 1  1+ . q

0

(7) For x, p ∈ C, Re (q) > 0, Re (x) > 0 and Re (p) > 0, one has [16] ∞ e

−pt q x−1

t

dt =

p

− qx



q

  x . q

0

(8) For x ∈ C, one obtains [17,18]  1 1 s −x es ds, =  (x) 2πi E

where the contour of integration E is the Hankel contour. (9) For x ∈ C and Re (x) > 1 one has [18–20] ∞

1  (x) = cos (2πx)

t x−1 cos tdt,

1 1 − + n < x < + n, n ∈ N0 . 2 2

0

(10) For x ∈ C and Re (x) > 1, one has [18–20]  (x) =

1 sin (2πx)

∞ t x−1 sin tdt,

−1 + n < x < 1 + n, n ∈ N0 .

0

1.1.2 Euler beta function In 1772, Euler considered the function given as [21] 1 B (x, y) =

t x−1 (1 − t)y−1 dt,

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

0

which is called the Euler beta function. In 1877, Prym presented the decomposition, given as [22] 1  (x) =

−t x−1

e t 0

∞ dt +

e−t t x−1 dt,

1

which is called the Prym decomposition.

(1.2)

Special functions

5

The property of the Euler beta function is presented as follows [18]: B (x, y) =

1.2

 (x)  (y) .  (x + y)

Laplace transform and properties

√ Let κ ∈ N, i = −1, λ ∈ C, α ∈ C, Re (λ) ≥ 0 and Re (α) ≥ 0. The Laplace transform of a function f is defined as [23–25] 

 L f (t) = f (s) =

∞

e−st f (t) dt,

(1.3)

0

and the corresponding inverse Laplace transform as −1

f (t) = L





1 f (s) = lim 2πi T →∞

r+iT 

est f (s) ds,

r−iT

where r ∈ R and s = r + iT . The properties of Laplace transform are given in Table 1.1 [23–25]. The Laplace transforms of the functions are given in Table 1.2 [25].

Table 1.1 The properties of the Laplace transform. Functions f (t) + g (t)

Laplace transforms f (s) + g (s)

tf (t) t κ f (t) f (1) (t)

−f (1) (s) (−1)κ f (κ) (s) sf (s) − f (0) κ s κ f (s) − s κ−j f (j −1) (0)

f (κ) (t) f (t) t

t

f (t) dt

0

f (t − λ) eλt f (t) f (at) (λ > 0)

t f (τ ) g (t − τ ) dτ 0

∞

j =1

f (ρ) dρ

s f (s) s

e−λs f (s) f (s − λ) f (s/λ) /λ f (s) g (s)

(1.4)

6

General Fractional Derivatives With Applications in Viscoelasticity

Table 1.2 Laplace transforms of the functions. Functions e−λt ϑ (τ ) δ (τ ) t -ν / (1 − ν) t ν / (1 + ν) t −1 t t 1/κ /  (1 + 1/κ) t ν e−λt / (1 + ν) sin (λt)

Laplace transforms  −1 s −1 1 + λs −1 s −1 1 sν s −ν s s −1 s −(1+1/κ) (s + λ)−(ν+1) λ s 2 +λ2 s

cos (λt)

s 2 +λ2 λ

sinh (λt)

s 2 −λ2 s

cosh (λt) t ν sinh (−λt) / (1 + ν) tν

cosh (−λt) / (1 + ν)

t ν sin (−λt) / (1 + ν) tν

cos (λt) / (1 + ν)

1.3

s 2−λ2  1 −(ν+1) + (s − λ)−(ν+1) 2 (s + λ)   −(ν+1) 1 s +λ − (s − λ)−(ν+1) 2   1 −(ν+1) + (s − iλ)−(ν+1) 2 (s + iλ)   1 −(ν+1) − (s − iλ)−(ν+1) 2 (s + iλ)

Regions of convergence Re (s) > −λ Re (s) > 0 Re (s) > 0 Re (s) > 0 Re (s) > 0 Re (s) > 0 Re (s) > 0 Re (s) > 0 Re (s) > −λ Re (s) > 0 Re (s) > 0 Re (s) > |λ| Re (s) > |λ| Re (s) > −λ Re (s) > −λ Re (s) > −λ Re (s) > −λ

Mittag-Leffler function

In 1902, G. M. Mittag-Leffler defined the following function [26]: Eα (t) =

∞  κ=0

tκ ,  (κα + 1)

(1.5)

which is called the Mittag-Leffler function. In this case, one has [14,25,27] ∞   Eα λt α = κ=0

λκ t κα ,  (κα + 1)

(1.6)

so that [25]  (α) :=

∞  κ=0

1 ,  (κα + 1)

(1.7)

which, for α = 1, yields [25] ∞  κ=0

1 = e =  (1) .  (κ + 1)

(1.8)

Special functions

7

The subsine function of Mittag-Leffler type, denoted by LSinα (t α ), is defined as [25, 28–30] ∞  α  LSinα t = κ=0

(−1)κ t (2κ+1)α ,  ((2κ + 1) α + 1)

(1.9)

which implies that ∞   (−1)κ λ2κ+1 t (2κ+1)α ; LSinα λt α =  ((2κ + 1) α + 1)

(1.10)

κ=0

the subcosine function of Mittag-Leffler type, denoted by LCosα (t α ), is defined as [25,28–31] ∞   (−1)κ t 2κα , LCosα t α =  (2κα + 1)

(1.11)

κ=0

which implies that ∞   (−1)κ λ2κ t 2κα LCosα λt α = ;  (2κα + 1)

(1.12)

κ=0

the hyperbolic subsine function of Mittag-Leffler type, denoted by LSinhα (t α ), is given as [25,30,31] ∞   LSinhα t α = κ=0

t (2κ+1)α ,  ((2κ + 1) α + 1)

(1.13)

which implies that ∞   LSinhα λt α = κ=0

λ2κ+1 t (2κ+1)α ;  ((2κ + 1) α + 1)

(1.14)

and the hyperbolic subcosine function of Mittag-Leffler type, denoted by LCoshα (t α ), is given as [25,30] ∞  α  LCoshα t = κ=0

t 2κα ,  (2κα + 1)

(1.15)

which implies that [25,30] ∞   LCoshα λt α = κ=0

λ2κ t 2κα .  (2κα + 1)

(1.16)

8

General Fractional Derivatives With Applications in Viscoelasticity

In fact, the Mittag-Leffler function with one-parameter constant can be written as [25, 28,29] Eα (λt α ) ∞ λκ t κα = (κα+1) =

κ=0 ∞ κ=0

λ2κ t 2κα (2κα+1)

+

∞ κ=0

λ2κ+1 t (2κ+1)α ((2κ+1)α+1)

(1.17)

= LCoshα (λt α ) + LSinhα (λt α ) , which yields that [25,28,29] Eα (iλt α ) ∞ (iλ)κ t κα = (κα+1) = =

κ=0 ∞ κ=0 ∞ κ=0

(iλ)2κ t 2κα (2κα+1)

+

κ=0

(−1)κ λ2κ t 2κα (2κα+1)

= LCosα

∞

+i

(iλ)2κ+1 t (2κ+1)α ((2κ+1)α+1) ∞ κ=0

(λt α ) + iLSin

(1.18)

(−1)κ λ2κ+1 t (2κ+1)α ((2κ+1)α+1)

α (λt

α)

and [25] Eα ((iλt)α ) ∞ (iλ)κα t κα = (κα+1) = =

κ=0 ∞ κ=0 ∞ κ=0

(iλt)2κα (2κα+1)

+

(−1)κα (λt)2κα (2κα+1)

∞ κ=0

(iλt)(2κ+1)α ((2κ+1)α+1)

+ iα

∞ κ=0

(1.19)

(−1)κα (λt)(2κ+1)α ((2κ+1)α+1) ,

which, due to Eq. (1.1), leads to [25] Eα ((iλt)α ) ∞ ∞ (−1)κα (λt)2κα α + i = (2κα+1) κ=0

κ=0

(−1)κα (λt)(2κ+1)α ((2κ+1)α+1)

(1.20)

= RLCosα (λt α ) + i α RLSinα (λt α ) , where [25,28,29] ∞   (−1)κα (λt)(2κ+1)α RLSinα (λt)α =  ((2κ + 1) α + 1) κ=0

(1.21)

Special functions

9

and [25,28,29] ∞   (−1)κα (λt)2κα . RLCosα (λt)α =  (2κα + 1)

(1.22)

κ=0

With the aid of [25] 

 t κα 1 L = κα ,  (κα + 1) s

(1.23)

we have [31] L {Eα (−λt α )}  ∞ (−1)κ λκ t κα =L (κα+1) =

∞ κ=0

=

κ=0

(−λ)κ s κα+1

1 1 s 1+λs −α

(1.24)  −α  λs  < 1

 −1  −α  λs  < 1 = s −1 1 + λs −α so that [25] L {LSinα (λt α )}  ∞ (−1)κ λ2κ+1 t (2κ+1)α =L ((2κ+1)α+1) =

∞ κ=0

=

κ=0

(−1)κ λ2κ+1 s (2κ+1)α+1

λ 1 s 1+α 1+λ2 s −2α

(1.25)  −α  λs  < 1

 −1  −α  λs  < 1 , = λs −(1+α) 1 + λ2 s −2α L {LCosα (λt α )}  ∞ (−1)κ λ2κ t 2κα =L (2κα+1) =

∞ κ=0

=

κ=0

(−1)κ λ2κ s 2κα+1

1 1 s 1+λ2 s −2α

(1.26)  −α  λs  < 1

 −1  −α  λs  < 1 , = s −1 1 + λ2 s −2α

10

General Fractional Derivatives With Applications in Viscoelasticity

L {LSinhα (λt α )}  ∞ λ2κ+1 t (2κ+1)α =L ((2κ+1)α+1) =

∞

κ=0

λ2κ+1 s (2κ+1)α+1

κ=0

=

1

λ s 1+α

1−λ2 s −2α

(1.27)  −α  λs  < 1

 −1  −α  λs  < 1 , = λs −(1+α) 1 − λ2 s −2α and [25] L {LCoshα (λt α )}  ∞ λ2κ t 2κα =L (2κα+1) =

∞ κ=0

=

κ=0

λ2κ s 2κα+1

1 1 s 1−λ2 s −2α

(1.28)  −α  λs  < 1

 −1  −α  λs  < 1 . = s −1 1 − λ2 s −2α It follows that [25]  Eα (iλt α ) − Eα (−iλt α ) , LSinα λt α = 2i  Eα (iλt α ) + Eα (−iλt α ) LCosα λt α = , 2  Eα (λt α ) − Eα (−λt α ) , LSinhα λt α = 2

(1.29) (1.30) (1.31)

and  Eα (λt α ) + Eα (−λt α ) . LCoshα λt α = 2 In this case, one gets [25,28,29]    Eα it α = LCosα t α + iLSinα t α .

(1.32)

(1.33)

Thus, one has [25,30]  Eα (it α ) − Eα (−it α ) , LSinα t α = 2i

(1.34)

 Eα (it α ) + Eα (−it α ) LCosα t α = , 2

(1.35)

Special functions

 Eα (t α ) − Eα (−t α ) LSinhα t α = , 2 and [25]  Eα (t α ) + Eα (−t α ) LCoshα t α = . 2 Thus, we have [25]    Eα −it α = LCosα t α − iLSinα t α

11

(1.36)

(1.37)

(1.38)

and [25]    Eα −iλt α = LCosα λt α − iLSinα λt α

(1.39)

so that [25]      Eα −it α = Re Eα −it α + iI m Eα −it α ,

(1.40)

where    Re Eα −it α = Cosα −t α

(1.41)

and [25]    I m Eα −it α = LSinα −t α .

(1.42)

In a similar way, one gets [25,30]    Eα it α = LCosα t α + iLSinα t α

(1.43)

and [25,30]    Eα iλt α = LCosα λt α + iLSinα λt α

(1.44)

so that [25]      Eα it α = Re Eα it α + iI m Eα it α ,

(1.45)

where [25]    Re Eα it α = Cosα t α

(1.46)

and [25]    I m Eα it α = LSinα t α . In this case, one obtains [25]   LCosα −λt α = LCosα λt α

(1.47)

(1.48)

12

General Fractional Derivatives With Applications in Viscoelasticity

and [25]

  LSinα −λt α = −LSinα λt α .

(1.49)

Thus, one gets [25] ϑ1 (α) := ϑ2 (α) := ϑ3 (α) :=

∞ 

1 ,  (2κα + 1)

(1.50) (1.51)

κ=0

t (2κ+1)α ,  ((2κ + 1) α + 1)

(1.52)

κ=0

(−1)κ ,  (2κα + 1)

(−1)κ .  ((2κ + 1) α + 1)

(1.53)

κ=0 ∞ 

∞ 

and [25] ϑ4 (α) :=

∞  κ=0

1.4

Miller–Ross function

In 1993, Miller and Ross considered the function defined as [25,32] Mα (λt α ) ∞ λκ t κ = tα (κ+1+α) κ=0 ∞ λκ t κ+α = (κ+1+α) κ=0

(1.54)

which is called the Miller–Ross function. The subsine function of Miller–Ross type, denoted by MRSinα (t α ), is defined as [25,32] MRSinα (t α ) ∞ (−1)κ t 2κ+1 = tα (2κ+2+α)

κ=0 ∞ (−1)κ t 2κ+1+α = (2κ+2+α) , κ=0

(1.55)

which yields that [25,32] ∞   (−1)κ λ2κ+1 t 2κ+1+α , MRSinα λt α =  (2κ + 2 + α) κ=0

(1.56)

Special functions

13

and the subcosine function of Miller–Ross type, denoted by MRCosα (t α ), is given as [25,32] MRCosα (t α ) ∞ (−1)κ t 2κ = tα (2κ+1+α)

(1.57)

κ=0 (−1)κ t 2κ+α = (2κ+1+α) , κ=0 ∞

which yields that [25,32] 

MRCosα λt

α



=

∞  (−1)κ λ2κ t 2κ+α κ=0

 (2κ + 1 + α)

;

(1.58)

the hyperbolic subsine function of Miller–Ross type, denoted by MRSinα (t α ), is defined as [25,32] MRSinhα (t α ) ∞ t 2κ+1 = tα (2κ+2+α)

(1.59)

κ=0 ∞ t 2κ+1+α = (2κ+2+α) , κ=0

which leads to [25,32] 

MRSinhα λt

α



∞  λ2κ+1 t 2κ+1+α = ,  (2κ + 2 + α)

(1.60)

κ=0

and the hyperbolic subcosine function of Miller–Ross type, denoted by MRCoshα (t α ), is provided as [25] MRCoshα (t α ) ∞ t 2κ = tα (2κ+1+α) =

(1.61)

κ=0 ∞

κ=0

t 2κ+α (2κ+1+α) ,

which leads to [25] ∞   MRCoshα λt α = κ=0

λ2κ t 2κ+α .  (2κ + 1 + α)

Thus, one has [25]    Mα iλt α = MRCosα λt α + iMRSinα λt α

(1.62)

(1.63)

14

General Fractional Derivatives With Applications in Viscoelasticity

and [25]    Mα λt α = MRCosα λt α + MRSinα λt α

(1.64)

so that [25]  Mα (iλt α ) − Mα (−iλt α ) , MRSinα λt α = 2i  Mα (iλt α ) + Mα (−iλt α ) , MRCosα λt α = 2  Mα (λt α ) − Mα (−λt α ) MRSinhα λt α = , 2 and [25]  Mα (λt α ) + Mα (−λt α ) . MRCoshα λt α = 2 With the aid of [25]   t κα 1 L = κα ,  (κα + 1) s

(1.65) (1.66) (1.67)

(1.68)

(1.69)

one has [25,32]    L Mα λt α  ∞ λκ t κ+α =L (κ+1+α) κ=0

=

∞

λκ s κ+α+1

(1.70)

κ=0 1 1 = s α+1 1−λs −1

 −1  λs  < 1  −1  −1  λs  < 1 = s −(α+1) 1 − λs −1 so that [25,32] L {MRSinα (t α )} ∞  (−1)κ λ2κ+1 t 2κ+1+α =L (2κ+2+α) =

∞

κ=0 (−1)κ λ2κ+1 s 2κ+2+α

κ=0  −1  λ 1 λs  < 1 = s 2+α 2 −2 1+λ s  −1  −1  λs  < 1 , = λs −(2+α) 1 + λ2 s −2

(1.71)

Special functions

15

L {MRSinhα (t α )} ∞  λ2κ+1 t 2κ+1+α =L (2κ+2+α) =

∞

κ=0

λ2κ+1 s 2κ+2+α

(1.72)

κ=0  −1  λ 1 λs  < 1 = s 2+α 1−λ2 s −2  −1  −1  λs  < 1 , = λs −(2+α) 1 − λ2 s −2

   L MRCosα λt α  ∞ (−1)κ λ2κ t 2κ+α =L (2κ+1+α) κ=0

=

∞

(−1)κ λ2κ s 2κ+1+α

(1.73)

κ=0 1 1 = s 1+α 1+λ2 s −2

 −1  λs  < 1  −1  −1  λs  < 1 = s −1 1 + λ2 s −2 and [25,32]    L MRCoshα λt α  ∞ λ2κ t 2κ+α =L (2κ+1+α) κ=0

=

∞

λ2κ s 2κ+1+α

(1.74)

κ=0 1 1 = s 1+α 1−λ2 s −2

 −1  λs  < 1  −1  −1  λs  < 1 . = s −1 1 − λ2 s −2 Thus, one has [25] μ0 (α) =

∞  κ=0

μ1 (α) =

∞  κ=0

μ2 (α) = μ3 (α) =

∞  κ=0 ∞  κ=0

1 ,  (κ + 1 + α)

(1.75)

(−1)κ ,  (2κ + 2 + α)

(1.76)

(−1)κ ,  (2κ + 1 + α)

(1.77)

t 2κ+1+α ,  (2κ + 2 + α)

(1.78)

16

General Fractional Derivatives With Applications in Viscoelasticity

and [25] μ4 (α) =

∞  κ=0

1 .  (2κ + 1 + α)

(1.79)

1.5 Rabotnov function In 1948, Rabotnov defined the function [25,33] ∞   (ιt)(κ+1)(α+1)−1 , α (ιt)α =  ((κ + 1) (α + 1))

(1.80)

κ=0

which can be given as [25] ∞   λκ t (κ+1)(α+1)−1 α λt α = ,  ((κ + 1) (α + 1))

(1.81)

κ=0

where λ = ια . It is called the Rabotnov function [25]. The subsine function of Rabotnov type, denoted by Sinα (t), is defined as [25] ∞   (−1)κ t 2(κ+1)(α+1)−1 Sinα t α = ,  (2 (κ + 1) (α + 1))

(1.82)

κ=0

which implies that [25] ∞   (−1)κ λ2κ+1 t 2(κ+1)(α+1)−1 Sinα λt α = ,  (2 (κ + 1) (α + 1))

(1.83)

κ=0

the subcosine function of Rabotnov type, denoted by Cosα (t), is given by [25] ∞   (−1)κ t (2κ+1)(α+1)−1 , Cosα t α =  ((2κ + 1) (α + 1))

(1.84)

κ=0

which leads to [25] ∞   (−1)κ λ2κ t (2κ+1)(α+1)−1 Cosα λt α = ;  ((2κ + 1) (α + 1))

(1.85)

κ=0

the hyperbolic subsine function of Rabotnov type, denoted by Sinhα (t), is defined as [25] ∞   Sinhα t α = κ=0

t 2(κ+1)(α+1)−1 ,  (2 (κ + 1) (α + 1))

(1.86)

Special functions

17

which yields that [25] ∞   λ2κ+1 t 2(κ+1)(α+1)−1 , Sinhα λt α =  (2 (κ + 1) (α + 1))

(1.87)

κ=0

and the hyperbolic subcosine function of Rabotnov type, denoted by Coshα (t), is obtained as [25] ∞   Coshα t α = κ=0

t (2κ+1)(α+1)−1 ,  ((2κ + 1) (α + 1))

(1.88)

which implies that [25] ∞   λ2κ t (2κ+1)(α+1)−1 Coshα λt α = .  ((2κ + 1) (α + 1))

(1.89)

κ=0

With the use of [25]  L

 1 t κα = κα ,  (κα + 1) s

(1.90)

one has [25,34] L { α (λt α )} ∞  λκ t (κ+1)(α+1)−1 =L ((κ+1)(α+1)) =

∞

κ=0

λκ s (κ+1)(α+1)

(1.91)

κ=0  −(α+1)  1 1  0 ,

  ϕ L Supercos1,1 (λt) ⎧ ⎫ ⎛ ⎞ ⎨ ⎬ ∞ −ϕ κ 2κ 2κ ⎝ ⎠ (−1) λ t =L (2κ+1) ⎩κ=0 2κ ⎭ =

∞

⎛ ⎝

κ=0

= s −1

∞

−ϕ 2κ ⎛

κ=0

⎝

⎞

κ 2κ

λ ⎠ (−1) 2κ+1 s

−ϕ 2κ

⎞ ⎠

 iλ 2κ s

(1.246)

Special functions

= = =

47





−ϕ  iλ −ϕ + 1 − iλ s s ϕ  ϕ  1+ iλ + 1− iλ s s s −1   2 ϕ 2 1+ λ2 s   −1 ϕ + 1−iλs −1 ϕ  2 −2  s −1 1+iλs λ s  < 1; Re (ϕ) > 0 , ϕ 2 −2 2 1+λ s s −1 2

1+

and [25]   ϕ L Super cosh1,1 (λt) ⎧ ⎫ ⎛ ⎞ ⎨ ⎬ ∞ −ϕ 2κ 2κ ⎝ ⎠ λ t =L (2κ+1) ⎭ ⎩κ=0 2κ ⎛ ⎞ ∞ −ϕ λ2κ ⎝ ⎠ 2κ+1 = s κ=0 2κ ⎛ ⎞ ∞ −ϕ  ⎝ ⎠ λ 2κ = s −1 s κ=0 2κ

−ϕ  −ϕ  −1  = s2 + 1 − λs 1 + λs = =

1.8

ϕ  ϕ  1+ λs + 1− λs   2 ϕ 1− λ2 s   −1 ϕ + 1−λs −1 ϕ s −1 1+λs ϕ 2 −2 2 1−λ s

(1.247)

s −1 2

 2 −2  λ s  < 1; Re (ϕ) > 0 .

Wiman function

Wiman considered the function defined as [41] ∞   Eα,υ t α = κ=0

t κα ,  (κα + υ)

(1.248)

which is called the Wiman function. The Wiman function with one-parameter constant λ is defined as [38] t υ−1 Eα,υ (λt α ) ∞ λκ t κα = t υ−1 (κα+υ) =

∞ κ=0

κ=0

λκ t κα+υ−1 (κα+υ)

.

(1.249)

48

General Fractional Derivatives With Applications in Viscoelasticity

The supersine function of Wiman type, denoted by Supersinα,υ (t α ), is defined as [25] ∞  α  (−1)κ t (2κ+1)α+υ−1 , Supersinα,υ t =  ((2κ + 1) α + υ)

(1.250)

κ=0

which implies that [25] ∞   (−1)κ λ2κ+1 t (2κ+1)α+υ−1 , Supersinα,υ λt α =  ((2κ + 1) α + υ)

(1.251)

κ=0

the supercosine function of Wiman type, denoted by Supercosα,υ (t α ), is obtained as [25] ∞   (−1)κ t 2κα+υ−1 , Supercosα,υ t α =  (2κα + υ)

(1.252)

κ=0

which implies that [25] 

Supercosα,υ λt

α



=

∞  (−1)κ λ2κ t 2κα+υ−1

 (2κα + υ)

κ=0

(1.253)

,

the hyperbolic supersine function of Wiman type, denoted by Supersinhα,υ (t α ), is defined as [25] ∞   Supersinhα,υ t α = κ=0

t (2κ+1)α+υ−1  ((2κ + 1) α + υ)

(1.254)

which implies that [25] ∞ 2κ+1 (2κ+1)α+υ−1   λ t , Supersinhα,υ λt α =  ((2κ + 1) α + υ)

(1.255)

κ=0

and the hyperbolic supercosine function Supercoshα,υ (t α ), is obtained as [25] ∞  α  t 2κα+υ−1 , Supercoshα,υ t =  (2κα + υ)

of

Wiman

type,

denoted

by

(1.256)

κ=0

which implies that [25] ∞ 2κ 2κα+υ−1   λ t . Supercoshα,υ λt α =  (2κα + υ) κ=0

(1.257)

Special functions

49

Therefore, one has [25]    t υ−1 Eα,υ λt α = Supercoshα,υ λt α + Supersinhα,υ λt α

(1.258)

and [25]    t υ−1 Eα,υ iλt α = Supercosα,υ λt α + iSupersinα,υ λt α ,

(1.259)

so that [25]  t υ−1 Eα,υ (it α ) − t υ−1 Eα,υ (−it α ) , Supersinα,υ t α = 2i

(1.260)

 t υ−1 Eα,υ (−iλt α ) + t υ−1 Eα,υ (−iλt α ) , Supercosα,υ t α = 2

(1.261)

 t υ−1 Eα,υ (t α ) − t υ−1 Eα,υ (−t α ) Supersinhα,υ t α = , 2

(1.262)

 t υ−1 Eα,υ (t α ) + t υ−1 Eα,υ (−t α ) Supercoshα,υ t α = , 2

(1.263)

 t υ−1 Eα,υ (iλt α ) − t υ−1 Eα,υ (−iλt α ) Supersinα,υ λt α = , 2i

(1.264)

 t υ−1 Eα,υ (iλt α ) + t υ−1 Eα,υ (−iλt α ) , Supercosα,υ λt α = 2

(1.265)

 t υ−1 Eα,υ (λt α ) − t υ−1 Eα,υ (−λt α ) Supersinhα,υ λt α = , 2 and [25]  t υ−1 Eα,υ (λt α ) + t υ−1 Eα,υ (−λt α ) Supercoshα,υ λt α = . 2

(1.266)

(1.267)

Thus, one has [25]   L t υ−1 Eα,υ (λt α ) ∞  λκ t κα+υ−1 =L (κα+υ) =

∞ κ=0

κ=0

(1.268)

λκ

s κα+υ

 = s −υ 1 − λs −α

 −α  λs  < 1; Re (υ) > 0

50

General Fractional Derivatives With Applications in Viscoelasticity

so that [25]   L Supersinα,υ (λt α )  ∞ (−1)κ λ2κ+1 t (2κ+1)α+υ−1 =L ((2κ+1)α+υ) =

∞ κ=0

κ=0

(1.269)

(−1)κ λ2κ+1 s (2κ+1)α+υ

 −1 = λs −(υ+α) 1+λ2 s −2α

 2 −2α   < 1; Re (υ) > 0 , λ s

  L Supersinhα,υ (λt α )  ∞ λ2κ+1 t (2κ+1)α+υ−1 =L ((2κ+1)α+υ) =

∞ κ=0

κ=0

(1.270)

λ2κ+1 s (2κ+1)α+υ

 −1 = λs −(υ+α) 1 − λ2 s −2α

 2 −2α  λ s  < 1; Re (υ) > 0 ,

  L Supercosα,υ (λt α )  ∞ (−1)κ λ2κ t 2κα+υ−1 =L (2κα+υ) =

∞ κ=0

κ=0

(1.271)

(−1)κ λ2κ s 2κα+υ

 −1 = s −υ 1 + λ2 s −2α

 2 −2α  λ s  < 1; Re (υ) > 0 ,

and [25]   L Supercoshα,υ (λt α )  ∞ λ2κ t 2κα+υ−1 =L (2κα+υ) =

∞ κ=0

κ=0

(1.272)

λ2κ s 2κα+υ

 −1 = s −υ 1 − λ2 s −2α

 2 −2α  λ s  < 1; Re (υ) > 0 .

As a special case, the Wiman function with one-parameter constant λ is defined as [38] t υ−1 E1,υ (λt) ∞ λκ t κ = t υ−1 (κ+υ) =

∞ κ=0

κ=0

λκ t κ+υ−1 (κ+υ) .

(1.273)

Special functions

51

The supersine function of Wiman type, denoted by Supersin1,υ (t), is defined as [25] ∞  (−1)κ t 2κ+υ , Supersin1,υ (t) =  (2κ + 1 + υ)

(1.274)

κ=0

which leads to [25] Supersin1,υ (λt) =

∞  (−1)κ λ2κ+1 t 2κ+υ

 (2κ + 1 + υ)

κ=0

,

(1.275)

the supercosine function of Wiman type, denoted by Supercos1,υ (t), is given as [25] Supercos1,υ (t) =

∞  (−1)κ t 2κ+υ−1 κ=0

 (2κ + υ)

(1.276)

,

which leads to [25] Supercos1,υ (λt) =

∞  (−1)κ λ2κ t 2α+υ−1

 (2κ + υ)

κ=0

,

(1.277)

the hyperbolic supersine function of Wiman type, denoted by Supersinh1,υ (t), is obtained as [25] Supersinh1,υ (t) =

∞  κ=0

t 2κ+υ ,  (2κ + 1 + υ)

(1.278)

which leads to [25] ∞  λ2κ+1 t 2κ+υ , Supersinh1,υ (λt) =  (2κ + 1 + υ)

(1.279)

κ=0

and the hyperbolic supercosine function of Wiman type, denoted by Supercosh1,υ (t), is defined as [25] Supercosh1,υ (t) =

∞  t 2κ+υ−1 ,  (2κ + υ)

(1.280)

κ=0

which leads to [25] Supercosh1,υ (λt) =

∞ 2κ 2κ+υ−1  λ t κ=0

 (2κ + υ)

.

(1.281)

Thus, one has [25] t υ−1 E1,υ (λt) = Supercosh1,υ (λt) + Supersinh1,υ (λt)

(1.282)

52

General Fractional Derivatives With Applications in Viscoelasticity

and [25] t υ−1 E1,υ (iλt) = Supercos1,υ (λt) + iSupersin1,υ (λt) ,

(1.283)

so that [25] Supersin1,υ (t) =

t υ−1 E1,υ (it) − t υ−1 E1,υ (−it) , 2i

t υ−1 E1,υ (−iλt) + t υ−1 E1,υ (−iλt) , 2 t υ−1 E1,υ (t) − t υ−1 E1,υ (−t) Supersinh1,υ (t) = , 2 t υ−1 E1,υ (t) + t υ−1 E1,υ (−t) Supercosh1,υ (t) = , 2 t υ−1 E1,υ (iλt) − t υ−1 E1,υ (−iλt) Supersin1,υ (λt) = , 2i t υ−1 E1,υ (iλt) + t υ−1 E1,υ (−iλt) Supercos1,υ (λt) = , 2 t υ−1 E1,υ (λt) − t υ−1 E1,υ (−λt) Supersinh1,υ (λt) = , 2 and [25] Supercos1,υ (t) =

Supercosh1,υ (λt) =

t υ−1 E1,υ (λt) + t υ−1 E1,υ (−λt) . 2

(1.284) (1.285) (1.286) (1.287) (1.288) (1.289) (1.290)

(1.291)

In this case, one has [25]   L t υ−1 E1,υ (λt) ∞  λκ t κ+υ−1 =L (κ+υ) =

∞

κ=0

(1.292)

λκ

κ=0 = s −υ

s κ+υ



1 − λs −1



 −1  λs  < 1; Re (υ) > 0

so that [25]   L Supersin1,υ (λt) ∞  (−1)κ λ2κ+1 t 2κ+υ =L (2κ+1+υ) =

∞ κ=0

κ=0 (−1)κ λ2κ+1 s 2κ+1+υ

 −1 = λs −(υ+1) 1+λ2 s −2

(1.293)  2 −2  λ s  < 1; Re (υ) > 0 ,

Special functions

53

  L Supersinh1,υ (λt) ∞  λ2κ+1 t 2κ+υ =L (2κ+1+υ) =

∞

κ=0

(1.294)

λ2κ+1

s 2κ+1+υ κ=0  −1 = λs −(υ+1) 1 − λ2 s −2

 2 −2  λ s  < 1; Re (υ) > 0 ,

  L Supercos1,υ (λt) ∞  (−1)κ λ2κ t 2κ+υ−1 =L (2κ+υ) =

∞

κ=0

κ=0 = s −υ

(1.295)

(−1)κ λ2κ s 2κ+υ



1 + λ2 s −2

−1

 2 −2  λ s  < 1; Re (υ) > 0 ,

and [25]   L Supercosh1,υ (λt) ∞  λ2κ t 2κ+υ−1 =L (2κ+υ) =

∞

κ=0

κ=0 = s −υ

(1.296)

λ2κ s 2κ+υ



1 − λ2 s −2

−1

 2 −2  λ s  < 1; Re (υ) > 0 .

1.9 The two-parameter Lorenzo–Hartley function In 2008, Lorenzo and Hartley considered the function defined as [36] ∞   λκ t (κ+1)α−1−υ , Rα,υ λt α =  ((κ + 1) α − υ)

(1.297)

κ=0

which is called the two-parameter Lorenzo–Hartley function. The subsine function of Lorenzo–Hartley type, denoted by RSinα (t α ), is defined as [25] ∞   (−1)κ t 2(κ+1)α−1−υ , RSinα,υ t α =  (2 (κ + 1) α − υ)

(1.298)

κ=0

which leads to [25] ∞   (−1)κ λ2κ+1 t 2(κ+1)α−1−υ , RSinα,υ λt α =  (2 (κ + 1) α − υ) κ=0

(1.299)

54

General Fractional Derivatives With Applications in Viscoelasticity

the subcosine function of Lorenzo–Hartley type, denoted by RCosα (t α ), is given as [25] ∞   (−1)κ t (2κ+1)α−1−υ RCosα,υ t α = ,  ((2κ + 1) α − υ)

(1.300)

κ=0

which implies that [25] ∞   (−1)κ λ2κ t (2κ+1)α−1−υ RCosα,υ λt α = ;  ((2κ + 1) α − υ)

(1.301)

κ=0

the hyperbolic subsine function of Lorenzo–Hartley type, denoted by RSinhα (t α ), is defined as [25] ∞   RSinhα,υ t α = κ=0

t 2(κ+1)α−1−υ ,  (2 (κ + 1) α − υ)

(1.302)

which implies that [25] ∞   λκ t 2(κ+1)α−1−υ RSinhα,υ λt α = ,  (2 (κ + 1) α − υ)

(1.303)

κ=0

and the hyperbolic subcosine function of Lorenzo–Hartley type, denoted by RCoshα (t α ), is obtained as [25] ∞   λκ t (2κ+1)α−1−υ , RCoshα,υ t α =  ((2κ + 1) α − υ)

(1.304)

κ=0

which implies that [25] ∞   λκ t (2κ+1)α−1−υ . RCoshα,υ λt α =  ((2κ + 1) α − υ)

(1.305)

κ=0

Here, one has [25]   L Rα,υ (λt α )  ∞ λκ t (κ+1)α−1−υ =L ((κ+1)α−υ) =

∞

κ=0

λκ s (κ+1)α−υ

κ=0 1 1 s α−υ 1−λs −α

 −α  λs  < 1  −1  −α  λs  < 1 = s −(α−υ) 1 − λs −α =

(1.306)

Special functions

55

so that [25]   L RSinα,υ (λt α ) ∞  (−1)κ λ2κ+1 t 2(κ+1)α−1−υ =L (2(κ+1)α−υ) =

∞

κ=0 (−1)κ λ2κ+1 s 2(κ+1)α−υ

(1.307)

κ=0  2 −2α  λ 1 λ s  0, g = [α] + 1, t ∈ [a, b] and let fg−α(t) = Ia+ f (t). If f (t) ∈ α α l, l ∈ L where Ia+ (Lκ ) = f : f = Ia+ κ (a, b) , then

 β α Ia+ Da+ f (t) = f (t) . (2.61)

α Ia+ (Lκ ),

g−α

(6C) Let α > 0, g = [α] + 1, t ∈ [a, b] and let fg−α (t) = Ia+ f (t). If f (t) ∈ α α α l, l ∈ L Ib− (Lκ ), where Ib− (Lκ ) = f : f = Ib− κ (a, b) , then  α α  Ib− Db− f (t) = f (t) . (2.62) κ−α f (t). If fκ−α (t) ∈ (7C) Let α > 0, κ = [α] + 1, t ∈ [a, b] and let fκ−α (t) = Ia+ κ AC [a, b], then



α α Ia+ Da+ f



κ−1 (κ−j )  fκ−α (a) (t) = f (t) − (t − a)α−j .  (α − 1 − j )

(2.63)

j =0

In this case, we have the following results: 1−α Let 1 > α > 0, t ∈ [a, b] and let f1−α (t) = Ia+ f (t). Then, we have [25,44,45]  α α  (t − a)α−1 f1−α (a) . Ia+ Da+ f (t) = f (t) −  (α) In particular, we have the following result.

(2.64)

Fractional derivatives with singular kernels

107

Let α > κ, t ∈ [a, b] and f (t) ∈ Lκ (a, b) (1 ≤ κ ≤ ∞). Then we have [25,44,45]  κ κ  α−κ f (t) (2.65) D Ia+ f (t) = Ia+ and  α−κ κ f (t) = Ib− f (t) . D κ Ib−



(2.66)

Property 2.8 ([25,44,45]). 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

κ

   (t − a)−α−j β α+β (β−j ) α Da+ f (t) = Da+ f (t) − . Da+ f (a+) Da+  (1 − a − j )

(2.67)

j =1

We now consider the rules for fractional integration by parts [25,44,45]. Property 2.9 ([25,44,45]). 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 b f1 (τ )



α f2 Ia+



b (τ ) dτ =

a

 α  f2 (τ ) Ib− f1 (τ ) dτ .

(2.68)

a

Property 2.10 ([25,44,45]). Let α > 0, g1 ≥ 1, g2 ≥ 1, and (1/g1 ) + (1/g  2 ) ≤ 1 + α. α α α l, l ∈ L If f1 (t) ∈ Ia+ b) , and f2 (t) ∈ (Lκ ), where Ia+ (Lκ ) = f : f = Ia+ (a, κ  α α α l, l ∈ L Ib− (Lκ ), where Ib− (Lκ ) = f : f = Ib− κ (a, b) , then b f1 (τ )



α f2 Da+



b (τ ) dτ =

a

  α f2 (τ ) Db− f2 (τ ) dτ .

(2.69)

a

Property 2.11 ([25,44,45]). If 0 < α < 1, then

 (Lf ) (s) β LI+ f (t) (s) = sα

(2.70)

and   α f (t) (ω) = s α (Lf ) (s) , LD+

(2.71)

where L {f (t)} (s) = (Lf ) (s).

2.3

Osler fractional calculus

In 1970, the ideas of the Riemann–Liouville fractional derivatives and fractional integrals with respect to another function on the complex domain were proposed by Osler

108

General Fractional Derivatives With Applications in Viscoelasticity

[25,65,66]. In 1993, the Riemann–Liouville fractional derivatives and fractional integrals with respect to another function on the real domain were suggested by Samko, Kilbas, and Marichev [25,44]. The present forms of the Riemann–Liouville fractional derivatives in the complex domain were investigated in 2006 by Kilbas, Srivastava, and Trujillo [25,45]. We now describe the theory of Riemann–Liouville fractional derivatives and integrals, which are called the Osler fractional derivatives and integrals (also called the Osler fractional calculus in his honor). Definition 2.8 ([25,44,45]). Let α > 0 and h(1) (t) > 0. The left-sided Osler fractional integral with respect to another function in the interval [a, b] is defined as follows: α f Ia+,h

1 (t) =  (α)

t a

f (τ ) (h (t) − h (τ ))1−α

h(1) (τ ) dτ ,

(2.72)

and the right-sided Osler with respect to another function in the interval [a, b] is given as α f Ib−,h

1 (t) =  (α)

b t

f (τ ) (h (τ ) − h (t))1−α

h(1) (τ ) dτ .

(2.73)

Definition 2.9 ([25,44,45]). Let α > 0, κ = [α] + 1 and h(1) (t) > 0. The left-sided Osler fractional derivative with respect to another function in the interval [a, b] is defined as follows: α f (t) Da+,h

 κ

κ−α d = h(1)1(t) dt f (t) Ia+,h

κ t d 1 1 = (n−α) (1) dt h (t) a

(2.74)

f (τ ) h(1) (τ ) dτ , (h(t)−h(τ ))α−κ+1

and the right-sided Osler fractional derivative with respect to another function in the interval [a, b] is given as α f (t) Db−,h

 κ

d α = − h(1)1(t) dt f (t) Ib−,h

κ b d 1 = (n−α) − h(1)1(t) dt t

(2.75)

f (τ ) h(1) (τ ) dτ . (h(τ )−h(t))α−κ+1

Property 2.12 ([25,44,45]). Let α > 0, β > 0 and h(1) (t) > 0. Then 



β α+β α Ia+,h f (t) = Ia+,h f (t) Ia+,h

(2.76)

Fractional derivatives with singular kernels

109

and



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

(2.77)

Property 2.13 ([25,44,45]). Let α > 0, f (t) ∈ Lκ (−∞, ∞) and h(1) (t) > 0. Then

 β α Ia+,h f (t) = f (t) Da+,h

(2.78)

 β α Db−,h Ib−,h f (t) = f (t) .

(2.79)

and

Property 2.14 ([25,44,45]). Let α > 0 and β > 0. β−1 (1L) If f (t) = (h(t)−h(a)) , then (β) α f (t) = Ia+,h

(h (t) − h (a))α+β−1 .  (α + β)

(2.80)

(2L) If f (t) = (h (b) − h (t))β−1 , then α Ib−,h f (t) =

(h (b) − h (t))α+β−1 .  (α + β)

(2.81)

Property 2.15 ([25,44,45]). Let α > 0 and β > 0. β−1 (1J) If f (t) = (h(t)−h(a)) , then (β) α f (t) = Ia+,h

(h (t) − h (a))β−α−1 .  (β − α)

(2.82)

(2J) If f (t) = (h (b) − h (t))β−1 , then α Ib−,h f (t) =

(h (b) − h (t))β−α−1 .  (β − α)

(2.83)

In particular, when h (t) = t α , one has [45] α Ia+,t αf

α (t) =  (α)

t a

α Ib−,t αf

1 (t) =  (α)

b t

τ α−1 (t α − τ α )1−α τ α−1 (τ α − t α )1−α

h(1) (τ ) dτ ,

(2.84)

h(1) (τ ) dτ ,

(2.85)

110

General Fractional Derivatives With Applications in Viscoelasticity

α Da+,t α f (t)  1−α d κ κ−α  = αt Ia+,h f (t) dt   t 1−κ d κ α τ α−1 = (κ−α) t 1−α dt α α α−κ+1 f (τ ) dτ ,

(2.86)

α Db−,t α f (t)

κ   d α f = − h(1)1(t) dt Ib− (t) b  1−α d κ  α 1−κ = (κ−α) −t α dt

(2.87)

a

(t −τ )

and

t

τ α−1

(τ −t α )α−κ+1

f (τ ) dτ .

Property 2.16. Let α > 0 and β > 0. Then



 β α+β α Ia+,t α Ia+,t α f (t) = Ia+,t α f (t) and





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

Property 2.17. Let α > 0 and f (t) ∈ Lκ (−∞, ∞). Then

 β α Da+,t α Ia+,t α f (t) = f (t) and



 β α Db−,t α Ib−,t α f (t) = f (t) .

(2.88)

(2.89)

(2.90)

(2.91)

Property 2.18. Let α > 0 and β > 0. α α )β−1 (1L) If f (t) = (t −a , then (β) α Ia+,t α f (t) =

(t α − a α )α+β−1 .  (α + β)

(2.92)

(2L) If f (t) = (bα − t α )β−1 , then α Ib−,t α f (t) =

(bα − t α )α+β−1 .  (α + β)

(2.93)

Property 2.19. Let α > 0 and β > 0. α α )β−1 (1J) If f (t) = (t −a , then (β) α Ia+,t α f (t) =

(t α − a α )β−α−1 .  (β − α)

(2.94)

Fractional derivatives with singular kernels

111

(2J) If f (t) = (bα − t α )β−1 , then

α Ib−,t α f (t) =

2.4

(bα − t α )β−α−1 .  (β − α)

(2.95)

Liouville–Weyl fractional calculus

Based on the Liouville’s work, Weyl developed the fractional calculus, which called the Liouville–Weyl fractional calculus (containing the Liouville–Weyl fractional integrals and derivatives) in the honor of Liouville [63,67] and Weyl [68].

2.4.1 Liouville–Weyl fractional integrals Let α > 0 and − ∞ < x < b < ∞. Definition 2.10 ([25,38]). The left-sided Liouville–Weyl fractional integral is defined as

LW α Cp I+ f

1 (t) =  (α)

t −∞

1 (t − τ )1−α

f (τ ) dτ

(2.96)

and the right-sided Liouville–Weyl fractional integral is given as

LW α Cp I− f

1 (t) =  (α)

∞ t

1 (τ − t)1−α

f (τ ) dτ .

(2.97)

2.4.2 Liouville–Weyl fractional derivatives Definition 2.11 ([25,38]). Let κ + 1 > α > κ and − ∞ < x < b. The left-sided Liouville–Weyl fractional derivative is defined as LW D α f + Cp

= =

dκ dt κ



(t)

  I+κ−α f (t)

dκ 1 (κ−α) dt κ

t −∞

1 f (t−τ )α−κ+1

(2.98) (τ ) dτ

112

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Weyl fractional derivative is given as LW D α f − Cp



(t)

  I−κ−α f (t) ∞ 1 dκ

=

dκ dt κ

=

1 (κ−α) dt κ

(2.99)

(τ −t)α−κ+1

t

f (τ ) dτ .

As the special cases (see [25]), the left-sided Liouville–Weyl fractional derivative is defined as LW D α f + Cp

= =

d dt



(t)

  I+1−α f (t)

d 1 (1−α) dt

t −∞

(2.100)

1 (t−τ )α f

(τ ) dτ ,

and the right-sided Liouville–Weyl fractional derivative is given as LW D α f − Cp

= =

d dt



(t)

  I−1−α f (t)

d 1 (1−α) dt

∞ t

1 (τ −t)α f

(2.101) (τ ) dτ .

In this case, one has from Eqs. (2.100) and (2.101) that LW D α f + Cp

= =

(t) t

1 (t−τ )α f (τ ) dτ −∞ ∞ f (t)−f (t−τ ) α dτ (1−α) τ α+1 0 d 1 (1−α) dt

(2.102)

and LW D α f − Cp

= =

(t)

∞ 1 d 1 (1−α) dt (τ −t)α f (τ ) dτ t ∞ f (t)−f (t+τ ) α dτ , (1−α) τ α+1 0

(2.103)

respectively. The properties of the Liouville–Weyl fractional derivatives and integrals are represented as follows.

Fractional derivatives with singular kernels

113

Property 2.20 ([25,38]). Suppose that α > 0 and β > 0, then 

LW α LW β LW α+β Cp I+ Cp I+ f (t) = Cp I+ f (t)

(2.104)

and

2.5

LW α LW β Cp I− Cp I− f



α+β

(t) = LW Cp I−

(2.105)

f (t) .

Samko–Kilbas–Marichev fractional calculus

In 1987, based on the Osler’s and Liouville–Weyl’s work, Samko, Kilbas and Marichev developed the Liouville–Weyl fractional calculus with respect to another function [44,69], which is called the Samko–Kilbas–Marichev fractional calculus in the honor of Samko, Kilbas, and Marichev [25].

2.5.1 Samko–Kilbas–Marichev fractional integrals Definition 2.12 ([25,44,69]). Let α > 0, −∞ < x < b < ∞ and h(1) (t) > 0. The left-sided Samko–Kilbas–Marichev fractional integral with respect to another function is defined as LW α Cp I+,h f

1 (t) =  (α)

t −∞

h(1) (τ ) (h (t) − h (τ ))1−α

f (τ ) dτ .

(2.106)

Let α > 0 and a < x < ∞. The right-sided Kilbas–Srivastava–Trujillo fractional integral with respect to another function is defined as LW α Cp I−,h f

1 (t) = −  (α)

∞

h(1) (τ ) (h (τ ) − h (t))1−α

t

f (τ ) dτ .

(2.107)

When h (t) = t α , one has [69] LW α Cp I+,t α f

α (t) =  (α)

t −∞

τ α−1 (t α − τ α )1−α

f (τ ) dτ

(2.108)

f (τ ) dτ .

(2.109)

and LW α Cp I−,h f

(t) = −

α  (α)

∞ t

τ α−1 (t α − τ α )1−α

114

General Fractional Derivatives With Applications in Viscoelasticity

2.5.2 Samko–Kilbas–Marichev fractional derivatives Definition 2.13 ([25,44,45,69]). Let h(1) (t) > 0. The left-sided Samko–Kilbas–Marichev fractional derivative with respect to another function in the interval [a, b] is defined as follows: LW D α f +,h Cp

= =



(t) κ 

d 1 h(1) (t) dt

1 (κ−α)



  κ−α I+,h f (t) κ t h(1) (t) d

1 h(1) (t) dt

−∞

(h(t)−h(τ ))α−κ+1

(2.110) f (τ ) dτ ,

and the right-sided Samko–Kilbas–Marichev fractional derivative with respect to another function in the interval [a, b] is give as LW D α f −,h Cp

=



1 h(1) (t)

(t) κ 

d dt

  κ−α I−,h f (t) κ ∞

d 1 − h(1)1(t) dt = (κ−α)

t

(2.111) h(1) (τ )

(h(τ )−h(t))α−κ+1

f (τ ) dτ .

Property 2.21 ([25]). Suppose that α > 0 and β > 0, then

 LW α LW β LW α+β Cp I+ Cp I+ f (t) = Cp I+ f (t) and



LW α LW β Cp I− Cp I− f



α+β

(t) = LW Cp I−

f (t) .

Property 2.22 ([25]). Suppose that α > 0, β > 0 and h(1) (t) > 0, then

 α+β LW α LW β I I f (t) = LW Cp +,h Cp +,h Cp I+,h f (t) and



LW α LW β Cp I−,h Cp I−,h f



α+β

(t) = LW Cp I−,h f (t) .

(2.112)

(2.113)

(2.114)

(2.115)

Property 2.23 ([25]). Suppose that α > 0, f (t) ∈ Lκ (−∞, ∞) and h(1) (t) > 0, then 

LW α LW β D I f (2.116) (t) = f (t) +,h Cp +,h Cp and



LW α LW β Cp D−,h Cp I−,h f



(t) = f (t) .

(2.117)

Fractional derivatives with singular kernels

115

In 2010, by considering a special case h (t) = t α , Kilbas defined the Samko– Kilbas–Marichev-type fractional derivative as [45,69] LW D α f +,t α Cp

= =



(t)

 d κ αt 1−α dt 

α 1−κ (κ−α)



 κ−α  I+,t α f (t)  t d κ τ α−1

t 1−α dt

−∞

(t α −τ α )α−κ+1

(2.118) f (τ ) dτ

and LW D α f −,t α Cp



(t)   d κ

 κ−α  = αt 1−α dt I+,t α f (t)  1−α d κ t α 1−κ −t = (κ−α) α dt −∞

(2.119)

τ α−1 f (τ −t α )α−κ+1

(τ ) dτ .

We easily have the following results: Property 2.24. Suppose that α > 0 and β > 0, then 

LW α LW β LW α+β Cp I+,t α Cp I+,t α f (t) = Cp I+,t α f (t) and



LW α LW β Cp I−,t α Cp I−,t α f



α+β

(t) = LW Cp I−,t α f (t) .

Property 2.25. Suppose that α > 0 and f (t) ∈ Lκ (−∞, ∞), then 

LW α LW β Cp D+,t α Cp I+,t α f (t) = f (t) and



LW α LW β Cp D−,t α Cp I−,t α f

2.6



(t) = f (t) .

(2.120)

(2.121)

(2.122)

(2.123)

Liouville–Sonine–Caputo fractional derivatives

2.6.1 History of Liouville–Sonine–Caputo fractional derivatives In 1832, Liouville considered the fractional derivative operator of the form [63] α L D0+ f

1 (t) = (−1)α  (α)

∞ 0

d κ f (t + τ ) α−1 τ dτ , dt κ

(2.124)

116

General Fractional Derivatives With Applications in Viscoelasticity

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

(2.125)

can be rewritten as α L D0+ f

1 (t) = α (−1)  (α)

∞ 0

d κ f (t + τ ) α−1 τ dτ . d (t + τ )κ

(2.126)

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

(2.127)

so from Eqs. (2.126) and (2.127) we have α L Dt+ f

(t) =

∞

1 (−1)α  (α)

d κ f (h) (h − t)α−1 dh, dhκ

(2.128)

t

which yields α L Dt+ f

1 (t) = (−1)α  (α)

∞

d κ f (h) dhκ α dh =

(h − t)

t

1 (−1)α  (α)

∞ t

f (κ) (h) dh. (2.129) (h − t)α

From Eq. (2.129), one has  (z + κ) = (z)κ ,  (z) leading to α L Dt+ f

1 (t) =  (κ − α)

∞ t

f (κ) (h) (h − t)α−κ+1

(2.130)

dh,

which can used to consider problems in mathematical physics [25,70–73]. In 1832, Liouville presented the special case of the fractional derivative of the form (see [63]) h λ0

1 df (t) dt = λ√ (h − t) dt

h 0

df (t) dt

h dt = λ1

(h − t) 2

f (1) (t)

dt = λ 1

0

(h − t) 2

 1 1 2 L D0+ f (t) , 2 (2.131)

√ where λ- = 1/ 2g.

Fractional derivatives with singular kernels

When g = 1 √ π

h 0

1 2π ,

117

one easily has

1 1 df (t) dt =  √ (h − t) dt  12

h 0

1 1 df (t) 2 f (t) . (2.132) dt = L D0+ √ (h − t) dt

In 1871, Sonine showed that the fractional derivative can be presented as follows [74]: α S Da+ f

t

1 (t) =  (p + 1 − α)

df (t) (h − t)p−α dt, dt

(2.133)

a

where p < α < p + 1, which can further be expressed as follows: α S Da+ f (t) =

t

1  (p + 1 − α)

a

df (t) 1 dt, dt (h − t)α−p

(2.134)

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

1 (t) =  (1 − α)

t

df (t) 1 dt. dt (h − t)α

a

(2.135)

In 1967, Caputo considered the fractional derivative of the form [75] κ+α C D0+ f

t

1 (x) =  (1 − α)

0

1 d κ+1 f (t) dt, (h − t)α dt κ+1

(2.136)

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

1 (x) =  (p − h)

t 0

d p f (t) dt. (h − t)h−p+1 dt p 1

(2.137)

It is shown that α C D0+ f

1 (x) =  (κ − α)

t 0

f (κ) (t) (h − t)α−κ+1

dt.

(2.138)

Thus, one has α LS D+a f

1 (x) =  (1 − α)

t a

f (1) (τ ) dτ , (t − τ )α

(2.139)

118

General Fractional Derivatives With Applications in Viscoelasticity

which is due to Liouville [63] and Sonine [74], and α LSC D+∞ f

1 (x) =  (κ − α)

t ∞

f (κ) (t) (h − t)α−κ+1

(2.140)

dt,

which is due to Liouville (case a = ∞) [63], Sonine (case κ = 1) [74] and Gerasimov (case κ = 1 and a = 0) [70,76], α LSC D+∞ f

1 (x) =  (κ − α)

t 0

f (κ) (t) (h − t)α−κ+1

(2.141)

dt,

which is due to Liouville (cases κ = 1 and α = 1/2; a = ∞, see [63]) Sonine (case κ = 1) [74], Caputo (case a = 0) [75] and Smit and De Vries [77], α LSC D+a f

1 (x) =  (κ − α)

t a

f (κ) (t) (h − t)α−κ+1

(2.142)

dt,

which is due to the detailed information from the books by Podlubny (see [78]), Kilbas, Srivastava, Trujillo (see [45]) and Diethelm (see [79]). To honor the authors, the operator is called the Liouville–Sonine–Caputo fractional derivative. Definition 2.14 ([25]). Let α ≥ 0 and κ = [α] + 1. The left-sided Liouville–Sonine– Caputo fractional derivative is defined as α LSC D+a f

α (t) = I+a

 f

(κ)

 (t) =

1  (κ − α)

t a

f (κ) (τ ) (t − τ )α−κ+1

dτ ,

(2.143)

and the right-sided Liouville–Sonine–Caputo fractional derivative is given as α LSC Db− f

α (t) = Ib−

 f

(κ)

 (t) =

(−1)κ  (κ − α)

b t

f (κ) (τ ) (t − τ )α−κ+1

dτ .

(2.144)

The relations among the Riemann–Liouville and Liouville–Sonine–Caputo fractional derivatives are presented as follows. Property 2.26 ([25,79]). Let α ≥ 0, κ = [α] + 1 and f (t) ∈ AC κ (a, b). Then fracα f α f tional derivatives LSC Da+ (t) and LSC Db− (t) exist almost everywhere on (a, b) and can be represented as follows: α LSC D+a f

(t) =

κ−1  j =0

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

(2.145)

Fractional derivatives with singular kernels

119

and α LSC D−b f (t) =

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

(2.146)

j =0

respectively. As a special case, one has the following: α f Let 0 < α < 1 and f (t) ∈ AC (a, b). Then fractional derivatives Da+ (t) and α Db− f (t) exist almost everywhere on (a, b) and can be represented as follows [25,45,79]: α LS D+a f

α f (t) − (t) = Da+

(t − a)−α  (1 − α)

(2.147)

α LS D−b f

α f (t) − (t) = Db−

(b − t)−α ,  (1 − α)

(2.148)

and

respectively. There are the following relations: Property 2.27 ([25,45,79]). If f (κ−1) (a) = 0, where κ = [α] + 1, then α LSC D+a f

α f (t) . (t) = D+a

(2.149)

Property 2.28 ([25,45]). If f (κ−1) (b) = 0, where κ = [α] + 1, then α LSC D−b f

α f (t) . (t) = D−b

(2.150)

For 0 < β, one gets [25,45]: α (t LSC Da+

− a)β−1 (t − a)β−α−1 = ,  (β)  (β − α)

(2.151)

(b − t)β−α−1 ,  (β − α)

(2.152)

α (b − t) LSC Db−

β−1

 (β)

α LSC Da+ (t

=

− a)κ−1 = 0 (κ = [α] + 1) ,

(2.153)

= 0 (κ = [α] + 1) .

(2.154)

and α κ−1 LSC Db− (b − t)

Property 2.29 ([25,45]). The Laplace transforms of the Liouville–Sonine–Caputo fractional derivatives are given as follows: ⎛ ⎞ κ−1 (j )    f (0) ⎠ α f (t) (ω) = s α ⎝(Lf ) (s) − LLSC D+0 (0 < α; κ = [α] + 1) , s j −α+1 j =0

(2.155) where L {f (t)} (s) = (Lf ) (s).

120

2.7

General Fractional Derivatives With Applications in Viscoelasticity

Liouville fractional derivatives

In 1832, Liouville [63] defined the Liouville–Sonine–Caputo fractional derivative on the real line, which is called the Liouville fractional derivative [76]. In 1948, Gerasimov applied it to the theory of viscoelasticity [76]. Definition 2.15 ([63,25]). Let α ≥ 0 and κ = [α] + 1. The left-sided Liouville fractional derivative is defined as t   f (κ) (τ ) 1 α α (κ) dτ , (t) = LSC D+ f (t) = I+ f  (κ − α) (t − τ )α−κ+1

(2.156)

−∞

and the right-sided Liouville fractional derivative is given as (t) = I−α

α LSC D− f

2.8

 f

(κ)

 (t) =

(−1)κ  (κ − α)

∞ t

f (κ) (τ ) (τ − t)α−κ+1

dτ .

(2.157)

Almeida fractional derivatives with respect to another function

In 2017, based on the Liouville–Sonine–Caputo fractional derivative, Almeida defined the Liouville–Sonine–Caputo fractional derivative with respect to another function (called the Caputo fractional derivative with respect to another function in [80]), which is called the Almeida fractional derivative with respect to another function in the honor of Almeida [25]. Definition 2.16 ([80,25]). Let h(1) (t) > 0. The left-sided Almeida fractional derivative with respect to another function is defined as α LSC Da+,h f





(t)

κ−α = Ia+,h f (κ) (t)

κ t (κ) h(1) (τ ) d 1 1 fh (τ ) dτ, = (κ−α) α−κ+1 (1) dτ h (τ ) a

(2.158)

(h(t)−h(τ ))

and the right-sided Almeida fractional derivative with respect to another function is given as α LSC Db−,h f





(t)

κ−α = Ib−,h f (κ) (t)

=

1 (κ−α)

b t

h(1) (τ ) (h(τ )−h(t))α−κ+1



d − h(1)1(τ ) dτ

κ

(2.159) (κ)

fh (τ ) dτ .

Fractional derivatives with singular kernels

121

Property 2.30 ([80,25]). Let α > 0, f (t) ∈ ACκ (a, b) and h(1) (t) > 0. Then 

β α (2.160) LSC Da+,h Ia+,h f (t) = f (t) and



β α LSC Db−,h Ib−,h f



(t) = f (t) .

(2.161)

Property 2.31 ([80,25]). Let α > 0, f (t) ∈ AC1 (a, b) and h(1) (t) > 0. Then 

β α (2.162) LS Da+,h Ia+,h f (t) = f (t) and



β α LS Db−,h Ib−,h f



(t) = f (t) .

(2.163)

For the special case h (t) = t α , one has α LSC Da+,t α f (t) t  1−α d κ τ α−1 α 1−κ = (κ−α) τ α−κ+1 α α dτ (t −τ ) a t  τ (κ−1)(1−α) (κ) α 1−κ = (κ−α) f (τ ) dτ (t α −τ α )α−κ+1 a

f (τ ) dτ

(2.164)

and α LSC Db−,t α f (t) b  1−α d κ τ α−1 α 1−κ = (κ−α) −τ dτ (τ α −t α )α−κ+1 t b τ (κ−1)(1−α) (κ) α 1−κ f (τ ) dτ . = (κ−α) (t α −τ α )α−κ+1 t

f (τ ) dτ

Property 2.32 ([80,25]). Suppose that α > 0 and f (t) ∈ ACκ (a, b), then 

β α LSC Da+,t α Ia+,t α f (t) = f (t) and



β α LSC Db−,t α Ib−,t α f



(t) = f (t) .

Property 2.33 ([80,25]). Suppose that α > 0 and f (t) ∈ AC1 (a, b), then 

β α LS Da+,t α Ia+,t α f (t) = f (t)

(2.165)

(2.166)

(2.167)

(2.168)

122

General Fractional Derivatives With Applications in Viscoelasticity

and

2.9

β α LS Db−,t α Ib−,t α f



(t) = f (t) .

(2.169)

Liouville-type fractional derivative with respect to another function

Definition 2.17. Let α > 0, κ = [α] + 1 and h(1) (t) > 0. The left-sided Liouville-type fractional derivative with respect to another function is defined as L Dα f Cp +,h

(t)  κ−α = I+,h f (κ) (t)

κ t (κ) h(1) (τ ) 1 1 d = (κ−α) fh (τ ) dτ , α−κ+1 (1) dτ h (τ )

−∞

(2.170)

(h(t)−h(τ ))

and the right-sided Liouville-type fractional derivative with respect to another function is given as L Dα f Cp −,h

= =



κ−α I−,h

(t)  f (κ) (t) ∞  h(1) (τ )

1 (κ−α)

t

(h(τ )−h(t))α−κ+1

κ (κ) d − h(1)1(τ ) dτ fh (τ ) dτ .

(2.171)

Property 2.34. Suppose that α > 0, f (t) ∈ ACκ (−∞, ∞) and h(1) (t) > 0, then we have 

L α LW β D I f (2.172) (t) = f (t) Cp +,h Cp +,h and

L α LW β Cp D−,h Cp I−,h f



(t) = f (t) .

(2.173)

For the special case h (t) = t α , one has L Dα f Cp +,t α

=

α 1−κ (κ−α)

=

α 1−κ (κ−α)

(t) t

−∞ t −∞

τ α−1 (t α −τ α )α−κ+1

 1−α d κ τ f (τ ) dτ dτ

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

(2.174)

Fractional derivatives with singular kernels

123

and L Dα f Cp −,t α

= =

α 1−κ (κ−α) α 1−κ (κ−α)

(t) ∞ t ∞ t

τ α−1 (τ α −t α )α−κ+1 τ (κ−1)(1−α) (t α −τ α )α−κ+1



d −τ 1−α dτ

κ

f (τ ) dτ

(2.175)

f (κ) (τ ) dτ .

Property 2.35. Suppose that α > 0 and f (t) ∈ ACκ (a, b), then

 β α D I f (t) = f (t) α α LSC +,t a+,t

(2.176)

and

β α LSC D−,t α I−,t α f



(t) = f (t) .

(2.177)

Property 2.36. Suppose that α > 0 and f (t) ∈ AC1 (a, b), then

β α LS D+,t α I+,t α f



(t) = f (t)

(2.178)

(t) = f (t) .

(2.179)

and

2.10

β α LS D−,t α I−,t α f



Liouville–Grünwald–Letnikov fractional derivatives

2.10.1 History of the Liouville–Grünwald–Letnikov fractional derivatives In 1832, Liouville first established the fractional difference derivative formula of the form [63] αh f (t) , h→0 hα

α d D f (t) = lim

(2.180)

where αh f (t) is the fractional difference (see [63]), which is expressed as the sum of exponentials (see [81]). In 1847, Riemann (in the case α = 2) derived the fractional difference derivative of the form [64] dD

[α]

αh f (t) , h→0 hα

f (t) = lim

(2.181)

124

General Fractional Derivatives With Applications in Viscoelasticity

where αh f (t) =

∞ 

⎛ (−1)i ⎝

⎞ α

⎠ f (x − hi).

(2.182)

i

i=0

In 1867, based on the Liouville’s fractional difference operator, Grünwald proposed the fractional difference of the form [82] dD

α

αh f (t) , h→0 hα

f (t) = lim

(2.183)

where αh f (t) is a difference of fractional order, given by [81] αh f =

∞ 

⎛ (−1)i ⎝

⎞ α

⎠ f (x − hi).

(2.184)

i

i=0

We remark that this operator is called the Liouville–Grünwald fractional derivative by in the book [44]. In 1868, Letnikov proposed the fractional difference derivative of the form [83] αh f (t) , h→0 hα

α d D f (t) = lim

(2.185)

where αh f (t) is a difference of fractional order, whose appropriate interpretation as the fractional difference is related with the Riemann’s and Liouville’s expressions [25,44]. In the honor of Grünwald and Letnikov, the fractional difference derivative is called the Grünwald–Letnikov fractional derivative in the book [44]. Similarly, in the honor of Liouville, Grünwald and Letnikov, such derivatives are called the Liouville–Grünwald–Letnikov fractional derivatives [25]. Definition 2.18 ([25,44,45,82]). For α > 0, the left-sided fractional difference is defined as ⎛ ⎞ ∞  α ⎠ f (t − hi), αh f (t) = (2.186) (−1)i ⎝ i i=0 and the right-sided fractional difference is given as α−h f (t) =

∞  i=0

where h > 0.

⎛ (−1)i ⎝

⎞ α i

⎠ f (x + hi),

(2.187)

Fractional derivatives with singular kernels

125

Property 2.37 ([25,44,45,82]). If α > 0 and β > 0, then the semigroup properties, 

β α+β (2.188) αh h f (t) = h f (t) and



 β α+β α−h −h f (t) = −h f (t) ,

(2.189)

hold for any bounded function f (t).

2.10.2 Concepts of Liouville–Grünwald–Letnikov fractional derivatives Definition 2.19 ([25,44,45]). The left-sided Liouville–Grünwald–Letnikov fractional difference derivative is defined as α d D+ f

αh f (t) h→+0 hα

(t) = lim

(α > 0) ,

where the left-sided fractional difference is ⎛ ⎞ ∞  α ⎠ f (t − hi), αh f (t) = (−1)i ⎝ i i=0

(2.190)

(2.191)

and the right-sided fractional difference derivative is given as α d D− f

(t) = lim

h→+0

α−h f (t) hα

(α > 0) ,

where the right-sided fractional difference is ⎛ ⎞ ∞  α ⎠ f (x + hi). α−h f (t) = (−1)i ⎝ i i=0 Definition 2.20 ([25,44,45]). Taking ⎧ ⎨f (t) , t ∈ [a, b] , f (t) = ⎩0, t ∈ / [a, b] ,

(2.192)

(2.193)

(2.194)

the left-sided fractional difference on the interval [a, b] is defined as follows:  t−a  ⎛ ⎞ h  α α i⎝ ⎠ f (t − hi) (t ∈ R, h > 0, α > 0) h,+a f (t) = (2.195) (−1) i i=0

126

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided fractional difference on the interval [a, b] is given as 

b−t



h 

α−h,−b f (t) =

⎛

⎞ α

(−1)i ⎝

⎠ f (x + hi)

(t ∈ R, h > 0, α > 0) .

(2.196)

i

i=0

2.10.3 Liouville–Grünwald–Letnikov fractional derivatives on a bounded domain Definition 2.21 ([25,44,45]). The left-sided Liouville–Grünwald–Letnikov fractional derivative on the interval [a, b] is defined as α d D+a f (t) = lim

αh,+a f (t) hα

h→+0

(α > 0)

(2.197)

where the left-sided fractional difference on the interval [a, b] is  t−a  ⎛ ⎞ h  α ⎠ f (t − hi) (t ∈ R, h > 0, α > 0) , αh,+a f (t) = (−1)i ⎝ i i=0

(2.198)

and the right-sided Liouville–Grünwald–Letnikov fractional derivative on the interval [a, b] is given as α d D−b f

(t) = lim

α−h,−b f (t) hα

h→+0

(α > 0) ,

(2.199)

where the right-sided fractional difference on the interval [a, b] is 

α−h,−b f (t) =

b−t h





⎛ (−1)i ⎝

⎞ α

⎠ f (x + hi)

(t ∈ R, h > 0, α > 0) .

(2.200)

i

i=0

Property 2.38 ([25,44,45]). Let f (t) ∈ Lκ (R). Then α d Da+ f

α (t) =  (1 − α)

t

f (t) − f (τ ) (τ − t)1+α

a

dτ

(2.201)

dτ ,

(2.202)

and α d Db− f

α (t) =  (1 − α)

b t

f (t) − f (τ ) (t − τ )1+α

where 0 < α < 1 and 1 < κ < 1/α.

Fractional derivatives with singular kernels

2.11

127

Kilbas–Srivastava–Trujillo fractional difference derivatives

In 2006, Kilbas, Srivastava, and Trujillo considered the fractional difference derivatives, which are now called the Kilbas–Srivastava–Trujillo fractional difference derivatives [45]. Definition 2.22 ([25,45]). The left-sided Kilbas–Srivastava–Trujillo fractional difference derivative is defined as follows: α f (t) = Ma+

f (t) (t − a)−α α f (t) , + d Da+  (1 − α)

(2.203)

α f where d Da+ (t) is left-sided Liouville–Grünwald–Letnikov fractional derivative on the interval [a, b], and the right-sided Kilbas–Srivastava–Trujillo fractional difference derivative is given as α Mb− f (t) =

f (t) (b − t)−α α f (t) , + d Db−  (1 − α)

(2.204)

α f where d Db− (t) is the right-sided Liouville–Grünwald–Letnikov fractional derivatives on the interval [a, b].

Property 2.39 ([25,45]). Let α > 0. Then ⎞ ⎛ t 1 α ⎝f (t) t −α + α f (t) − f (τ ) dτ ⎠ M0+ f (t) =  (1 − α) (τ − t)1+α

(2.205)

0

and ⎛ α M0− f (t) =

1 ⎝f (t) (−t)−α + α  (1 − α)

0 t

⎞ f (t) − f (τ ) (t − τ )1+α

dτ ⎠ .

(2.206)

Property 2.40 ([25,45]). Let 1 > α > 0. Then α d D+a l α d D−b l

= 0,

(2.207)

= 0,

(2.208)

α l= M+a

l (t − a)−α ,  (1 − α)

(2.209)

l (b − t)−α ,  (1 − α)

(2.210)

and α M−b l=

where l is a constant.

128

General Fractional Derivatives With Applications in Viscoelasticity

Property 2.41 ([25,45]). Let 1 > α > 0 and h ∈ R. The left-sided Liouville– Grünwald–Letnikov fractional difference derivative can be represented as α d D+ f

(t)

αh f (t) α h→+0 |h| t  f (t)−f (τ ) α = (1−α) dτ (τ −t)1+α −∞ ∞ f (τ )−f (t−τ ) α dτ , = (1−α) τ 1+α 0

= lim

(2.211)

and the right-sided Liouville–Grünwald–Letnikov fractional difference derivative can be given as α d D− f

(t)

α−h f (t) α h→+0 |h| ∞  f (t)−f (τ ) α = (1−α) dτ (t−τ )1+α t ∞ f (τ )−f (t+τ ) α = (1−α) dτ , τ 1+α 0

= lim

(2.212)

where |·| denotes the absolute value.

2.12

Riesz fractional calculus

In this section, we present the Riesz fractional calculus [25]. In 1949, Riesz [84] defined the fractional calculus based on the Fourier’s work [85], which is called the Riesz fractional calculus [25,86].

2.12.1 Riesz fractional calculus Definition 2.23 ([25,85,86]). Let 1 > α > 0. The Riesz fractional integral in the onedimensional space is defined as α Rz IR f (t)   α I f +L I−α f (t) = L 2+cos(πα/2)

= =

1 2 cos(πα/2)(α) 1 2(α) cos(πα/2)



× ∞ −∞

t

−∞

f (τ ) dτ (τ −t)1−α

f (τ ) dτ, |τ −t|1−α

+

∞ t

 f (τ ) dτ (t−τ )1−α

(2.213)

Fractional derivatives with singular kernels

129

where α L I+ f

1 (t) =  (α)

t

f (τ )

−∞

(τ − t)1−α

dτ

and α L I− f

1 (t) =  (α)

∞

f (τ ) (t − τ )1−α

t

dτ .

Definition 2.24 ([25,85,86]). Let 1 > α > 0. The Riesz fractional derivative in the one-dimensional space is defined as α Rz DR f (t)   α α f (t) D f +L D− = L 2+cos(πα/2)



=

1 2 cos(πα/2)(1−α)

×

=

1 2 cos(πα/2)(1−α)

×

d dt d dt

t −∞

∞

−∞

f (τ ) (τ −t)α dτ

−

d dt

∞ t

 f (τ ) (t−τ )α dτ

(2.214)

f (τ ) |τ −t|α dτ ,

where α L D+ f

1 d (t) =  (1 − α) dt

t −∞

f (τ ) dτ (τ − t)α

and α L D− f

1 d (t) = −  (1 − α) dt

∞ t

f (τ ) dτ . (t − τ )α

Property 2.42. Let cos (πα/2) =

eiπα/2 + e−iπα/2 . 2

(2.215)

Then α Rz IR f (t)  α  I f +L I−α f (t) = L 2+cos(πα/2)  α  I+ f +L I−α f (t) = Leiπα/2 +e−iπα/2

=

1   eiπα/2 +e−iπα/2 (α)

(2.216) ∞ −∞

f (τ ) dτ |τ −t|1−α

130

General Fractional Derivatives With Applications in Viscoelasticity

and α Rz DR f (t)  α  α f (t) D f +L D− = L 2+cos(πα/2)   α α f (t) D+ f +L D− = Leiπα/2 +e−iπα/2

=

(2.217)

1   eiπα/2 +e−iπα/2 (1−α)

×

d dt

∞ −∞

f (τ ) |τ −t|α dτ .

For more details of the Riesz fractional calculus, see [25,85–87].

2.12.2 Riesz-type fractional calculus Based on the Riesz fractional calculus, we will present the Riesz-type fractional derivative and integral, which is called the Riesz-type fractional calculus. Definition 2.25. Let 1 > α > 0. The Riesz-type fractional integral on a bounded domain is defined as α Rz I[a,b] f (t)  α  α L I f +L Ib− f (t) = a+ 2 cos(πα/2)

= =

1 2 cos(πα/2)(α) 1 2(α) cos(πα/2)

×

 t a

b a

f (τ ) dτ (τ −t)1−α

+

b t

 f (τ ) dτ (t−τ )1−α

f (τ ) dτ, |τ −t|1−α

where α L Ia+ f

1 (t) =  (α)

t a

f (τ ) (τ − t)1−α

dτ

and α L Ib− f

1 (t) =  (α)

b t

f (τ ) (t − τ )1−α

dτ .

Definition 2.26. Let 1 > α > 0. The Riesz-type fractional derivative on a bounded domain is defined as α Rz D[a,b] f (t)  α  α L Da+ f +L Db− f (t) = 2 cos(πα/2)

(2.218)

Fractional derivatives with singular kernels

 = =

1 2 cos(πα/2)(1−α)

×

1 2 cos(πα/2)(1−α)

×

t

d dt

f (τ ) (τ −t)α dτ

a

b

d dt

131

a

−

d dt

b t

 f (τ ) (t−τ )α dτ

(2.219)

f (τ ) |τ −t|α dτ ,

where α L Da+ f

d 1 (t) =  (1 − α) dt

t

f (τ ) dτ (τ − t)α

a

and α L Db− f

d 1 (t) = −  (1 − α) dt

b t

f (τ ) dτ . (t − τ )α

2.12.3 Liouville–Sonine–Caputo–Riesz-type fractional derivatives Based on the Riesz and Liouville–Sonine–Caputo fractional derivatives, the Liouville– Sonine–Caputo–Riesz-type fractional derivatives will be first represented as follows. Definition 2.27. Let 1 > α > 0. The Liouville–Sonine–Caputo–Riesz-type fractional derivative in the one-dimensional space is defined as α RzC DR f (t)   α f (t) D α f +LSC D− = LSC 2+cos(πα/2) 

= =

1 2 cos(πα/2)(1−α) 1 2 cos(πα/2)(1−α)

× ×

t

−∞

∞

−∞

f (1) (τ ) (τ −t)α dτ

f (1) (τ ) |τ −t|α

−

∞ f (1) (τ ) t

(t−τ )α dτ

dτ ,

where α LSC D+ f

1 (t) =  (1 − α)

t −∞

f (1) (τ ) dτ (τ − t)α

and α LSC D− f (t) = −

1  (1 − α)

∞ t

f (1) (τ ) dτ . (t − τ )α

 (2.220)

132

General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.28. Let 1 > α > 0. The Liouville–Sonine–Caputo–Riesz-type fractional derivative on a bounded domain is defined as α RzC D[a,b] f (t)   α α LSC Da+ f +LSC Db− f (t) = 2 cos(πα/2) 

= =

1 2 cos(πα/2)(1−α) 1 2 cos(πα/2)(1−α)

t

×

a

×

b a

f (1) (τ ) (τ −t)α dτ

f (1) (τ ) |τ −t|α

−

b t

 f (1) (τ ) (t−τ )α dτ

(2.221)

dτ ,

where α LSC Da+ f

1 (t) =  (1 − α)

t

f (1) (τ ) dτ (τ − t)α

a

and α LSC Db− f

2.13

1 (t) = −  (1 − α)

b t

f (1) (τ ) dτ . (t − τ )α

Feller fractional calculus

As a general version of the Riesz fractional calculus, Feller proposed the fractional derivative and fractional integral in 1952, which is called the Feller fractional derivative and integral, respectively (see [88]).

2.13.1 Feller fractional calculus We now present the concept and properties of the Feller fractional derivative and integral, which are called the Feller fractional calculus in the honor of Feller (see [25, 86]). Definition 2.29 ([25,86,88]). Let 1 > α > 0. The Feller fractional integral is defined as follows: α F IR,ϑ f

(t)

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

sin((α+ϑ)π/2) sin(πϑ)

×

1 (α)

−∞ ∞ t

(τ −t)

f (τ ) dτ , (t−τ )1−α

(2.222)

Fractional derivatives with singular kernels

133

where the two components are given by H− (ϑ, α) =

sin ((α − ϑ) π/2) sin (πϑ)

(2.223)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.224)

and

Definition 2.30 ([25,86,88]). Let 1 > α > 0. The Feller fractional derivative is defined as follows: α F DR,ϑ f



(t)

 αf αf = − H+ (ϑ, α) L D+ (t) + H− (ϑ, α) L D− (t) t f (τ ) d 1 × (1−α) = − sin((α+ϑ)π/2) sin(πϑ) dt (τ −t)α dτ

− sin((α−ϑ)π/2) sin(πϑ)

×

d 1 (1−α) dt

∞ t

(2.225)

−∞

f (τ ) (t−τ )α dτ ,

where the two components are given as H− (ϑ, α) =

sin ((α − ϑ) π/2) sin (πϑ)

(2.226)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.227)

and

Property 2.43 ([25,86,88]). Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.228)

Then α F IR,0 f

(t) = Rz IRα f (t)

(2.229)

and α F DR,0 f

(t) = Rz DRα f (t) .

(2.230)

134

General Fractional Derivatives With Applications in Viscoelasticity

Proof. By definition, one has α F IR,0 f

(t)

= H− (0, α) L I+α f (t) + H+ (0, α) L I−α f (t) =



 (t) 2 cos(πα/2)

α α L I+ f +L I− f

(2.231)

= Rz IRα f (t) and α F DR,0 f

(t)   αf αf = − H+ (0, α) L D+ (t) + H− (0, α) L D− (t) = −L

α f (t)+ D α f (t) D+ L − 2 cos(απ/2)

(2.232)

= Rz DRα f (t) , completing the proof.

2.13.2 Feller-type fractional calculus Based on the Feller fractional calculus, we will present the Feller-type fractional derivative and integral, which is called the Feller-type fractional calculus. Definition 2.31. The Feller type fractional integral on a bounded domain is defined as follows: α F I[a,b],ϑ f

(t)

α f α f = H− (ϑ, α) L Ia+ (t) + H+ (ϑ, α) L Ib− (t) t  f (τ ) 1 = sin((α−ϑ)π/2) × (α) 1−α dτ sin(πϑ) a

+ sin((α+ϑ)π/2) sin(πϑ)

×

1 (α)

b t

(2.233)

(τ −t)

f (τ ) dτ . (t−τ )1−α

Definition 2.32. Let 1 > α > 0. The Feller-type fractional derivative on a bounded domain is defined as follows: α F D[a,b],ϑ f

(t)   α f α f = − H+ (ϑ, α) L Da+ (t) + H− (ϑ, α) L Db− (t) t f (τ ) d 1 = − sin((α+ϑ)π/2) × (1−α) sin(πϑ) dt (τ −t)α dτ

× − sin((α−ϑ)π/2) sin(πϑ)

d 1 (1−α) dt

b t

a

f (τ ) (t−τ )α dτ .

(2.234)

Fractional derivatives with singular kernels

135

Property 2.44. Consider 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.235)

Then α F I[a,b],0 f

α f (t) (t) = Rz I[a,b]

(2.236)

and α F D[a,b],0 f

α f (t) . (t) = Rz D[a,b]

(2.237)

Proof. We have α F I[a,b],0 f

(t)

α f α f = H− (0, α) L Ia+ (t) + H+ (0, α) L Ib− (t) 

=

α α L Ia+ f +L Ib− f



(t)

(2.238)

2 cos(πα/2)

α f = Rz I[a,b] (t)

and α F D[a,b],0 f

(t)   α f α f = − H+ (0, α) L Da+ (t) + H− (0, α) L Db− (t) =−

α α L Da+ f (t)+L Db− f (t)

(2.239)

2 cos(απ/2)

α = Rz D[a,b] f (t) ,

finishing the proof.

2.13.3 Liouville–Sonine–Caputo–Feller-type fractional derivatives Based on the Feller and Liouville–Sonine–Caputo fractional derivatives, the Liouville– Sonine–Caputo–Feller-type fractional derivatives will be first presented as follows. Definition 2.33. Let 1 > α > 0. The Liouville–Sonine–Caputo–Feller-type fractional derivative is defined as follows:

136

General Fractional Derivatives With Applications in Viscoelasticity

α F C DR,ϑ f

(t)   αf αf = − H+ (ϑ, α) LSC D+ (t) + H− (ϑ, α) LSC D− (t) t f (1) (τ ) 1 = − sin((α+ϑ)π/2) × sin(πϑ) (1−α) (τ −t)α dτ

(2.240)

−∞

− sin((α−ϑ)π/2) sin(πϑ)

×

∞ f (1) (τ )

1 (1−α)

(t−τ )α dτ .

t

Definition 2.34. Let 1 > α > 0. The Liouville–Sonine–Caputo–Feller-type fractional derivative on a bounded domain is defined as follows: α F C D[a,b],ϑ f

(t)   α f α f = − H+ (ϑ, α) LSC Da+ (t) + H− (ϑ, α) LSC Db− (t) t f (1) (τ ) 1 = − sin((α+ϑ)π/2) × (1−α) sin(πϑ) (τ −t)α dτ

× − sin((α−ϑ)π/2) sin(πϑ)

1 (1−α)

b t

(2.241)

a

f (1) (τ ) (t−τ )α dτ .

Property 2.45. Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.242)

Then and α F C DR,0 f

(t) = RzC DRα f (t) .

(2.243)

Proof. By definition, one has α F C DR,0 f

(t)   αf αf = − H+ (0, α) LSC D+ (t) + H− (0, α) LSC D− (t) = − LSC

α f (t)+ α D+ LSC D− f (t) 2 cos(απ/2)

(2.244)

= RzC DRα f (t) . This completes the proof. Property 2.46. Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.245)

Fractional derivatives with singular kernels

137

Then α F C DR,0 f

(t) = RzC DRα f (t) .

(2.246)

Proof. According the definition, one has α F C D[a,b],0 f

(t)   α f α f = − H+ (0, α) LSC Da+ (t) + H− (0, α) LSC Db− (t)

=−

α α LSC Da+ f (t)+LSC Db− f (t)

(2.247)

2 cos(απ/2)

α = RzC D[a,b] f (t) ,

completing the proof.

2.14

Herrmann fractional calculus

In 2012, based on the Feller-type fractional calculus, Herrmann introduced another fractional integral and derivative (see [86]), which is called the Herrmann fractional calculus (also called the Richard fractional calculus) in the honor of Herrmann [25].

2.14.1 Herrmann fractional calculus Definition 2.35 ([25,86]). Let 1 > α > 0 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.248)

The Herrmann fractional integral is defined as α H IR f

(t)

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



 (t) 2 sin(απ/2)

α α L I+ f +L I− f

=

1 2 sin(πα/2)(α)

=

1 2(α) sin(πα/2)



× ∞ −∞

t

−∞

f (τ ) dτ (τ −t)1−α

f (τ ) dτ , |τ −t|1−α

where α L I+ f

1 (t) =  (α)

t a

f (τ ) (τ − t)1−α

dτ

+

∞ t

 f (τ ) dτ (t−τ )1−α

(2.249)

138

General Fractional Derivatives With Applications in Viscoelasticity

and

α L I− f

1 (t) =  (α)

b

f (τ ) (t − τ )1−α

t

dτ .

Definition 2.36 ([25,86]). The Herrmann fractional derivative is defined as α H DR f

(t)   αf αf = − H+ (1, α) L D+ (t) + H− (1, α) L D− (t) =

= =



α α L D+ f +L D− f



(t)



2 sin(απ/2)

1 2 cos(πα/2)(1−α)

×

1 2 cos(πα/2)(1−α)

×

d dt

t

f (τ ) (τ −t)α dτ

−∞ ∞  f (τ ) d dt |τ −t|α dτ , −∞

−

d dt

∞ t



(2.250)

f (τ ) (t−τ )α dτ

where

α L D+ f

1 d (t) =  (1 − α) dt

t −∞

f (τ ) dτ (τ − t)α

and α L D− f

1 d (t) = −  (1 − α) dt

∞ t

f (τ ) dτ . (t − τ )α

Property 2.47 ([25,86]). Let 1 > α > 0 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.251)

Then α F IR,1 f

(t) = H IRα f (t)

(2.252)

and α F DR,1 f

(t) = H DRα f (t) .

(2.253)

Fractional derivatives with singular kernels

139

Proof. According the definition of the Feller fractional calculus, one has α F IR,1 f

(t)

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



 (t) 2 sin(πα/2)

(2.254)

α α L I+ f +L I− f

= H IRα f (t) and α F DR,1 f

(t)   αf αf = − H+ (1, α) L D+ (t) + H− (1, α) L D− (t) = −L

(2.255)

α f (t)+ D α f (t) D+ L − 2 sin(απ/2)

= H DRα f (t) , completing the proof.

2.14.2 Herrmann-type fractional calculus Based on the Herrmann fractional calculus, we will present the Herrmann-type fractional derivative and integral, which is called the Herrmann-type fractional calculus. Definition 2.37. Let 1 > α > 0 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.256)

The Herrmann-type fractional integral is defined as α H I[a,b] f

(t)

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

= = =

α α L Ia+ f +L Ib− f



(t)



2 sin(απ/2) 1 2 sin(πα/2)(α) 1 2(α) sin(πα/2)

×

t a

b a

f (τ ) dτ (τ −t)1−α

+

t

f (τ ) dτ , |τ −t|1−α

where α L Ia+ f

1 (t) =  (α)

t a

f (τ ) (τ − t)1−α

b

dτ

 f (τ ) dτ (t−τ )1−α

(2.257)

140

General Fractional Derivatives With Applications in Viscoelasticity

and

α L Ib− f

1 (t) =  (α)

b t

f (τ ) (t − τ )1−α

dτ .

Definition 2.38. The Herrmann-type fractional derivative is defined as α H D[a,b] f

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

= = =

α α L Da+ f +L Db− f



(t)



2 sin(απ/2) 1 2 cos(πα/2)(1−α) 1 2 cos(πα/2)(1−α)

× ×

t

d dt d dt

f (τ ) (τ −t)α dτ

a

b a

−

d dt

b t



(2.258)

f (τ ) (t−τ )α dτ

f (τ ) |τ −t|α dτ ,

where

α L Da+ f

d 1 (t) =  (1 − α) dt

t

f (τ ) dτ (τ − t)α

a

and

α L Db− f

d 1 (t) = −  (1 − α) dt

b t

f (τ ) dτ . (t − τ )α

Property 2.48. Let 1 > α > 0 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.259)

Then α F I[a,b],1 f

α f (t) (t) = H I[a,b]

(2.260)

and α F D[a,b],1 f

α f (t) . (t) = H D[a,b]

(2.261)

Fractional derivatives with singular kernels

141

Proof. According the definition of the Feller-type fractional calculus, one has α F I[a,b],1 f

(t)

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

=

α α L Ia+ f +L Ib− f



(2.262)

(t)

2 sin(πα/2)

α f = H I[a,b] (t)

and α F D[a,b],1 f

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

α α L Da+ f (t)+L Db− f (t)

(2.263)

2 sin(απ/2)

α = H D[a,b] f

(t) ,

finishing the proof.

2.14.3 Liouville–Sonine–Caputo–Herrmann-type fractional derivatives Based on the Herrmann and Liouville–Sonine–Caputo fractional derivatives, the Liouville–Sonine–Caputo–Herrmann-type fractional derivative will be first proposed as follows. Definition 2.39. The Liouville–Sonine–Caputo–Herrmann-type fractional derivative is defined as α H C DR f

(t)   αf αf = − H+ (1, α) LSC D+ (t) + H− (1, α) LSC D− (t) =

= =



α α LSC D+ f +LSC D− f



(t)



2 sin(απ/2)

1 2 cos(πα/2)(1−α)

×

t

f (1) (τ ) (τ −t)α dτ

−∞ −∞  f (1) (τ ) 1 2 cos(πα/2)(1−α) × |τ −t|α dτ , −∞

−

∞ f (1) (τ ) t

where α LSC D+ f

1 (t) =  (1 − α)

t −∞

f (1) (τ ) dτ (τ − t)α

(t−τ )α dτ



(2.264)

142

General Fractional Derivatives With Applications in Viscoelasticity

and α LSC D− f (t) = −

1  (1 − α)

∞ t

f (1) (τ ) dτ . (t − τ )α

It follows that α H C DR f

(t)   αf αf = − H+ (1, α) LSC D+ (t) + H− (1, α) LSC D− (t) =



α α LSC D+ f +LSC D− f



(t)

(2.265)

2 sin(απ/2)

α f = F C DR,1 (t) .

Based on Herrmann’s ideas, we have the following: Definition 2.40. The Liouville–Sonine–Caputo–Herrmann-type fractional derivative on a bounded domain is defined as α F C D[a,b] f (t)   α α LSC Da+ f +LSC Db− f (t) = 2 sin(απ/2) 

=

1 2 cos(πα/2)(1−α)

=

1 2 cos(πα/2)(1−α)

t

×

a

×

b a

f (1) (τ ) (τ −t)α dτ

−

b t

 f (1) (τ ) (t−τ )α dτ

(2.266)

f (1) (τ ) |τ −t|α dτ ,

where α LSC Da+ f

1 (t) =  (1 − α)

t

f (1) (τ ) dτ (τ − t)α

a

and α LSC Db− f (t) = −

1  (1 − α)

b t

f (1) (τ ) dτ . (t − τ )α

Similarly, α H C D[a,b] f

(t)   αf α f = − Ha+ (1, α) LSC D+ (t) + H− (1, α) LSC Db− (t) 

=

α α LSC Da+ f +LSC Db− f

2 sin(απ/2)

α = F C D[a,b],1 f

(t) .



(t)

(2.267)

Fractional derivatives with singular kernels

143

2.15 Samko–Kilbas–Marichev symmetric fractional difference derivative In 1993, in view of the Grünwald’s, Letnikov’s, Riesz’s works, Samko, Kilbas and Marichev proposed the symmetric fractional difference derivative, which is called the Grünwald–Letnikov–Riesz fractional derivative [44]. In our book [25], we call it as the Samko–Kilbas–Marichev symmetric fractional difference derivative in the honor of Samko, Kilbas, and Marichev. Definition 2.41 ([25,44]). Let 2 > α > 1. The Samko–Kilbas–Marichev symmetric fractional difference derivative is defined as follows: α T a Dt f

= =

(t)



α f (t) αh f (t) + lim −h α α |h| h→+0 h→+0 |h|   t f (t)−f (τ ) ∞ f (t)−f (τ ) α dτ + dτ . 2(1−α) cos(απ/2) (τ −t)1+α (t−τ )1+α t −∞ 1 2 cos(απ/2)

lim

(2.268)

It follows from Eq. (2.268) that [25,44] α T a Dt f

(t)

=

α 2(1−α) cos(απ/2)

=

α 2(1−α) cos(απ/2)

 ×

t

f (t)−f (τ ) dτ (τ −t)1+α

∞ f (t)−f (τ )

+ dτ (t−τ )1+α t −∞  ∞ 2f (t)−f (t+τ )−f (t−τ ) × dτ . τ 1+α

 (2.269)

0

With the aid of sin (απ ) α =  (1 + α)  (1 − α) π

(2.270)

sin (απ) = 2 sin (απ/2) cos (απ/2)

(2.271)

and

(see [25,44]), the Samko–Kilbas–Marichev fractional difference derivative can be expressed in the form α T a Dt f

(t)

= − 2(1−α)αcos(απ/2) =

∞ f (τ )−f (t+τ )

dτ

τ 1+α 0 ∞ (1+α) sin(απ/2)  f (t+τ )−f (τ ) dτ . π τ 1+α 0

(2.272)

144

General Fractional Derivatives With Applications in Viscoelasticity

2.16 Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative In 2018, based on the Grünwald’s, Letnikov’s, and Herrmann’s research, Yang proposed the symmetric fractional difference derivative, which is called the Grünwald– Letnikov–Herrmann-type symmetric fractional difference derivative for the first time [25]. Definition 2.42 ([25]). The Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative is defined as follows:

 α−h f (t) αh f (t) 1 α + lim × lim . (2.273) L Dt f (t) = h→+0 |h|α h→+0 |h|α 2 sin (απ/2) It follows from Eq. (2.273) that α L Dt f

=

(t)

α 2(1−α) sin(απ/2)

 ×

∞ f (τ )−f (t−τ ) 0

τ 1+α

dτ +

∞ f (τ )−f (t+τ ) 0

τ 1+α



(2.274)

dτ

which leads to α L Dt f

α × (t) = 2 (1 − α) sin (απ/2)

∞

f (τ ) − f (t + τ ) dτ τ 1+α

(2.275)

0

and α L Dt f

 (1 + α) cos (απ/2) (t) = π

∞

f (t + τ ) − f (τ ) dτ . τ 1+α

(2.276)

0

2.17 Grünwald–Letnikov–Feller-type symmetric fractional difference derivative In 2018, based on the Grünwald’s, Letnikov’s, and Feller’s works, Yang proposed the symmetric fractional difference derivative, which is called the Grünwald–Letnikov– Feller-type symmetric fractional difference derivative for the first time [25]. Definition 2.43 ([25]). The Grünwald–Letnikov–Feller-type symmetric fractional difference derivative is defined as follows: α F V Dt;ϑ f

α−h f (t) αh f (t) + H− (ϑ, α) × lim , (2.277) α h→+0 |h| h→+0 |h|α

(t) = H+ (ϑ, α) × lim

Fractional derivatives with singular kernels

145

where the two components are H− (ϑ, α) =

sin ((α − ϑ) π/2) sin (πϑ)

(2.278)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.279)

and

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

(2.280)

one has α F V Dt;ϑ f

(t)

= H+ (ϑ, α) ×

α (1−α)

×

+H− (ϑ, α) ×

α (1−α)

×

t

f (t)−f (τ ) dτ (τ −t)1+α −∞ ∞ f (t)−f (τ ) dτ (t−τ )1+α t

(2.281)

which leads to [25] α F V Dt;ϑ f

=

(t)

sin((α+ϑ)π/2) sin(πϑ)

×

α (1−α)

×

× + sin((α−ϑ)π/2) sin(πϑ)

α (1−α)

×

t

f (t)−f (τ ) dτ (τ −t)1+α

−∞ ∞ f (t)−f (τ ) dτ . (t−τ )1+α t

(2.282)

It follows from Eq. (2.278) that [25] α F V Dt;ϑ f

(t)

sin((α+ϑ)π/2) sin(πϑ)

×

(1+α) sin(πα) π

×

× + sin((α−ϑ)π/2) sin(πϑ)

(1+α) sin(πα) π

×

=

t

f (t)−f (τ ) dτ (τ −t)1+α

(2.283)

f (t)−f (τ ) dτ (τ −t)1+α

(2.284)

−∞ ∞ f (t)−f (τ ) dτ , (t−τ )1+α t

which leads to α F V Dt;ϑ f

(t) ×

(1+α) sin((α+ϑ)π/2) π

×

sin(πα) × + sin(πϑ)

(1+α) sin((α−ϑ)π/2) π

×

=

sin(πα) sin(πϑ)

t

−∞ ∞ f (t)−f (τ ) dτ. (t−τ )1+α t

146

General Fractional Derivatives With Applications in Viscoelasticity

Property 2.49. Let ϑ = 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.285)

Then α F V Dt;0 f

(t) = T a Dτα f (t) .

(2.286)

Proof. According to the definition of the Grünwald–Letnikov–Feller-type symmetric fractional difference derivative, one has α F V Dt;0 f

(t) αh f (t) |h|α

= H+ (0, α) × lim h→+0

1 = 2 cos(απ/2) × lim

h→+0

+ H− (0, α) × lim h→+0  α−h f (t) αh f (t) |h|α + lim |h|α

α−h f (t) |h|α

(2.287)

h→+0

= T a Dtα f (t) , finishing the proof. Property 2.50. Let ϑ = 1 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.288)

Then α F V Dt;1 f

(t) = L Dtα f (t) .

(2.289)

Proof. In a similar way, making use of the definition of the Grünwald–Letnikov– Feller-type symmetric fractional difference derivative, one has α F V Dt;1 f

(t) α f (t)

α f (t)

h = H+ (1, α) × lim |h| + H− (1, α) × lim −h α |h|α h→+0 h→+0 

α α  f (t) h f (t) 1 = 2 sin(απ/2) × lim |h| + lim −h α |h|α h→+0 h→+0   t f (t)−f (τ ) ∞ f (t)−f (τ ) α = 2(1−α) sin(απ/2) 1+α dτ + 1+α dτ

−∞ ∞

(τ −t)

f (τ )−f (t+τ ) dτ τ 1+α 0 ∞ f (t+τ )−f (τ ) = (1+α) cos(απ/2) dτ π τ 1+α 0 = L Dtα f (t) ,

=

α 2(1−α) sin(απ/2)

completing the proof.

×

t

(t−τ )

(2.290)

Fractional derivatives with singular kernels

147

2.18 Samko–Kilbas–Marichev symmetric fractional difference derivative on a bounded domain In 2018, in view of the Grünwald’s, Letnikov’s, Riesz’s research, Yang proposed the symmetric fractional difference derivative on a bounded domain [a, b] [25]. Definition 2.44 ([25]). Let 2 > α > 1. The Samko–Kilbas–Marichev symmetric fractional difference derivative in the interval [a, b] is defined as follows: 1 2 cos (απ/2)

α T a Dt;[a,b] f (t) =

lim

αh,+a f (t)

h→+0

|h|α

α−h,−b f (t)

+ lim

|h|α

h→+0

 . (2.291)

It follows from Eq. (2.291) that [25] α T a Dt;[a,b] f

=

(t)

1 2 cos(απ/2)

×

α (1−α)

×

 t a

f (t)−f (τ ) dτ (τ −t)1+α

+

b t

 f (t)−f (τ ) dτ (t−τ )1+α

(2.292) .

With the aid of sin (απ ) α =  (1 + α)  (1 − α) π

(2.293)

sin (απ) = 2 sin (απ/2) cos (απ/2)

(2.294)

and

(see [25]), the Samko–Kilbas–Marichev fractional difference derivative can be expressed in the form: α T a Dt;[a,b] f

(t)

= (1+α) πsin(απ/2) ×



t a

f (t)−f (τ ) dτ (τ −t)1+α

+

b t

 f (t)−f (τ ) dτ (t−τ )1+α

(2.295) .

2.19 Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative on a bounded domain In 2018, based on the Grünwald’s, Letnikov’s, and Herrmann’s works, Yang proposed the symmetric fractional difference derivative on a bounded domain for the first time [25].

148

General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.45 ([25]). The Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative in the interval [a, b] is defined as follows: α L Dt;[a,b] f

1 × (t) = 2 sin (απ/2)

lim

αh,+a f (t)

h→+0

|h|α

+ lim

α−h,−b f (t)



|h|α

h→+0

. (2.296)

It follows from Eq. (2.296) that α L Dt;[a,b] f

=

(t)

1 2 sin(απ/2)

×

 α (1−α)

×

t a

f (t)−f (τ ) dτ (τ −t)1+α

+

b t



(2.297)

f (t)−f (τ ) dτ (t−τ )1+α

which leads to α L Dt;[a,b] f

=

(t)

(1+α) cos(απ/2) π

 ×

t a

f (t)−f (τ ) dτ (τ −t)1+α

+

b t

 f (t)−f (τ ) dτ (t−τ )1+α

(2.298) .

2.20 Grünwald–Letnikov–Feller-type symmetric fractional difference derivative on a bounded domain In 2018, based on the Grünwald’s, Letnikov’s, and Feller’s works, Yang proposed the symmetric fractional difference derivative on a bounded domain for the first time [25]. Definition 2.46 ([25]). The Grünwald–Letnikov–Feller-type symmetric fractional difference derivative in the interval [a, b] is defined as follows: α F V Dt;[a,b];ϑ f (t) = H+ (ϑ, α)× lim

h→+0

αh,+a f (t) |h|α

+H− (ϑ, α)× lim

h→+0

α−h,−b f (t)

, |h|α (2.299)

where the two components are H− (ϑ, α) =

sin ((α − ϑ) π/2) sin (πϑ)

(2.300)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) . sin (πϑ)

(2.301)

and

Fractional derivatives with singular kernels

149

In a similar way, making use of sin (απ ) α =  (1 + α) ,  (1 − α) π

(2.302)

one has α F V Dt;[a,b];ϑ f

(t)

= H+ (ϑ, α) ×

α (1−α)

×

+H− (ϑ, α) ×

α (1−α)

×

t

a b t

f (t)−f (τ ) dτ (τ −t)1+α

(2.303)

f (t)−f (τ ) dτ (t−τ )1+α

which leads to [25] α F V Dt;[a,b];ϑ f

=

(t)

sin((α+ϑ)π/2) sin(πϑ)

×

× + sin((α−ϑ)π/2) sin(πϑ)

α (1−α)

×

α (1−α)

×

t

a b t

f (t)−f (τ ) dτ (τ −t)1+α

(2.304)

f (t)−f (τ ) dτ . (t−τ )1+α

It follows from Eq. (2.278) that α F V Dt;ϑ f

=

(t)

sin((α+ϑ)π/2) sin(πϑ)

×

× + sin((α−ϑ)π/2) sin(πϑ)

(1+α) sin(πα) π

×

(1+α) sin(πα) π

×

t

a b t

f (t)−f (τ ) dτ (τ −t)1+α

(2.305)

f (t)−f (τ ) dτ , (t−τ )1+α

which leads to α F V Dt;ϑ f

=

sin(πα) sin(πϑ)

(t) ×

sin(πα) × + sin(πϑ)

(1+α) sin((α+ϑ)π/2) π

×

(1+α) sin((α−ϑ)π/2) π

×

t

a b t

f (t)−f (τ ) dτ (τ −t)1+α

(2.306)

f (t)−f (τ ) dτ . (t−τ )1+α

Property 2.51. Let ϑ = 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.307)

Then α F V Dt;[a,b];0 f

α f (t) . (t) = T a Dt;[a,b]

(2.308)

150

General Fractional Derivatives With Applications in Viscoelasticity

Proof. According to the definition of the Grünwald–Letnikov–Feller-type symmetric fractional difference derivative on a bounded domain, one has α F V Dt;[a,b];0 f

(t) αh,+a f (t) |h|α

α−h,−b f (t) α h→+0 |h| α α f (t) h,+a f (t)  + lim −h,−b |h|α |h|α h→+0 h→+0

= H+ (0, α) × lim h→+0

1 = 2 cos(απ/2) × lim

+ H− (0, α) × lim

(2.309)

α = T a Dt;[a,b] f (t) ,

finishing the proof. Property 2.52. Let ϑ = 1 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.310)

Then α F V Dt;[a,b];1 f

α f (t) . (t) = L Dt;[a,b]

(2.311)

Proof. In a similar way, making use of the definition of the Grünwald–Letnikov– Feller-type symmetric fractional difference derivative on a bounded domain, one has α F V Dt;[a,b];1 f

(t) α

α

f (t)

f (t)

= H+ (1, α) × lim h,+a + H− (1, α) × lim −h,−b α |h|α h→+0 h→+0 |h|

α α   f (t) f (t) 1 = 2 sin(απ/2) × lim h,+a + lim −h,−b |h|α |h|α h→+0 h→+0   t f (t)−f (τ ) b f (t)−f (τ ) α = 2(1−α) sin(απ/2) 1+α dτ + 1+α dτ a

(τ −t)

t

(2.312)

(t−τ )

α = L Dt;[a,b] f (t) ,

completing the proof.

2.21

Erdelyi–Kober-type calculus

In fact, the Erdelyi–Kober-type operators of fractional integration and fractional derivative are extensions of the Riemann–Liouville fractional calculus (see [25,44, 45,89,90]).

Fractional derivatives with singular kernels

151

2.21.1 Erdelyi–Kober-type fractional integrals Let  = (a, b) (−∞ ≤ a < b ≤ ∞) be a finite interval on the real axis R. Definition 2.47 ([25,44,45]). Let α > 0, σ > 0, t > 0, and η ∈ R. The left-sided Erdelyi–Kober-type fractional integral is defined as α Ia+;σ,η f

t

σ t −σ (α+η) (t) =  (α)

τ σ (η+1)−1 f (τ ) (t σ − τ σ )1−α

a

dτ ,

(2.313)

and the right-sided Erdelyi–Kober-type fractional integral is given as α f Ib−;σ,η

σ t −σ η (t) =  (α)

b

τ σ (1−α−η)−1 f (τ ) (τ σ − t σ )1−α

t

dτ .

(2.314)

Let a = −∞ and b = +∞. Definition 2.48 ([25,44,45]). The left-sided Erdelyi–Kober-type fractional integral is defined as α E I+;σ,η f

σ t −σ (α+η) (t) =  (α)

t −∞

τ σ (η+1)−1 f (τ ) (t σ − τ σ )1−α

dτ ,

(2.315)

and the right-sided Erdelyi–Kober-type fractional integral is given as α E I−;σ,η f

σ t −σ η (t) =  (α)

∞

τ σ (1−α−η)−1 f (τ ) (τ σ − t σ )1−α

t

dτ .

(2.316)

For a special case, for σ = 1, the left-sided Erdelyi–Kober-type fractional integral can be written as [25,44,45] α f Ia+;1,η

t −(α+η) (t) =  (α)

t a

τ (η+1)−1 f (τ ) (t − τ )1−α

dτ,

(2.317)

and the right-sided Erdelyi–Kober-type fractional integral as α f Ib−;1,η

t −η (t) =  (α)

b

τ −(α+η) f (τ ) (τ − t)1−α

t

dτ .

(2.318)

Taking σ = 1, a = −∞ and b = ∞, one has [25,44,45] α f I0+;1,η

t −(α+η) (t) =  (α)

t 0

τ η f (τ ) (t − τ )1−α

dτ

(2.319)

152

General Fractional Derivatives With Applications in Viscoelasticity

and α f I−;1,η

t −η (t) =  (α)

∞

τ −(α+η) f (τ ) (τ − t)1−α

t

(2.320)

dτ .

Taking η = 0, we get [25,44,45] α f (t) = Ia+;σ,0

t

σ t −σ α  (α)

τ σ −1 f (τ ) (t σ − τ σ )1−α

a

dτ

(2.321)

and α f Ib−;σ,0

σ (t) =  (α)

b

τ σ (1−α)−1 f (τ ) (τ σ − t σ )1−α

t

dτ .

(2.322)

Taking η = 0, a = −∞ and b = −∞, we obtain [25,44,45] α Y I+;σ,0 f

t

σ t −σ α (t) =  (α)

τ σ −1 f (τ )

−∞

(t σ − τ σ )1−α

dτ

(2.323)

and α Y I−;σ,0 f

∞

σ (t) :=  (α)

τ σ (1−α)−1 f (τ ) (τ σ − t σ )1−α

t

dτ .

(2.324)

Taking η = 0 and σ = 1, we have [25,44,45] α Ia+;1,0 f

t −α (t) =  (α)

t a

f (τ ) (t − τ )1−α

dτ

(2.325)

dτ .

(2.326)

and α Ib−;1,0 f

1 (t) =  (α)

b

τ −α f (τ ) (τ − t)1−α

t

Taking η = 0, σ = 1, a = −∞ and b = −∞ yields [25,44,45] α L I+;1,0 f

t −α (t) =  (α)

t −∞

f (τ ) (t − τ )1−α

dτ

(2.327)

Fractional derivatives with singular kernels

153

and α L I−;1,0 f

1 (t) =  (α)

∞ t

τ −α f (τ ) (τ − t)1−α

dτ .

(2.328)

2.21.2 Erdelyi–Kober-type fractional derivatives Let  = (a, b) (−∞ ≤ a < b ≤ ∞) be a finite interval on the real axis R. Definition 2.49 ([25,44,45]). Let α > 0, κ = [α] + 1, σ > 0, t > 0, η ∈ R, and D = d/dt. The left-sided Erdelyi–Kober-type fractional derivative is defined as κ



1 α σ (α+η) α f (t) = t −σ η D t f (2.329) I Da+;σ,η (t) , a+;σ,η+α σ t σ −1 and the right-sided Erdelyi–Kober-type fractional derivative is given as κ



1 α σ (κ+η−α) α f (t) = t −σ (η+α) D t f I Db−;σ,η (t) . b−;σ,η+α−κ σ t σ −1

(2.330)

Let α > 0, κ = [α] + 1, σ = 1, t > 0, η ∈ R, and D = d/dt. Definition 2.50 ([25,44,45]). The left-sided Erdelyi–Kober-type fractional derivative is defined as κ



α −η 1 α f (t) (2.331) D t α+η Ia+;1,η+α Da+;1,η f (t) = t σ and the left-sided Erdelyi–Kober-type fractional derivative is given as κ



α −(η+α) 1 α f (t) . D t κ+η−α Ib−;1,η+α−κ Db−;1,η f (t) = t σ

(2.332)

Let α > 0, κ = [α] + 1, σ = 1, t > 0, D = d/dt and η ∈ R. Definition 2.51 ([25,44,45]). The left-sided Erdelyi–Kober-type fractional derivative is defined as κ



α −η 1 α D+;1,η f (t) = t f (t) (2.333) D t α+η I+;1,η+α σ and the right-sided Erdelyi–Kober-type fractional derivative is κ



α −(η+α) 1 α f (t) . D t κ+η−α I−;1,η+α−κ D−;1,η f (t) = t σ In particular, we have the following results [25,44,45]:

(2.334)

154

General Fractional Derivatives With Applications in Viscoelasticity

(1) Suppose that κ = [α] + 1, a = −∞ and b = −∞, then κ



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

(t) = t

−σ (η+α)

κ

1 σt

D σ −1



α t σ (κ+η−α) I−;σ,η+α−κ f (t) . (2.336)

(2) Suppose that κ = [α] + 1, η = 0, then

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

κ

1 σt

D σ −1

  α t σ (κ−α) Ib−;σ,α−κ f (t) .

(3) Suppose that κ = [α] + 1 and η = 0, then

κ  α  1 α D+;σ,0 f (t) = D t σ α I+;σ,α f (t) σ t σ −1 and α D−;σ,0 f (t) = t −σ α

1 σt

κ

D σ −1

 α  t σ (κ−α) I−;σ,α−κ f (t) .

(4) When 0 < Re (α) < 1 and η = 0, the following relations hold:

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

(t) = t

−σ α

1 σ t σ −1

(2.335)

(2.337)

(2.338)

(2.339)

(2.340)

(2.341)



 α  D t σ (1−α) Ib−;σ,α−1 f (t) .

(2.342)

Property 2.53 ([25,44,45]). Suppose that α > 0 and −∞ ≤ a < b ≤ ∞, then 

α α Ia+;σ,η f (t) = f (t) , (2.343) Da+;σ,η  α α Ib−;σ,η f (t) = f (t) , Db−;σ,η 

α α I+;σ,η f (t) = f (t) , D+;σ,η 

α α I−;σ,η f (t) = f (t) . D−;σ,η

(2.344) (2.345) (2.346)

Fractional derivatives with singular kernels

155

2.21.3 Erdelyi–Kober–Riesz-type fractional difference derivative on the real line In 2018, based on the Riesz’s, Erdelyi’s and Kober’s results, Yang proposed the definition of the Erdelyi–Kober–Riesz-type fractional integral and derivative on the real line R (see [25]). Definition 2.52 ([25]). The Erdelyi–Kober–Riesz-type fractional integral on the real line R is defined as α Rw IR f



1 α α f (t) + E I−;σ,η f (t) . × E I+;σ,η 2 cos (πα/2)

(t) =

(2.347)

It follows that [25] α Rw IR f

=

(t)



σ t −σ η 2 cos(πα/2)(α)

t −σ α

×

t −∞

τ σ (η+1)−1 f (τ ) dτ (t σ −τ σ )1−α

+

∞ τ σ (1−α−η)−1 f (τ ) t

(τ σ −t σ )1−α



(2.348)

dτ .

Definition 2.53 ([25]). The Erdelyi–Kober–Riesz-type fractional derivative on the real line R is defined as α Rw DR f

(t) =



1 α α f (t) + E D−;σ,η f (t) . × E D+;σ,η 2 cos (πα/2)

(2.349)

It follows from Eq. (2.349) that α Rw DR f

(t)

 κ



σ (α+η) I α −(η+α) 1 D t f + t (t) +;σ,η+α 2 cos(πα/2) σ κ  



1 α −σ α σ (κ+η−α) +t D t I−;σ,η+α−κ f (t) . σ t σ −1

=

t −σ η

×



1

σ t σ −1

κ

D

(2.350)

2.21.4 Erdelyi–Kober–Herrmann-type fractional difference derivative on the real line In 2018, based on the Herrmann’s, Erdelyi’s and Kober’s research, Yang proposed the definition of the Erdelyi–Kober–Herrmann-type fractional derivative on the real line R (see [25]). Definition 2.54 ([25]). The Erdelyi–Kober–Herrmann-type fractional derivative on the real line R is defined as α Rv DR f (t) =



1 α α f (t) + E I−;σ,η f (t) . × E I+;σ,η 2 sin (πα/2)

(2.351)

156

General Fractional Derivatives With Applications in Viscoelasticity

It follows from Eq. (2.351) that α Rv DR f

=

(t) =

σ t −σ η 2 sin(πα/2)(α)

 × t −σ α

t −∞

τ σ (η+1)−1 f (τ ) dτ (t σ −τ σ )1−α

+

∞ τ σ (1−α−η)−1 f (τ ) t

(τ σ −t σ )1−α



(2.352)

dτ .

2.21.5 Erdelyi–Kober–Feller-type fractional difference derivative on the real line In 2018, based on the Feller’s, Erdelyi’s and Kober’s results, Yang proposed the definition of the Erdelyi–Kober–Feller-type fractional integral and derivative on the real line R (see [25]). Definition 2.55 ([25]). The Erdelyi–Kober–Feller-type fractional derivative on the real line R is defined as α Ro DR;ϑ f

α α f (t) + H− (ϑ, α) × E D+;σ,η f (t) . (t) = H+ (ϑ, α) × E I−;σ,η

(2.353)

It follows from Eq. (2.353) that  κ 



1 α f −σ η × H (ϑ, α) σ (α+η) I α D = t D t f (t) (t) Ro R;ϑ + a+;σ,η+α σ t σ −1 κ  



α f (t) , +H− (ϑ, α) × t −σ α σ t σ1−1 D t σ (κ+η−α) Ib−;σ,η+α−κ

(2.354)

which leads to α Ro DR;ϑ f

(t)  κ 



1 σ (α+η) I α = −t −σ η × sin((α+ϑ)π/2) D t f (t) + +;σ,η+α sin(πϑ) σ t σ −1 κ  



1 −σ α σ (κ+η−α) I α × t D t f + sin((α−ϑ)π/2) (t) . σ −1 −;σ,η+α−κ sin(πϑ) σt

(2.355)

Property 2.54. Let ϑ = 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.356)

Then α Ro DR;0 f

(t) = Rw IRα f (t) .

(2.357)

Fractional derivatives with singular kernels

157

Proof. According to the definition of the Erdelyi–Kober–Feller-type fractional derivative on the real line, one has α Ro DR;0 f

(t)

α α f (t) + H− (0, α) × E D+;σ,η f (t) = H+ (0, α) × E I−;σ,η

 1 α α = 2 cos(πα/2) × E I+;σ,η f (t) + E I−;σ,η f (t)

(2.358)

= Rw IRα f (t) , which completes the proof. Property 2.55. Let ϑ = 1 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.359)

Then α Ro DR;1 f

(t) = Rv IRα f (t) .

(2.360)

Proof. In a similar way, making use of the Erdelyi–Kober–Feller-type fractional derivative on the real line, one has α Ro DR;1 f

(t)

α α = H+ (1, α) × E I−;σ,η f (t) + H− (1, α) × E D+;σ,η f (t) 

1 α α × E I+;σ,η f (t) + E I−;σ,η f (t) = 2 sin(πα/2)

(2.361)

= Rv IRα f (t) , which completes the proof.

2.21.6 Erdelyi–Kober–Riesz-type fractional difference derivative on a bounded domain Based on the Riesz’s, Erdelyi’s and Kober’s research, we will propose the definitions of the Erdelyi–Kober–Riesz-type fractional derivative and integral in the interval [a, b]. Definition 2.56. The Erdelyi–Kober–Riesz-type fractional integral in the interval [a, b] is defined as α Rw I[a,b] f (t) =



1 α α × E Ia+;σ,η f (t) + E Ib−;σ,η f (t) . 2 cos (πα/2)

(2.362)

158

General Fractional Derivatives With Applications in Viscoelasticity

It follows that α Rw I[a,b] f

=

(t)



σ t −σ η 2 cos(πα/2)(α)

t −σ α

×

t a

τ σ (η+1)−1 f (τ ) dτ (t σ −τ σ )1−α

+

b t

 τ σ (1−α−η)−1 f (τ ) dτ (τ σ −t σ )1−α

(2.363) .

The Erdelyi–Kober–Riesz-type fractional derivative in the interval [a, b] is defined as α Rw D[a,b] f

(t) =



1 α α f (t) + E Db−;σ,η f (t) . × E Da+;σ,η 2 cos (πα/2)

(2.364)

It follows from Eq. (2.364) that α Rw D[a,b] f

(t)

 κ



σ (α+η) I α −(η+α) 1 D t f + t (t) a+;σ,η+α 2 cos(πα/2) σ κ  



1 α +t −σ α σ t σ −1 D t σ (κ+η−α) Ib−;σ,η+α−κ f (t) . t −σ η

=

×



1

κ

σ t σ −1

D

(2.365)

2.21.7 Erdelyi–Kober–Herrmann-type fractional difference derivative on a bounded domain Based on the Herrmann’s, Erdelyi’s and Kober’s results, we will give the definition of the Erdelyi–Kober–Herrmann-type fractional derivative on a bounded domain. Definition 2.57. The Erdelyi–Kober–Herrmann-type fractional derivative in the interval [a, b] is defined as α Rv D[a,b] f

(t) =



1 α α f (t) + E Ib−;σ,η f (t) . × E Ia+;σ,η 2 sin (πα/2)

(2.366)

It follows from Eq. (2.366) that α Rv D[a,b] f

=

(t)

σ t −σ η 2 sin(πα/2)(α)

 ×

t −σ α

t a

τ σ (η+1)−1 f (τ ) dτ (t σ −τ σ )1−α

+

b t

 τ σ (1−α−η)−1 f (τ ) dτ (τ σ −t σ )1−α

(2.367) .

2.21.8 Erdelyi–Kober–Feller-type fractional difference derivative on a bounded domain Based on the Feller’s, Erdelyi’s and Kober’s results, we will present the definition of the Erdelyi–Kober–Feller-type fractional derivative on a bounded domain.

Fractional derivatives with singular kernels

159

Definition 2.58. The Erdelyi–Kober–Feller-type fractional derivative on the interval [a, b] is defined as α Ro D[a,b];ϑ f

α α f (t) + H− (ϑ, α) × E Da+;σ,η f (t) . (2.368) (t) = H+ (ϑ, α) × E Ib−;σ,η

It follows from Eq. (2.368) that  κ 



1 α −σ η × H (ϑ, α) σ (α+η) I α D t f (t) Ro D[a,b];ϑ f (t) = t + a+;σ,η+α σ t σ −1 κ  



α +H− (ϑ, α) × t −σ α σ t σ1−1 D t σ (κ+η−α) Ib−;σ,η+α−κ f (t) ,

(2.369)

which leads to α Ro D[a,b];ϑ f

= −t −σ η

×

(t) 

sin((α+ϑ)π/2) sin(πϑ)

× t −σ α + sin((α−ϑ)π/2) sin(πϑ)







α t σ (α+η) Ia+;σ,η+α f (t) +  

α t σ (κ+η−α) Ib−;σ,η+α−κ f (t) .

1 D σ t σ −1 κ

1 D σ t σ −1

κ

(2.370)

Property 2.56. Let ϑ = 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.371)

Then α Ro D[a,b];0 f

α f (t) . (t) = Rw I[a,b]

(2.372)

Proof. According to the definition of the Erdelyi–Kober–Feller-type fractional derivative on an interval, one has α Ro D[a,b];0 f

(t)

α α = H+ (0, α) × E Ib−;σ,η f (t) + H− (0, α) × E Da+;σ,η f (t)

 1 α α = 2 cos(πα/2) × E Ia+;σ,η f (t) + E Ib−;σ,η f (t)

(2.373)

α f = Rw I[a,b] (t) ,

completing the proof. Property 2.57. Let ϑ = 1 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.374)

Then α Ro D[a,b];1 f

α f (t) . (t) = Rv I[a,b]

(2.375)

160

General Fractional Derivatives With Applications in Viscoelasticity

Proof. In a similar way, making use of the Erdelyi–Kober–Feller-type fractional derivative on an interval, one has α Ro D[a,b];1 f

(t)

α α = H+ (1, α) × E Ib−;σ,η f (t) + H− (1, α) × E Da+;σ,η f (t)

 1 α α = 2 sin(πα/2) × E Ia+;σ,η f (t) + E Ib−;σ,η f (t)

(2.376)

α f = Rv I[a,b] (t) ,

completing the proof.

2.22

Hadamard fractional calculus

In 1892, Hadamard investigated the fractional integrals and derivatives (see [91]), which are now called the Hadamard fractional calculus [25,44,45].

2.22.1 Hadamard fractional calculus We now introduce the definitions and some properties of the Hadamard fractional integrals and derivatives [25,44,45]. Definition 2.59 ([25,44,45]). Let α > 0. The left-sided Hadamard-type fractional integral is defined as α H Ia+ f

1 (t) =  (α)

 t dτ t σ f (τ ) , log τ τ

(2.377)

a

and the right-sided Hadamard-type fractional integral is given as

α H Ib− f

1 (t) =  (α)

b

log

dτ τ σ f (τ ) . t τ

(2.378)

t

Let α > 0 and δ = tD, where D =

d dt .

Definition 2.60 ([25,44,45]). The left-sided Hadamard fractional derivative is defined as α H Da+ f

 t  1 dτ d κ t σ f (τ ) t , log (t) =  (α) dt τ τ a

(2.379)

Fractional derivatives with singular kernels

161

and the right-sided Hadamard fractional derivative is α H Db− f

 b 1 dτ d κ

τ σ f (τ ) t . log (t) =  (α) dt t τ

(2.380)

t

Property 2.58 ([25,44,45]). Suppose that 0 ≤ a < b ≤ ∞, α > 0 and β > 0, then

α H Ia+

β

t log a

  (1 + β) t β+α−1 = , log  (α + β) a

(2.381)





b β+α−1 b β  (1 + β) = , log log t  (α + β) t



β  (1 + β) t β−α−1 t α = , log H Da+ a  (β − α) a 



b β−α−1 b β  (1 + β) α = . log H Db− log t  (β − α) t α H Ib−

(2.382) (2.383) (2.384)

Property 2.59 ([25,44,45]). Suppose that 0 ≤ a < b ≤ ∞, α > 0 and Lκ (a, b) (1 ≤ κ ≤ ∞), then   α α (2.385) H Da+ H Ia+ (t) = f (t) and 

α α H Db− H Ib−



(t) = f (t) .

(2.386)

2.22.2 Hadamard–Riesz-type fractional difference derivative on a bounded domain Based on the Riesz’s and Hadamard’s investigations, we will propose the definitions of the Hadamard–Riesz-type fractional derivative and integral in the interval [a, b]. Definition 2.61. The Hadamard–Riesz-type fractional integral in the interval [a, b] is defined as α RH u I[a,b] f

(t) =

  α 1 α f (t) + H Ib− f (t) . × H Ia+ 2 cos (πα/2)

(2.387)

It follows that α RH u I[a,b] f

=

(t)

1 2 cos(πα/2)

×

 1 (α)

t 

log

a

 t σ τ

f (τ )

dτ τ

+

1 (α)

b 

log

t

 τ σ t

 f (τ )

dτ τ

(2.388) .

162

General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.62. The Hadamard–Riesz-type fractional derivative in the interval [a, b] is defined as α RH u D[a,b] f

(t) =

  1 α α f (t) + H Db− f (t) . × H Da+ 2 cos (πα/2)

(2.389)

It follows from Eq. (2.389) that α RH u D[a,b] f

=

1 2 cos(πα/2)

(t)



×

 d 2 1 + (α) t dt

1 (α)

b  t

log τt



d t dt

σ

2 t  a

log τt 

σ

f (τ ) dτ τ

(2.390)

f (τ ) dτ . τ

2.22.3 Hadamard–Herrmann-type fractional difference derivative on a bounded domain Based on the Herrmann’s and Hadamard’s research, we will give the definition of the Hadamard–Herrmann-type fractional derivative on a bounded domain. Definition 2.63. The Hadamard–Herrmann-type fractional derivative in the interval [a, b] is defined as α RH v D[a,b] f

(t) =

  1 α α f (t) + H Db− f (t) . × H Da+ 2 sin (πα/2)

(2.391)

It follows from Eq. (2.366) that  α RH v D[a,b] f 1 + (α)



d t dt

(t) =

2 b 

1 2 sin(πα/2)

log

t

 τ σ t

×

f (τ )

1 (α)



dτ τ



d t dt

2 t  a

log τt

σ

f (τ ) dτ τ (2.392)

.

2.22.4 Hadamard–Feller-type fractional difference derivative on a bounded domain Based on the Feller’s and Hadamard’s results, we will present the definition of the Hadamard–Feller-type fractional derivative on the bounded domain. Definition 2.64. The Hadamard–Feller-type fractional derivative on the real line [a, b] is defined as α RH w D[a,b];ϑ f

α α f (t) + H− (ϑ, α) × H Da+ f (t) . (t) = H+ (ϑ, α) × H Db−

(2.393)

Fractional derivatives with singular kernels

163

It follows from Eq. (2.368) that α RH w D[a,b];ϑ f

(t)

= H+ (ϑ, α) ×

1 (α)

+H− (ϑ, α) ×

1 (α)



2 b 



t t

d t dt

d t dt

2 

log τt

a

log τt

σ σ

f (τ ) dτ τ

(2.394)

f (τ ) dτ τ .

Property 2.60. Let ϑ = 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.395)

Then α RH w D[a,b];0 f

α f (t) . (t) = RH u D[a,b]

(2.396)

Proof. According to the definition of the Hadamard–Feller-type fractional derivative on an interval, one has α RH w D[a,b];0 f

(t)

α f α f = H+ (0, α) × H Db− (t) + H− (0, α) × H Da+ (t)   1 α f α f = 2 cos(πα/2) × H Da+ (t) + H Db− (t)

(2.397)

α f (t) , = RH u D[a,b]

completing the proof. Property 2.61. Let ϑ = 1 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.398)

Then α RH w D[a,b];1 f

α f (t) . (t) = RH v D[a,b]

(2.399)

Proof. In a similar way, making use of the Hadamard–Feller-type fractional derivative on an interval, one has α RH w D[a,b];1 f

(t)

α f α f = H+ (1, α) × H Db− (t) + H− (1, α) × H Da+ (t)   1 α f α f = 2 sin(πα/2) × H Da+ (t) + H Db− (t) α f (t) , = RH v D[a,b]

which completes the proof.

(2.400)

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General Fractional Derivatives With Applications in Viscoelasticity

2.22.5 Hadamard fractional calculus on the real line We now present the Hadamard fractional integrals and derivatives on the real line. Let  = (−∞, ∞). Definition 2.65. Let α > 0. The left-sided Hadamard fractional integral on the real line is defined as α H I+ f

1 (t) =  (α)

t log −∞

t τ

σ f (τ )

dτ , τ

(2.401)

and the right-sided Hadamard fractional integral on the real line is α H I− f

1 (t) =  (α)

∞

log

dτ τ σ f (τ ) . t τ

(2.402)

t

Definition 2.66. Let α > 0. The left-sided Hadamard fractional derivative on the real line is defined as α H D+ f

 t  1 dτ d κ t σ f (τ ) t , log (t) =  (α) dt τ τ

(2.403)

−∞

and the right-sided Hadamard fractional derivative on the real line is α H D− f

 ∞ 1 dτ d κ

τ σ f (τ ) t . log (t) =  (α) dt t τ

(2.404)

t

Property 2.62. Suppose that 0 ≤ a < b ≤ ∞, α > 0 and Lκ (−∞, ∞) (1 ≤ κ ≤ ∞), then 

α α H D+ H I+



(t) = f (t)

(2.405)

(t) = f (t) .

(2.406)

and 

α α H D− H I−



2.22.6 Hadamard-type fractional calculus In 2018, based on the Hadamard fractional integrals and derivatives, Yang first proposed the Hadamard-type fractional integrals and derivatives, which are called the Hadamard-type fractional calculus [25].

Fractional derivatives with singular kernels

165

Definition 2.67 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional integral is defined as α H v Ia+ f

 t  1 dτ d κ t σ κλτ e f (τ ) t , log (t) =  (α) dt τ τ

(2.407)

a

and the right-sided Hadamard-type fractional integral is α H v Ib− f

 b 1 dτ d κ

τ σ κλτ e f (τ ) t . log (t) =  (α) dt t τ

(2.408)

t

Definition 2.68 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional derivative is defined as α H v Da+ f

κ  t  1 dτ t σ −λt d f (τ ) e , log (t) =  (α) dt τ τ

(2.409)

a

and the right-sided Hadamard-type fractional derivative is α H v Db− f (t) =

 b d κ

1 dτ τ σ f (τ ) e−λt . log  (α) dt t τ

(2.410)

t

Property 2.63 ([25]). Suppose that 0 ≤ a < b ≤ ∞, α > 0 and Lκ (a, b) (1 ≤ κ ≤ ∞), then   α α (2.411) H v Da+ H v Ia+ (t) = f (t) and 

α α H v Db− H v Ib−



(t) = f (t) .

(2.412)

Definition 2.69 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional integral is defined as α H v I+ f

 t  1 dτ d κ t σ κλτ e f (τ ) t , log (t) =  (α) dt τ τ

(2.413)

−∞

and the right-sided Hadamard type fractional integral is α H v I− f (t) =

 ∞ 1 dτ d κ

τ σ κλτ e f (τ ) t . log  (α) dt t τ t

(2.414)

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General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.70 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional derivative is defined as α H v D+ f

κ  t  1 dτ t σ −λt d f (τ ) e , log (t) =  (α) dt τ τ

(2.415)

−∞

and the right-sided Hadamard-type fractional derivative is α H v D− f

κ ∞

1 dτ τ σ −λt d f (τ ) e . log (t) =  (α) dt t τ

(2.416)

t

Property 2.64 ([25]). Suppose that 0 ≤ a < b ≤ ∞, α > 0 and Lκ (−∞, ∞) (1 ≤ κ ≤ ∞), then   α α (2.417) H v D+ H v I+ (t) = f (t) and 

α α H v D− H v I−



(t) = f (t) .

(2.418)

In fact, the Hadamard and Hadamard-type fractional calculi were discussed in the book [25].

2.22.7 Hadamard-type fractional calculus involving the exponential function In 2018, based on the Hadamard type fractional integrals and derivatives, Yang first proposed the Hadamard-type fractional integrals and derivatives involving the exponential function, which is called the Hadamard-type fractional calculus involving the exponential function [25]. Definition 2.71 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional integral involving the exponential function is defined as α Sv Ia+ f

1 (t) =  (α)

t a

eκλτ (τ − t)α−κ+1

f (τ ) dτ ,

(2.419)

and the right-sided Hadamard-type fractional integral involving the exponential function is α Sv Ib− f (t) =

1  (α)

b t

eκλτ (t − τ )α−κ+1

f (τ ) dτ .

(2.420)

Fractional derivatives with singular kernels

167

In a special case, when 1 > α > 0, the left-sided Hadamard-type fractional integral involving the exponential function can be represented as α Sv Ia+ f

1 (t) =  (α)

t a

eλτ f (τ ) dτ , (τ − t)α

(2.421)

and the right-sided Hadamard-type fractional integral involving the exponential function is given by α Sv Ib− f

1 (t) =  (α)

b t

eλτ f (τ ) dτ . (t − τ )α

(2.422)

Definition 2.72 ([25]). Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional derivative involving the exponential function is defined as α Sv Da+ f

κ  t f (τ ) 1 −λt d dτ , e (t) =  (α) dt (τ − t)α−κ+1

(2.423)

a

and the right-sided Hadamard-type fractional derivative involving the exponential function is α Sv Db− f

κ b f (τ ) 1 −λt d dτ . e (t) =  (α) dt (t − τ )α−κ+1

(2.424)

t

As direct results, one has the following: Let 1 > α > 0. The left-sided Hadamard-type fractional derivative involving the exponential function can be expressed as α Sv Da+ f

 t f (τ ) 1 −λt d dτ , e (t) =  (α) dt (τ − t)α−κ+1

(2.425)

a

and the right-sided Hadamard type fractional derivative involving the exponential function is given by α Sv Db− f

 b f (τ ) 1 −λt d dτ . e (t) =  (α) dt (t − τ )α−κ+1

(2.426)

t

Property 2.65. Suppose that 0 ≤ a < b ≤ ∞, 1 > α > 0, and β > 0, then    β−1 − a) (t − a)β+α−1 (t α −λτ , (t) = Sv Ia+ e  (β)  (β + α)

(2.427)

168

General Fractional Derivatives With Applications in Viscoelasticity





 (t − a)β−1 (t − a)β−α−1 , e (t) =  (β)  (β − α)   β−1 − a) (t (b − t)β+α−1 α , (t) = e−λt Sv Da+  (β)  (β + α) α Sv Ib−

and



α (t Sv Db−

−λτ

− a)β−1  (β)

 (t) = e−λt

(b − t)β−α−1 .  (β − α)

(2.428)

(2.429)

(2.430)

Property 2.66. Suppose that 0 ≤ a < b ≤ ∞, α > 0, and Lκ (a, b) (1 ≤ κ ≤ ∞), then   α α (2.431) Sv Da+ Sv Ia+ (t) = f (t) and 

α α Sv Db− Sv Ib−



(t) = f (t) .

(2.432)

2.22.8 Hadamard-type fractional calculus involving the exponential function on the real line Based on the Hadamard-type fractional calculus involving the exponential function on a bounded domain, we will propose the Hadamard-type fractional calculus involving the exponential function on the real line. Definition 2.73. Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional integral involving the exponential function on real the line is defined as α Sv I+ f

1 (t) =  (α)

t −∞

eκλτ (τ − t)α−κ+1

f (τ ) dτ ,

(2.433)

and the right-sided Hadamard-type fractional integral involving the exponential function on the real line is α Sv I− f

1 (t) =  (α)

∞ t

eκλτ (t − τ )α−κ+1

f (τ ) dτ .

(2.434)

In a special case, when 1 > α > 0, the left-sided Hadamard-type fractional integral involving the exponential function on the real line can be represented as α Sv I+ f

1 (t) =  (α)

t −∞

eλτ f (τ ) dτ , (τ − t)α

(2.435)

Fractional derivatives with singular kernels

169

and the right-sided Hadamard-type fractional integral involving the exponential function on the real line can be given by α Sv I− f

1 (t) =  (α)

∞ t

eλτ f (τ ) dτ . (t − τ )α

(2.436)

Definition 2.74. Let α > 0 and κ = [α] + 1. The left-sided Hadamard-type fractional derivative involving the exponential function on the real line is defined as α Sv D+ f

κ  t f (τ ) 1 −λt d e dτ , (t) =  (α) dt (τ − t)α−κ+1

(2.437)

−∞

and the right-sided Hadamard-type fractional derivative involving the exponential function on the real line is α Sv D− f

κ ∞ f (τ ) 1 −λt d −e dτ . (t) =  (α) dt (t − τ )α−κ+1

(2.438)

t

As immediate results, one has the following: Let 1 > α > 0. The left-sided Hadamard-type fractional derivative involving the exponential function on the real line can be expressed as

α Sv D+ f

 t f (τ ) 1 −λt d dτ , e (t) =  (α) dt (τ − t)α

(2.439)

−∞

and the right-sided Hadamard-type fractional derivative involving the exponential function on the real line can be given by α Sv D− f

 ∞ f (τ ) 1 −λt d −e dτ . (t) =  (α) dt (t − τ )α

(2.440)

t

Property 2.67. Suppose that 0 ≤ a < b ≤ ∞, α > 0, and Lκ (a, b) (1 ≤ κ ≤ ∞), then 







α α Sv D+ Sv I+

(t) = f (t)

(2.441)

(t) = f (t) .

(2.442)

and α α Sv D− Sv I−

170

General Fractional Derivatives With Applications in Viscoelasticity

2.22.9 Extended Hadamard-type fractional derivatives Definition 2.75. The left-sided extended Hadamard-type fractional derivative involving the exponential function is defined as α Mm Da+ f (t) =

e−λt  (α)

t

f (1) (τ ) dτ , (τ − t)α

a

(2.443)

and the right-sided extended Hadamard-type fractional derivative involving the exponential function is α Mm Db− f

e−λt (t) = −  (α)

b

f (1) (τ ) dτ . (t − τ )α

t

(2.444)

It follows that α Mmv Da+ f (t) =

e−λt  (α)

t

f (κ) (τ ) (τ − t)α−κ+1

a

(2.445)

dτ

and α Mmv Db− f

b

(−1)κ e−λt (t) =  (α)

t

f (κ) (τ ) (t − τ )α−κ+1

dτ,

(2.446)

where κ = [α] + 1. Similarly, one has the extended Hadamard-type fractional derivatives involving the exponential function on the real line as follows: α Mm D+ f

α Mm D− f

e−λt (t) =  (α)

t −∞

e−λt (t) = −  (α)

α Mmv D+ f

f (1) (τ ) dτ , (τ − t)α

e−λt (t) =  (α)

∞

t ∞

t

(2.447)

f (1) (τ ) dτ , (t − τ )α f (κ) (τ )

(τ − t)α−κ+1

(2.448)

(2.449)

dτ ,

and α Mmv D− f (t) =

(−1)κ e−λt  (α)

∞ t

f (κ) (τ ) (t − τ )α−κ+1

dτ .

(2.450)

Fractional derivatives with singular kernels

2.23

171

Marchaud fractional derivatives

In 1927, Marchaud presented the concepts of the fractional derivatives [92], which are called the Marchaud fractional derivatives in the honor of Marchaud [25,44,45].

2.23.1 Definition of Marchaud fractional derivatives Definition 2.76 ([25,44,45]). Let 1 > α > 0. The left-sided Marchaud fractional derivative is defined as α M D+ f

α (t) =  (1 − α)

t

f (t) − f (τ ) (τ − t)α+1

−∞

dτ ,

(2.451)

and the right-sided Marchaud fractional derivative is α M D− f

α (t) =  (1 − α)

∞

f (t) − f (τ ) (t − τ )α+1

t

dτ ,

(2.452)

where −∞ < t < ∞. In this case, we get [25,44,45] α M D+ f

=

(t) ∞ f (τ )−f (t−τ )

α (1−α)

α = (1−α)

τ α+1

0 t

−∞

dτ

(2.453)

f (t)−f (τ ) dτ (τ −t)α+1

and α M D− f

=

(t) ∞ f (τ )−f (t+τ )

α (1−α)

α = (1−α)

0 ∞ t

τ α+1

dτ

(2.454)

f (t)−f (τ ) dτ . (t−τ )α+1

2.23.2 Definition of Marchaud-type fractional derivatives Definition 2.77. Let 1 > α > 0. The left-sided Marchaud-type fractional derivative is defined as α M Da+ f

α (t) =  (1 − α)

t a

f (t) − f (τ ) (τ − t)α+1

dτ ,

(2.455)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Marchaud-type fractional derivative is α M Db− f

α (t) =  (1 − α)

b

f (t) − f (τ ) (t − τ )α+1

t

(2.456)

dτ ,

where t ∈ [a, b].

2.23.3 Marchaud-type fractional derivatives with respect to another function Based on the above, we will propose the Marchaud type fractional derivatives with respect to another function. Definition 2.78. Let 1 > α > 0 and h(1) (t) > 0. The left-sided Marchaud-type fractional derivative with respect to another function is defined as α M Da+,h f

α (t) =  (1 − α)

t

f (t) − f (τ ) (h (τ ) − h (t))α+1

a

h(1) (τ ) dτ ,

(2.457)

and the right-sided Marchaud-type fractional derivative with respect to another function is α M Db− f

α (t) =  (1 − α)

b t

f (t) − f (τ ) (h (t) − h (τ ))α+1

h(1) (τ ) dτ ,

(2.458)

where t ∈ [a, b]. As special cases, we have the following results: (1E) When h (t) = t α , one has α M Da+,t α f

α M Db− f

α2 (t) =  (1 − α)

α2 (t) =  (1 − α)

t

τ α−1 (τ α − t α )α+1

a

b t

τ α−1 (t α − τ α )α+1

(f (t) − f (τ )) dτ ,

(f (t) − f (τ )) dτ ,

(2.459)

(2.460)

(2E) When h (t) = ln (t), one has α M Da+,ln(t) f

α M Db−,ln(t) f

α (t) =  (1 − α) (t) =

α  (1 − α)

t a

b t

f (t) − f (τ ) dτ ,   τ α+1 τ ln t

(2.461)

f (t) − f (τ ) dτ dτ .   t α+1 τ ln τ

(2.462)

Fractional derivatives with singular kernels

173

Definition 2.79. Let 1 > α > 0, a = −∞, b = ∞ and h(1) (t) > 0. The left-sided Marchaud-type fractional derivative with respect to another function is defined as α (t) =  (1 − α)

α M D+,h f

t

f (t) − f (τ ) (h (τ ) − h (t))α+1

−∞

h(1) (τ ) dτ ,

(2.463)

and the right-sided Marchaud-type fractional derivative with respect to another function is α M D−.h f

α (t) =  (1 − α)

∞

f (t) − f (τ ) (h (t) − h (τ ))α+1

t

h(1) (τ ) dτ ,

(2.464)

where −∞ < t < ∞. Based on the above, we have the following results: (1F) When h (t) = t α , one has α M D+,t α f

α M D− f

α2 (t) =  (1 − α)

(t) =

α2  (1 − α)

t −∞

τ α−1 (τ α − t α )α+1

∞ t

τ α−1 (t α − τ α )α+1

(f (t) − f (τ )) dτ ,

(f (t) − f (τ )) dτ .

(2.465)

(2.466)

(2F) When h (t) = ln (t), one has α M D+,ln(t) f

α M D−,ln(t) f

2.24

α (t) =  (1 − α) α (t) =  (1 − α)

t −∞

∞ t

f (t) − f (τ ) dτ dτ ,   τ α+1 τ ln t f (t) − f (τ ) dτ dτ .   t α+1 τ ln τ

(2.467)

(2.468)

Riemann–Liouville-type tempered fractional calculus

In 1884, Sonine presented the fractional integral involving the exponential function for the first time [93]. Rafalson introduced the fractional integral and derivative involving the exponential function in 1971 [94]. In 1979, Gearhart proposed the fractional integral involving the exponential function [95]. In 2015, Zayernouri et al. presented the fractional derivative involving the exponential function [96] and Sabzikar proposed the fractional integral and derivative involving the exponential function [97], respectively. In 2016, Li and Deng presented the fractional derivative involving the

174

General Fractional Derivatives With Applications in Viscoelasticity

exponential function [98]. In the same year, Atangana presented the fractional derivative involving the exponential function [99]. In 2017, Terres introduced the fractional derivative involving the exponential function [100]. Also in 2017, Dehghan et al. gave the fractional derivative involving the exponential function with Riesz structure [101]. In 2018, Yang proposed the fractional derivative involving the exponential function and its extended versions [25]. We start with the Riemann–Liouville-type tempered fractional derivatives.

2.24.1 Riemann–Liouville-type tempered fractional derivatives Definition 2.80 ([25]). Let α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R+ . The left-sided Riemann–Liouville-type tempered fractional derivative is defined as RL D α,λ f Cp a+

= =

1 (κ−α)

(t)  d κ t dt

t

dκ 1 (κ−α) dt κ

a

a

f (τ ) e−λ(t−τ ) dτ (t−τ )α−κ+1

(2.469)

f (τ ) e−λ(t−τ ) dτ , (t−τ )α−κ+1

and the right-sided Riemann–Liouville-type tempered fractional derivative is RL D α,λ f Cp b−

=

(−1)κ (κ−α) κ

(t)  d κ b dt κ

(−1) d = (κ−α) dt κ

b t

t

f (τ ) e−λ(τ −t) dτ (τ −t)α−κ+1

(2.470)

f (τ ) e−λ(τ −t) dτ . (τ −t)α−κ+1

2.24.2 Riemann–Liouville-type tempered fractional integrals Definition 2.81 ([25]). Let α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R+ . The left-sided Riemann–Liouville-type tempered fractional integral is defined as t RL α,λ Cp Ia+ f

(t) =

1 (t − τ )

α+1

a

κ−α E1,−α (−λ (t − τ )) f (τ ) dτ ,

(2.471)

and the right-sided Riemann–Liouville-type tempered fractional integral is b RL α,λ Cp Ib− f

(t) = t

1 (τ − t)α+1

κ−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Property 2.68 ([25]). Suppose that α > 0 and β > 0, then 

α+β,λ RL α,λ RL β,λ I I f f (t) (t) = RL Cp a+ Cp a+ Cp Da+

(2.472)

(2.473)

Fractional derivatives with singular kernels

175

and

RL α,λ RL β,λ Cp Ib− Cp Ib− f



α+β,λ

(t) = RL Cp Db−

f (t) ,

(2.474)

where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞). Property 2.69 ([25]). If α > 0, then

RL α,λ RL α,λ Cp Da+ Cp Ia+ f



(t) = f (t)

(2.475)

(t) = f (t) ,

(2.476)

and

RL α,λ RL α,λ Cp Db− Cp Ib− f



where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞). Property 2.70 ([25]). Suppose that α > 0, then

RL α,λ Cp D0+ f

L





α−κ s α f (s) (t) = s −1 λ + s

(2.477)

and L

2.25

RL α,λ Cp I0+ f



α−κ s α f (s) . (t) = s −1 λ + s

(2.478)

Liouville–Weyl-type tempered fractional calculus

2.25.1 Liouville–Weyl-type tempered fractional derivatives on the real line In 2018, the Liouville–Weyl-type tempered fractional derivatives on the real line were first proposed in the book [25]. Definition 2.82 ([25]). Let α ∈ R, κ < α < κ + 1 and λ ∈ R+ . The left-sided Liouville–Weyl-type tempered fractional derivative on the real line is defined as RL D α,λ f Cp +

= =

1 (κ−α)

(t)  d κ t dt

dκ 1 (κ−α) dt κ

t

−∞

−∞

f (τ ) e−λ(t−τ ) dτ (t−τ )α−κ+1

f (τ ) e−λ(t−τ ) dτ, (t−τ )α−κ+1

(2.479)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Weyl-type tempered fractional derivative on the real line is RL D α,λ f Cp −

=

(t)  d κ ∞

(−1)κ (κ−α) (−1)κ

= (κ−α)

dt dκ dt κ

∞ t

t

f (τ ) e−λ(τ −t) dτ (τ −t)α−κ+1

(2.480)

f (τ ) e−λ(τ −t) dτ . (τ −t)α−κ+1

2.25.2 Liouville–Weyl-type tempered fractional integrals on the real line In 2018, Yang first reported the Liouville–Weyl-type tempered fractional derivatives on the real line in the book [25]. Definition 2.83 ([25]). Let α > 0, κ = [α] +1 and λ ∈ R+ . The left-sided Liouville-type tempered fractional integral on the real line is defined as t RL α,λ Cp I+ f

1

(t) =

(t − τ )α+1

−∞

κ−α E1,−α (−λ (t − τ )) f (τ ) dτ ,

(2.481)

and the right-sided Liouville-type tempered fractional integral on the real line is ∞ RL α,λ Cp I− f

(t) =

1 (τ − t)α+1

t

κ−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Property 2.71 ([25]). Suppose that α > 0 and β > 0, then

 RL α,λ RL β,λ RL α+β,λ f (t) Cp I+ Cp I+ f (t) = Cp D+ and



RL α,λ RL β,λ Cp I− Cp I− f



α+β,λ

(t) = RL Cp D−

f (t) ,

(2.482)

(2.483)

(2.484)

where f (t) ∈ Lκ (−∞, ∞) (1 ≤ κ ≤ ∞). Property 2.72 ([25]). If α > 0, then 

RL α,λ RL α,λ Cp D+ Cp I+ f (t) = f (t) and



RL α,λ RL α,λ Cp D− Cp I− f



(t) = f (t) ,

where f (t) ∈ Lκ (−∞, ∞) (1 ≤ κ ≤ ∞).

(2.485)

(2.486)

Fractional derivatives with singular kernels

177

2.25.3 Liouville–Sonine–Caputo-type tempered fractional derivatives In 2018, the Liouville–Sonine–Caputo-type tempered fractional derivatives were addressed for the first time in the book [25]. Definition 2.84 ([25]). Let α > 0, κ = [α] +1, − ∞ < a < b < ∞ and λ ∈ R+ . The left-sided Liouville–Sonine–Caputo-type tempered fractional derivative is defined as LSC α,λ Cp Da+ f

1 (t) =  (κ − α)

t

e−λ(t−τ ) (t − τ )α−κ+1

a

f (κ) (τ ) dτ ,

(2.487)

and the right-sided Liouville–Sonine–Caputo-type tempered fractional derivative is LSC α,λ Cp Db− f

(−1)κ (t) =  (κ − α)

b t

e−λ(τ −t) (τ − t)α−κ+1

f (κ) (τ ) dτ .

(2.488)

Property 2.73 ([25]). Suppose that 1 > α > 0, −∞ < a < b < ∞, λ ∈ R+ and f (t) ∈ AC [a, b], then LSC α,λ Cp Da+ f

α,λ (t) = RL Cp Da+ f (t) +

1 e−λ(t−a) f (a)  (1 − α) (t − a)α

(2.489)

LSC α,λ Cp Db− f

α,λ (t) = RL Cp Db− f (t) −

1 e−λ(b−t) f (b) .  (1 − α) (b − t)α

(2.490)

and

Property 2.74 ([25]). Suppose that α > 0, κ = [α] + 1, −∞ < a < b < ∞ and λ ∈ R+ , then LSC α,λ Cp Da+ f

α,λ (t) = RL Cp Da+ f (t) +

κ−1 −λ(t−a)  e (t − a)j −α j =0

 (1 + j − α)

f (j ) (a)

(2.491)

and LSC α,λ Cp Db− f

α,λ (t) = RL Cp Db− f (t) +

κ−1  (−1)j e−λ(t−a) (b − t)j −α

f (j ) (b).

(2.492)

Property 2.75 ([25]). Suppose that α > 0, then ⎞ ⎛ κ−1

α−κ  α,λ −1 s α−κ ⎝s κ f (s) − f (κ−j ) (0)⎠ . L LSC Cp D0+ f (t) = s λ + s

(2.493)

j =0

 (1 + j − α)

j =0

178

General Fractional Derivatives With Applications in Viscoelasticity

2.25.4 Liouville–Sonine–Caputo-type tempered fractional derivatives In 2018, the Liouville–Sonine–Caputo-type tempered fractional derivatives (also called the Liouville-type tempered fractional derivatives) were addressed for the first time in the book [25]. Definition 2.85 ([25]). Let α > 0, κ = [α] +1 and λ ∈ R+ . The left-sided Liouville–Sonine–Caputo-type tempered fractional derivative is defined as LSC α,λ Cp D+ f

1 (t) =  (κ − α)

t

e−λ(t−τ )

−∞

(t − τ )α−κ+1

f (κ) (τ ) dτ ,

(2.494)

and the right-sided Liouville–Sonine–Caputo-type tempered fractional derivative is LSC α,λ Cp D− f

(−1)κ (t) =  (κ − α)

∞ t

e−λ(τ −t) (τ − t)α−κ+1

f (κ) (τ ) dτ .

(2.495)

2.25.5 Liouville–Weyl–Riesz type tempered fractional calculus Based on the Riesz and Liouville–Weyl-type tempered fractional calculi, we will present the Liouville–Weyl–Riesz-type tempered fractional integrals and derivatives for the first time, which will be called the Liouville–Weyl–Riesz-type fractional calculus. Definition 2.86. Let 1 > α > 0. The Liouville–Weyl–Riesz-type tempered fractional integral in the one-dimensional space is defined as α RzT IR f (t)

 RL I α,λ f +RL I α,λ f (t) Cp + Cp −

=

2 cos(πα/2) t 1 1 = 2 cos(πα/2) × E κ−α (−λ (t − τ )) f (τ ) dτ (t−τ )α+1 1,−α −∞ ∞ 1 1 × E κ−α (−λ (τ − t)) f (τ ) dτ + 2 cos(πα/2) (τ −t)α+1 1,−α t ∞ 1 1 E κ−α (−λ |τ − t|) f (τ ) dτ = 2(α) cos(πα/2) |τ −t|α+1 1,−α −∞

where t RL α,λ Cp I+ f

(t) = −∞

1 (t − τ )

α+1

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

(2.496)

Fractional derivatives with singular kernels

179

and ∞ RL α,λ Cp I− f

(t) =

1

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

(τ − t)α+1

t

Definition 2.87. Let 1 > α > 0. The Liouville–Weyl–Riesz-type fractional tempered derivative in the one-dimensional space is defined as α RzT DR f (t) 

RL D α,λ f +RL D α,λ f (t) Cp + Cp −

=

=

2 cos(πα/2)



1 2 cos(πα/2)

×

d 1 (1−α) dt

t

 f (τ ) −λ(t−τ ) dτ (t−τ )α e

−∞ ∞  f (τ ) −λ(τ −t) d 1 1 − 2 cos(πα/2) × (1−α) dτ dt (τ −t)α e t ∞ e−λ|τ −t| 1 d = 2 cos(πα/2)(1−α) × dt |τ −t|α f (τ ) dτ , −∞

(2.497)

where RL α,λ Cp D+ f

d 1 (t) =  (1 − α) dt

t −∞

f (τ ) −λ(t−τ ) e dτ (t − τ )α

and RL α,λ Cp D− f

d −1 (t) =  (1 − α) dt

∞ t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

2.25.6 Riemann–Liouville–Riesz-type fractional calculus Based on the Riesz and Riemann–Liouville-type tempered fractional calculi, we will present the Riemann–Liouville–Riesz-type fractional derivative and integral, which will be called the Riemann–Liouville–Riesz-type fractional calculus. Definition 2.88. Let 1 > α > 0. The Riemann–Liouville–Riesz-type tempered fractional integral on a bounded domain is defined as: α Rz I[a,b] f (t) 

RL I α,λ f +RL I α,λ f (t) Cp a+ Cp b−

=

=

2 cos(πα/2) t κ−α 1 1 2 cos(πα/2) × (t−τ )α+1 E1,−α (−λ (t a

− τ )) f (τ ) dτ

(2.498)

180

General Fractional Derivatives With Applications in Viscoelasticity

+ =

1 2 cos(πα/2)

×

b

1 E κ−α (−λ (τ (τ −t)α+1 1,−α

t

1 2(α) cos(πα/2)

b a

− t)) f (τ ) dτ

1 E κ−α (−λ |τ |τ −t|α+1 1,−α

− t|) f (τ ) dτ

where t RL α,λ Cp Ia+ f

1

(t) =

(t − τ )

α+1

a

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

and b RL α,λ Cp Ib− f

1

(t) =

(τ − t)α+1

t

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Definition 2.89. Let 1 > α > 0. The Riemann–Liouville–Riesz-type tempered fractional derivative on a bounded domain is defined as α RzT D[a,b] f (t) 

RL D α,λ f +RL D α,λ f (t) Cp a+ Cp b−

=

=

2 cos(πα/2)

1 2 cos(πα/2)

×

1 × − 2 cos(πα/2)

=

d 1 (1−α) dt d 1 (1−α) dt

1 2 cos(πα/2)(1−α)

×

d dt

b t

b a

t a

f (τ ) −λ(t−τ ) dτ (t−τ )α e

 (2.499)

f (τ ) −λ(τ −t) dτ (τ −t)α e e−λ|τ −t| |τ −t|α f

(τ ) dτ ,

where RL α,λ Cp Da+ f

d 1 (t) =  (1 − α) dt

t a

f (τ ) −λ(t−τ ) e dτ (t − τ )α

and RL α,λ Cp Db− f

d −1 (t) =  (1 − α) dt

b t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

2.25.7 Liouville–Sonine–Caputo–Riesz-type tempered fractional derivatives Based on the Riesz and Liouville–Sonine–Caputo tempered fractional derivatives, the Liouville–Sonine–Caputo–Riesz-type tempered fractional derivatives will be first presented as follows.

Fractional derivatives with singular kernels

181

Definition 2.90. Let 1 > α > 0. The Liouville–Sonine–Caputo–Riesz-type tempered fractional derivative in the one-dimensional space is defined as α RzT C DR f (t) 

LSC D α,λ f +LCS D α,λ f (t) + − Cp Cp

=

=

2 cos(πα/2)



1 2 cos(πα/2)

×



t

1 (1−α)

f (1) (τ ) −λ(t−τ ) dτ (t−τ )α e

−∞ ∞ (1)  f (τ ) −λ(τ −t) 1 1 − 2 cos(πα/2) × (1−α) dτ (τ −t)α e t ∞ e−λ|τ −t| (1) 1 = 2 cos(πα/2)(1−α) × (τ ) dτ , |τ −t|α f −∞

(2.500)

where LSC α,λ Cp D+ f

d 1 (t) =  (1 − α) dt

t −∞

f (1) (τ ) −λ(t−τ ) e dτ (t − τ )α

and LSC α,λ Cp D− f

−1 (t) =  (1 − α)

∞ t

f (1) (τ ) −λ(τ −t) e dτ . (τ − t)α

Definition 2.91. Let 1 > α > 0. The Liouville–Sonine–Caputo–Riesz-type tempered fractional derivative on a bounded domain is defined as α RzT C D[a,b] f (t)

 LSC D α,λ f +LCS D α,λ f (t) a+ Cp Cp b−

=

=

2 cos(πα/2)

1 2 cos(πα/2)

×

1 × − 2 cos(πα/2)

=

1 (1−α) 1 (1−α)

1 2 cos(πα/2)(1−α)

×

b t

b a

t a

f (1) (τ ) −λ(t−τ ) dτ (t−τ )α e



f (1) (τ ) −λ(τ −t) dτ (τ −t)α e e−λ|τ −t| (1) (τ ) dτ , |τ −t|α f

where LSC α,λ Cp Da+ f

d 1 (t) =  (1 − α) dt

t a

f (1) (τ ) −λ(t−τ ) e dτ (t − τ )α

(2.501)

182

General Fractional Derivatives With Applications in Viscoelasticity

and LSC α,λ Cp Db− f

(t) =

−1  (1 − α)

b

f (1) (τ ) −λ(τ −t) e dτ . (τ − t)α

t

2.25.8 Liouville–Weyl–Feller tempered fractional calculus Based on the Feller and Liouville–Weyl-type tempered fractional calculi, we will introduce the theory of the new tempered fractional calculus, which will be called the Liouville–Weyl–Feller tempered fractional calculus. Definition 2.92. Let 1 > α > 0. The Liouville–Weyl–Feller tempered fractional integral is defined as follows: α F T IR,ϑ f

(t)

α,λ RL α,λ = H− (ϑ, α) RL Cp I+ f (t) + H+ (ϑ, α) Cp I− f (t) t κ−α 1 = sin((α−ϑ)π/2) × α+1 E1,−α (−λ (t − τ )) f (τ ) dτ sin(πϑ) −∞

+ sin((α+ϑ)π/2) sin(πϑ)

×

∞ t

(2.502)

(t−τ )

1 E κ−α (−λ (τ (τ −t)α+1 1,−α

− t)) f (τ ) dτ,

where t RL α,λ Cp I+ f

(t) = −∞

1 (t − τ )α+1

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

and ∞ RL α,λ Cp I− f

(t) =

1 (τ − t)

α+1

t

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Definition 2.93. Let 1 > α > 0. The Liouville–Weyl–Feller tempered fractional derivative is defined as follows: α F T DR,ϑ f

(t)

 α,λ RL α,λ = − H− (ϑ, α) RL + f (t) + H+ (ϑ, α) Cp D− f (t) Cp D   t f (τ ) −λ(t−τ ) sin((α+ϑ)π/2) d 1 = − sin(πϑ) × (1−α) dt dτ (t−τ )α e

1 d + sin((α−ϑ)π/2) sin(πϑ) (1−α) dt

∞ t

−∞

f (τ ) −λ(τ −t) dτ , (τ −t)α e

(2.503)

Fractional derivatives with singular kernels

183

where RL α,λ Cp D+ f

d 1 (t) =  (1 − α) dt

t −∞

f (τ ) −λ(t−τ ) e dτ (t − τ )α

and RL α,λ Cp D− f

d −1 (t) =  (1 − α) dt

∞ t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

Property 2.76. Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.504)

Then α F T IR,0 f

(t) = RzT IRα f (t)

(2.505)

and α F T DR,0 f

(t) = RzT DRα f (t) .

(2.506)

Proof. According the definition of the operator, one has α F T IR,0 f

(t)

α,λ RL α,λ = H− (0, α) RL Cp I+ f (t) + H+ (0, α) Cp I− f (t)

=

RL I α,λ f +RL I α,λ f Cp + Cp −

 (t)

(2.507)

2 cos(πα/2)

= RzT IRα f (t) and α F T DR,0 f

(t)

 α,λ RL D α,λ f (t) = − H− (0, α) RL D f + H α) (t) (0, + + − Cp Cp



=−

RL D α,λ f +RL D α,λ f Cp + Cp −

2 cos(απ/2)

= RzT DRα f (t) , which completes the proof.

 (t)

(2.508)

184

General Fractional Derivatives With Applications in Viscoelasticity

2.25.9 Riemann–Liouville–Feller-type tempered fractional calculus Based on the Feller and Riemann–Liouville tempered fractional calculi, we will present the Riemann–Liouville–Feller-type tempered fractional derivative and integral, which will be called the Riemann–Liouville–Feller-type tempered fractional calculus. Definition 2.94. The Riemann–Liouville–Feller-type tempered fractional integral on a bounded domain is defined as follows: α F T I[a,b],ϑ f

(t)

α,λ RL α,λ = H− (ϑ, α) RL Cp Ia+ f (t) + H+ (ϑ, α) Cp Ib− f (t) t κ−α 1 = sin((α−ϑ)π/2) × α+1 E1,−α (−λ (t − τ )) f (τ ) dτ sin(πϑ) a

+ sin((α+ϑ)π/2) sin(πϑ)

×

b t

(2.509)

(t−τ )

1 E κ−α (−λ (τ (τ −t)α+1 1,−α

− t)) f (τ ) dτ,

where t RL α,λ Cp Ia+ f

(t) = a

1 (t − τ )α+1

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

and b RL α,λ Cp Ib− f

(t) = t

1 (τ − t)α+1

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Definition 2.95. Let 1 > α > 0. The Riemann–Liouville–Feller-type tempered fractional derivative on a bounded domain is defined as follows: α F T D[a,b],ϑ f

(t)

 α,λ RL α,λ = − H− (ϑ, α) RL Cp Da+ f (t) + H+ (ϑ, α) Cp Db− f (t)

 t f (τ ) −λ(t−τ ) d 1 = − sin((α+ϑ)π/2) × e dτ α sin(πϑ) (1−α) dt (t−τ )

1 d + sin((α−ϑ)π/2) sin(πϑ) (1−α) dt

b t

a

f (τ ) −λ(τ −t) dτ , (τ −t)α e

where RL α,λ Cp Da+ f

d 1 (t) =  (1 − α) dt

t a

f (τ ) −λ(t−τ ) e dτ (t − τ )α

(2.510)

Fractional derivatives with singular kernels

185

and RL α,λ Cp Db− f

d −1 (t) =  (1 − α) dt

b t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

As a direct result, we have the following property. Property 2.77. Consider 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.511)

Then α F T I[a,b],0 f

α f (t) (t) = RzT I[a,b]

(2.512)

and α F T D[a,b],0 f

α f (t) . (t) = RzT D[a,b]

(2.513)

2.25.10 Liouville–Sonine–Caputo–Feller-type tempered fractional derivatives Based on the Feller and Liouville–Sonine–Caputo tempered fractional derivatives, the Liouville–Sonine–Caputo–Feller-type tempered fractional derivatives will be first introduced as follows. Definition 2.96. Let 1 > α > 0. The Liouville–Sonine–Caputo–Feller-type tempered fractional derivative is defined as follows: α F T C DR,ϑ f

(t)

 α,λ α,λ = − H− (ϑ, α) LSC D+ f (t) + H+ (ϑ, α) LSC Cp  Cp D− f (t) t f (1) (τ ) −λ(t−τ ) 1 = − sin((α+ϑ)π/2) × (1−α) dτ sin(πϑ) (t−τ )α e

−∞

1 + sin((α−ϑ)π/2) sin(πϑ) (1−α)

∞ f (1) (τ ) t

(τ −t)α e

−λ(τ −t) dτ ,

where LSC α,λ Cp D+ f

1 (t) =  (1 − α)

t −∞

f (1) (τ ) −λ(t−τ ) e dτ (t − τ )α

and LSC α,λ Cp D− f

−1 (t) =  (1 − α)

∞ t

f (1) (τ ) −λ(τ −t) e dτ . (τ − t)α

(2.514)

186

General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.97. Let 1 > α > 0. The Liouville–Sonine–Caputo–Feller-type tempered fractional derivative on a bounded domain is defined as follows: α F T C D[a,b],ϑ f

(t)

 α,λ LSC D α,λ f = − H− (ϑ, α) LSC D f + H α) (t) (t) (ϑ, + a+ Cp Cp

b− t f (1) (τ ) −λ(t−τ ) sin((α+ϑ)π/2) 1 = − sin(πϑ) × (1−α) (t−τ )α e dτ

1 + sin((α−ϑ)π/2) sin(πϑ) (1−α)

b t

(2.515)

a

f (1) (τ ) −λ(τ −t) dτ , (τ −t)α e

where LSC α,λ Cp Da+ f

1 (t) =  (1 − α)

t a

f (1) (τ ) −λ(t−τ ) e dτ (t − τ )α

and LSC α,λ Cp Db− f

−1 (t) =  (1 − α)

b t

f (1) (τ ) −λ(τ −t) e dτ . (τ − t)α

As the direct results, we have the following: Property 2.78. Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.516)

Then α F C DR,0 f

(t) = RzC DRα f (t) .

(2.517)

Property 2.79. Let 1 > α > 0 and H+ (0, α) = H− (0, α) =

1 . 2 cos (απ/2)

(2.518)

Then α F C DR,0 f

(t) = RzC DRα f (t) .

(2.519)

2.25.11 Liouville–Weyl–Herrmann tempered fractional calculus Based on the Herrmann and Liouville–Weyl-type tempered fractional calculi, we will address the theory of the new tempered fractional calculus, which will be called the Liouville–Weyl–Herrmann tempered fractional calculus.

Fractional derivatives with singular kernels

187

Definition 2.98. Consider 1 > α > 0 and let H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.520)

The Liouville–Weyl–Herrmann tempered fractional integral is defined as α H T IR f

(t) f (t)

α,λ RL α,λ = H− (1, α) RL Cp I+ f (t) + H+ (1, α) Cp I− f (t) t κ−α 1 1 = 2 sin(απ/2) × α+1 E1,−α (−λ (t − τ )) f (τ ) dτ −∞

1 + 2 sin(απ/2)

∞

×

t

(2.521)

(t−τ )

1 E κ−α (−λ (τ (τ −t)α+1 1,−α

− t)) f (τ ) dτ,

where t RL α,λ Cp I+ f

(t) = −∞

1 (t − τ )α+1

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

and ∞ RL α,λ Cp I− f

(t) = t

1 (τ − t)α+1

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

Definition 2.99. The Liouville–Weyl–Herrmann tempered fractional derivative is defined as α H T DR f

(t)

 α,λ RL D α,λ f (t) = − H− (1, α) RL D f + H α) (t) (1, + a+ Cp Cp b− t f (τ ) −λ(t−τ ) d 1 1 = − 2 sin(απ/2) × (1−α) dt (t−τ )α e dτ

1 × + 2 sin(απ/2)

d 1 (1−α) dt

b t

a

f (τ ) −λ(τ −t) dτ , (τ −t)α e

where RL α,λ Cp D+ f

d 1 (t) =  (1 − α) dt

t −∞

f (τ ) −λ(t−τ ) e dτ (t − τ )α

and RL α,λ Cp D− f

(t) =

d −1  (1 − α) dt

∞ t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

(2.522)

188

General Fractional Derivatives With Applications in Viscoelasticity

Property 2.80. Let 1 > α > 0 and 1 . 2 sin (απ/2)

H+ (1, α) = H− (1, α) =

(2.523)

Then α F T IR,1 f

(t) = H T IRα f (t)

(2.524)

and α F T DR,1 f

(t) = H T DRα f (t) .

(2.525)

2.25.12 Riemann–Liouville–Herrmann-type tempered fractional calculus Based on the Herrmann and Riemann–Liouville tempered fractional calculi, we will present the Riemann–Liouville–Herrmann-type tempered fractional derivative and integral, which will be called the Riemann–Liouville–Herrmann-type tempered fractional calculus. Definition 2.100. Let 1 > α > 0 and 1 . 2 sin (απ/2)

H+ (1, α) = H− (1, α) =

(2.526)

The Riemann–Liouville–Herrmann-type tempered fractional integral is defined as α H T I[a,b] f

(t) f (t)

α,λ RL α,λ = H− (1, α) RL Cp Ia+ f (t) + H+ (1, α) Cp Ib− f (t)

=

1 2 sin(απ/2)

×

1 × + 2 sin(απ/2)

t

a b  t

1 E κ−α (−λ (t (t−τ )α+1 1,−α

− τ )) f (τ ) dτ

1 E κ−α (−λ (τ (τ −t)α+1 1,−α

− t)) f (τ ) dτ,

where t RL α,λ Cp Ia+ f

(t) = a

1 (t − τ )α+1

1−α E1,−α (−λ (t − τ )) f (τ ) dτ

and b RL α,λ Cp Ib− f

(t) =

1 (τ − t)

α+1

t

1−α E1,−α (−λ (τ − t)) f (τ ) dτ .

(2.527)

Fractional derivatives with singular kernels

189

Definition 2.101. The Riemann–Liouville–Herrmann-type tempered fractional derivative is defined as α H T D[a,b],ϑ f

(t)

 α,λ RL D α,λ f (t) D f + H = − H− (1, α) RL α) (t) (1, + Cp a+ Cp b−

 t  f (τ ) −λ(t−τ ) d 1 1 × (1−α) e dτ = − 2 sin(απ/2) dt (t−τ )α

(2.528)

a

1 d 1 + 2 sin(απ/2) (1−α) dt

b t

f (τ ) −λ(τ −t) dτ , (τ −t)α e

where RL α,λ Cp Da+ f

d 1 (t) =  (1 − α) dt

t a

f (τ ) −λ(t−τ ) e dτ (t − τ )α

and RL α,λ Cp Db− f

d −1 (t) =  (1 − α) dt

b t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

As an immediate result, we have the following property: Property 2.81. Let 1 > α > 0 and H+ (1, α) = H− (1, α) =

1 . 2 sin (απ/2)

(2.529)

Then α F T I[a,b],1 f

α f (t) (t) = H T I[a,b]

(2.530)

and α F T D[a,b],1 f

α f (t) . (t) = H T D[a,b]

(2.531)

2.25.13 Liouville–Sonine–Caputo–Herrmann-type tempered fractional derivatives Based on the Herrmann and Liouville–Sonine–Caputo tempered fractional derivatives, the Liouville–Sonine–Caputo–Herrmann-type tempered fractional derivative will be first proposed as follows.

190

General Fractional Derivatives With Applications in Viscoelasticity

Definition 2.102. The Liouville–Sonine–Caputo–Herrmann-type tempered fractional derivative is defined as α H T C DR,ϑ f

(t)

 α,λ LSC D α,λ f = − H− (1, α) LSC D f + H α) (t) (t) (1, + + − Cp Cp

1 = − 2 sin(απ/2) ×

1 (1−α)

1 1 + 2 sin(απ/2) (1−α)

t −∞

f (1) (τ ) −λ(t−τ ) dτ (t−τ )α e

∞ f (1) (τ ) t

(τ −t)α e

(2.532)

−λ(τ −t) dτ ,

where LSC α,λ Cp D+ f

(t) =

d 1  (1 − α) dt

t −∞

f (1) (τ ) −λ(t−τ ) e dτ (t − τ )α

and LSC α,λ Cp D− f

∞

d −1 (t) =  (1 − α) dt

t

f (1) (τ ) −λ(τ −t) e dτ . (τ − t)α

It follows that α H T C DR f

α f (t) . (t) = F T C DR,1

(2.533)

Based on the Herrmann’s ideas, we have the following. Definition 2.103. The Liouville–Sonine–Caputo–Herrmann-type tempered fractional derivative on a bounded domain is defined as α H T C D[a,b],ϑ f

(t)

 α,λ RL D α,λ f (t) D f + H = − H− (1, α) RL α) (t) (1, + a+ Cp Cp b−

 t  f (τ ) −λ(t−τ ) d 1 1 = − 2 sin(απ/2) × (1−α) e dτ α dt (t−τ )

a

d 1 1 + 2 sin(απ/2) (1−α) dt

b t

f (τ ) −λ(τ −t) dτ , (τ −t)α e

where RL α,λ Cp Da+ f

d 1 (t) =  (1 − α) dt

t a

f (τ ) −λ(t−τ ) e dτ (t − τ )α

(2.534)

Fractional derivatives with singular kernels

191

and RL α,λ Cp Db− f

−1 d (t) =  (1 − α) dt

b t

f (τ ) −λ(τ −t) e dτ . (τ − t)α

Similarly, α H T C D[a,b] f

2.26

α f (t) . (t) = F T C D[a,b],1

(2.535)

Riemann–Liouville-type tempered fractional calculus with respect to another function

In 2018, Yang proposed the Riemann–Liouville-type tempered fractional calculus with respect to another function for the first time in his book [25].

2.26.1 Riemann–Liouville-type tempered fractional integrals with respect to another function Definition 2.104 ([25]). Let h(1) (t) > 0, α > 0, κ = [α] +1, − ∞ < a < b < ∞ and λ ∈ R+ . The left-sided Riemann–Liouville-type tempered fractional derivative with respect to another function is defined as RL α,λ Cp Da+,h f

(t) =

1  (κ − α)

1

d h(1) (t) dt

κ  t a

e−λ(h(t)−h(τ )) h(1) (τ ) f (τ ) dτ , (h (t) − h (τ ))α (2.536)

and the right-sided Riemann–Liouville-type tempered fractional derivative with respect to another function is RL α,λ Cp Db−,h f

(−1)κ (t) =  (κ − α)

1

d (1) h (t) dt

κ b t

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

f (τ ) dτ. (2.537)

2.26.2 Riemann–Liouville-type tempered fractional integrals with respect to another function In 2018, the Riemann–Liouville-type tempered fractional integrals with respect to another function were first reported in the book [25]. Definition 2.105 ([25]). Let h(1) (t) > 0, α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R+ .

192

General Fractional Derivatives With Applications in Viscoelasticity

The left-sided Riemann–Liouville-type tempered fractional integral with respect to another function is defined as t RL α,λ Cp Ia+,h f

(t) =

κ−α E1,−α (−λ (h (t) − h (τ ))) h(1) (τ )

(h (t) − h (τ ))α+1

a

f (τ ) dτ,

(2.538)

and the right-sided Riemann–Liouville-type tempered fractional integral with respect to another function is b RL α,λ Cp Ib− f

(t) =

κ−α h(1) (τ ) E1,−α (−λ (h (τ ) − h (t)))

(h (τ ) − h (t))α+1

t

f (τ ) dτ .

Property 2.82 ([25]). Suppose that h(1) (t) > 0, α > 0 and β > 0. Then 

α+β,λ RL α,λ RL β,λ I I f (t) = RL Cp a+,h Cp a+,h Cp Da+,h f (t) and



RL α,λ RL β,λ Cp Ib−,h Cp Ib−,h f



α+β,λ

(t) = RL Cp Db−,h f (t) ,

(2.539)

(2.540)

(2.541)

where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞). Property 2.83. If h(1) (t) > 0, α > 0, then

 RL α,λ RL α,λ Cp Da+,h Cp Ia+,h f (t) = f (t) and



RL α,λ RL α,λ Cp Db−,h Cp Ib−,h f



(2.542)

(t) = f (t) ,

(2.543)

where f (t) ∈ Lκ [a, b] (1 ≤ κ ≤ ∞).

2.26.3 Riemann–Liouville-type tempered fractional derivatives with respect to another function on the real line In 2018, the Riemann–Liouville-type tempered fractional derivatives with respect to another function on the real line were first proposed in the book [25]. Definition 2.106 ([25]). Let h(1) (t) > 0, α > 0, κ = [α] +1 and λ ∈ R+ . The left-sided Riemann–Liouville-type tempered fractional derivative with respect to another function on the real line is defined as RL α,λ Cp D+,h f

1 (t) =  (κ − α)

1

d h(1) (t) dt

κ  t −∞

e−λ(h(t)−h(τ )) (h (t) − h (τ ))α−κ+1

f (τ ) dτ, (2.544)

Fractional derivatives with singular kernels

193

and the right-sided Riemann–Liouville-type tempered fractional derivative with respect to another function on the real line is RL α,λ Cp D−,h f

(−1)κ (t) =  (κ − α)

1

d h(1) (t) dt

κ  t −∞

e−λ(h(t)−h(τ )) (h (τ ) − h (t))α−κ+1

f (τ ) dτ . (2.545)

2.26.4 Liouville-type tempered fractional integrals with respect to another function on the real line In fact, the Liouville-type tempered fractional integrals with respect to another function on the real line were addressed in the book [25]. Definition 2.107. Let h(1) (t) > 0, α > 0, κ = [α] +1 and λ ∈ R+ . The left-sided Liouville-type tempered fractional integral with respect to another function on the real line is defined as t L α,λ Cp I+,h f

κ−α E1,−α (−λ (h (t) − h (τ ))) h(1) (τ )

(t) =

(h (t) − h (τ ))α+1

−∞

f (τ ) dτ ,

(2.546)

and the right-sided Liouville-type tempered fractional integral with respect to another function on the real line is ∞ L α,λ Cp I−,h f

(t) =

κ−α h(1) (τ ) E1,−α (−λ (h (τ ) − h (t)))

(h (τ ) − h (t))α+1

t

f (τ ) dτ .

Property 2.84. Suppose that h(1) (t) > 0, α > 0 and β > 0, then

 α+β,λ RL α,λ RL β,λ I I f (t) = RL Cp +,h Cp +,h Cp D+,h f (t) and



RL α,λ RL β,λ Cp I−,h Cp I−,h f



α+β,λ

(t) = RL Cp D−,h

f (t) ,

(2.547)

(2.548)

(2.549)

where f (t) ∈ Lκ (−∞, ∞) (1 ≤ κ ≤ ∞). Property 2.85. If h(1) (t) > 0 and α > 0, then 

RL α,λ RL α,λ D I f (t) = f (t) Cp +,h Cp +,h and



RL α,λ RL α,λ Cp D−,h Cp I−,h f



(t) = f (t) ,

where f (t) ∈ Lκ (−∞, ∞) (1 ≤ κ ≤ ∞).

(2.550)

(2.551)

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General Fractional Derivatives With Applications in Viscoelasticity

2.26.5 Liouville–Sonine–Caputo-type tempered fractional derivatives with respect to another function In 2018, the Liouville–Sonine–Caputo-type tempered fractional derivatives with respect to another function were first proposed in the book [25]. Definition 2.108 ([25]). Let h(1) (t) > 0, α > 0, κ = [α] +1, − ∞ < a < b < ∞ and λ ∈ R+ . The left-sided Liouville–Sonine–Caputo-type tempered fractional derivative with respect to another function is defined as LSC α,λ Cp Da+,h f

1 (t) =  (κ − α)

t a

e−λ(h(t)−h(τ )) (h (t) − h (τ ))α−κ+1

κ

1 d (1) h (τ ) dτ

f (τ ) dτ, (2.552)

and the right-sided Liouville–Sonine–Caputo-type tempered fractional derivative with respect to another function is LSC α,λ Cp Db− f

(−1)κ (t) =  (κ − α)

b t

e−λ(h(τ )−h(t)) (h (τ ) − h (t))α−κ+1

1 d (1) dτ h (τ )

κ f (τ ) dτ . (2.553)

Property 2.86. Suppose that h(1) (t) > 0, 1 > α > 0, −∞ < a < b < ∞, λ ∈ R+ and f (t) ∈ AC [a, b], then LSC α,λ Cp Da+,h f

α,λ (t) = RL Cp Da+,h f (t) +

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

(2.554)

LSC α,λ Cp Db−,h f

α,λ (t) = RL Cp Db−,h f (t) −

1 e−λ(h(b)−h(t)) f (b) .  (1 − α) (h (b) − h (t))α

(2.555)

and

Property 2.87. Suppose that h(1) (t) > 0, κ + 1 > α > κ, −∞ < a < b < ∞ and λ ∈ R+ , then LSC D α,λ f Cp a+,h

(t)

α,λ = RL Cp Da+,h f (t) +

κ−1 j =0

e−λ(h(t)−h(a)) (h(t)−h(a))j −α (1+j −α)



d 1 h(1) (t) dt

κ

f (j ) (t)



(2.556)

t=a

and LSC D α,λ f Cp b−,h

(t)

α,λ = RL Cp Db−,h f (t) +

κ−1 j =0

(−1)j e−λ(h(t)−h(a)) (h(b)−h(t))j −α (1+j −α)



d 1 h(1) (t) dt

κ

f (j ) (t)

 t=b

.

(2.557)

Fractional derivatives with singular kernels

195

2.26.6 Liouville-type tempered fractional derivatives with respect to another function In fact, the Liouville-type tempered fractional derivatives with respect to another function were first reported in 2018 (see [25]). Definition 2.109 ([25]). Let h(1) (t) > 0, α > 0, κ = [α] +1 and λ ∈ R+ . The left-sided Liouville-type tempered fractional derivative with respect to another function is defined as α,λ L Cp D+,h f

1 (t) =  (κ − α)

t −∞

e−λ(h(t)−h(τ )) (h (t) − h (τ ))α−κ+1

1 d (1) h (τ ) dτ

κ f (τ ) dτ, (2.558)

and the right-sided Liouville-type tempered fractional derivative with respect to another function is α,λ L Cp D−,h f

2.27

(−1)κ (t) =  (κ − α)

∞ t

e−λ(h(τ )−h(t)) (h (τ ) − h (t))α−κ+1

1 d h(1) (τ ) dτ

κ f (τ ) dτ . (2.559)

Hilfer derivatives

In 2000, based on the Riemann–Liouville and Liouville–Sonine–Caputo derivatives, Hilfer introduced the mixed fractional derivative [102], which is called the Hilfer derivative in his honor [25]. Definition 2.110 ([25,102]). Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided Hilfer fractional derivative is defined as



α,β β(1−α) d (1−β)(1−α) Da+ f (t) = Ia+ f (2.560) I (t) , dt a+ (1−β)(1−α)

where Ia+

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



α,β β(1−α) d (1−β)(1−α) f Ib− Db− f (t) = −Ib− (t) , dt (1−β)(1−α)

where Ib−

(2.561)

f (t) ∈ AC1 (−∞, ∞).

Definition 2.111 ([25,102]). Let κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) and −∞ < a < b < ∞.

196

General Fractional Derivatives With Applications in Viscoelasticity

The left-sided Hilfer fractional derivative is defined as

 κ

α,β,κ β(κ−α) d (1−β)(κ−α) I Da+ f (t) = Ia+ f (t) , dt κ a+ (1−β)(κ−α)

where Ia+

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

 κ

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

where Ib−

(2.562)

(2.563)

f (t) ∈ ACκ (a, b).

In particular, for β = 1 and β = 0, we have the following cases: (1G) When κ < α < 1 + κ, κ = [α] + 1, β = 1, f ∈ Lκ (a, b), −∞ < a < b < ∞ and f (t) ∈ ACκ (a, b), the left-sided Liouville–Sonine–Caputo fractional derivative is defined as

κ  α,1,κ α,κ κ−α d Da+ f (t) = Ia+ f (t) = LSC Da+ f (t) , (2.564) dt κ and the right-sided Liouville–Sonine–Caputo fractional derivative is

κ  α,1,κ α,κ κ−α d Db− f (t) = (−1)κ Ib− f (t) = LSC Db− f (t) . dt κ

(2.565)

(2G) When κ < α < 1 + κ, κ = [α] + 1, β = 0, f ∈ Lκ (a, b), −∞ < a < b < ∞ and f (t) ∈ ACκ (a, b), the left-sided Riemann–Liouville fractional derivative is defined as α,0,κ f (t) = Da+

d κ (κ−α)  α,κ I f (t) = Da+ f (t) , dt κ a+

(2.566)

and the right-sided Riemann–Liouville fractional derivative is α,0,κ f (t) = (−1)κ Db−

d κ  κ−α  α,κ I f (t) = Db− f (t) . dt κ b−

(2.567)

In this case, we have the following relations: Property 2.88. Let 0 < α < 1, 0 ≤ β ≤ 1 and f ∈ L1 (a, b). Then α,β

α,β,1

(2.568)

α,β

α,β,1

(2.569)

Da+ f (t) = Da+ f (t) and Db− f (t) = Db− f (t) .

Fractional derivatives with singular kernels

197

Property 2.89 ([25,102]). (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 α,β (1−β)(1−α) f (0) . (2.570) L D0+ f (t) (s) = s α f (s) − s β(1−α) I0+ (2) Suppose that κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b), −∞ < a < (1−β)(1−α) f (t) ∈ ACκ (a, b), then b < ∞ and I0+ α,β,κ (1−β)(κ−α) f (0) . L D0+ f (t) (s) = s α f (s) − s β(κ−α) I0+

(2.571)

2.27.1 Liouville–Weyl–Hilfer-type derivatives In 2018, based on the Liouville, Liouville–Weyl and Hilfer fractional derivatives, the Liouville–Weyl–Hilfer-type derivatives were proposed in the book [25]. Definition 2.112 ([25]). Let 0 < α < 1, 0 ≤ β ≤ 1 and f ∈ L1 (−∞, ∞). The left-sided Liouville–Weyl–Hilfer-type fractional derivative is defined as



α,β β(1−α) d (1−β)(1−α) D+ f (t) = I+ f (2.572) I (t) , dt + (1−β)(1−α)

f (t) ∈ AC1 (−∞, ∞), and the right-sided Liouville–Weyl–Hilferwhere I+ type fractional derivative is



α,β β(1−α) d (1−β)(1−α) f (2.573) I− D− f (t) = −I− (t) , dt (1−β)(1−α)

where I−

f (t) ∈ AC1 (−∞, ∞).

Definition 2.113 ([25]). Let κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, and f ∈ Lκ (−∞, ∞). The left-sided Liouville–Weyl–Hilfer type fractional derivative is given as

 κ

α,β,κ β(1−α) d (1−β)(κ−α) D+ f (t) = I+ f (2.574) I (t) , dt κ + (1−β)(κ−α)

f (t) ∈ ACκ (−∞, ∞), and the right-sided Liouville–Weyl–Hilferwhere I+ type fractional derivative is

 κ

α,β,κ (1−β)(κ−α) κ β(1−α) d D− f (t) = (−1) I− f (2.575) I (t) , dt κ − (1−β)(κ−α)

where I−

f (t) ∈ ACκ (−∞, ∞).

In a similar way, for β = 1 and β = 0, we have the following cases:

198

General Fractional Derivatives With Applications in Viscoelasticity

(1H) When κ < α < 1 + κ, κ = [α] + 1, β = 1, f ∈ Lκ (−∞, ∞) and f (t) ∈ ACκ (−∞, ∞), the left-sided Liouville fractional derivative is defined as

 dκ α,1,κ α f (t) = I+κ−α κ f (t) = LSC D+ f (t) , (2.576) D+ dt and the right-sided Liouville fractional derivative is

 dκ α,1,κ α D− f (t) = (−1)κ I−κ−α κ f (t) = LSC D− f (t) . dt

(2.577)

(2H) When κ < α < 1 + κ, κ = [α] + 1, β = 0, f ∈ Lκ (−∞, ∞) and f (t) ∈ ACκ (−∞, ∞), the left-sided Liouville–Weyl fractional derivative is defined as α,0,κ f (t) = D+

d κ (κ−α)  α f (t) = LW I Cp D+ f (t) , dt κ +

(2.578)

and the right-sided Liouville–Weyl fractional derivative is α,0,κ f (t) = (−1)κ D−

2.28

d κ  κ−α  α I f (t) = LW Cp D− f (t) . dt κ −

(2.579)

Mixed fractional derivatives

2.28.1 Riesz–Hilfer-type fractional derivative In 2018, based on the Riesz and Hilfer fractional derivatives, Yang first proposed the Riesz–Hilfer-type fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+ f (t) ∈ (1−β)(1−α) AC1 (a, b) and Ib− f (t) ∈ AC1 (a, b). Definition 2.114 ([25]). The Riesz–Hilfer-type fractional derivative is defined as

 α,β α,β Da+ f + Db− f (t) H i α,β (2.580) Rz D[a,b] f (t) = − 2 cos (πα/2) where



α,β β(1−α) d (1−β)(1−α) f Ia+ Da+ f (t) = Ia+ (t) dt

(2.581)



α,β β(1−α) d (1−β)(1−α) Db− f (t) = −Ib− f Ib− (t) . dt

(2.582)

and

Fractional derivatives with singular kernels

199

2.28.2 Feller–Hilfer-type fractional derivative In 2018, based on the Feller and Hilfer fractional derivatives, Yang first proposed the Feller–Hilfer type fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+ f (t) ∈ (1−β)(1−α) AC1 (a, b) and Ib− f (t) ∈ AC1 (a, b). Definition 2.115 ([25]). The Feller–Hilfer-type fractional derivative is defined as F e D α,β,θ f Rz [a,b]



(t)

 α,β α,β = − H+ (ϑ, α) Da+ f (t) + H− (ϑ, α) Db− f (t) α,β

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

sin((α−ϑ)π/2) sin(πϑ)

(2.583)

α,β

× Db− f (t) ,

where H− (ϑ, α) =

sin ((α − ϑ) π/2) , sin (πϑ)

sin ((α + ϑ) π/2) , sin (πϑ)



α,β β(1−α) d (1−β)(1−α) Da+ f (t) = Ia+ f Ia+ (t) , dt H+ (ϑ, α) =

(2.584) (2.585) (2.586)

and α,β Db− f



β(1−α) d (1−β)(1−α) f I (t) = −Ib− (t) . dt b−

(2.587)

2.28.3 Herrmann–Hilfer-type fractional derivative In 2018, based on the Herrmann and Hilfer fractional derivatives, Yang first proposed the Herrmann–Hilfer type fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+ f (t) ∈ (1−β)(1−α) AC1 (a, b) and Ib− f (t) ∈ AC1 (a, b). Definition 2.116 ([25]). The Herrmann–Hilfer-type fractional derivative is defined as 

α,β α,β Da+ f + Db− f (t) Ri α,β (2.588) Rz D[a,b] f (t) = − 2 sin (πα/2) where



α,β β(1−α) d (1−β)(1−α) Da+ f (t) = Ia+ f Ia+ (t) dt

(2.589)

200

General Fractional Derivatives With Applications in Viscoelasticity

and



α,β β(1−α) d (1−β)(1−α) f Ib− Db− f (t) = −Ib− (t) . dt

(2.590)

It is clear that F e α,β,0 Rz D[a,b] f

i (t) = H Rz D[a,b] f (t)

α,β

(2.591)

F e α,β,1 Rz D[a,b] f

(t) = Ri Rz D[a,b] f (t) .

α,β

(2.592)

and

2.28.4 Riesz–Hilfer fractional derivative In 2018, based on the Riesz and Hilfer fractional derivatives, Yang first proposed the Riesz–Hilfer fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+ f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α) f (t) ∈ AC1 (−∞, ∞). and I− Definition 2.117 ([25]). The Riesz–Hilfer type fractional derivative is defined as

 α,β α,β D+ f + D− f (t) H i α,β (2.593) Rz DR f (t) = − 2 cos (πα/2) where α,β D+ f

and



β(1−α) d (1−β)(1−α) f I (t) = I+ (t) dt +

(2.594)



α,β β(1−α) d (1−β)(1−α) D− f (t) = −I− f I− (t) . dt

(2.595)

2.28.5 Feller–Hilfer fractional derivative In 2018, based on the Feller and Hilfer fractional derivatives, Yang first proposed the Feller–Hilfer fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+ f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α) f (t) ∈ AC1 (−∞, ∞). and I− Definition 2.118 ([25]). The Feller–Hilfer fractional derivative is defined as F e D α,β,θ f Rz R

(t) α,β

α,β

H+ (ϑ, α) D+ f (t) + H− (ϑ, α) D− f (t) α,β

= − sin((α+ϑ)π/2) × D+ f (t) − sin(πϑ)

sin((α−ϑ)π/2) sin(πϑ)

(2.596) α,β

× D− f (t) ,

Fractional derivatives with singular kernels

201

where H− (ϑ, α) =

sin ((α − ϑ) π/2) , sin (πϑ)

sin ((α + ϑ) π/2) , sin (πϑ)



α,β β(1−α) d (1−β)(1−α) D+ f (t) = I+ f I (t) , dt + H+ (ϑ, α) =

(2.597)

(2.598) (2.599)

and



α,β β(1−α) d (1−β)(1−α) f I− D− f (t) = −I− (t) . dt

(2.600)

2.28.6 Herrmann–Hilfer fractional derivative In 2018, based on the Herrmann and Hilfer fractional derivatives, Yang first proposed the Herrmann–Hilfer fractional derivative in his book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+ f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α) f (t) ∈ AC1 (−∞, ∞). and I− Definition 2.119 ([25]). The Herrmann–Hilfer fractional derivative is defined as

Ri α,β Rz DR f

(t) = −

 α,β α,β D+ f + D− f (t) 2 sin (πα/2)

(2.601)

where



α,β β(1−α) d (1−β)(1−α) f D+ f (t) = I+ I+ (t) dt

(2.602)

and α,β D− f



β(1−α) d (1−β)(1−α) f I (t) = −I− (t) . dt −

(2.603)

It is clear that F e α,β,0 f Rz DR

i (t) = H Rz DR f (t)

α,β

(2.604)

F e α,β,1 f Rz DR

(t) = Ri Rz DR f (t) .

α,β

(2.605)

and

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General Fractional Derivatives With Applications in Viscoelasticity

2.28.7 Sousa–de Oliveira fractional derivative with respect to another function In 2018, Sousa and de Oliveira introduced the Hilfer fractional derivative with respect to another function [103], which is called the Sousa–de Oliveira fractional derivative with respect to another function [25]. This work is first elaborated in the book [25]. Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞ and h(1) (t) > 0. Definition 2.120 ([25]). The left-sided Sousa–de Oliveira fractional derivative with respect to another function is defined as

 

1 d (1−β)(1−α)  α,β β(1−α) f (2.606) Ia+,h Da+,h f (t) = Ia+,h (t) , h(1) (t) dt (1−β)(1−α)

f (t) ∈ AC1 (a, b), and the right-sided Sousa–de Oliveira fractional where Ia+,h derivative with respect to another function is

 

1 d (1−β)(1−α)  α,β β(1−α) Db−,h f (t) = −Ib−,h f (2.607) Ib−,h (t) , h(1) (t) dt (1−β)(1−α)

where Ib−,h

f (t) ∈ AC1 (−∞, ∞).

Let κ < α < 1 + κ, κ = [α] + 1, 0 ≤ β ≤ 1, f ∈ Lκ (a, b) , −∞ < a < b < ∞ and h(1) (t) > 0. Definition 2.121 ([25,103]). The left-sided Sousa–de Oliveira fractional derivative with the respect to another function is defined as

α,β,κ Da+,h f

(t) =

β(κ−α) Ia+,h

1

d (1) h (t) dt

κ

(1−β)(κ−α) f Ia+,h

 (t) ,

(2.608)

(1−β)(1−α)

f (t) ∈ ACκ (a, b), and the right-sided Sousa–de Oliveira fractional where Ia+,h derivative with the respect to another function is

α,β,κ β(κ−α) Db−,h f (t) = (−1)κ Ib−,h (1−β)(1−α)

where Ib−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) f Ib−,h (t) ,

(2.609)

f (t) ∈ ACκ (a, b).

2.28.8 Liouville–Weyl–Sousa–de Oliveira-type fractional derivatives Based on the Sousa–de Oliveira fractional derivative with respect to another function and Liouville–Weyl fractional derivative, the Liouville–Weyl–Sousa–de Oliveira-type fractional derivatives will be proposed for the first time.

Fractional derivatives with singular kernels

203

Definition 2.122. Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞) and h(1) (t) > 0. The left-sided Liouville–Weyl–Sousa–de Oliveira-type fractional derivative with respect to another function is defined as

 

1 d (1−β)(1−α)  α,β β(1−α) f (2.610) I D+,h f (t) = I+,h (t) , +,h h(1) (t) dt (1−β)(1−α)

where I+,h f (t) ∈ AC1 (−∞, ∞), and the right-sided Liouville–Weyl–Sousa– de Oliveira-type fractional derivative with respect to another function is

 

1 d (1−β)(1−α)  α,β β(1−α) f (2.611) I D−,h f (t) = −I−,h (t) , −,h h(1) (t) dt (1−β)(1−α)

where I−,h

f (t) ∈ AC1 (−∞, ∞).

Definition 2.123. Let κ < α < 1 + κ, 0 ≤ β ≤ 1, f ∈ Lκ (−∞, ∞) and h(1) (t) > 0. The left-sided Liouville–Weyl–Sousa–de Oliveira type fractional derivative with the respect to another function is defined as

 

1 d κ (1−β)(κ−α)  α,β,κ β(κ−α) f (2.612) I D+,h f (t) = I+,h (t) , +,h h(1) (t) dt (1−β)(κ−α)

f (t) ∈ ACκ (−∞, ∞), and the right-sided Liouville–Weyl–Sousa– where I+,h de Oliveira-type fractional derivative with the respect to another function is

 

1 d κ (1−β)(κ−α)  α,β,κ β(κ−α) f I D−,h f (t) = (−1)κ I−,h (t) , (2.613) −,h h(1) (t) dt (1−β)(κ−α)

where I−,h

f (t) ∈ ACκ (−∞, ∞).

2.28.9 Hilfer–Riesz type fractional derivative with respect to another function In 2018, generalizing the Hilfer fractional derivative with respect to another function and Riesz derivatives, the Hilfer–Riesz-type fractional derivative with respect to another function was proposed in the book [25]. (1−β)(1−α) f (t) ∈ Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+,h (1−β)(1−α)

AC1 (a, b) and Ib−,h

f (t) ∈ AC1 (a, b).

Definition 2.124 ([25]). The Hilfer–Riesz-type fractional derivative with respect to another function is defined as

 α,β α,β Da+,h f + Db−,h f (t) H i α,β , (2.614) Rz D[a,b],h f (t) = − 2 cos (πα/2)

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General Fractional Derivatives With Applications in Viscoelasticity

where

α,β,κ Da+,h f

β(κ−α) Ia+,h

(t) =

1

d h(1) (t) dt

κ

(1−β)(κ−α) f Ia+,h

 (t)

(2.615)

and

α,β,κ β(κ−α) Db−,h f (t) = (−1)κ Ib−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) f Ib−,h (t) .

(2.616)

2.28.10 Hilfer–Feller-type fractional derivative with respect to another function In 2018, using the Hilfer fractional derivative with respect to another function and Feller derivatives, the Hilfer–Feller-type fractional derivative with respect to another function was proposed in the book [25]. (1−β)(1−α) f (t) ∈ Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+,h (1−β)(1−α)

AC1 (a, b) and Ib−,h

f (t) ∈ AC1 (a, b).

Definition 2.125 ([25]). The Hilfer–Feller type fractional derivative with respect to another function is defined as F e D α,β,θ f Rz [a,b],h



(t) α,β

α,β

= − H+ (ϑ, α) Da+,h f (t) + H− (ϑ, α) Db−,h f (t) α,β

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

sin((α−ϑ)π/2) sin(πϑ)



(2.617)

α,β

× Db−,h f (t) ,

where H− (ϑ, α) =

sin ((α − ϑ) π/2) , sin (πϑ)

(2.618)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) , sin (πϑ)

(2.619)

α,β,κ Da+,h f

(t) =

β(κ−α) Ia+,h

1

d h(1) (t) dt

κ

(1−β)(κ−α) f Ia+,h

 (t) ,

(2.620)

and

α,β,κ β(κ−α) Db−,h f (t) = (−1)κ Ib−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) f Ib−,h (t) .

(2.621)

Fractional derivatives with singular kernels

205

2.28.11 Hilfer–Herrmann-type fractional derivative with respect to another function In 2018, generalizing the Hilfer fractional derivative with respect to another function and Herrmann derivatives, the Hilfer–Herrmann-type fractional derivative with respect to another function was proposed in the book [25]. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b), −∞ < a < b < ∞, Ia+,h f (t) ∈ (1−β)(1−α)

AC1 (a, b) and Ib−,h

f (t) ∈ AC1 (a, b).

Definition 2.126 ([25]). The Hilfer–Herrmann-type fractional derivative with respect to another function is defined as

 α,β α,β Da+,h f + Db−,h f (t) Ri α,β (2.622) Rz D[a,b],h f (t) = − 2 sin (πα/2) where

α,β,κ Da+,h f

(t) =

and

β(κ−α) Ia+,h

1

d (1) h (t) dt

α,β,κ Db−,h f

(t) = (−1)

κ

β(κ−α) Ib−,h

κ

1

(1−β)(κ−α) f Ia+,h

d h(1) (t) dt

 (t)

κ

 (1−β)(κ−α) f Ib−,h (t) .

(2.623)

(2.624)

Clearly, one has F e α,β,0 Rz D[a,b],h f

i (t) = H Rz D[a,b],h f (t)

α,β

(2.625)

F e α,β,1 Rz D[a,b],h f

(t) = Ri Rz D[a,b],h f (t) .

α,β

(2.626)

and

2.28.12 Hilfer–Riesz fractional derivative with respect to another function Based on the Hilfer fractional derivative with respect to another function and Riesz derivatives, the Hilfer–Riesz fractional derivative with respect to another function will be proposed for the first time. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+,h f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α)

and I−,h

f (t) ∈ AC1 (−∞, ∞).

Definition 2.127. The Hilfer–Riesz fractional derivative with respect to another function is defined as

 α,β α,β D+,h f + D−,h f (t) H i α,β , (2.627) Rz DR,h f (t) = − 2 cos (πα/2)

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General Fractional Derivatives With Applications in Viscoelasticity

where

α,β,κ β(κ−α) D+,h f (t) = I+,h

1

d h(1) (t) dt

κ

(1−β)(κ−α)

I+,h

 f

(t)

(2.628)

and

α,β,κ D−,h f

(t) = (−1)

κ

β(κ−α) I−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) I−,h f (t) .

(2.629)

2.28.13 Hilfer–Feller type fractional derivative with respect to another function Based on the Hilfer fractional derivative with respect to another function and Feller derivatives, the Hilfer–Feller fractional derivative with respect to another function will be proposed. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+,h f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α)

and I−,h

f (t) ∈ AC1 (−∞, ∞).

Definition 2.128. The Hilfer–Feller fractional derivatives with respect to another function is defined as F e D α,β,θ f Rz R,h



(t) α,β

α,β

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

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

sin((α−ϑ)π/2) sin(πϑ)



(2.630)

α,β

× D−,h f (t) ,

where H− (ϑ, α) =

sin ((α − ϑ) π/2) , sin (πϑ)

(2.631)

H+ (ϑ, α) =

sin ((α + ϑ) π/2) , sin (πϑ)

(2.632)



α,β,κ D+,h f

(t) =

β(κ−α) I+,h

1

d (1) h (t) dt

κ

(1−β)(κ−α) I+,h f

 (t) ,

(2.633)

and

α,β,κ β(κ−α) D−,h f (t) = (−1)κ I−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) I−,h f (t) .

(2.634)

Fractional derivatives with singular kernels

207

2.28.14 Hilfer–Herrmann-type fractional derivative with respect to another function Based on the Hilfer fractional derivative with respect to another function and Herrmann derivatives, the Hilfer–Herrmann fractional derivative with respect to another function will be proposed. (1−β)(1−α) Let 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (−∞, ∞), I+,h f (t) ∈ AC1 (−∞, ∞) (1−β)(1−α)

and I−,h

f (t) ∈ AC1 (−∞, ∞).

Definition 2.129. The Hilfer–Herrmann fractional derivative with respect to another function is defined as

 α,β α,β D+,h f + D−,h f (t) Ri α,β , (2.635) Rz DR,h f (t) = − 2 sin (πα/2) where

α,β,κ D+,h f

(t) =

β(κ−α) I+,h

1

d (1) h (t) dt

κ

(1−β)(κ−α) f I+,h

 (t)

(2.636)

and

α,β,κ β(κ−α) D−,h f (t) = (−1)κ I−,h

1

d (1) h (t) dt

κ

 (1−β)(κ−α) f I−,h (t) .

(2.637)

It is clear that F e α,β,0 Rz DR,h f

i (t) = H Rz DR,h f (t)

α,β

(2.638)

F e α,β,1 Rz DR,h f

(t) = Ri Rz DR,h f (t) .

α,β

(2.639)

and

For the theory and application of the fractional calculus, the reader is further referred to the references [104–117].

Fractional derivatives with nonsingular kernels

3

Contents 3.1 History of fractional derivatives with nonsingular kernels 211 3.2 Sonine general fractional calculus with nonsingular kernels 224 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8

Sonine general fractional integrals with nonsingular kernel 224 Sonine general fractional derivatives with nonsingular kernel 225 Sonine–Choudhary-type general fractional derivative with Sonine nonsingular kernel 226 Sonine–Wick–Choudhary-type general fractional calculus with Sonine nonsingular kernel 227 Sonine–Wick–Choudhary-type general fractional derivatives with Sonine nonsingular kernel 228 Sonine–Choudhary-type general fractional derivative with Sonine nonsingular kernel 229 Hilfer–Sonine–Choudhary-type general fractional derivative with nonsingular kernel 229 Sonine general fractional calculus with respect to another function 232

3.3 General fractional derivatives with Mittag-Leffler nonsingular kernel 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8

3.4 General fractional derivatives with Wiman nonsingular kernel 3.4.1 3.4.2 3.4.3 3.4.4

252

General fractional derivatives via Wiman function 252 Prabhakar general fractional integrals on the real line 255 Hilfer type general fractional derivative with Wiman nonsingular kernel 256 General fractional derivatives with respect to another function via Wiman function

3.5 General fractional derivatives with Prabhakar nonsingular kernel 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6 3.5.7 3.5.8

237

Hille–Tamarkin general fractional derivative 238 Hille–Tamarkin general fractional integrals 240 Liouville–Weyl–Hille–Tamarkin-type general fractional calculus 241 Hilfer–Hille–Tamarkin-type general fractional derivative with nonsingular kernel 243 General fractional derivatives with respect to another function via Mittag-Leffler nonsingular kernel 245 Hille–Tamarkin general fractional integrals with respect to another function 247 Liouville–Weyl–Hille–Tamarkin type general fractional calculus with respect to another function 248 Hilfer–Hille–Tamarkin-type general fractional derivative with respect to another function 250

258

263

Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel 264 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel 265 Prabhakar-type general fractional integrals 266 Kilbas–Saigo–Saxena type general fractional derivative with Prabhakar nonsingular kernel on the real line 268 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line 269 Prabhakar-type general fractional integrals on the real line 271 Hilfer-type general fractional derivative with Prabhakar nonsingular kernel 271 General fractional derivatives with respect to another function via Prabhakar nonsingular kernel 274

General Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00008-X Copyright © 2020 Elsevier Inc. All rights reserved.

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General Fractional Derivatives With Applications in Viscoelasticity

3.5.9 Prabhakar-type general fractional integrals with respect to another function 277 3.5.10 Kilbas–Saigo–Saxena-type general fractional derivative on the real line 278 3.5.11 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line 280 3.5.12 Prabhakar-type general fractional integrals with respect to another function on the real line 282 3.5.13 Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel 283

3.6 General fractional derivatives with Gorenflo–Mainardi nonsingular kernel 3.6.1 3.6.2 3.6.3

3.7 General fractional derivatives with Miller–Ross nonsingular kernel 3.7.1 3.7.2 3.7.3

285

Prabhakar-type general fractional integrals 287 Hilfer-type general fractional derivatives with Gorenflo–Mainardi nonsingular kernel 289 Riemann–Liouville–Hilfer general fractional derivatives with Gorenflo–Mainardi nonsingular kernel 290

293

Riemann–Liouville-type general fractional derivative with Miller–Ross nonsingular kernel Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel 294 Hilfer-type general fractional derivatives with Miller–Ross nonsingular kernel 297

293

3.8 General fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 298 3.8.1 3.8.2 3.8.3

Riemann–Liouville-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel 298 Liouville–Sonine–Caputo-type general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 299 Hilfer-type general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 302

3.9 General fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel 303 3.9.1 3.9.2 3.9.3

Riemann–Liouville-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel 303 Liouville–Sonine–Caputo-type general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel 305 Hilfer-type general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel 307

In this chapter, we investigate the history of the fractional derivatives with nonsingular kernels, which are called the general fractional derivatives with nonsingular kernels. We introduce the family of the general fractional derivatives with Sonine, Mittag-Leffler, Wiman, Prabhakar, Gorenflo–Mainardi, Miller–Ross nonsingular kernel, general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel, and general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel. Moreover, we propose the family of Hilfer-type general fractional derivatives based on them.

Fractional derivatives with nonsingular kernels

211

3.1 History of fractional derivatives with nonsingular kernels In 1884, Russian mathematician Sonine presented a generalized version of the Abeltype equation in the form [93,118] t G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.1)

a

with the exact solution given as follows [93,118]: d J (t) = dt

t G2 (t − τ ) F (τ ) dτ ,

(3.2)

a

where t G1 (τ ) G2 (t − τ ) dt = 1.

(3.3)

0

Moreover, Sonine showed [44,118,119] that G1 (t) =

t α−1  (α)

(3.4)

G2 (t) =

t −α ,  (1 − α)

(3.5)

and

where 0 < α < 1. It is called the Sonine integral, or Sonine condition [44]. For the more details of the Sonine integral, see [120,121]. Similarly, b G1 (τ − t) J (τ ) dτ = F (t) ,

(3.6)

t

where d J (t) = − dt

b G2 (τ − t) F (τ ) dτ , t

(3.7)

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General Fractional Derivatives With Applications in Viscoelasticity

with t G1 (τ ) G2 (t − τ ) dt = 1.

(3.8)

0

In 1973, Choudhary considered the extended version of the generalized Abel-type equation of the form [122] ∞ G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.9)

t

with the exact solution given as follows: d J (t) = − dt

∞ G2 (τ − t) F (τ ) dτ ,

(3.10)

t

where t G1 (τ ) G2 (t − τ ) dt = 1.

(3.11)

0

It is called the Sonine–Choudhary integral. Similarly, one has t G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.12)

−∞

where d J (t) = dt

t G2 (t − τ ) F (τ ) dτ ,

(3.13)

−∞

with t G1 (τ ) G2 (t − τ ) dt = 1.

(3.14)

0

In fact, one has from Eq. (3.14) that t

∞ G1 (τ ) G2 (t − τ ) dt =

lim

t→∞ 0

G1 (τ ) G2 (t − τ ) dt, 0

(3.15)

Fractional derivatives with nonsingular kernels

213

which leads to ∞ G1 (τ ) G2 (t − τ ) dt = 1.

(3.16)

0

Thus, we have the following relations: (1A) ∞ G1 (τ − t) J (τ ) dτ = F (t)

(3.17)

t

and d J (t) = − dt

∞ G2 (t − τ ) F (τ ) dτ ,

(3.18)

t

where ∞ G1 (τ ) G2 (t − τ ) dt = 1,

(3.19)

0

(1B) t G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.20)

−∞

and d J (t) = dt

t G2 (t − τ ) F (τ ) dτ ,

(3.21)

−∞

with ∞ G1 (τ ) G2 (t − τ ) dt = 1.

(3.22)

0

In a similar way, we have the following relations: (2A) t G1 (t − τ ) J (τ ) dτ = F (t) , 0

(3.23)

214

General Fractional Derivatives With Applications in Viscoelasticity

and d J (t) = dt

t G2 (t − τ ) F (τ ) dτ ,

(3.24)

0

where t G1 (τ ) G2 (t − τ ) dt = 1.

(3.25)

G1 (τ − t) J (τ ) dτ = F (t) ,

(3.26)

0

(2B) b t

where d J (t) = − dt

b G2 (τ − t) F (τ ) dτ ,

(3.27)

t

and t G1 (τ ) G2 (t − τ ) dt = 1.

(3.28)

0

Taking the Laplace transforms of Eqs. (3.23) and (3.24), we obtain   t L G1 (t − τ ) J (τ ) dτ 0

= G1 (s) J (s)

(3.29)

= L {F (t)} = F (s) and L {J (t)}  =L

d dt

t

 G2 (t − τ ) F (τ ) dτ

0

= sG2 (s) F (s) = J (s) ,

(3.30)

Fractional derivatives with nonsingular kernels

215

where L {J (t)} = J (s), L {G1 (t)} = G1 (s), L {G2 (t)} = G2 (s), and L {F (t)} = F (s). Thus, we have G1 (s) J (s) = F (s)

(3.31)

sG2 (s) F (s) = J (s) ,

(3.32)

and

so that sG1 (s) G2 (s) = 1,

(3.33)

which is the Laplace transform of Eq. (3.28), i.e.,  L

t

 G1 (τ ) G2 (t − τ ) dt

0

= L {1}

(3.34)

= G1 (s) G2 (s) = 1s . Thus, we obtain G2 (s) =

1 sG1 (s)

,

(3.35)

.

(3.36)

or G1 (s) =

1 sG2 (s)

In this case, we have the following results: (3A) Let −∞ < a < b < ∞. The left-sided Sonine integral is defined as t G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.37)

a

and the right-sided Sonine integral is b G1 (τ − t) J (τ ) dτ = F (t) , t

(3.38)

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General Fractional Derivatives With Applications in Viscoelasticity

(3B) Let −∞ < a < b < ∞. The left-sided Sonine derivative is defined as t

d J (t) = dt

G2 (t − τ ) F (τ ) dτ ,

(3.39)

a

and the right-sided Sonine derivative is b

d J (t) = − dt

G2 (τ − t) F (τ ) dτ .

(3.40)

t

(3C) Let −∞ < a < b < ∞. The left-sided Sonine derivative is defined as t J (t) =

G2 (t − τ ) F (1) (τ ) dτ ,

(3.41)

a

and the right-sided Sonine derivative is b G2 (τ − t) F (1) (τ ) dτ .

J (t) = −

(3.42)

t

In a similar way, we get the following results: (4A) Let −∞ < a < b < ∞. The left-sided Sonine–Choudhary integral is defined as t G1 (t − τ ) J (τ ) dτ = F (t) ,

(3.43)

a

and the right-sided Sonine–Choudhary integral is b G1 (τ − t) J (τ ) dτ = F (t) .

(3.44)

t

(4B) Let −∞ < a < b < ∞. The left-sided Sonine–Choudhary derivative is defined as d J (t) = dt

t G2 (t − τ ) F (τ ) dτ ,

(3.45)

a

and the right-sided Sonine–Choudhary derivative is dy J (τ ) = dt

b G2 (τ − t) F (τ ) dτ . t

(3.46)

Fractional derivatives with nonsingular kernels

217

(4C) Let −∞ < a < b < ∞. The left-sided Sonine–Choudhary derivative is defined as t G2 (t − τ ) F (1) (τ ) dτ ,

J (t) =

(3.47)

a

and the right-sided Sonine–Choudhary derivative is b G2 (τ − t) F (1) (τ ) dτ .

J (t) = −

(3.48)

t

We note that Rubin discussed in Eqs. (3.41) and (3.42) (see [120,121]). In 1968, Wick showed [123] G1 (t) = t α−1 1 (t)

(3.49)

G2 (t) = t −α 2 (t) ,

(3.50)

and

where 1 (t) =

∞ 

ak t k

(3.51)

bk t k

(3.52)

k=0

and 2 (t) =

∞  k=0

with the coefficients ak = 0 and bk = 0. Thus, we have the following Sonine–Wick definitions: (4A) The left-sided Sonine–Wick integral can be expressed as F (t) = =

t a t a

(t − τ )α−1 1 (t − τ ) J (τ ) dτ 1 (t−τ ) J (t−τ )1−α

(τ ) dτ,

(3.53)

218

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Sonine–Wick integral is F (t) b = (τ − t)α−1 1 (τ − t) J (τ ) dτ =

t b t

1 (τ −t) J (τ −t)1−α

(3.54)

(τ ) dτ ,

where 1 (t) =

∞ 

ak t k

(3.55)

k=0

with the coefficients ak = 0. (4B) The left-sided Sonine–Wick derivative can be expressed as J (t) = =

t

d dt

a t

d dt

a

(t − τ )−α 2 (t − τ ) F (τ ) dτ 2 (t−τ ) (t−τ )α F

(3.56)

(τ ) dτ,

and the right-sided Sonine–Wick derivative is J (t) d = − dt d = − dt

b t b t

(τ − t)−α 2 (τ − t) F (τ ) dτ 2 (τ −t) (τ −t)α F

(3.57)

(τ ) dτ ,

where 2 (t) =

∞ 

bk t k

(3.58)

k=0

with the coefficients bk = 0. (4C) The left-sided Sonine–Wick-type derivative can be expressed as J (t) t = (t − τ )−α 2 (t − τ ) F (1) (τ ) dτ =

a t a

2 (t−τ ) (1) (τ ) dτ, (t−τ )α F

(3.59)

Fractional derivatives with nonsingular kernels

219

and the right-sided Sonine–Wick-type derivative can be written as J (t) d = − dt

=−

b t

b

(τ − t)−α 2 (τ − t) F (1) (τ ) dτ

t

(3.60)

2 (τ −t) (1) (τ ) dτ , (τ −t)α F

where 2 (t) =

∞ 

bk t k

(3.61)

k=0

with the coefficients bk = 0. Similarly, one has the following results [122]: (4A) The left-sided Sonine–Wick–Choudhary integral can be expressed as F (t) t = (t − τ )α−1 1 (t − τ ) J (τ ) dτ =

−∞ t −∞

1 (t−τ ) J (t−τ )1−α

(3.62)

(τ ) dτ,

and the right-sided Sonine–Wick–Choudhary integral is F (t) ∞ = (τ − t)α−1 1 (τ − t) J (τ ) dτ =

t ∞ t

1 (τ −t) J (τ −t)1−α

(3.63)

(τ ) dτ ,

where 1 (t) =

∞ 

ak t k

(3.64)

k=0

with the coefficients ak = 0. (4B) The left-sided Sonine–Wick–Choudhary derivative can be expressed as J (t) =

d dt

=

d dt

t −∞ t −∞

(t − τ )−α 2 (t − τ ) F (τ ) dτ 2 (t−τ ) (t−τ )α F

(τ ) dτ,

(3.65)

220

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Sonine–Wick–Choudhary derivative can be obtained as J (t) d = − dt d = − dt

∞ t ∞ t

(τ − t)−α 2 (τ − t) F (τ ) dτ 2 (τ −t) (τ −t)α F

(3.66)

(τ ) dτ ,

where 2 (t) =

∞ 

bk t k

(3.67)

k=0

with the coefficients bk = 0. (4C) The left-sided Sonine–Wick–Choudhary-type derivative can be expressed as J (t) t = (t − τ )−α 2 (t − τ ) F (1) (τ ) dτ =

−∞ t −∞

(3.68)

2 (t−τ ) (1) (τ ) dτ, (t−τ )α F

and the right-sided Sonine–Wick–Choudhary-type derivative can be given as J (t) d = − dt

=−

∞

(τ − t)−α 2 (τ − t) F (1) (τ ) dτ

t

∞ 2 (τ −t) t

(τ −t)α

(3.69)

F (1) (τ ) dτ ,

where 2 (t) =

∞ 

bk t k

(3.70)

k=0

with the coefficients bk = 0. We now present two examples for the fractional and tempered fractional calculi. Taking the Laplace transforms, we have  α−1  t L {G1 (t)} = L = s 1−α (3.71)  (α) and

 L {G2 (t)} = L

 t −α = sα  (1 − α)

(3.72)

Fractional derivatives with nonsingular kernels

221

so that L {G1 (t)} L {G2 (t)} = s 1−α × s α = s.

(3.73)

From Eqs. (3.71), (3.72), and (3.73), we may derive the Riemann–Liouville and Liouville–Weyl fractional calculi. Taking the Laplace transforms, we have  

1−α 1 1−α E s −1 (3.74) L {G1 (t)} = L − τ = s −1 λ + s (−λ (t )) α+1 1,−α (t − τ ) and





α−1 = s −1 λ + s

(3.75)

α−1 

1−α L {G1 (t)} L {G2 (t)} = s −1 λ + s × s −1 λ + s s = s.

(3.76)

e−λt L {G2 (t)} = L tα so that

From Eqs. (3.74), (3.75), and (3.76), we may derive the Riemann–Liouville and Liouville–Weyl tempered fractional calculi. In 1884, Sonine showed that [93] S (λt p ) ∞

= t −p =

∞

n=0

n=0

(λt)n (n+1)(n−p+1)

(3.77)

λn t n−p (n+1)(n−p+1)

and S (−λt q ) ∞

= t −q =

∞

n=0

(−λt)n (n+1)(n−q+1)

n=0 (−λ)n t n−q (n+1)(n−q+1) ,

(3.78)

which are called the first Sonine functions (see [25]), where p + q = 1 and λ is a constant. When p > 0, one gets S1 (λt p ) ∞

= t −p =

∞

n=0

n=0

(λt)n (n+1)(n−p+1) λn t n−p

(n+1)(n−p+1)

(3.79)

222

General Fractional Derivatives With Applications in Viscoelasticity

and S2 (λt p )

∞

= t −(1−p) = =

∞

n=0 ∞

n=0

(−λt)n (n+1)(n−(1−p)+1)

n=0 (−λ)n t n−(1−p) (n+1)(n−(1−p)+1)

(3.80)

(−λ)n t n+p−1 (n+1)(n+p) ,

which are called the first Sonine function, denoted as S1 (λt p ), and the second Sonine function, denoted as S2 (λt p ), respectively. In fact, we find that ∞    Y λt α =

λn t α ,  (n + 1)  (α + 1)

n=0

(3.81)

where α ∈ R+ and λ is a constant. Taking the Laplace transforms of Eqs. (3.79), (3.80) and (3.81), we have L {S1 (λt p )} ∞ 

λn t n−p =L (n+1)(n−p+1) =

n=0

∞

n=0

= sp

λn s −(n−p) (n+1)

∞

n=0 λ p =s es

(3.82)

λn s −n (n+1)

  λ    0 and λ ∈ R. The left-sided Sonine–Wick–Choudhary-type general fractional integral is defined as t α,λ S I+ f

  S1 λ (t − τ )α f (τ ) dτ ,

(t) =

(3.113)

−∞

and the right-sided Sonine–Wick–Choudhary-type general fractional integral is ∞ α,λ S I− f

(t) =

  S1 λ (τ − t)α f (τ ) dτ ,

(3.114)

t

where the nonsingular kernel is ∞    S1 λt α = n=0

λn t n−α .  (n + 1)  (n − α + 1)

(3.115)

3.2.5 Sonine–Wick–Choudhary-type general fractional derivatives with Sonine nonsingular kernel Definition 3.5. Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine–Wick–Choudhary-type general fractional derivative is defined as α,λ S D+ f

=

d dt

=

d dt



(t)

α,λ I f (t) SH + t    S2 λ (t − τ )α+1 f (τ ) dτ,

(3.116)

−∞

where t α,λ SH I+ f

(t) =



S2 λ (t − τ )α+1 f (τ ) dτ ,

(3.117)

−∞

and the right-sided Sonine–Wick–Choudhary-type general fractional derivative is α,λ S D− f d = − dt d = − dt

(t)

α,λ SH I− f

∞ t

(t)

  S2 λ (τ − t)α+1 f (τ ) dτ,

(3.118)

Fractional derivatives with nonsingular kernels

229

where ∞ α,λ SH I− f

(t) =



S2 λ (τ − t)α+1 f (τ ) dτ ,

(3.119)

t

with the nonsingular kernel being ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.120)

3.2.6 Sonine–Choudhary-type general fractional derivative with Sonine nonsingular kernel Definition 3.6. Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine–Choudhary-type general fractional derivative is defined as α,λ SC D+ f

(t)

= SH I+α,λ f (1) (t) t 

=

−∞

S2 λ (t − τ )

 α+1

(3.121) f (1) (τ ) dτ,

and the right-sided Sonine–Choudhary-type general fractional derivative is α,λ SC D− f

(t)

= −SH I−α,λ f (1) (t) ∞   = − S2 λ (τ − t)α+1 f (1) (τ ) dτ,

(3.122)

t

with the nonsingular kernel being ∞  α  S2 λt = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.123)

3.2.7 Hilfer–Sonine–Choudhary-type general fractional derivative with nonsingular kernel Based on Hilfer’s idea, as well as definitions of Sonine general fractional derivative and Sonine–Choudhary-type general fractional derivative, we propose new general fractional derivatives with nonsingular kernel, which are called the Hilfer–Sonine– Choudhary-type general fractional derivatives with nonsingular kernel.

230

General Fractional Derivatives With Applications in Viscoelasticity

Definition 3.7. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer–Sonine–Choudhary-type general fractional derivative is defined as   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.124) S Da+ f (t) = SH Ia+ SH Ia+ dt (1−β)(1−α),λ

where SH Ia+

β(1−α),λ f SH Ia+

f (t) ∈ AC1 (a, b), t

(t) =



S2 λ (t − τ )β(1−α)+1 f (τ ) dτ ,

(3.125)

a

and (1−β)(1−α),λ f SH Ia+

t (t) =



S2 λ (t − τ )(1−β)(1−α)+1 f (τ ) dτ ,

(3.126)

a

and the right-sided Hilfer–Sonine–Choudhary-type general fractional derivative is   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.127) S Db− f (t) = − SH Ib− SH Ib− dt (1−β)(1−α),λ

where SH Ib−

β(1−α),λ f SH Ib−

f (t) ∈ AC1 (a, b), b

(t) =



S2 λ (τ − t)β(1−α)+1 f (τ ) dτ ,

(3.128)

t

and (1−β)(1−α),λ f SH Ib−

b (t) =



S2 λ (τ − t)(1−β)(1−α)+1 f (τ ) dτ ,

(3.129)

t

with ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.130)

Property 3.3 ([25,102]). Suppose 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and (1−β)(1−α),λ f (t) ∈ AC1 (a, b), then SH I0+  L

α,β,λ S D0+ f

 λ (1−β)(1−α),λ f (0) . (t) (s) = s α f (s) − s 1−β(1−α) e− s SH I0+

(3.131)

Fractional derivatives with nonsingular kernels

231

Proof. Direct computation gives  L

α,β,λ S D0+ f

=L



 (t) (s)

β(1−α),λ d (1−β)(1−α),λ SH Ia+ dt SH Ia+



f (t)



λ λ (1−β)(1−α),λ = s 1−β(1−α) e− s s × s 1−(1−β)(1−α) e− s f (s) − SH I0+ f (0) λ

λ

λ

= s 1−β(1−α) e− s × s × s 1−(1−β)(1−α) e− s f (s) − s 1−β(1−α) e− s SH I0+ = s α f (s) − s 1−β(1−α) e

− λs SH

(1−β)(1−α),λ I0+ f

(1−β)(1−α),λ

f (0)

(0) . (3.132)

Thus, we finish the proof. Definition 3.8. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer–Sonine–Choudhary-type general fractional derivative on the real line is defined as   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.133) D f = I I (t) S + SH + SH + dt (1−β)(1−α),λ

where SH I+

β(1−α),λ f SH I+

f (t) ∈ AC1 (−∞, ∞), t



S2 λ (t − τ )β(1−α)+1 f (τ ) dτ ,

(t) =

(3.134)

−∞

and (1−β)(1−α),λ

SH I+

t f (t) =



S2 λ (t − τ )(1−β)(1−α)+1 f (τ ) dτ ,

(3.135)

−∞

and the right-sided Hilfer–Sonine–Choudhary-type general fractional derivative on the real line is   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.136) D f = − I I (t) S − SH − SH − dt (1−β)(1−α),λ

where SH I−

β(1−α),λ f SH I−

f (t) ∈ AC1 (−∞, ∞), ∞

(t) = t



S2 λ (τ − t)β(1−α)+1 f (τ ) dτ ,

(3.137)

232

General Fractional Derivatives With Applications in Viscoelasticity

and (1−β)(1−α),λ f SH I−

∞ (t) =



S2 λ (τ − t)(1−β)(1−α)+1 f (τ ) dτ .

(3.138)

t

3.2.8 Sonine general fractional calculus with respect to another function In this section, we present the general fractional calculus with respect to another function. Definition 3.9. Let h(1) (t) > 0, α > 0 and λ ∈ R. The left-sided Sonine general fractional integral with respect to another function in the interval [a, b] is defined as t α,λ S Ia+,h f

(t) =

  S1 λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ ,

(3.139)

a

and the right-sided Sonine general fractional integral with respect to another function in the interval [a, b] is b α,λ S Ib− f

  S1 λ (h (τ ) − h (t))α h (τ ) f (τ ) dτ ,

(t) =

(3.140)

t

where the nonsingular kernel is ∞    S1 λt α = n=0

λn t n−α .  (n + 1)  (n − α + 1)

(3.141)

Definition 3.10. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine general fractional derivative with respect to another function in the interval [a, b] is defined as α,λ S Da+,h f

= =





1 h(1) (t)

(t)

d dt

1 d h(1) (t) dt

t

α,λ SH Ia+,h f

(t)

(3.142)

  S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) f (τ ) dτ,

a

where t α,λ SH Ia+,h f

(t) = a



S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) f (τ ) dτ ,

(3.143)

Fractional derivatives with nonsingular kernels

233

and the right-sided Sonine general fractional derivative with respect to another function in the interval [a, b] is α,λ S Db−,h f



(t)



α,λ d = − h(1)1(t) dt SH Ib−,h f (t)

b   d S2 λ (h (τ ) − h (t))α+1 h(1) (τ ) f (τ ) dτ = − h(1)1(t) dt

(3.144)

t

where b α,λ SH Ib−,h f

(t) =



S2 λ (h (τ ) − h (t))α+1 h(1) (τ ) f (τ ) dτ ,

(3.145)

t

with the nonsingular kernel being ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.146)

Definition 3.11. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine–Choudhary general fractional derivative with respect to another function in the interval [a, b] is defined as α,λ SC Da+,h f α,λ = SH Ia+,h t 

(t)

1

h(1) (t)

d dt

f (t)



 d = S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) h(1)1(τ ) dτ f (1) (τ ) dτ,

(3.147)

a

and the right-sided Sonine–Choudhary general fractional derivative with respect to another function in the interval [a, b] is α,λ SC Db−,h f

(t)

α,λ d = SH Ib−,h − h(1)1(t) dt f (t)

b    d = S2 λ (h (τ ) − h (t))α+1 h(1) (τ ) − h(1)1(τ ) dτ f (1) (τ ) dτ,

(3.148)

t

with the nonsingular kernel being ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.149)

234

General Fractional Derivatives With Applications in Viscoelasticity

Definition 3.12. Let h(1) (t) > 0, α > 0 and λ ∈ R. The left-sided Sonine–Wick– Choudhary-type general fractional integral with respect to another function is defined as t α,λ S I+,h f

  S1 λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ ,

(t) =

(3.150)

−∞

and the right-sided Sonine–Wick–Choudhary-type general fractional integral with respect to another function is ∞ α,λ S I−,h f

(t) =

  S1 λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ ,

(3.151)

t

where the nonsingular kernel is ∞  α  S1 λt = n=0

λn t n−α .  (n + 1)  (n − α + 1)

(3.152)

Definition 3.13. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine–Wick–Choudhary-type general fractional derivative with respect to another function is defined as α,λ S D+,h f

= =





(t)

1 h(1) (t)

d dt

1 d h(1) (t) dt



α,λ SH I+,h f

t

−∞

(t)

(3.153)

  S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) f (τ ) dτ,

where t α,λ SH I+,h f

(t) =



S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) f (τ ) dτ ,

(3.154)

−∞

and the right-sided Sonine–Wick–Choudhary-type general fractional derivative with respect to another function is α,λ S D−,h f

(t)



α,λ d = − h(1)1(t) dt SH I−,h f (t)

∞   d S2 λ (h (τ ) − h (t))α+1 h(1) (τ ) f (τ ) dτ , = − h(1)1(t) dt

t

(3.155)

Fractional derivatives with nonsingular kernels

235

where ∞ α,λ SH I−,h f

(t) =



S2 λ (h (τ ) − h (t))α+1 h(1) (τ ) f (τ ) dτ ,

(3.156)

t

with the nonsingular kernel being ∞  α  S2 λt = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.157)

Definition 3.14. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Sonine–Choudhary-type general fractional derivative with respect to another function is defined as α,λ SC D+,h f

(t)

α,λ 1 d = SH I+,h f (t) (1) h (t) dt

t   d S2 λ (h (t) − h (τ ))α+1 h(1) (τ ) h(1)1(τ ) dτ f (τ ) dτ , =

(3.158)

−∞

and the right-sided Sonine–Choudhary-type general fractional derivative with respect to another function is α,λ SC D−,h f

(t)

α,λ = −SH I−,h ∞ 

=−

t

d 1 f h(1) (t) dt

(t)

(3.159)



 d S2 λ (τ − t)α+1 h(1) (τ ) h(1)1(τ ) dτ f (τ ) dτ,

with the nonsingular kernel being ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.160)

Definition 3.15. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer–Sonine–Choudhary-type general fractional derivative with respect to another function is defined as 

 α,β,λ

S Da+,h f (t) =

β(1−α),λ

SH Ia+,h

1

d h(1) (t) dt



 (1−β)(1−α),λ

SH Ia+,h

f (t) ,

(3.161)

236

General Fractional Derivatives With Applications in Viscoelasticity (1−β)(1−α),λ

where SH Ia+,h

β(1−α),λ f SH Ia+,h

f (t) ∈ AC1 (a, b), t

(t) =



S2 λ (h (t) − h (τ ))β(1−α)+1 h(1) (τ ) f (τ ) dτ ,

(3.162)

a

and (1−β)(1−α),λ f SH Ia+,h

t (t) =



S2 λ (h (t) − h (τ ))(1−β)(1−α)+1 h(1) (τ ) f (τ ) dτ ,

a

(3.163) and the right-sided Hilfer–Sonine–Choudhary type general fractional derivative with respect to another function is 

 α,β,λ S Db−,h f

β(1−α),λ SH Ib−,h

(t) = −

(1−β)(1−α),λ

where SH Ib−,h

β(1−α),λ f SH Ib−,h

1

d (1) h (t) dt



 (1−β)(1−α),λ SH Ib−,h

f (t) ,

(3.164)



S2 λ (h (τ ) − h (t))β(1−α)+1 h(1) (τ ) f (τ ) dτ ,

(3.165)

f (t) ∈ AC1 (a, b), b

(t) = t

and

(1−β)(1−α),λ f SH Ib−,h

b (t) =



S2 λ (h (τ ) − h (t))(1−β)(1−α)+1 h(1) (τ ) f (τ ) dτ ,

t

(3.166) with ∞    S2 λt α = n=0

(−λ)n t n+α−1 .  (n + 1)  (n + α)

(3.167)

Definition 3.16. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer–Sonine–Choudhary-type general fractional derivative with respect to another function on the real line is defined as 

 α,β,λ

S D+,h f (t) =

β(1−α),λ

SH I+,h

1

d h(1) (t) dt



 (1−β)(1−α),λ

SH I+,h

f (t) ,

(3.168)

Fractional derivatives with nonsingular kernels (1−β)(1−α),λ

where SH I+,h

β(1−α),λ f SH I+,h

237

f (t) ∈ AC1 (−∞, ∞), t

(t) =



S2 λ (h (t) − h (τ ))β(1−α)+1 h(1) (τ ) f (τ ) dτ ,

(3.169)

−∞

and t

(1−β)(1−α),λ f SH I+,h

(t) =



S2 λ (h (t) − h (τ ))(1−β)(1−α)+1 h(1) (τ ) f (τ ) dτ ,

−∞

(3.170) and the right-sided Hilfer–Sonine–Choudhary-type general fractional derivative with respect to another function on the real line is  α,β,λ

(1−β)(1−α),λ

where SH I−,h

β(1−α),λ f SH I−,h

 β(1−α),λ

S D−,h f (t) =

SH I−,h

1

d (1) h (t) dt



 (1−β)(1−α),λ

SH I−,h

f (t) ,

(3.171)

f (t) ∈ AC1 (−∞, ∞), ∞

(t) =



S2 λ (h (τ ) − h (t))β(1−α)+1 h(1) (τ ) f (τ ) dτ ,

(3.172)

t

and (1−β)(1−α),λ f SH I−,h

∞ (t) =



S2 λ (h (τ ) − h (t))(1−β)(1−α)+1 h(1) (τ ) f (τ ) dτ .

t

(3.173)

3.3

General fractional derivatives with Mittag-Leffler nonsingular kernel

In this section, the Hille–Tamarkin general fractional derivative involving the MittagLeffler function kernel with power law in the sense of Riemann–Liouville will be considered. This derivative was proposed in 1930 by Hille and Tamarkin, and now it is called the Hille–Tamarkin general fractional derivative [124]. In fact, some of the presented forms of the general fractional derivatives via Mittag-Leffler function were given in [25].

238

General Fractional Derivatives With Applications in Viscoelasticity

3.3.1 Hille–Tamarkin general fractional derivative Definition 3.17 ([124,25]). Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Hille–Tamarkin general fractional derivative is defined as RL α,λ Ml Da+ f

d (t) = dt

t

  Eα −λ (t − τ )α f (τ ) dτ ,

(3.174)

a

and the right-sided Hille–Tamarkin general fractional derivative is RL α,λ Ml Db− f

b

d (t) = − dt

  Eα −λ (τ − t)α f (τ ) dτ .

(3.175)

t

Definition 3.18. Let α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Hille–Tamarkin general fractional derivative is defined as (for λ = ±1, see [130–135]; also see [25]) RL α,κ,λ Ml Da+ f

dκ (t) = κ dt

t

  Eα −λ (t − τ )α f (τ ) dτ ,

(3.176)

a

and the right-sided Hille–Tamarkin general fractional derivative is RL α,κ,λ Ml Db− f

dκ (t) = (−1) dt κ

b

κ

  Eα −λ (τ − t)α f (τ ) dτ .

(3.177)

t

For κ = 1 and a = 0, we have d dt

t

Eα (−λ (t − τ )α ) f (τ ) dτ

0 t

(3.178)

= Eα (−λ (t − τ )α ) f (1) (τ ) dτ − Eα (−λt α ) f (0) , 0

such that the Gorenflo–Mainardi formulation [136] can be obtained as d dt

t

Eα (−λ (t − τ )α ) f (τ ) dτ

0

(3.179)

t

= Eα (−λ (t − τ )

α

) f (1) (τ ) dτ

+ Eα

(−λt α ) f

(0)

0

due to t



Eα −λτ 0

α



t f

(1)

(t − τ ) dτ = 0

  Eα −λ (t − τ )α f (1) (τ ) dτ .

(3.180)

Fractional derivatives with nonsingular kernels

239

Gorenflo and Mainardi did not continue to investigate the Hille–Tamarkin general fractional derivative [136]. Definition 3.19 ([25]). Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is defined as t LSC α,λ Ml Da+ f

  Eα −λ (t − τ )α f (1) (τ ) dτ ,

(t) =

(3.181)

a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is b LSC α,λ Ml Db− f

(t) = −

  Eα −λ (τ − t)α f (1) (τ ) dτ .

(3.182)

t

Definition 3.20. Let α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is defined as t LSC α,κ,λ Ml Da+ f

(t) =

  Eα −λ (t − τ )α f (κ) (τ ) dτ ,

(3.183)

a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is b LSC α,κ,λ Ml Db− f

(t) = (−1)

κ

  Eα −λ (τ − t)α f (κ) (τ ) dτ .

(3.184)

t

However, we have to repeat the following cases: In 2016, the left-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is defined as [137] LSC α,λ MlA Da+ f

 (α) (t) = 1−α



t Eα a

 α α − (t − τ ) f (1) (τ ) dτ , 1−α

(3.185)

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is [137] LSC α,λ MlA Db− f

 (α) (t) = 1−α



b Eα t

 α α − (τ − t) f (1) (τ ) dτ , 1−α

(3.186)

240

General Fractional Derivatives With Applications in Viscoelasticity

the left-sided general fractional derivative via Mittag-Leffler function is defined as [137]

α,λ RL MlA Da+ f

(t) =

 (α) d 1 − α dt

t a

  α Eα − (t − τ )α f (τ ) dτ , 1−α

(3.187)

and the right-sided general fractional derivative via Mittag-Leffler function is [137]

α,κ,λ RL MlA Db− f

 (α) d (t) = (−1) 1 − α dt

b

κ

t

  α α Eα − (τ − t) f (τ ) dτ , 1−α

(3.188)

where  (α) is the normalization constant such that  (0) = 1 and  (1) = 1.

3.3.2 Hille–Tamarkin general fractional integrals As the inverse operator of the Hille–Tamarkin general fractional derivative, we have the following results for integrals: Definition 3.21 ([25]). Let α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Hille–Tamarkin general fractional integral, as the inverse operator of the Hille–Tamarkin general fractional derivative, is defined as t RL α,κ,λ Ml Ia+ f

(t) =

1 (t − τ )

α+1−κ

a

  −1 −λ (t − τ )α f (τ ) dτ , Eα,κ−α

(3.189)

and the right-sided Hille–Tamarkin general fractional integral is b RL α,κ,λ Ml Ib− f

(t) = (−1)

κ

1 (τ − t)

α+1−κ

t

  −1 −λ (τ − t)α f (τ ) dτ . Eα,κ−α

(3.190)

For κ = 1, the left-sided Hille–Tamarkin general fractional integral, as the inverse operator of the Hille–Tamarkin general fractional derivative, is defined as [124,25] RL I α,λ f (t) Ml a+ t 1 −1 α = (t−τ )α Eα,1−α (−λ (t − τ ) ) f a t f (τ ) λ = f (t) + (α) dτ , (t−τ )1−α a

(τ ) dτ

(3.191)

Fractional derivatives with nonsingular kernels

241

and the right-sided Hille–Tamarkin general fractional integral is RL I α,λ f Ml b− b

=−

t 

(t)

−1 1 (τ −t)α Eα,1−α (−λ (τ

= − f (t) +

λ (α)

b t

− t)α ) f (τ ) dτ 

f (τ ) dτ (τ −t)1−α

(3.192)

.

We have the following properties of the Hille–Tamarkin general fractional calculus. Property 3.4 ([25]). Let α > 0, κ = [α] +1 and λ ∈ R. Then     α,λ α,1,λ −α −1 f (s) − RL L RL Ml D0+ f (t) = 1 + λs Ml I0+ f (+0) .

(3.193)

Property 3.5 ([25]). Let 1 > α > 0 and λ ∈ R. Then     α,λ α,λ −α −1 L RL f (s) − RL Ml D0+ f (t) = 1 + λs Ml I0+ f (+0) .

(3.194)

Property 3.6 ([25]). Let α > 0, κ = [α] +1 and λ ∈ R. Then  L

RL D α,κ,λ f Ml 0+

  −1 f (s) (t) = s κ−1 1 + λs −α 

 κ−1

κ−μ−1 α,κ,λ − s I f . D μ RL (+0) Ml 0+

(3.195)

κ=0

Property 3.7 ([25]). Let α > 0, κ = [α] +1 and λ ∈ R. Then ⎞ ⎛ κ      −1 α,κ,λ −1 ⎝s κ f (s) − L LSC s κ−j f (j −1) (0)⎠ . (3.196) 1 + λs −α Ml D0+ f (t) = s j =1

3.3.3 Liouville–Weyl–Hille–Tamarkin-type general fractional calculus Based on the Liouville–Weyl fractional calculus, the Liouville–Weyl–Hille–Tamarkintype general fractional calculus was proposed in the book [25]. Definition 3.22 ([25]). Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative is defined as α,λ RL Mlr D+ f

d (t) = dt

t −∞

  Eα −λ (t − τ )α f (τ ) dτ ,

(3.197)

242

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative is α,λ RL Mlr D− f

d (t) = − dt

∞

  Eα −λ (τ − t)α f (τ ) dτ .

(3.198)

t

Definition 3.23 ([25]). Let α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative is defined as dκ (t) = κ dt

α,κ,λ RL f Mlr D+

t

  Eα −λ (t − τ )α f (τ ) dτ ,

(3.199)

−∞

and the right-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative is RL α,κ,λ f Ml D−

dκ (t) = (−1) dt κ

∞

κ

  Eα −λ (τ − t)α f (τ ) dτ .

(3.200)

t

Definition 3.24 ([25]). Let 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function on the real line is defined as t LSC α,λ Ml D+ f

(t) =

  Eα −λ (t − τ )α f (1) (τ ) dτ ,

(3.201)

−∞

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function on the real line is ∞ LSC α,λ Ml D− f

(t) = (−1)

κ

  Eα −λ (τ − t)α f (1) (τ ) dτ .

(3.202)

t

Definition 3.25 ([25]). Let α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is defined as t LSC α,κ,λ f Ml D+

(t) = −∞

  Eα −λ (t − τ )α f (κ) (τ ) dτ ,

(3.203)

Fractional derivatives with nonsingular kernels

243

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function is ∞ LSC α,κ,λ f Ml D−

(t) = (−1)

κ

  Eα −λ (τ − t)α f (κ) (τ ) dτ .

(3.204)

t

Definition 3.26 ([25]). Let α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin-type general fractional integral, as the inverse operator of the Liouville–Weyl–Hille–Tamarkin-type general fractional derivative, is defined as t RL α,κ,λ f Ml I+

1

(t) =

(t − τ )

α+1−κ

−∞

  −1 −λ (t − τ )α f (τ ) dτ , Eα,κ−α

(3.205)

and the right-sided Liouville–Weyl–Hille–Tamarkin-type general fractional integral is ∞ RL α,κ,λ f Ml I−

(t) = (−1)

1

κ

(τ − t)

α+1−κ

t

  −1 −λ (τ − t)α f (τ ) dτ . Eα,κ−α

(3.206)

3.3.4 Hilfer–Hille–Tamarkin-type general fractional derivative with nonsingular kernel Based on Hilfer’s idea, as well as definitions of Hille–Tamarkin general fractional derivative, and Liouville–Sonine–Caputo-type general fractional derivative via Mittag-Leffler function, we propose new general fractional derivatives with MittagLeffler kernel, which are called the Hilfer–Hille–Tamarkin-type general fractional derivatives with nonsingular kernel. Definition 3.27. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer–Hille–Tamarkin-type general fractional derivative is defined as  α,β,λ

H H T Da+ f (t) =

(1−β)(1−α),λ

where ML Ia+

β(1−α),λ f ML Ia+

β(1−α),λ

ML Ia+

 d (1−β)(1−α),λ I f (t) , ML a+ dt

(3.207)

f (t) ∈ AC1 (a, b), t

(t) = a



Eα −λ (t − τ )β(1−α) f (τ ) dτ ,

(3.208)

244

General Fractional Derivatives With Applications in Viscoelasticity

and (1−β)(1−α),λ f ML Ia+

t (t) =



Eα −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.209)

a

and the right-sided Hilfer–Hille–Tamarkin-type general fractional derivative is   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.210) D f = − I I (t) H H T b− ML b− ML b− dt (1−β)(1−α),λ

where ML Ib−

β(1−α),λ f ML Ib−

f (t) ∈ AC1 (a, b), b

(t) =



Eα −λ (τ − t)β(1−α) f (τ ) dτ ,

(3.211)

t

and (1−β)(1−α),λ f ML Ib−

b (t) =



Eα −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.212)

t

Property 3.8. (1) Suppose 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and (1−β)(1−α),λ f (t) ∈ AC1 (a, b), then ML I0+   λ α,β,λ (1−β)(1−α),λ L H H T D0+ f (t) (s) = s α f (s)−s 1−β(1−α) e− s SH I0+ f (0) . (3.213) Proof. By straightforward computation,  L

α,β,λ H H T D0+ f

=L



 (t) (s)

β(1−α),λ d (1−β)(1−α),λ ML I0+ dt ML I0+



 f (t)

 −1  −1 = s −1 1 + λs −β(1−α) s × s −1 1 + λs −(1−β)(1−α) f (s)

(1−β)(1−α),λ −SH I0+ f (0)

 −1  −1 (1−β)(1−α),λ = s −1 1 + λs −β(1−α) 1 + λs −(1−β)(1−α) f (s) − ML I0+ f (0) , (3.214) which finishes the proof. Definition 3.28. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer–Hille–Tamarkin-type general fractional derivative on the real line is defined as   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.215) D f = I I (t) HHT + ML + ML + dt

Fractional derivatives with nonsingular kernels (1−β)(1−α),λ

where ML I+

β(1−α),λ

ML I+

245

f (t) ∈ AC1 (−∞, ∞), t



Eα −λ (t − τ )β(1−α) f (τ ) dτ ,

f (t) =

(3.216)

−∞

and (1−β)(1−α),λ f ML I+

t



Eα −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(t) =

(3.217)

−∞

and the right-sided Hilfer–Hille–Tamarkin-type general fractional derivative on the real line is   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.218) D f = − I I (t) HHT − ML − ML − dt (1−β)(1−α),λ

where ML I−

β(1−α),λ f ML I−

f (t) ∈ AC1 (−∞, ∞), ∞

(t) =



Eα −λ (τ − t)β(1−α) f (τ ) dτ ,

(3.219)

t

and (1−β)(1−α),λ f ML I−

∞ (t) =



Eα −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.220)

t

3.3.5 General fractional derivatives with respect to another function via Mittag-Leffler nonsingular kernel In this section, we present the general fractional derivatives with respect to another function via Mittag-Leffler nonsingular kernel. Definition 3.29. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Hille–Tamarkin general fractional derivative with respect to another function is defined as  RL α,λ Ml Da+,h f

(t) =

1

d (1) h (t) dt

 t

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

a

(3.221)

246

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Hille–Tamarkin general fractional derivative is 

RL α,λ Ml Db−,h f

1

d (t) = − (1) h (t) dt

 b

  Eα −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.222) Definition 3.30. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Hille–Tamarkin general fractional derivative with respect to another function is defined as  RL α,κ,λ Ml Da+,h f

1

d h(1) (t) dt

(t) =

κ  t

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

a

(3.223) and the right-sided Hille–Tamarkin general fractional derivative with respect to another function is  RL α,κ,λ Ml Db−,h f

−1 d h(1) (t) dt

(t) =

κ b

  Eα −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.224) Definition 3.31. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is defined as t LSC α,λ Ml Da+,h f

(t) =

  Eα −λ (h (t) − h (τ ))α h(1) (τ )



1 d (1) h (τ ) dτ

 f (τ ) dτ ,

a

(3.225) and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is b LSC α,λ Ml Db−,h f

(t) = −

  Eα −λ (h (τ ) − h (t))α h(1) (τ )



1 d (1) h (τ ) dτ

 f (τ ) dτ .

t

(3.226) Definition 3.32. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is defined as

Fractional derivatives with nonsingular kernels

LSC D α,κ,λ f Ml a+,h t

247

(t)

= Eα (−λ (h (t) − h (τ ))α ) h(1) (τ ) a



1 d h(1) (τ ) dτ

κ

(3.227) f (τ ) dτ ,

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is LSC D α,κ,λ f b−,h Ml b κ

= (−1)

(t)

Eα (−λ (h (τ ) − h (t))α ) h(1) (τ )

t



1 h(1) (τ )

d dτ

κ

(3.228) f (τ ) dτ .

3.3.6 Hille–Tamarkin general fractional integrals with respect to another function For the inverse operator of the Hille–Tamarkin general fractional derivative with respect to another function, we have the following results: Definition 3.33. Let h(1) (t) > 0, α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Hille–Tamarkin general fractional integral with respect to another function, as the inverse operator of the Hille–Tamarkin general fractional derivative with respect to another function, is defined as t RL α,κ,λ Ml Ia+,h f

(t) =

h(1) (τ ) (h (t) − h (τ ))

α+1−κ

a

  −1 Eα,κ−α −λ (h (t) − h (τ ))α f (τ ) dτ , (3.229)

and the right-sided Hille–Tamarkin general fractional integral with respect to another function is b RL α,κ,λ Ml Ib−,h f

(t) = (−1)

κ

h(1) (τ ) (h (τ ) − h (t))

α+1−κ

t

  −1 −λ (h (τ ) − h (t))α f (τ ) dτ . Eα,κ−α (3.230)

For h(1) (t) > 0 and κ = 1, the left-sided Hille–Tamarkin general fractional integral with respect to another function, as the inverse operator of the Hille–Tamarkin general fractional derivative with respect to another function, is defined as RL α,λ Ml Ia+,h f

λ (t) = f (t) +  (α)

t a

h(1) (τ ) f (τ ) (h (t) − h (τ ))1−α

dτ ,

(3.231)

248

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Hille–Tamarkin general fractional integral with respect to another function as ⎡ ⎤ b (1) h (τ ) f (τ ) λ RL α,λ ⎣ dτ ⎦ . (3.232) Ml Ib−,h f (t) = − f (t) +  (α) (h (τ ) − h (t))1−α t

3.3.7 Liouville–Weyl–Hille–Tamarkin type general fractional calculus with respect to another function Based on the Liouville–Weyl fractional calculus, we present the Liouville–Weyl– Hille–Tamarkin-type general fractional calculus with respect to another function. Definition 3.34. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin type general fractional derivative with respect to another function is defined as  α,λ RL Mlr D+,h f

(t) =

1

d h(1) (t) dt

 t

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

−∞

(3.233) and the right-sided Liouville–Weyl–Hille–Tamarkin type general fractional derivative with respect to another function is 

α,λ RL Mlr D− f

1

d (t) = − (1) h (t) dt

 ∞

  Eα −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.234) Definition 3.35. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative with respect to another function is defined as  α,κ,λ RL Mlr D+,h f

(t) =

1

d (1) h (t) dt

κ  t

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

−∞

(3.235) and the right-sided Liouville–Weyl–Hille–Tamarkin-type general fractional derivative with respect to another function is  RL α,κ,λ Ml D−,h f

(t) =

−1 d h(1) (t) dt

κ ∞

  Eα −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.236)

Fractional derivatives with nonsingular kernels

249

Definition 3.36 ([25]). Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function on the real line is defined as t LSC α,λ Ml D+,h f

  Eα −λ (h (t) − h (τ ))α h(1) (τ )

(t) =



−∞

1 d h(1) (τ ) dτ

 f (τ ) dτ , (3.237)

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function on the real line is ∞ LSC α,λ Ml D−,h f



Eα −λ (h (τ ) − h (t))

(t) =

α





(1)

h

t

−1 d (τ ) (1) h (τ ) dτ

 f (τ ) dτ . (3.238)

Definition 3.37. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is defined as t LSC α,κ,λ Ml D+,h f

(t) =

  Eα −λ (h (t) − h (τ ))α h(1) (τ )



−∞

1 d (1) h (τ ) dτ

κ f (τ ) dτ , (3.239)

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with respect to another function via Mittag-Leffler function is ∞ LSC α,κ,λ Ml D−,h f



Eα −λ (h (τ ) − h (t))

(t) =

α





(1)

h

−1 d (τ ) (1) h (τ ) dτ

κ f (τ ) dτ .

t

(3.240) Definition 3.38. Let h(1) (t) > 0, α > 0, κ = [α] +1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Weyl–Hille–Tamarkin-type general fractional integral with respect to another function, as the inverse operator of the Liouville–Weyl–Hille– Tamarkin-type general fractional derivative with respect to another function, is defined as t RL α,κ,λ Ml I+,h f

(t) = −∞

h(1) (τ ) (h (t) − h (τ ))

α+1−κ

  −1 −λ (h (t) − h (τ ))α f (τ ) dτ , Eα,κ−α (3.241)

250

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Weyl–Hille–Tamarkin-type general fractional integral with respect to another function is ∞ RL α,κ,λ Ml I−,h f

(t) = (−1)

h(1) (τ )

κ

(h (τ ) − h (t))

α+1−κ

t

  −1 −λ (h (τ ) − h (t))α f (τ ) dτ . Eα,κ−α (3.242)

3.3.8 Hilfer–Hille–Tamarkin-type general fractional derivative with respect to another function Based on Hilfer’s idea, as well as the definitions of Hille–Tamarkin general fractional derivative and Liouville–Sonine–Caputo-type general fractional derivative via MittagLeffler function, we propose new general fractional derivatives with Mittag-Leffler kernel with respect to another function, which are called the Hilfer–Hille–Tamarkintype general fractional derivatives with respect to another function. Definition 3.39. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer–Hille–Tamarkin-type general fractional derivative with respect to another function is defined as  α,β,λ

H H T Da+,h f (t) =

 β(1−α),λ

(1−β)(1−α),λ

where ML Ia+,h

β(1−α),λ

ML Ia+,h

1

d h(1) (t) dt



 (1−β)(1−α),λ

f (t) ,

(3.243)



Eα −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ ,

(3.244)

ML Ia+,h

ML Ia+,h

f (t) ∈ AC1 (a, b), t

f (t) = a

and

(1−β)(1−α),λ f ML Ia+,h

t (t) =



Eα −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ ,

a

(3.245) and the right-sided Hilfer–Hille–Tamarkin-type general fractional derivative with respect to another function is 

 α,β,λ

H H T Db−,h f (t) = −

β(1−α),λ

ML Ib−

1

d h(1) (t) dt



 (1−β)(1−α),λ

ML Ib−

f (t) , (3.246)

Fractional derivatives with nonsingular kernels (1−β)(1−α),λ

where ML Ib−,h

β(1−α),λ f ML Ib−,h

251

f (t) ∈ AC1 (a, b), b

(t) =



Eα −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ ,

(3.247)

t

and (1−β)(1−α),λ f ML Ib−,h

b (t) =



Eα −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ .

t

(3.248) Definition 3.40. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer–Hille–Tamarkin type general fractional derivative with respect to another function on the real line is defined as     1 d α,β,λ β(1−α),λ (1−β)(1−α),λ f (t) , (3.249) D f = I I (t) H H T +,h ML +,h ML +,h h(1) (t) dt (1−β)(1−α),λ

where ML I+,h

β(1−α),λ f ML I+,h

f (t) ∈ AC1 (−∞, ∞),

t (t) =



Eα −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ ,

(3.250)

−∞

and (1−β)(1−α),λ f ML I+,h

t (t) =



Eα −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ ,

−∞

(3.251) and the right-sided Hilfer–Hille–Tamarkin-type general fractional derivative with respect to another function on the real line is     1 d α,β,λ β(1−α),λ (1−β)(1−α),λ f (t) , (3.252) H H T D−,h f (t) = − ML I−,h ML I−,h h(1) (t) dt (1−β)(1−α),λ

where ML I−,h

β(1−α),λ f ML I−,h

f (t) ∈ AC1 (−∞, ∞), ∞

(t) = t



Eα −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ ,

(3.253)

252

General Fractional Derivatives With Applications in Viscoelasticity

and (1−β)(1−α),λ f ML I−,h

∞ (t) =



Eα −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ .

t

(3.254)

3.4

General fractional derivatives with Wiman nonsingular kernel

In this section, the general fractional derivative via Wiman function is the new general fractional derivative with Wiman nonsingular kernel. In fact, some of the presented forms of the general fractional derivatives via Wiman function were given in [25]; for λ = ±1, see [133,134].

3.4.1 General fractional derivatives via Wiman function Definition 3.41 ([25]). Let 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with Wiman nonsingular kernel is defined as RL D α,υ,λ f GW a+

= =



(t)



α,υ,λ d RL dt GW O Ia+ f (t) t d υ−1 Eα,υ (−λ (t dt (t − τ ) a

(3.255) − τ )α ) f (τ ) dτ ,

where t α,υ,λ RL GW O Ia+ f

(t) =

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

(3.256)

a

and the right-sided general fractional derivative with Wiman nonsingular kernel is RL D α,υ,λ f GW b−



(t)

α,υ,λ d RL = − dt GW O Ib− f (t)

d = − dt

b t

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

(3.257)

Fractional derivatives with nonsingular kernels

253

where b α,υ,λ RL GW O Ib− f

  (τ − t)υ−1 Eα,υ −λ (τ − t)α f (τ ) dτ .

(t) =

(3.258)

t

Definition 3.42 ([25]). Let α ≥ 0, κ = [α] + 1, and υ, λ ∈ R. The left-sided general fractional derivative with Wiman nonsingular kernel is defined as α,υ,κ,λ RL f GW Da+

dκ (t) = κ dt

t

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

(3.259)

a

and the right-sided general fractional derivative with Wiman nonsingular kernel is α,υ,κ,λ RL f GW Db−

dκ (t) = (−1) dt κ

b

κ

  (τ − t)υ−1 Eα,υ −λ (τ − t)α f (τ ) dτ .

(3.260)

t

Definition 3.43 ([25]). Let 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with Wiman nonsingular kernel is defined as t LSC α,υ,λ GW Da+ f

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

(t) =

(3.261)

a

and the right-sided general fractional derivative with Wiman nonsingular kernel is b LSC α,υ,λ GW Db− f

(t) = −

  (τ − t)υ−1 Eα,υ −λ (τ − t)α f (1) (τ ) dτ .

(3.262)

t

Definition 3.44 ([25]). Let α ≥ 0, κ = [α] + 1, and υ, λ ∈ R. The left-sided general fractional derivative with Wiman nonsingular kernel is defined as t LSC α,υ,κ,λ f GW Da+

(t) =

  (t − τ )υ−1 Eα,υ −λ (t − τ )α f (κ) (τ ) dτ ,

(3.263)

a

and the right-sided general fractional derivative with Wiman nonsingular kernel is b LSC α,υ,κ,λ f GW Db−

(t) = (−1)κ t

  (τ − t)υ−1 Eα,υ −λ (τ − t)α f (κ) (τ ) dτ .

(3.264)

254

General Fractional Derivatives With Applications in Viscoelasticity

For the inverse operator of the general fractional derivative with Wiman nonsingular kernel, we have the following results for the Prabhakar general fractional integrals. Definition 3.45. Let 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided Prabhakar general fractional integral is defined as t α,υ,λ R GW Ia+ f

(t) =

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

(3.265)

a

and the right-sided Prabhakar general fractional integral is b α,υ,λ R GW Ib− f

(t) =

  −1 −λ (τ − t)α f (τ ) dτ . (τ − t)−υ Eα,1−υ

(3.266)

t

Definition 3.46 ([25]). Let α ≥ 0, κ = [α] + 1, and υ, λ ∈ R. The left-sided Prabhakar general fractional integral is defined as t α,υ,κ,λ R f GW Ia+

(t) =

  −1 −λ (t − τ )α f (τ ) dτ (t − τ )κ−υ−1 Eα,κ−υ

(3.267)

a

and the right-sided Prabhakar general fractional integral is b α,υ,κ,λ R f GW Ib−

(t) =

  −1 −λ (τ − t)α f (τ ) dτ . (τ − t)κ−υ−1 Eα,κ−υ

(3.268)

t

The Laplace transforms of the general fractional calculus with Wiman nonsingular kernel are presented as follows [25]: Property 3.9 ([25]). If κ = [α] + 1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and υ, λ ∈ R, then  L

α,υ,κ,λ R f GW I0+

   (t) = s υ−κ 1+λs −α f (s) .

(3.269)

Property 3.10 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and υ, λ ∈ R, then  L

RL D α,υ,λ f GW 0+

 = s κ−1 1 + λs

 (t)  −α −1

f (s) −

κ−1

μ=0



 α,υ,κ,λ s κ−μ−1 D μ R f (0) . GW I0+

(3.270)

Fractional derivatives with nonsingular kernels

255

Property 3.11 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and υ, λ ∈ R, then ⎞ ⎛ κ     −1 ⎝ κ α,υ,κ,λ f (t) = s −υ 1+λs −α s κ−j f (j −1) (0)⎠ . s f (s) − L LSC GW D0+ 

j =1

(3.271) Property 3.12 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and υ, λ ∈ R, then  L

α,υ,λ RL GW D0+ f

  −1 α,υ,λ f (s) − R (t) = 1 + λs −α GW I0+ f (0) .

(3.272)

3.4.2 Prabhakar general fractional integrals on the real line We now present the Prabhakar type general fractional integrals on the real line. Definition 3.47 ([25]). Let 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided Prabhakar general fractional integral on the real line is defined as t α,υ,λ R f GW I+

(t) =

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

(3.273)

−∞

and the right-sided Prabhakar general fractional integral on the real line is α,ϕ,υ,λ R f GW I−

∞ (t) =

  −1 −λ (τ − t)α f (τ ) dτ . (τ − t)−υ Eα,1−υ

(3.274)

t

Definition 3.48 ([25]). Let α ≥ 0, κ = [α] + 1, and υ, λ ∈ R. The left-sided Prabhakar general fractional integral on the real line is defined as t α,υ,κ,λ R f GW I+

  −1 −λ (t − τ )α f (τ ) dτ (t − τ )κ−υ−1 Eα,κ−υ

(t) =

(3.275)

−∞

and the right-sided Prabhakar general fractional integral on the real line is ∞ α,υ,κ,λ R f GW I−

(t) = t

  −1 −λ (τ − t)α f (τ ) dτ . (τ − t)κ−υ−1 Eα,κ−υ

(3.276)

256

General Fractional Derivatives With Applications in Viscoelasticity

3.4.3 Hilfer type general fractional derivative with Wiman nonsingular kernel In this section, we suggest the Hilfer-type general fractional derivative with Wiman nonsingular kernel. Definition 3.49. Let 0 < α < 1, 0 ≤ β ≤ 1, υ, λ ∈ R and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer type general fractional derivative with Wiman nonsingular kernel is defined as   β(1−α),υ,λ d RL (1−β)(1−α),υ,λ α,υ,λ RL f (t) , (3.277) D f = I I (t) H GW a+ GW O a+ dt GW O a+ (1−β)(1−α),υ,λ

where RL GW O Ia+

β(1−α),υ,λ RL f GW O Ia+

f (t) ∈ AC1 (a, b), t

(t) =



(t − τ )υ−1 Eβ(1−α),υ −λ (t − τ )β(1−α) f (τ ) dτ ,

a

(3.278) and RL I (1−β)(1−α),υ,λ f (t) GW O a+ t = (t − τ )υ−1 E(1−β)(1−α),υ a



 −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.279)

and the right-sided Hilfer type general fractional derivative with Wiman nonsingular kernel is   β(1−α),υ,λ d RL (1−β)(1−α),υ,λ α,υ,λ RL f (t) , (3.280) I H GW Db− f (t) = GW O Ib− dt GW O b− (1−β)(1−α),υ,λ

where RL GW O Ib−

β(1−α),υ,λ RL f GW O Ib−

f (t) ∈ AC1 (a, b), b

(t) =



(τ − t)υ−1 Eβ(1−α),υ −λ (τ − t)β(1−α) f (τ ) dτ ,

t

(3.281) and RL I (1−β)(1−α),υ,λ f (t) GW O b− b = (τ − t)υ−1 E(1−β)(1−α),υ t



 −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.282)

Fractional derivatives with nonsingular kernels

257

Property 3.13. (1) Suppose that 0 < α < 1, 0 ≤ β ≤ 1, υ, λ ∈ R, f ∈ L1 (a, b) (1−β)(1−α),υ,λ f (t) ∈ AC1 (a, b), then (−∞ < a < b < ∞) and RL GW O I0+ 

 (t) (s)  −β(1−α) −1

α,υ,λ H GW D0+ f

L

 = s −1 1 + λs

 −1 (1−β)(1−α),υ,λ . s 1−υ 1+λs −α f (s) − RL I f (0) GW O 0+ (3.283)

Proof. By direct calculation, we get  L

α,υ,λ H GW D0+ f



 (t) f (t) (s)





RL I β(1−α),υ,λ d RL I (1−β)(1−α),υ,λ f (t) GW O 0+ dt GW O 0+

 −1  −1 (1−β)(1−α),υ,λ −υ −α =s s × s −υ 1+λs −α 1+λs f (s) − RL I f (0) GW O 0+

 −1 1−υ  −1 (1−β)(1−α),υ,λ −1 −β(1−α) −α , =s s f (s) − RL I f 1 + λs 1+λs (0) GW O 0+

=L

(3.284) competing the proof. Definition 3.50. Let 0 < α < 1, 0 ≤ β ≤ 1, υ, λ ∈ R, and f ∈ L1 (−∞, ∞). The left-sided Hilfer-type general fractional derivative with Wiman nonsingular kernel on the real line is defined as   β(1−α),υ,λ d RL (1−β)(1−α),υ,λ α,υ,λ RL f (t) = GW O I+ I f (t) , (3.285) H GW D+ dt GW O + (1−β)(1−α),υ,λ

where RL GW O I+

β(1−α),υ,λ RL f GW O I+

f (t) ∈ AC1 (−∞, ∞), t

(t) =



(t − τ )υ−1 Eβ(1−α),υ −λ (t − τ )β(1−α) f (τ ) dτ ,

−∞

(3.286) and (1−β)(1−α),υ,λ RL f GW O I+ t

=

(t)



(t − τ )υ−1 E(1−β)(1−α),υ −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.287)

−∞

and the right-sided Hilfer-type general fractional derivative with Wiman nonsingular kernel on the real line is   β(1−α),υ,λ d RL (1−β)(1−α),υ,λ α,υ,λ RL f (t) , (3.288) D f = I I (t) H GW − GW O − dt GW O −

258

General Fractional Derivatives With Applications in Viscoelasticity (1−β)(1−α),υ,λ

where RL GW O I−

β(1−α),υ,λ RL f GW O I−

f (t) ∈ AC1 (−∞, ∞), ∞

(t) =



(τ − t)υ−1 Eβ(1−α),υ −λ (τ − t)β(1−α) f (τ ) dτ ,

t

(3.289) and RL I (1−β)(1−α),υ,λ f (t) GW O − ∞ = (τ − t)υ−1 E(1−β)(1−α),υ t

  −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.290)

3.4.4 General fractional derivatives with respect to another function via Wiman function Definition 3.51. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with respect to another function via Wiman nonsingular kernel is defined as RL D α,υ,λ f GW a+,h

= =



1 h(1) (t)



1 h(1) (t)

(t)

d dt

d dt

t

RL I α,υ,λ f GW O a+,h

(t)

(h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ ,

a

(3.291) where t α,υ,λ RL GW O Ia+,h f

(t) =

  (h (t) − h (τ ))υ−1 Eα,υ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ ,

a

(3.292) and the right-sided general fractional derivative with respect to another function via Wiman nonsingular kernel as RL D α,υ,λ f GW b−,h

= =



(t)

d − h(1)1(t) dt

b −1 d h(1) (t) dt



RL I α,υ,λ f GW O b−,h

(t)

(h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ,

t

(3.293)

Fractional derivatives with nonsingular kernels

259

where b α,υ,λ RL GW O Ib−,h f

  (h (τ ) − h (t))υ−1 Eα,υ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

(t) = t

(3.294) Definition 3.52. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is defined as RL D α,υ,λ f GW +,h

= =





(t)

RL I α,υ,λ f GW O +,h t

1 d h(1) (t) dt

1 d h(1) (t) dt



−∞

(t)

(h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ , (3.295)

where t α,υ,λ RL GW O I+,h f

(t) =

  (h (t) − h (τ ))υ−1 Eα,υ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ ,

−∞

(3.296) and the right-sided general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is RL D α,υ,λ f GW −,h

(t)



α,υ,λ d RL = − h(1)1(t) dt I f (t) GW O −,h

∞ −1 d = h(1) (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ, (t) dt t

(3.297) where ∞ α,υ,λ RL GW O I−,h f

(t) =

  (h (τ ) − h (t))υ−1 Eα,υ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.298) Definition 3.53. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with respect to another function via Wiman nonsingular kernel is defined as

260

General Fractional Derivatives With Applications in Viscoelasticity

LSC D α,υ,λ f GW a+,h t

(t)

= (h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ )



a

1 d h(1) (τ ) dτ

f (τ ) dτ , (3.299)

and the right-sided general fractional derivative with respect to another function via Wiman nonsingular kernel is LSC D α,υ,λ f GW b−,h b

(t)

= − (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ )



1 h(1) (τ )

t

d dτ

f (τ ) dτ . (3.300)

Definition 3.54. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is defined as LSC D α,υ,λ f GW +,h t

=

−∞

(t)

(h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ )



1 h(1) (τ )

d dτ

f (τ ) dτ , (3.301)

and the right-sided general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is LSC D α,υ,λ f GW −,h ∞

=−

(t)

(h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ )

t



1 h(1) (τ )

d dτ

f (τ ) dτ . (3.302)

Definition 3.55. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided Prabhakar general fractional integral with respect to another function is defined as t α,υ,λ R GW Ia+,h f

(t) =

  −1 −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ, (h (t) − h (τ ))−υ Eα,1−υ

a

(3.303)

Fractional derivatives with nonsingular kernels

261

and the right-sided Prabhakar general fractional integral with respect to another function is b α,υ,λ R GW Ib−,h f

(t) =

  −1 −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))−υ Eα,1−υ

t

(3.304) Definition 3.56. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and υ, λ ∈ R. The left-sided Prabhakar general fractional integral with respect to another function on the real line is defined as t α,υ,λ R GW I+,h f

(t) =

  −1 −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ, (h (t) − h (τ ))−υ Eα,1−υ

−∞

(3.305) and the right-sided Prabhakar general fractional integral with respect to another function on the real line is ∞ α,υ,λ R GW Ib−−,h f

(t) =

  −1 −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))−υ Eα,1−υ

t

(3.306) Definition 3.57. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, υ, λ ∈ R and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with respect to another function via Wiman nonsingular kernel is defined as 

 α,υ,λ H GW Da+,h f

(t) =

β(1−α),υ,λ RL GW O Ia+,h

1

d h(1) (t) dt



 (1−β)(1−α),υ,λ RL GW O Ia+,h

f (t) , (3.307)

(1−β)(1−α),υ,λ

where RL GW O Ia+,h

RL I β(1−α),υ,λ f GW O a+,h t

f (t) ∈ AC1 (a, b),

(t)

  = (h (t) − h (τ ))υ−1 Eβ(1−α),υ −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ , a

(3.308)

262

General Fractional Derivatives With Applications in Viscoelasticity

and RL I (1−β)(1−α),υ,λ f (t) GW O a+,h t = (h (t) − h (τ ))υ−1 E(1−β)(1−α),υ a

  −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ , (3.309)

and the right-sided Hilfer-type general fractional derivative with respect to another function via Wiman nonsingular kernel is     1 d RL β(1−α),υ,λ (1−β)(1−α),υ,λ α,υ,λ RL f (t) , D f = I I H GW b−,h (t) GW O b−,h h(1) (t) dt GW O b−,h (3.310) (1−β)(1−α),υ,λ

where RL GW O Ib−,h

RL I β(1−α),υ,λ f GW O b− b

f (t) ∈ AC1 (a, b),

(t)

  = (h (τ ) − h (t))υ−1 Eβ(1−α),υ −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ, t

(3.311) and RL I (1−β)(1−α),υ,λ f (t) GW O b−,h b = (h (τ ) − h (t))υ−1 E(1−β)(1−α),υ t

  −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ . (3.312)

Definition 3.58. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, υ, λ ∈ R and f ∈ L1 (−∞, ∞). The left-sided Hilfer-type general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is defined as     1 d RL β(1−α),υ,λ (1−β)(1−α),υ,λ α,υ,λ RL f (t) , I H GW D+,h f (t) = GW O I+,h h(1) (t) dt GW O +,h (3.313) (1−β)(1−α),υ,λ

where RL GW O I+,h

RL I β(1−α),υ,λ f GW O +,h t

=

−∞

f (t) ∈ AC1 (−∞, ∞),

(t)

  (h (t) − h (τ ))υ−1 Eβ(1−α),υ −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ , (3.314)

Fractional derivatives with nonsingular kernels

263

and RL I (1−β)(1−α),υ,λ f (t) GW O +,h t υ−1

=

−∞

(h (t) − h (τ ))

  E(1−β)(1−α),υ −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ , (3.315)

and the right-sided Hilfer-type general fractional derivative with respect to another function via Wiman nonsingular kernel on the real line is  α,υ,λ H GW D−,h f

(t) =

 β(1−α),υ,λ RL GW O I−,h

1

d h(1) (t) dt



 (1−β)(1−α),υ,λ RL GW O I−,h

f (t) , (3.316)

(1−β)(1−α),υ,λ

where RL GW O I−,h

RL I β(1−α),υ,λ f GW O − ∞

f (t) ∈ AC1 (a, b),

(t)

  (h (τ ) − h (t))υ−1 Eβ(1−α),υ −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ,

=

t

(3.317) and RL I (1−β)(1−α),υ,λ f (t) GW O −,h ∞ = (h (τ ) − h (t))υ−1 E(1−β)(1−α),υ t



 −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ . (3.318)

3.5

General fractional derivatives with Prabhakar nonsingular kernel

In 2004, Kilbas et al. suggested the general fractional derivative via Prabhakar function [128,25,133,134]. After ten years, Garra et al. introduced another form of the general fractional derivative via Prabhakar function [25,129,133,134]. In fact, some of the presented forms of the general fractional derivatives and integrals via Prabhakar function, which are called the general fractional calculus with Prabhakar nonsingular kernel, were proposed in [25]; for λ = ±1, see [129,133,134].

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General Fractional Derivatives With Applications in Viscoelasticity

3.5.1 Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel Definition 3.59 ([25]). Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel is defined as RL D α,υ,ϕ,λ f (t) GP a+

α,υ,ϕ,λ d = dt f (t) GP Ia+ t ϕ d = dt (t − τ )υ−1 Eα,υ (−λ (t a

(3.319) − τ )α ) f (τ ) dτ ,

and the right-sided Kilbas–Saigo–Saxena type general fractional derivative with Prabhakar nonsingular kernel is RL D α,υ,ϕ,λ f (t) GP b− α,υ,ϕ,λ d = − dt f GP Ib− d = − dt

b

(t)

(3.320)

ϕ

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

t

where the left-sided Prabhakar general fractional integral is [25,37] α,υ,ϕ,λ

GP Ia+

t f (t) =

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(3.321)

a

and the right-sided Prabhakar general fractional integral is

α,υ,ϕ,λ f GP Ib−

b (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.322)

t

Definition 3.60 ([25]). Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel is defined as RL D α,υ,ϕ,κ,λ f (t) GP a+

α,υ,ϕ,λ dκ = dt I f (t) κ GP a+ t  κ ϕ d = dt (t − τ )υ−1 Eα,υ (−λ (t κ a

(3.323) − τ )α ) f (τ ) dτ ,

Fractional derivatives with nonsingular kernels

265

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel is RL D α,υ,ϕ,κ,λ f (t) GP b−

 d κ α,υ,ϕ,λ = − dt f (t) GP Ib− b ϕ dκ = (−1)κ dt (τ − t)υ−1 Eα,υ κ t

(3.324) (−λ (τ − t)α ) f (τ ) dτ ,

where the left-sided Prabhakar general fractional integral is [25,37]

α,υ,ϕ,λ f GP Ia+

t (t) =

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(3.325)

a

and the right-sided Prabhakar general fractional integral is

α,υ,ϕ,λ f GP Ib−

b (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.326)

t

3.5.2 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel Definition 3.61 ([25]). Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel is defined as LS D α,υ,ϕ,λ f (t) GP a+ α,υ,ϕ,λ (1) = GP Ia+ f (t) t ϕ = (t − τ )υ−1 Eα,υ (−λ (t a

(3.327) − τ )α ) f (1) (τ ) dτ ,

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel is LS D α,υ,ϕ,λ f (t) GP b−  α,υ,ϕ,λ  = GP Ib− −f (1) (t) b ϕ = − (τ − t)υ−1 Eα,υ (−λ (τ t

(3.328) − t)α ) f (1) (τ ) dτ ,

266

General Fractional Derivatives With Applications in Viscoelasticity

where the left-sided Prabhakar general fractional integral is [25,37] α,υ,ϕ,λ f GP Ia+

t (t) =

  ϕ −λ (t − τ )α f (τ ) dτ, (t − τ )υ−1 Eα,υ

(3.329)

a

and the right-sided Prabhakar general fractional integral is α,υ,ϕ,λ f GP Ib−

b (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.330)

t

Definition 3.62 ([25]). Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel is defined as LSC D α,ϕ,υ,κ,λ f (t) a+ GP  α,ϕ,υ,λ  (κ) = GP Ia+ f (t) t ϕ = (t − τ )υ−1 Eα,υ (−λ (t a

(3.331) − τ )α ) f (κ) (τ ) dτ ,

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel is LSC D α,ϕ,υ,κ,λ f (t) GP b−  α,ϕ,υ,κ,λ  = GP Ib− (−1)κ f (κ) (t) b ϕ = (−1)κ (τ − t)υ−1 Eα,υ (−λ (τ t

(3.332) − t)α ) f (κ) (τ ) dτ ,

where the left-sided Prabhakar general fractional integral is [25,37] α,υ,ϕ,λ f GP Ia+

t (t) =

  ϕ −λ (t − τ )α f (τ ) dτ, (t − τ )υ−1 Eα,υ

(3.333)

a

and the right-sided Prabhakar general fractional integral is α,υ,ϕ,λ

GP Ib−

b f (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.334)

t

3.5.3 Prabhakar-type general fractional integrals In 2018, Yang proposed the Prabhakar-type general fractional integrals with Prabhakar nonsingular kernel in his book [25].

Fractional derivatives with nonsingular kernels

267

Definition 3.63 ([25]). Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as α,ϕ,υ,λ R f GP Ia+

t (t) =

  −ϕ (t − τ )−υ Eα,1−υ −λ (t − τ )α f (τ ) dτ,

(3.335)

a

and the right-sided Prabhakar-type general fractional integral is α,ϕ,υ,λ R f GP Ib−

b (t) =

  −ϕ (τ − t)−υ Eα,1−υ −λ (τ − t)α f (τ ) dτ .

(3.336)

t

Definition 3.64 ([25]). Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as α,ϕ,υ,κ,λ R f GP Ia+

t (t) =

  −ϕ (t − τ )κ−υ−1 Eα,κ−υ −λ (t − τ )α f (τ ) dτ,

(3.337)

a

and the right-sided Prabhakar general fractional integral is α,ϕ,υ,κ,λ R f GP Ib−

b (t) =

  −ϕ (τ − t)κ−υ−1 Eα,κ−υ −λ (τ − t)α f (τ ) dτ .

(3.338)

t

The Laplace transforms of the general fractional calculus with Prabhakar nonsingular kernel are presented as follows [25]: Property 3.14 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and λ, υ, ϕ ∈ R, then    ϕ α,υ,ϕ,κ,λ L R f (t) = s υ−κ 1+λs −α f (s) . (3.339) GP I0+ Property 3.15 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and λ, υ, ϕ ∈ R, then   α,υ,ϕ,λ f (t) L RL GP D0+  (3.340) κ−1  −ϕ

κ−μ−1 μ R α,ϕ,υ,κ,λ = s κ−1 1 + λs −α f (s) − s f (0) . D GP I0+ μ=0

Property 3.16 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and λ, υ, ϕ ∈ R, then ⎞ ⎛ κ      −ϕ ⎝ κ α,υ,ϕ,κ,λ L LSC f (t) = s −υ 1+λs −α s κ−j f (j −1) (0)⎠ . s f (s) − GP D0+ j =1

(3.341)

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General Fractional Derivatives With Applications in Viscoelasticity

Property 3.17 ([25]). If κ = [α]+1, −∞ < a < b < ∞, f (t) ∈ Ll (a, b) (1 ≤ l < ∞) and λ, υ, ϕ ∈ R, then  L

RL α,υ,ϕ,λ f GP D0+

  −1 α,υ,ϕ,λ f (s) − R f (0) . (t) = 1 + λs −α GP I0+

(3.342)

3.5.4 Kilbas–Saigo–Saxena type general fractional derivative with Prabhakar nonsingular kernel on the real line Definition 3.65. Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel on the real line is defined as RL D α,υ,ϕ,λ f (t) GP +

α,υ,ϕ,λ d = dt f (t) GP I+ t ϕ d = dt (t − τ )υ−1 Eα,υ −∞

(3.343) (−λ (t − τ )α ) f (τ ) dτ ,

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel on the real line is RL D α,υ,ϕ,λ f (t) GP −

α,υ,ϕ,λ d = − dt f (t) GP I− ∞ ϕ d = − dt (τ − t)υ−1 Eα,υ (−λ (τ t

(3.344) − t)α ) f (τ ) dτ ,

where the left-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ f GP I+

t

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(t) =

(3.345)

−∞

and the right-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ f GP I−

∞ (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.346)

t

Definition 3.66 ([25]). Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel on the real line is defined as

Fractional derivatives with nonsingular kernels

RL D α,υ,ϕ,κ,λ f (t) GP +

α,υ,ϕ,λ dκ = dt f (t) κ GP I+ t ϕ dκ = dt (t − τ )υ−1 Eα,υ κ −∞

269

(3.347) (−λ (t − τ )α ) f (τ ) dτ ,

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with Prabhakar nonsingular kernel on the real line is RL D α,υ,ϕ,κ,λ f (t) GP −

 d κ α,υ,ϕ,λ = − dt f (t) GP I− ∞ ϕ dκ = (−1)κ dt (τ − t)υ−1 Eα,υ κ t

(3.348) (−λ (τ − t)α ) f (τ ) dτ ,

where the left-sided Prabhakar general fractional integral on the real line is

α,υ,ϕ,λ f GP I+

t

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(t) =

(3.349)

−∞

and the right-sided Prabhakar general fractional integral on the real line is

α,υ,ϕ,λ f GP I−

∞ (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.350)

t

3.5.5 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line Definition 3.67. Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line is defined as LS D α,υ,ϕ,λ f (t) GP + α,υ,ϕ,λ (1) = GP I+ f (t) t ϕ = (t − τ )υ−1 Eα,υ −∞

(3.351) (−λ (t − τ )α ) f (1) (τ ) dτ ,

270

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line is LS D α,υ,ϕ,λ f (t) GP −  α,υ,ϕ,λ  = GP I− −f (1) (t) ∞ ϕ = − (τ − t)υ−1 Eα,υ (−λ (τ t

(3.352) − t)α ) f (1) (τ ) dτ ,

where the left-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ f GP I+

t

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(t) =

(3.353)

−∞

and the right-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ

GP I−

∞ f (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.354)

t

Definition 3.68. Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line is defined as LSC D α,ϕ,υ,κ,λ f (t) + GP   α,ϕ,υ,λ = GP I+ f (κ) (t) t ϕ = (t − τ )υ−1 Eα,υ (−λ (t −∞

(3.355) − τ )α ) f (κ) (τ ) dτ ,

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel on the real line is LSC D α,ϕ,υ,κ,λ f (t) − GP α,ϕ,υ,κ,λ  = GP I− (−1)κ ∞ κ υ−1

(τ − t)

= (−1)

f (κ) (t) ϕ Eα,υ



(3.356)

(−λ (τ − t)α ) f (κ) (τ ) dτ ,

t

where the left-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ f GP I+

t (t) = −∞

  ϕ −λ (t − τ )α f (τ ) dτ (t − τ )υ−1 Eα,υ

(3.357)

Fractional derivatives with nonsingular kernels

271

and the right-sided Prabhakar general fractional integral on the real line is α,υ,ϕ,λ f GP I−

∞ (t) =

  ϕ −λ (τ − t)α f (τ ) dτ . (τ − t)υ−1 Eα,υ

(3.358)

t

3.5.6 Prabhakar-type general fractional integrals on the real line We now present the Prabhakar type general fractional integrals with Prabhakar nonsingular kernel on the real line. Definition 3.69. Let 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Prabhakar type general fractional integral on the real line is defined as α,ϕ,υ,λ R f GP I+

t

  −ϕ (t − τ )−υ Eα,1−υ −λ (t − τ )α f (τ ) dτ

(t) =

(3.359)

−∞

and the right-sided Prabhakar type general fractional integral on the real line is α,ϕ,υ,λ R f GP I−

∞ (t) =

  −ϕ (τ − t)−υ Eα,1−υ −λ (τ − t)α f (τ ) dτ .

(3.360)

t

Definition 3.70. Let α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Prabhakar type general fractional integral on the real line is defined as α,ϕ,υ,κ,λ R f GP I+

t

  −ϕ (t − τ )κ−υ−1 Eα,κ−υ −λ (t − τ )α f (τ ) dτ

(t) =

(3.361)

−∞

and the right-sided Prabhakar type general fractional integral on the real line is α,ϕ,υ,κ,λ R f GP I−

∞ (t) =

  −ϕ (τ − t)κ−υ−1 Eα,κ−υ −λ (τ − t)α f (τ ) dτ .

(3.362)

t

3.5.7 Hilfer-type general fractional derivative with Prabhakar nonsingular kernel Based on Hilfer’s idea, as well as using the definitions of the Kilbas–Saigo–Saxenatype general fractional derivative with Prabhakar nonsingular kernel and Garra– Gorenflo–Polito–Tomovski-type general fractional derivative with Prabhakar nonsingular kernel, we propose a new general fractional derivative with Prabhakar nonsingular kernel which is called the Hilfer-type general fractional derivative with Prabhakar nonsingular kernel.

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General Fractional Derivatives With Applications in Viscoelasticity

Definition 3.71. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with Prabhakar nonsingular kernel is defined as   β(1−α),υ,ϕ,λ d (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ f (t) , (3.363) D f = I I (t) H GP a+ GP a+ GP a+ dt (1−β)(1−α),υ,ϕ,λ

where GP Ia+

β(1−α),υ,ϕ,λ f GP Ia+

f (t) ∈ AC1 (a, b), t

(t) =



ϕ (t − τ )υ−1 Eβ(1−α),υ −λ (t − τ )β(1−α) f (τ ) dτ ,

a

(3.364) and (1−β)(1−α),υ,ϕ,λ f (t) GP Ia+ t ϕ = (t − τ )υ−1 E(1−β)(1−α),υ a



 −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.365)

and the right-sided Hilfer type general fractional derivative with Prabhakar nonsingular kernel is   β(1−α),υ,ϕ,λ d (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ f (t) , (3.366) D f = I I (t) H GP b− GP b− GP b− dt (1−β)(1−α),υ,ϕ,λ

where GP Ib−

β(1−α),υ,ϕ,λ f GP Ib−

f (t) ∈ AC1 (a, b), b

(t) =



ϕ (τ − t)υ−1 Eβ(1−α),υ −λ (τ − t)β(1−α) f (τ ) dτ ,

t

(3.367) and (1−β)(1−α),υ,ϕ,λ f (t) GP Ib− b  ϕ = (τ − t)υ−1 E(1−β)(1−α),υ t



 −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.368)

Property 3.18. (1) Suppose that 0 < α < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) (−∞ < a < (1−β)(1−α),υ,ϕ,λ f (t) ∈ AC1 (a, b), then b < ∞) and GP I0+  (t) (s)

 −1 1−υ  −ϕ (1−β)(1−α),υ,ϕ,λ = s −1 1 + λs −β(1−α) s 1+λs −α f (s) − GP I0+ f (0) . 

L

α,υ,ϕ,λ f H GP D0+

(3.369)

Fractional derivatives with nonsingular kernels

273

Proof. By direct computation,  L

α,υ,ϕ,λ f H GP D0+

=L



 (t) f (t) (s)

β(1−α),υ,ϕ,λ d (1−β)(1−α),υ,ϕ,λ GP I0+ dt GP I0+



 f (t)

 −ϕ  −ϕ (1−β)(1−α),υ,ϕ,λ = s −υ 1+λs −α s × s −υ 1+λs −α f (s) − GP I0+ f (0)

 −ϕ 1−υ  −ϕ (1−β)(1−α),υ,ϕ,λ s 1+λs −α f (s) − GP I0+ f (0) , = s −1 1 + λs −β(1−α) (3.370) finishing the proof. Definition 3.72. Let 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer-type general fractional derivative with Prabhakar nonsingular kernel on the real line is defined as   β(1−α),υ,ϕ,λ d (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ f (t) = GP I+ f (t) , (3.371) H GP D+ GP I+ dt (1−β)(1−α),υ,ϕ,λ

where GP I+

β(1−α),υ,ϕ,λ f GP I+

f (t) ∈ AC1 (−∞, ∞), t



ϕ (t − τ )υ−1 Eβ(1−α),υ −λ (t − τ )β(1−α) f (τ ) dτ ,

(t) = −∞

(3.372) and (1−β)(1−α),υ,ϕ,λ f (t) GP I+ t ϕ = (t − τ )υ−1 E(1−β)(1−α),υ −∞

  −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.373)

and the right-sided Hilfer-type general fractional derivative with Prabhakar nonsingular kernel on the real line is   β(1−α),υ,ϕ,λ d (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ f (t) , (3.374) f (t) = GP I− H GP D− GP I− dt (1−β)(1−α),υ,ϕ,λ

where GP I−

β(1−α),υ,ϕ,λ f GP I−

f (t) ∈ AC1 (−∞, ∞), ∞

(t) =



ϕ (τ − t)υ−1 Eβ(1−α),υ −λ (τ − t)β(1−α) f (τ ) dτ ,

t

(3.375)

274

General Fractional Derivatives With Applications in Viscoelasticity

and (1−β)(1−α),υ,ϕ,λ f (t) GP I− ∞  ϕ = (τ − t)υ−1 E(1−β)(1−α),υ t



 −λ (τ − t)(1−β)(1−α) f (τ ) dτ ,

(3.376)

3.5.8 General fractional derivatives with respect to another function via Prabhakar nonsingular kernel In this section, the Kilbas–Saigo–Saxena type general fractional derivatives with respect to another function (also called the Kilbas–Saigo–Saxena-type general fractional derivatives with respect to another function via Prabhakar nonsingular kernel) were reported in the book [25]. Definition 3.73 ([25]). Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function is defined as RL D α,υ,ϕ,λ f GP a+,h

= =



1 h(1) (t)

d dt

d 1 h(1) (t) dt

(t)



t

α,υ,ϕ,λ GP Ia+,h f

(t) ϕ

(h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ ,

a

(3.377) and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function is RL D α,υ,ϕ,λ f GP b−,h

= =



1 h(1) (t)

d dt

d 1 h(1) (t) dt

(t)

α,υ,ϕ,λ GP Ib−,h f

b

(t) ϕ

(h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ ,

t

(3.378) where the left-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ia+,h f

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

a

(3.379)

Fractional derivatives with nonsingular kernels

275

and the right-sided Prabhakar general fractional with respect to another function integral is α,υ,ϕ,λ GP Ib−,h f

b (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.380) Definition 3.74 ([25]). Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function is defined as RL D α,υ,ϕ,κ,λ f (t) GP a+,h

κ α,υ,ϕ,λ d = h(1)1(t) dt GP Ia+,h f (t)

κ t ϕ d = h(1)1(t) dt (h (t) − h (τ ))υ−1 Eα,υ a

(−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ , (3.381)

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function is RL D α,υ,ϕ,κ,λ f (t) GP b−,h

κ α,υ,ϕ,λ −1 d = h(1) GP Ib−,h f (t) dt

=



−1 d h(1) (t) dt

κ b

(t) ϕ

(h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ ,

t

(3.382) where the left-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ia+,h f

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

a

(3.383) and the right-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ib−,h f

b (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.384)

276

General Fractional Derivatives With Applications in Viscoelasticity

In the book [25], the Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function (also called the Garra–Gorenflo–Polito– Tomovski-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel) was proposed for the first time. Definition 3.75 ([25]). Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function is defined as LS D α,υ,ϕ,λ f GP a+,h α,υ,ϕ,λ

= GP Ia+,h

(t)

d 1 h(1) (t) dt

f (t)

t ϕ d = (h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) h(1)1(τ ) dτ f (τ ) dτ , a

(3.385) and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function is LS D α,υ,ϕ,λ f GP b−,h α,υ,ϕ,λ

= GP Ib−,h

(t)

−1

h(1) (t)

b

d dt

f (t) ϕ

= (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) t



−1 d h(1) (τ ) dτ

f (τ ) dτ , (3.386)

where the left-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ia+,h f

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

a

(3.387) and the right-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ib−,h f

b (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.388) Definition 3.76 ([25]). Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function is defined as

Fractional derivatives with nonsingular kernels

LSC D α,ϕ,υ,κ,λ f GP a+,h α,ϕ,υ,λ

= GP Ia+,h



(t)

d 1 h(1) (t) dt

277

κ f (t)

κ t ϕ d = (h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) h(1)1(τ ) dτ f (τ ) dτ , a

(3.389) and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function is LSC D α,ϕ,υ,κ,λ f GP b−,h α,ϕ,υ,κ,λ

= GP Ib−,h

(t)

−1 d h(1) (t) dt



κ f (t)

κ b ϕ −1 d f (τ ) dτ , = (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) h(1) dτ (τ ) t

(3.390) where the left-sided Prabhakar general fractional integral with respect to another function is α,υ,ϕ,λ GP Ia+,h f

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

a

(3.391) and the right-sided Prabhakar general fractional integral with respect to another function is

α,υ,ϕ,λ GP Ib−,h f

b (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.392)

3.5.9 Prabhakar-type general fractional integrals with respect to another function In 2018, Yang proposed the Prabhakar-type general fractional integrals with respect to another function in his book [25]. Definition 3.77 ([25]). Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral with respect to another function is defined as

278

General Fractional Derivatives With Applications in Viscoelasticity

α,ϕ,υ,λ R GP Ia+,h f

t

  −ϕ (h (t) − h (τ ))−υ Eα,1−υ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ

(t) = a

(3.393) and the right-sided Prabhakar-type general fractional integral with respect to another function is

α,ϕ,υ,λ R GP Ib−,h f

b (t) =

  −ϕ (h (τ ) − h (t))−υ Eα,1−υ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.394) Definition 3.78 ([25]). Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral with respect to another function is defined as R I α,ϕ,υ,κ,λ f GP a+,h t

(t) −ϕ

= (h (t) − h (τ ))κ−υ−1 Eα,κ−υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ

(3.395)

a

and the right-sided Prabhakar general fractional integral with respect to another function is R I α,ϕ,υ,κ,λ f GP b−,h

(t)

(3.396) b −ϕ = (h (τ ) − h (t))κ−υ−1 Eα,κ−υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ . t

3.5.10 Kilbas–Saigo–Saxena-type general fractional derivative on the real line We now present the Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel on the real line, which is called the Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function on the real line. Definition 3.79. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function on the real line is defined as

Fractional derivatives with nonsingular kernels

RL D α,υ,ϕ,λ f GP +,h

= =





1 d h(1) (t) dt 1 d h(1) (t) dt

(t)



α,υ,ϕ,λ

GP I+,h

t

279

f (t) ϕ

−∞

(h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ , (3.397)

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function on the real line is RL D α,υ,ϕ,λ f GP −,h

= =





−1

h(1) (t)

d dt

−1 d h(1) (t) dt

(t)



α,υ,ϕ,λ f GP I−,h

∞

(t) ϕ

(h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ ,

t

(3.398) where the left-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ

GP I+,h

t f (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

−∞

(3.399) and the right-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I−,h

∞ (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.400) Definition 3.80. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function on the real line is defined as RL D α,υ,ϕ,κ,λ f (t) GP +,h

κ α,υ,ϕ,λ d = h(1)1(t) dt f (t) GP I+,h

κ t ϕ d = h(1)1(t) dt (h (t) − h (τ ))υ−1 Eα,υ −∞

(−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ , (3.401)

280

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Kilbas–Saigo–Saxena-type general fractional derivative with respect to another function on the real line is RL D α,υ,ϕ,κ,λ f (t) GP −,h



κ α,υ,ϕ,λ −1 d = h(1) I f (t) GP −,h (t) dt

κ ∞ ϕ −1 d = h(1) (h (τ ) − h (t))υ−1 Eα,υ (t) dt t

(−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ , (3.402)

where the left-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I+,h

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

−∞

(3.403) and the right-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I−,h

∞ (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.404)

3.5.11 Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line We now present the Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel on the real line, which is called the Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line. Definition 3.81. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function is defined as LS D α,υ,ϕ,λ f (t) GP +,h

α,υ,ϕ,λ 1 d = GP I+,h f (t) (1) dt h (t)

t ϕ d = f (h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) h(1)1(τ ) dτ −∞

(τ ) dτ , (3.405)

Fractional derivatives with nonsingular kernels

281

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line is LS D α,υ,ϕ,λ f (t) GP −,h

α,υ,ϕ,λ −1 d = GP I−,h f (t) (1) h (t) dt

∞ ϕ −1 d = (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) h(1) f (τ ) dτ t

(τ ) dτ , (3.406)

where the left-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I+,h

t (t) =

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

−∞

(3.407) and the right-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I−,h

∞ (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.408) Definition 3.82. Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line is defined as LSC D α,ϕ,υ,κ,λ f (t) GP +,h



κ α,ϕ,υ,λ d 1 = GP I+,h f (t) (1) h (t) dt

κ t ϕ d = (h (t) − h (τ ))υ−1 Eα,υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) h(1)1(τ ) dτ −∞

f (τ ) dτ , (3.409)

and the right-sided Garra–Gorenflo–Polito–Tomovski-type general fractional derivative with respect to another function on the real line is LSC D α,ϕ,υ,κ,λ f (t) GP −,h



κ α,ϕ,υ,κ,λ −1 d = GP I−,h f (t) (1) h (t) dt

κ ∞ ϕ −1 d = (h (τ ) − h (t))υ−1 Eα,υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) h(1) (τ ) dτ t

f (τ ) dτ , (3.410)

282

General Fractional Derivatives With Applications in Viscoelasticity

where the left-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I+,h

t

  ϕ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ (h (t) − h (τ ))υ−1 Eα,υ

(t) = −∞

(3.411) and the right-sided Prabhakar general fractional integral with respect to another function on the real line is α,υ,ϕ,λ f GP I−,h

∞ (t) =

  ϕ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ . (h (τ ) − h (t))υ−1 Eα,υ

t

(3.412)

3.5.12 Prabhakar-type general fractional integrals with respect to another function on the real line We now present the Prabhakar type general fractional integrals with respect to another function via Prabhakar nonsingular kernel on the real line. Definition 3.83. Let h(1) (t) > 0, 1 ≥ α ≥ 0 and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral with respect to another function on the real line is defined as α,ϕ,υ,λ R f GP I+,h

t (t) =

  −ϕ (h (t) − h (τ ))−υ Eα,1−υ −λ (h (t) − h (τ ))α h(1) (τ ) f (τ ) dτ

−∞

(3.413) and the right-sided Prabhakar-type general fractional integral with respect to another function on the real line is α,ϕ,υ,λ R f GP I−,h

∞ (t) =

  −ϕ (h (τ ) − h (t))−υ Eα,1−υ −λ (h (τ ) − h (t))α h(1) (τ ) f (τ ) dτ .

t

(3.414) Definition 3.84 ([25]). Let h(1) (t) > 0, α ≥ 0, κ = [α] + 1, and λ, υ, ϕ ∈ R. The left-sided Prabhakar-type general fractional integral with respect to another function on the real line is defined as R I α,ϕ,υ,κ,λ f GP +,h t

=

−∞

(t) −ϕ

(h (t) − h (τ ))κ−υ−1 Eα,κ−υ (−λ (h (t) − h (τ ))α ) h(1) (τ ) f (τ ) dτ

(3.415)

Fractional derivatives with nonsingular kernels

283

and the right-sided Prabhakar type general fractional integral with respect to another function on the real line is R I α,ϕ,υ,κ,λ f GP −,h ∞

=

t

(t) −ϕ

(h (τ ) − h (t))κ−υ−1 Eα,κ−υ (−λ (h (τ ) − h (t))α ) h(1) (τ ) f (τ ) dτ .

(3.416)

3.5.13 Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel Definition 3.85. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel is defined as 

 α,υ,ϕ,λ H GP Da+,h f

β(1−α),υ,ϕ,λ GP Ia+,h

(t) =

1

d h(1) (t) dt



 (1−β)(1−α),υ,ϕ,λ GP Ia+,h

f (t) , (3.417)

(1−β)(1−α),υ,ϕ,λ

where GP Ia+,h

β(1−α),υ,ϕ,λ f GP Ia+,h

f (t) ∈ AC1 (a, b),

(t)

t

  ϕ = (h (t) − h (τ ))υ−1 Eβ(1−α),υ −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ , a

(3.418) and (1−β)(1−α),υ,ϕ,λ f GP Ia+,h

(t)

t   ϕ = (h (t) − h (τ ))υ−1 E(1−β)(1−α),υ −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ , a

(3.419) and the right-sided Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel is 

 α,υ,ϕ,λ

H GP Db−,h f (t) =

β(1−α),υ,ϕ,λ

GP Ib−,h

−1 d h(1) (t) dt



 (1−β)(1−α),υ,ϕ,λ

GP Ib−,h

f (t) , (3.420)

284

General Fractional Derivatives With Applications in Viscoelasticity (1−β)(1−α),υ,ϕ,λ

where GP Ib−,h

β(1−α),υ,ϕ,λ f GP Ib−,h

f (t) ∈ AC1 (a, b),

(t)

b   ϕ = (h (τ ) − h (t))υ−1 Eβ(1−α),υ −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ , t

(3.421) and (1−β)(1−α),υ,ϕ,λ f GP Ib−,h

(t)

b

  ϕ = (h (τ ) − h (t))υ−1 E(1−β)(1−α),υ −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ , t

(3.422) Definition 3.86. Let h(1) (t) > 0, 0 < α < 1, 0 ≤ β ≤ 1, and f ∈ L1 (−∞, ∞). The left-sided Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel on the real line is defined as     1 d β(1−α),υ,ϕ,λ (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ D f = I I f (t) (t) , GP + GP + H GP + h(1) (t) dt (3.423) (1−β)(1−α),υ,ϕ,λ

where GP I+,h

β(1−α),υ,ϕ,λ f GP I+,h

=

t

−∞

f (t) ∈ AC1 (−∞, ∞),

(t)

  ϕ (h (t) − h (τ ))υ−1 Eβ(1−α),υ −λ (h (t) − h (τ ))β(1−α) h(1) (τ ) f (τ ) dτ , (3.424)

and (1−β)(1−α),υ,ϕ,λ f GP I+,h

=

t

−∞

(t)

  ϕ (h (t) − h (τ ))υ−1 E(1−β)(1−α),υ −λ (h (t) − h (τ ))(1−β)(1−α) h(1) (τ ) f (τ ) dτ , (3.425)

and the right-sided Hilfer-type general fractional derivative with respect to another function via Prabhakar nonsingular kernel on the real line is     −1 d β(1−α),υ,ϕ,λ (1−β)(1−α),υ,ϕ,λ α,υ,ϕ,λ f (t) , D f = I I (t) H GP −,h GP −,h GP −,h h(1) (t) dt (3.426)

Fractional derivatives with nonsingular kernels (1−β)(1−α),υ,ϕ,λ

where GP I−,h

β(1−α),υ,ϕ,λ f GP I−,h ∞ 

=

t

285

f (t) ∈ AC1 (−∞, ∞),

(t)

  ϕ (h (τ ) − h (t))υ−1 Eβ(1−α),υ −λ (h (τ ) − h (t))β(1−α) h(1) (τ ) f (τ ) dτ , (3.427)

and (1−β)(1−α),υ,ϕ,λ f (t) GP I−,h ∞  ϕ = (h (τ ) − h (t))υ−1 E(1−β)(1−α),υ t



 −λ (h (τ ) − h (t))(1−β)(1−α) h(1) (τ ) f (τ ) dτ . (3.428)

3.6

General fractional derivatives with Gorenflo–Mainardi nonsingular kernel

In 2018, the general fractional derivatives with Gorenflo–Mainardi nonsingular kernel were proposed in the book [25]. Definition 3.87 ([25]). Let 1 > α > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with Gorenflo– Mainardi nonsingular kernel is defined as RL D α,γ ,λ f (t) GM a+

α,γ ,λ d = dt GM Ia+ f (t) t d = dt Gα,γ (−λ (t − τ )α ) f a

(3.429) (τ ) dτ ,

and the right-sided Riemann–Liouville type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is RL D α,γ ,λ f GM b−

(t)

 d  α,γ ,λ = − dt GM Ib− f (t)

d = − dt

b

(3.430)

Gα,γ (−λ (τ − t)α ) f (τ ) dτ ,

t

where α,γ ,λ GM Ia+ f

t (t) = a

  Gα,γ −λ (t − τ )α f (τ ) dτ

(3.431)

286

General Fractional Derivatives With Applications in Viscoelasticity

and α,γ ,λ GM Ib− f

b (t) =

  Gα,γ −λ (τ − t)α f (τ ) dτ ,

(3.432)

t

with ∞    (−λ)κ t κα−1+υ . Gα,γ −λt α =  (κα + υ) κ=0

Definition 3.88. Let α > 0, κ = [α] + 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with Gorenflo– Mainardi nonsingular kernel is defined as RL D α,γ ,λ f GM a+

= =

dκ dt κ dκ dt κ

t

(t)

α,γ ,λ I f (t) GM a+

(3.433)

Gα,γ (−λ (t − τ )α ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is RL D α,γ ,λ f (t) GM b−  d κ α,γ ,λ = − dt GM Ib− f



d = − dt

κ b

(t)

(3.434)

Gα,γ (−λ (τ − t)α ) f (τ ) dτ .

t

Definition 3.89 ([25]). Let 1 > α > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as LS D α,γ ,λ f (t) GM a+   α,γ ,λ = GM Ia+ f (1) (t) t = Gα,γ (−λ (t − τ )α ) f (1) (τ ) dτ, a

(3.435)

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is

Fractional derivatives with nonsingular kernels

LS D α,γ ,λ f (t) GM b−  α,γ ,λ  = GM Ib− −f (1) (t) b

287

(3.436)

= − Gα,γ (−λ (τ − t)α ) f (1) (τ ) dτ . t

Definition 3.90 ([25]). Let α > 0, κ = [α] + 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as LSC D α,γ ,λ f (t) GM a+  α,γ ,λ  (κ) = GM Ia+ f (t) t = Gα,γ (−λ (t − τ )α ) f (κ) (τ ) dτ, a

(3.437)

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is LSC D α,γ ,λ f (t) GM b−  α,γ ,λ  = GM Ib− (−1)κ f (κ) (t) b = (−1)κ Gα,γ (−λ (τ − t)α ) f (κ) (τ ) dτ . t

(3.438)

3.6.1 Prabhakar-type general fractional integrals For the inverse operator of the Riemann–Liouville-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel, we have the following results for the Prabhakar-type general fractional integrals: Definition 3.91 ([25]). Let α > 0, κ = [α] + 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as α,γ ,λ R GM Ia+ f

t (t) =

  −1 −λ (t − τ )α f (τ ) dτ , (t − τ )κ−γ −1 Eα,κ−γ

(3.439)

a

and the right-sided Prabhakar-type general fractional integral is α,γ ,λ R GM Ib− f

b = (−1)

κ

  −1 −λ (τ − t)α f (τ ) dτ . (τ − t)κ−γ −1 Eα,κ−γ

t

The properties of the general fractional calculus are presented as follows.

(3.440)

288

General Fractional Derivatives With Applications in Viscoelasticity

Property 3.19 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then   α,γ ,λ L RL D f (t) GM 0+ 

(3.441) κ−1  −1

κ−μ−1 μ R α,γ ,λ = s κ−γ 1 + λs −α f (s) − s D GM I0+ f (+0) . μ=0

Property 3.20 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then    −1 α,γ ,λ α,γ ,λ 1−γ 1 + λs −α f (s) − R (3.442) L RL GM D0+ f (t) = s GM I0+ f (+0) . Property 3.21 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then ⎞ ⎛ κ      −1 ⎝ κ α,γ ,λ −γ s κ−j f (j −1) (0)⎠ . 1 + λs −α s f (s) − L LSC GM D0+ f (t) = s j =1

(3.443) Property 3.22 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then     α,γ ,λ γ −κ 1 + λs −α f (s) . (3.444) L R GM I0+ f (t) = s Property 3.23 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then

α,γ ,λ R α,γ ,λ RL D I f (3.445) (t) = f (t) (1 > α > 0) , GM a+ GM a+ and



α,γ ,λ R α,γ ,λ RL GM Db− GM Ib− f α,γ ,λ R α,γ ,λ RL GM Da+ GM Ia+ f

α,γ ,λ R α,γ ,λ RL GM Db− GM Ib− f





(t) = f (t)

(1 > α > 0) ,

(3.446)

(t) = f (t)

(κ + 1 > α > κ) ,

(3.447)

(t) = f (t)

(κ + 1 > α > κ) .

(3.448)

Property 3.24 ([25]). If −∞ < a < b < ∞, γ ∈ R, λ ∈ R and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞), then

α,γ ,λ R α,γ ,λ LS (3.449) GM Da+ GM Ia+ f (t) = f (t) (1 > α > 0) ,

α,γ ,λ R α,γ ,λ LS GM Db− GM Ib− f



(t) = f (t)

(1 > α > 0) ,

(3.450)

Fractional derivatives with nonsingular kernels

and



α,γ ,λ LSC α,γ ,λ R GM Da+ GM Ia+ f

α,γ ,λ LSC α,γ ,λ R GM Db− GM Ib− f



(t) = f (t)

289

(κ + 1 > α > κ) ,

(t) = f (t) (κ + 1 > α > κ) .

(3.451)

(3.452)

3.6.2 Hilfer-type general fractional derivatives with Gorenflo–Mainardi nonsingular kernel We now present the Hilfer type general fractional derivatives with Gorenflo–Mainardi nonsingular kernel. Definition 3.92. Let 0 < α < 1, 0 ≤ β ≤ 1, γ , λ ∈ R, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as   α,γ ,λ β(1−α),γ ,λ d (1−β)(1−α),γ ,λ f (t) , (3.453) Y GM Da+ f (t) = GM Ia+ GM Ia+ dt where β(1−α),γ ,λ f GM Ia+

t (t) =



Gβ(1−α),γ −λ (t − τ )β(1−α) f (τ ) dτ

(3.454)

a

and (1−β)(1−α),γ ,λ f GM Ia+

t (t) =



G(1−β)(1−α),γ −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

a

(3.455) and the right-sided Hilfer-type general fractional derivative with Gorenflo–Mainardi nonsingular kernel is   α,γ ,λ β(1−α),γ ,λ d (1−β)(1−α),γ ,λ f (t) , (3.456) D f = I I (t) Y GM b− GM b− GM b− dt where β(1−α),γ ,λ f GM Ib−

b (t) =



Gβ(1−α),γ −λ (τ − t)β(1−α) f (τ ) dτ

(3.457)

t

and (1−β)(1−α),γ ,λ f GM Ib−

b (t) =



G(1−β)(1−α),γ −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

t

(3.458)

290

General Fractional Derivatives With Applications in Viscoelasticity

Property 3.25. Let 0 < α < 1, 0 ≤ β ≤ 1, γ , λ ∈ R, f ∈ L1 (a, b) (−∞ < a < b < ∞) (1−β)(1−α),γ ,λ and GM I0+ f (t) ∈ AC1 (a, b). Then 

 (t)  −α −1

α,γ ,λ Y GM D0+ f

L

 = s −γ 1 + λs

 −1 (1−β)(1−α),γ ,λ s 1−γ 1 + λs −α f (s) − GM I0+ f (0) . (3.459)

Proof. By direct calculation,  L

α,γ ,λ Y GM D0+ f

=L

 (t)



β(1−α),γ ,λ d (1−β)(1−α),γ ,λ GM I0+ dt GM I0+



 f (t)

 −1  −1 (1−β)(1−α),γ ,λ = s −γ 1 + λs −α s × s −γ 1 + λs −α f (s) − GM I0+ f (0)

 −1 1−γ  −1 (1−β)(1−α),γ ,λ s f (s) − GM I0+ f (0) , 1 + λs −α = s −γ 1 + λs −α (3.460)

which finishes the proof.

3.6.3 Riemann–Liouville–Hilfer general fractional derivatives with Gorenflo–Mainardi nonsingular kernel We now suggest the new general fractional derivatives with Gorenflo–Mainardi nonsingular kernel, which is called the Riemann–Liouville–Hilfer general fractional derivatives with Gorenflo–Mainardi nonsingular kernel. Definition 3.93. Let 1 > α > 0, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville–Hilfer general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as H i D α,β,γ ,λ f GM a+ t

(t)

α,β = Gα,γ (−λ (t − τ )α ) Da+ f (τ ) dτ a t

(3.461)



β(1−α) d (1−β)(1−α) = Gα,γ (−λ (t − τ )α ) Ia+ f (τ ) dτ , dτ Ia+ a

where the left-sided Hilfer fractional derivative is 

 α,β β(1−α) d (1−β)(1−α) Ia+ f Da+ f (τ ) = Ia+ (τ ) , dτ

(3.462)

Fractional derivatives with nonsingular kernels

291

with β(1−α) Ia+ f

1 (t) =  (β (1 − α))

t a

f (τ ) (t − τ )1−β(1−α)

(3.463)

dτ

and (1−β)(1−α) Ia+ f

1 (t) =  ((1 − β) (1 − α))

t

f (τ ) (t − τ )

1−(1−β)(1−α)

a

dτ ,

(3.464)

and the right-sided Riemann–Liouville–Hilfer general fractional derivative with Gorenflo–Mainardi nonsingular kernel is H i D α,β,γ ,λ f GM b− b

(t)

α,β = − Gα,γ (−λ (τ − t)α ) Db− f (τ ) dτ t b

= − Gα,γ t

(3.465)



β(1−α) d (1−β)(1−α) I f (−λ (τ − t)α ) Ib− (τ ) dτ , b− dτ

where the right-sided Hilfer fractional derivative is α,β

β(1−α)

Db− f (τ ) = −Ib−

d (1−β)(1−α)

I f , dτ b−

(3.466)

with β(1−α) Ib− f

1 (t) =  (β (1 − α))

b t

f (τ ) (τ − t)1−β(1−α)

(3.467)

dτ

and (1−β)(1−α) Ib− f

1 (t) =  ((1 − β) (1 − α))

b

f (τ ) (τ − t)

1−(1−β)(1−α)

t

dτ .

(3.468)

Definition 3.94. Let κ + 1 > α > κ, 0 ≤ β ≤ 1, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville–Hilfer general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as H ir D α,β,γ ,κ,λ f GM a+ t

(t)

α,β,κ = Gα,γ (−λ (t − τ )α ) Da+ f (τ ) dτ a

t β(κ−α) d κ (1−β)(κ−α) = Gα,γ (−λ (t − τ )α ) Ia+ I f (τ ) dτ , κ a+ dτ a

(3.469)

292

General Fractional Derivatives With Applications in Viscoelasticity

where d κ (1−β)(κ−α)

I f dτ κ a+

α,β,κ

β(κ−α)

β(κ−α) Ia+ f

1 (t) =  (β (κ − α))

Da+ f (τ ) = Ia+

(3.470)

with t a

f (τ ) (t − τ )1−β(κ−α)

(3.471)

dτ

and (1−β)(κ−α) Ia+ f

1 (t) =  ((1 − β) (κ − α))

t a

f (τ ) (t − τ )

1−(1−β)(κ−α)

dτ ,

(3.472)

and the right-sided Riemann–Liouville–Hilfer general fractional derivative with Gorenflo–Mainardi nonsingular kernel is H ir D α,β,γ ,κ,λ f GM b−

(t)

b α,β,κ = Gα,γ (−λ (τ − t)α ) Db− f (τ ) dτ t

= (−1)κ

b t

(3.473)



β(κ−α) d κ (1−β)(κ−α) I Gα,γ (−λ (τ − t)α ) Ib− f (τ ) dτ , κ b− dτ

where  α,β,κ Db− f

β(κ−α) (τ ) = Ib−

d − dτ

κ

(1−β)(κ−α) Ib− f ,

(3.474)

with β(κ−α) Ib− f

1 (t) =  (β (κ − α))

b t

f (τ ) (τ − t)1−β(κ−α)

(3.475)

dτ

and (1−β)(κ−α)

Ib−

f (t) =

1  ((1 − β) (κ − α))

b t

f (τ ) (τ − t)

1−(1−β)(κ−α)

dτ .

(3.476)

Fractional derivatives with nonsingular kernels

3.7

293

General fractional derivatives with Miller–Ross nonsingular kernel

In fact, the general fractional derivatives with Miller–Ross nonsingular kernel were proposed in the book [25]. We now investigate the general fractional calculus with Miller–Ross nonsingular kernel in this section.

3.7.1 Riemann–Liouville-type general fractional derivative with Miller–Ross nonsingular kernel Definition 3.95 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and γ , λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with Miller– Ross nonsingular kernel is defined as RL D α,λ f MR a+

= =

d dt d dt



(t)

α,λ MR Ia+ f

t

(t)

(3.477)

Mα (−λ (t − τ )α ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type general fractional derivative with Miller– Ross nonsingular kernel is RL D α,λ f MR b− d = − dt d = − dt



(t)

α,λ MR Ib− f

b

(t)

(3.478)

Mα (−λ (τ − t)α ) f (τ ) dτ ,

t

where t α,λ MR Ia+ f

(t) =

  Mα −λ (t − τ )α f (τ ) dτ

(3.479)

  Mα −λ (τ − t)α f (τ ) dτ ,

(3.480)

a

and b α,λ MR Ib− f (t) = t

with the Miller–Ross function ∞    (−λ)κ t κ+α Mα −λt α = .  (κ + 1 + α) κ=0

(3.481)

294

General Fractional Derivatives With Applications in Viscoelasticity

Definition 3.96 ([25]). Let α > 0, κ = [α] +1, −∞ < a < b < ∞ and γ , λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with Miller– Ross nonsingular kernel is defined as RL D α,κ,λ f MR a+

= =

dκ dt κ dκ dt κ

t

(t)

α,λ MR Ia+ f

(t)

(3.482)

Mα (−λ (t − τ )α ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type general fractional derivative with Miller– Ross nonsingular kernel is RL D α,κ,λ f MR b−

 d = − dt

κ

 d κ = − dt

(t)

α,λ I f (t) MR b−

b

(3.483)

Mα (−λ (τ − t)α ) f (τ ) dτ .

t

3.7.2 Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel Definition 3.97 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and γ , λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel is defined as LSC D α,λ f MR a+

(t)

 α,λ  (1) f (t) = MR Ia+

(3.484)

t = Mα (−λ (t − τ )α ) f (1) (τ ) dτ, a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel is LSC D α,λ f MR b−

(t)

 α,λ  −f (1) (t) = MR Ib− b = − Mα (−λ (τ − t)α ) f (1) (τ ) dτ . t

(3.485)

Fractional derivatives with nonsingular kernels

295

Definition 3.98 ([25]). Let α > 0, κ = [α] + 1, −∞ < a < b < ∞ and γ , λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel is defined as LSC D α,λ f MR a+ (t)  α,λ  (κ) = MR Ia+ f (t) t

(3.486)

= Mα (−λ (t − τ )α ) f (κ) (τ ) dτ , a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with Miller–Ross nonsingular kernel is LSC D α,λ f MR b−

(t)

 α,λ  = MR Ib− (−1)κ f (κ) (t) = (−1)

b κ

(3.487)

Mα (−λ (τ − t)α ) f (κ) (τ ) dτ .

t

We now consider the inverse operator of the general fractional derivatives with Miller–Ross nonsingular kernel as follows. Definition 3.99 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞ and γ , λ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as t α,κ,λ R MR Ia+ f

(t) = a

1 (t − τ )α+2−κ

E1,κ−α−1 (−λ (t − τ )) f (τ ) dτ ,

(3.488)

and the right-sided Prabhakar-type general fractional integral is b α,κ,λ R MR Ib− f

(t) = t

1 (t − τ )α+2−κ

E1,κ−α−1 (−λ (τ − t)) f (τ ) dτ .

(3.489)

The properties of the general fractional calculus with Miller–Ross nonsingular kernel are given as follows. Property 3.26 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then  L

RL α,κ,λ MR D0+ f

 (t)

κ−1 

   −1 α,λ f (s) − s κ−μ−1 D μ R = s κ−γ 1 + λs −α MR I0+ f (+0) . μ=0

(3.490)

296

General Fractional Derivatives With Applications in Viscoelasticity

Property 3.27 ([25]). Let 1 > α > 0, −∞ < a < b < ∞, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then    −1 α,λ α,λ 1−γ f (s) − R (3.491) 1 + λs −α L RL MR D0+ f (t) = s MR I0+ f (+0) . Property 3.28 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then ⎞ ⎛ κ 

−1  α,κ,λ −α−1 ⎝s κ f (s) − L LSC s κ−j f (j −1) (0)⎠ . 1 + λs −1 MR D0+ f (t) = s 

j =1

(3.492) Property 3.29 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, λ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then 

 α,κ,λ α+1−κ 1 + λs −1 f (s) . (3.493) L R MR I0+ f (t) = s Property 3.30 ([25]). Let −∞ < a < b < ∞, λ ∈ C and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then

α,λ RL α,λ R (3.494) MR Da+ MR Ia+ f (t) = f (t) (1 > α > 0) ,

α,λ RL α,λ R MR Db− MR Ib− f



(t) = f (t) (1 > α > 0) ,

α,κ,λ RL α,κ,λ R MR Da+ MR Ia+ f (t) = f (t) (κ + 1 > α > κ) ,

(3.495) (3.496)

and

α,κ,λ RL α,κ,λ R MR Db− MR Ib− f



(t) = f (t) (κ + 1 > α > κ) .

(3.497)

Property 3.31 ([25]). Let −∞ < a < b < ∞, λ ∈ C and f (t) ∈ ACκ (a, b) (1 ≤ κ < ∞). Then

α,λ R α,λ LS D I f (3.498) (t) = f (t) (1 > α > 0) , MR a+ MR a+

α,λ R α,λ LS MR Db− MR Ib− f



(t) = f (t) (1 > α > 0) ,

α,κ,λ LSC α,κ,λ R D I f (t) = f (t) (κ + 1 > α > κ) , a+ a+ MR MR

(3.499) (3.500)

and

α,κ,λ LSC α,κ,λ R MR Db− MR Ib− f



(t) = f (t) (κ + 1 > α > κ) .

(3.501)

Fractional derivatives with nonsingular kernels

297

3.7.3 Hilfer-type general fractional derivatives with Miller–Ross nonsingular kernel We now present the Hilfer-type general fractional derivatives with Miller–Ross nonsingular kernel. Definition 3.100. Let 0 < α < 1, 0 ≤ β ≤ 1, λ ∈ R, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with Miller–Ross nonsingular kernel is defined as   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.502) Y MR Da+ f (t) = MR Ia+ MR Ia+ dt where β(1−α),λ f MR Ia+

t (t) =



Mβ(1−α) −λ (t − τ )β(1−α) f (τ ) dτ ,

(3.503)

a

and (1−β)(1−α),λ f MR Ia+

t (t) =



M(1−β)(1−α) −λ (t − τ )(1−β)(1−α) f (τ ) dτ , (3.504)

a

and the right-sided Hilfer-type general fractional derivative with Miller–Ross nonsingular kernel is   α,β,λ β(1−α),λ d (1−β)(1−α),λ f (t) , (3.505) MR Db− f (t) = MR Ib− MR Ib− dt where β(1−α),λ f MR Ib−

b (t) =



Mβ(1−α) −λ (τ − t)β(1−α) f (τ ) dτ ,

(3.506)

t

and (1−β)(1−α),λ f MR Ib−

b (t) =



M(1−β)(1−α) −λ (τ − t)(1−β)(1−α) f (τ ) dτ . (3.507)

t

Property 3.32. Let 0 < α < 1, 0 ≤ β ≤ 1, λ ∈ R, f ∈ L1 (a, b) (−∞ < a < b < ∞) (1−β)(1−α),λ and MR I0+ f (t) ∈ AC1 (a, b). Then  L

α,γ ,λ Y MR D0+ f

 = s −α−1 1 + λs

 (t)  −1 −1

(3.508)  −1 (1−β)(1−α),λ s −α 1 + λs −1 f (s) − MR I0+ f (0) .

298

General Fractional Derivatives With Applications in Viscoelasticity

Proof. Using the definitions, we get  L

α,γ ,λ Y GM D0+ f

 (t)

 β(1−α),λ d (1−β)(1−α),λ I I f (t) MR a+ dt MR a+

   −1 −1 (1−β)(1−α),λ f (s) − MR I0+ f (0) s × s −α−1 1 + λs −1 = s −α−1 1 + λs −1

 −1 −α  −1 (1−β)(1−α),λ 1 + λs −1 f (s) − MR I0+ f (0) . s = s −α−1 1 + λs −1 =L



(3.509) Therefore, the proof is completed.

3.8

General fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel

In 2018, the general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel were proposed in the book [25]. In this section, we present the general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel.

3.8.1 Riemann–Liouville-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel Definition 3.101 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and γ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with oneparameter Lorenzo–Hartley nonsingular kernel is defined as RL D α,γ f a+ A

= =

(t)

 α,γ  d dt A Ia+ f (t) t d α dt Fα (−γ (t − τ ) ) f a

(3.510) (τ ) dτ ,

and the right-sided Riemann–Liouville-type general fractional derivative with oneparameter Lorenzo–Hartley nonsingular kernel is RL D α,γ f A b− d = − dt d = − dt



(t)

α,γ A Ib− f

b t

 (t)

Fα (−γ (τ − t)α ) f (τ ) dτ ,

(3.511)

Fractional derivatives with nonsingular kernels

299

where α,γ A Ia+ f

t (t) =

  Fα −γ (t − τ )α f (τ ) dτ

(3.512)

  Fα −γ (τ − t)α f (τ ) dτ ,

(3.513)

a

and α,γ A Ib− f

b (t) = t

with ∞    (−λ)κ t (κ+1)α−1 Fα −γ t α = .  ((κ + 1) α)

(3.514)

κ=0

Definition 3.102 ([25]). Let α > 0, κ+1 > α > κ, −∞ < a < b < ∞ and γ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with oneparameter Lorenzo–Hartley nonsingular kernel is defined as RL D α,κ,γ f a+ A

= =

(t)

 α,γ  dκ dt κ A Ia+ f (t) t dκ Fα (−γ (t − τ )α ) f κ dt a

(3.515) (τ ) dτ ,

and the right-sided Riemann–Liouville-type general fractional derivative with oneparameter Lorenzo–Hartley nonsingular kernel is RL D α,κ,γ f b− A

(t)

 d κ  α,γ  = − dt A Ib− f (t)

(3.516)

 d κ b = − dt Fα (−γ (τ − t)α ) f (τ ) dτ . t

3.8.2 Liouville–Sonine–Caputo-type general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel Definition 3.103 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and γ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as

300

General Fractional Derivatives With Applications in Viscoelasticity

LS D α,γ f a+ A

(t)



 α,γ = A Ia+ f (1) (t)

(3.517)

t

= Fα (−γ (t − τ )α ) f (1) (τ ) dτ, a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is LS D α,λ f A b− α,γ

= A Ib−

(t)  (1)  −f (t)

(3.518)

b

= − Fα (−γ (τ − t)α ) f (1) (τ ) dτ . t

Definition 3.104 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞ and γ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as LSC D α,κ,γ f a+ A

(t)

 α,γ  = A Ia+ f (κ) (t)

(3.519)

t

= Fα (−γ (t − τ )α ) f (κ) (τ ) dτ, a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is LSC D α,κ,λ f A b− α,γ

(t)



= A Ib− (−1)κ f (κ) (t) = (−1)κ

b



(3.520)

Fα (−γ (τ − t)α ) f (κ) (τ ) dτ .

t

As the inverse operator of the general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel, the Prabhakar-type general fractional integrals are presented as follows: Definition 3.105 ([25]). Let α ∈ C, κ + 1 > α > κ, −∞ < a < b < ∞ and γ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as R α,κ,γ A Ia+ f

t (t) = a

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

(3.521)

Fractional derivatives with nonsingular kernels

301

and the right-sided Prabhakar-type general fractional integral is R α,κ,γ A Ib− f

b (t) =

  −1 −γ (τ − t)α f (τ ) dτ . (τ − t)κ−α−1 Eα,κ−α

(3.522)

t

The properties of the general fractional derivatives with one-parameter Lorenzo– Hartley nonsingular kernel are given as follows: Property 3.33 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then  L

RL α,κ,γ A D0+ f

κ−1  

    α,γ κ−α −α −1 1+γs f (s)− s κ−μ−1 D μ R (t) = s A I0+ f (+0) . μ=0

(3.523) Property 3.34 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then    −1 α,γ α,γ 1−α L RL 1 + γ s −α f (s) − R (3.524) A D0+ f (t) = s A I0+ f (+0) . Property 3.35 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then ⎞ ⎛ κ      −1 α,γ −α ⎝s κ f (s) − 1 + γ s −α s κ−j f (j −1) (0)⎠ . (3.525) L LSC A D0+ f (t) = s j =1

Property 3.36 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then     α,κ,γ (3.526) I f = s α−κ 1 + γ s −α f (s) . L R (t) A 0+ Property 3.37 ([25]). Let −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1≤ l < ∞). Then

RL α,γ R α,γ (3.527) A Da+ A Ia+ f (t) = f (t) (1 > α > 0) , and



RL α,γ R α,γ A Db− A Ib− f RL α,γ R α,γ A Da+ A Ia+ f

RL α,γ R α,γ A Db− A Ib− f





(t) = f (t)

(1 > α > 0) ,

(3.528)

(t) = f (t)

(κ + 1 > α > κ) ,

(3.529)

(t) = f (t)

(κ + 1 > α > κ) .

(3.530)

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General Fractional Derivatives With Applications in Viscoelasticity

Property 3.38 ([25]). Let −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then

LS α,γ R α,γ D I f (3.531) (t) = f (t) (1 > α > 0) , a+ A a+ A

LS α,γ R α,γ A Db− A Ib− f



LSC α,γ R α,γ A Da+ A Ia+ f

(t) = f (t)

(1 > α > 0) ,

(3.532)

(t) = f (t)

(κ + 1 > α > κ) ,

(3.533)

(t) = f (t)

(κ + 1 > α > κ) .

(3.534)

and

LSC α,γ R α,γ A Db− A Ib− f



3.8.3 Hilfer-type general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel We now present the Hilfer-type general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel. Definition 3.106. Let 0 < α < 1, 0 ≤ β ≤ 1, λ ∈ R, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as  α,γ Y LH Da+ f

β(1−α),γ A Ia+

(t) =

 d (1−β)(1−α),γ f (t) , AI dt a+

(3.535)

where β(1−α),γ f A Ia+

t (t) =



Fβ(1−α) −γ (t − τ )β(1−α) f (τ ) dτ

(3.536)



F(1−β)(1−α) −γ (t − τ )(1−β)(1−α) f (τ ) dτ ,

(3.537)

a

and (1−β)(1−α),γ f A Ia+

t (t) = a

and the right-sided Hilfer-type general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is  α,γ

Y LH Db− f (t) =

β(1−α),γ

A Ib−

 d (1−β)(1−α),γ I f (t) , A dt b−

(3.538)

Fractional derivatives with nonsingular kernels

303

where β(1−α),γ f A Ib−

b (t) =



Fβ(1−α) −γ (τ − t)β(1−α) f (τ ) dτ

(3.539)

t

and (1−β)(1−α),γ f A Ib−

b (t) =



F(1−β)(1−α) −γ (τ − t)(1−β)(1−α) f (τ ) dτ .

(3.540)

t

Property 3.39. Let 0 < α < 1, 0 ≤ β ≤ 1, γ ∈ R, f ∈ L1 (a, b) (−∞ < a < b < ∞) (1−β)(1−α),γ and A I0+ f (t) ∈ AC1 (a, b). Then  (t)

(3.541)  −1 2−α  −1 (1−β)(1−α),γ 1 + γ s −α f (s) − A I0+ f (0) . s = s 1−α 1 + γ s −α

L



α,γ Y LH D0+ f

Proof. By direct calculation, L



α,γ Y LH D0+ f



 (t)

 f (t)

 −1  −1 (1−β)(1−α),γ = s 1−α 1 + γ s −α f (s) − A I0+ f (0) s × s 1−α 1 + γ s −α

 −1 2−α  −1 (1−β)(1−α),γ = s 1−α 1 + γ s −α 1 + γ s −α f (s) − A I0+ f (0) . s =L

β(1−α),γ d (1−β)(1−α),γ A I0+ dt A I0+



(3.542) Therefore, the proof is finished.

3.9

General fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel

In 2018, the general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel were proposed in the book [25]. In this section, we present the general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel.

3.9.1 Riemann–Liouville-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel Definition 3.107 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and υ, λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with twoparameter Lorenzo–Hartley nonsingular kernel is defined as

304

General Fractional Derivatives With Applications in Viscoelasticity

RL D α,υ,λ f H L a+

= =

d dt d dt



(t)

α,υ,λ H L Ia+ f

t

(t)

(3.543)

Rα,υ (−λ (t − τ )α ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type general fractional derivative with twoparameter Lorenzo–Hartley nonsingular kernel is RL D α,υ,λ f H L b− d = − dt d = − dt



(t)

α,υ,λ H L Ib− f

b

(t)

(3.544)

Rα,υ (−λ (τ − t)α ) f (τ ) dτ ,

t

where t α,υ,λ H L Ia+ f

(t) =

  Rα,υ −λ (t − τ )α f (τ ) dτ

(3.545)

  Rα,υ −λ (τ − t)α f (τ ) dτ ,

(3.546)

a

and b α,υ,λ H L Ib− f

(t) = t

with ∞    (−λ)κ t (κ+1)α−1−υ Rα,υ −γ t α = .  ((κ + 1) α − υ)

(3.547)

κ=0

Definition 3.108 ([25]). Let α > 0, κ+1 > α > κ, −∞ < a < b < ∞ and υ, λ ∈ R. The left-sided Riemann–Liouville-type general fractional derivative with twoparameter Lorenzo–Hartley nonsingular kernel is defined as RL D α,υ,κ,λ f H L a+

= =

dκ dt κ dκ dt κ

t

(t)

α,υ,λ H L Ia+ f (t)

(3.548)

Rα,υ (−λ (t − τ )α ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type general fractional derivative with twoparameter Lorenzo–Hartley nonsingular kernel is

Fractional derivatives with nonsingular kernels

RL D α,υ,κ,λ f H L b−



d = − dt

(t)

κ

= (−1)κ

α,υ,λ H L Ib− f

dκ dt κ

b

(t)

305

(3.549)

Rα,υ (−λ (τ − t)α ) f (τ ) dτ .

t

3.9.2 Liouville–Sonine–Caputo-type general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel Definition 3.109 ([25]). Let 1 > α > 0, −∞ < a < b < ∞ and υ, λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is defined as LSC D α,υ,λ f a+ HL

(t)  α,υ,λ  (1) f (t) = H L Ia+

(3.550)

t

= Rα,υ (−λ (t − τ )α ) f (1) (τ ) dτ , a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is LSC D α,υ,λ f HL b−

(t)   α,υ,λ −f (1) (t) = H L Ib−

(3.551)

b

= − Rα,υ (−λ (τ − t)α ) f (1) (τ ) dτ . t

Definition 3.110 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞ and υ, λ ∈ R. The left-sided Liouville–Sonine–Caputo-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is defined as LSC D α,υ,κ,λ f a+ HL

(t)

 α,υ,λ  (κ) = H L Ia+ f (t)

(3.552)

t

= Rα,υ (−λ (t − τ )α ) f (κ) (τ ) dτ, a

and the right-sided Liouville–Sonine–Caputo-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is

306

General Fractional Derivatives With Applications in Viscoelasticity

LSC D α,υ,κ,λ f (t) HL b−  α,υ,λ  = H L Ib− (−1)κ f (κ) (t) b = (−1)κ Rα,υ (−λ (τ − t)α ) f (κ) (τ ) dτ . t

(3.553)

As the inverse operators of the general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel, we give the Prabhakar-type general fractional integrals as follows: Definition 3.111 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞ and υ, λ ∈ R. The left-sided Prabhakar-type general fractional integral is defined as t α,υ,κ,λ R f H L Ia+

(t) =

  −1 −γ (t − τ )α f (τ ) dτ (t − τ )κ−(α−υ)−1 Eα,κ−(α−υ)

(3.554)

a

and the right-sided Prabhakar-type general fractional integral is b α,υ,κ,λ R f H L Ib−

(t) =

  −1 −γ (τ − t)α f (τ ) dτ . (3.555) (τ − t)κ−(α−υ)−1 Eα,κ−(α−υ)

t

The properties of the general fractional derivatives with two-parameter Lorenzo– Hartley nonsingular kernel are presented as follows: Property 3.40 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, υ, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then 

RL D α,υ,κ,λ f H L 0+

L

 (t)

 (3.556) κ−1  −1

κ−μ−1 μ R α,υ,κ,λ f (s) − s f (0) . D H L I0+ = s κ−(α−υ) 1 + λs −α μ=0

Property 3.41 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, υ, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then    −1 α,υ,λ α,υ,λ D f = s 1−(α−υ) 1 + λs −α f (s) − R L RL (t) H L 0+ H L I0+ f (0) . (3.557) Property 3.42 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, υ, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then  L

LSC α,υ,κ,λ f H L D0+



(t) = s

 −(α−υ)

1 − λs

 −α −1

⎛ ⎝s κ f (s) −

κ 

⎞ s κ−j f (j −1) (0)⎠ .

j =1

(3.558)

Fractional derivatives with nonsingular kernels

307

Property 3.43 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞, υ, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then     α,υ,κ,λ (3.559) I f = s (α−υ)−κ 1 − λs −α f (s) . L R (t) H L 0+ Property 3.44 ([25]). Let −∞ < a < b < ∞, υ, γ ∈ R and f (t) ∈ Ll (a, b) (1 ≤ l < ∞). Then

α,υ,λ RL α,υ,λ R (3.560) H L Da+ H L Ia+ f (t) = f (t) (1 > α > 0) ,

α,υ,λ RL α,υ,λ R H L Db− H L Ib− f



(t) = f (t) (1 > α > 0) ,

α,υ,κ,λ RL α,υ,κ,λ R f (t) = f (t) (κ + 1 > α > κ) , H L Da+ H L Ia+

(3.561) (3.562)

and

α,υ,κ,λ RL α,υ,κ,λ R f H L Db− H L Ib−



(t) = f (t) (κ + 1 > α > κ) .

(3.563)

Property 3.45 ([25]). Let −∞ < a < b < ∞, γ ∈ R and f (t) ∈ Ll (a, b) (1≤ l < ∞). Then

α,υ,λ LSC α,υ,λ R (3.564) H L Da+ H L Ia+ f (t) = f (t) (1 > α > 0) ,

α,υ,λ LSC α,υ,λ R H L Db− H L Ib− f



(t) = f (t) (1 > α > 0) ,

α,υ,κ,λ LSC α,υ,κ,λ R D I f (t) = f (t) (κ + 1 > α > κ) , a+ H L a+ HL

(3.565) (3.566)

and

α,υ,κ,λ LSC α,υ,κ,λ R f H L Ib− H L Db−



(t) = f (t)

(κ + 1 > α > κ) .

(3.567)

3.9.3 Hilfer-type general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel We now present the Hilfer-type general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel. Definition 3.112. Let 0 < α < 1, 0 ≤ β ≤ 1, λ ∈ R, and f ∈ L1 (a, b) (−∞ < a < b < ∞). The left-sided Hilfer-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is defined as   β(1−α),υ,λ d (1−β)(1−α),υ,λ α,υ,λ D f = I I f (t) , (3.568) (t) Y T LH a+ H L a+ H L a+ dt

308

General Fractional Derivatives With Applications in Viscoelasticity

where β(1−α),υ,λ f H L Ia+

t (t) =



Rβ(1−α),υ −λ (t − τ )β(1−α) f (τ ) dτ

(3.569)

a

and (1−β)(1−α),υ,λ f H L Ia+

t (t) =



R(1−β)(1−α),υ −λ (t − τ )(1−β)(1−α) f (τ ) dτ ,

a

(3.570) and the right-sided Hilfer-type general fractional derivative with two-parameter Lorenzo–Hartley nonsingular kernel is   β(1−α),υ,λ d (1−β)(1−α),υ,λ α,υ,λ f (t) , (3.571) Y T LH Db− f (t) = H L Ib− H L Ib− dt where β(1−α),υ,λ f H L Ib−

b (t) =



Rβ(1−α),υ −λ (τ − t)β(1−α) f (τ ) dτ

(3.572)

t

and (1−β)(1−α),υ,λ f H L Ib−

b (t) =



R(1−β)(1−α),υ −λ (τ − t)(1−β)(1−α) f (τ ) dτ .

t

(3.573) Property 3.46. Let 0 < α < 1, 0 ≤ β ≤ 1, γ ∈ R, f ∈ L1 (a, b) (−∞ < a < b < ∞) (1−β)(1−α),υ,λ and H L I0+ f (t) ∈ AC1 (a, b). Then  L

α,υ,λ Y T LH D0+ f

 (t)

 −1 1−(α−υ)  −1 (1−β)(1−α),υ,λ = s −(α−υ) 1 − λs −α s 1 − λs −α f (s) − H L I0+ f (0) . (3.574) Proof. By direct computation, we obtain   α,υ,λ L Y T LH D0+ f (t)    β(1−α),υ,λ d (1−β)(1−α),υ,λ f = L H L I0+ I (t) H L 0+ dt

Fractional derivatives with nonsingular kernels

309

 −1  −1 (1−β)(1−α),υ,λ = s −(α−υ) 1 − λs −α f (s) − H L I0+ f (0) s × s −(α−υ) 1 − λs −α

 −1 1−(α−υ)  −1 (1−β)(1−α),υ,λ f (s) − H L I0+ f (0) , 1 − λs −α s = s −(α−υ) 1 − λs −α (3.575) finishing the proof. Moreover, there are some open problems involving the nonsingular power-law kernels [25,138–140].

Variable-order fractional derivatives with singular kernels

4

Contents 4.1 Riemann–Liouville-type variable-order fractional calculus with singular kernel 311 4.1.1 4.1.2 4.1.3

History of the variable-order fractional calculus with singular kernel Riemann–Liouville-type variable-order fractional integrals 312 Riemann–Liouville-type variable-order fractional derivatives 313

311

4.2 Variable-order Hilfer-type fractional derivatives with singular kernel 4.3 Liouville–Weyl-type variable-order fractional calculus 323 4.3.1 4.3.2

318

Liouville–Weyl-type variable-order fractional integrals 323 Liouville–Weyl-type variable-order fractional derivatives 324

4.4 Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel 330 4.4.1 4.4.2 4.4.3

Riesz-type variable-order fractional derivative with singular kernel 330 Feller-type variable-order fractional derivative with singular kernel 333 Herrmann-type variable-order fractional derivative with singular kernel 334

4.5 Variable-order tempered fractional derivatives with weakly singular kernel 340

In this chapter, we investigate the variable-order fractional derivatives with singular kernels. We main present the concepts of the Riemann–Liouville-type variable-order fractional integrals and derivatives with singular kernel, variable-order Hilfer-type fractional derivatives with singular kernel, Liouville–Weyl-type variable-order fractional integrals and derivatives with singular kernel, as well as Riesz- Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel.

4.1

Riemann–Liouville-type variable-order fractional calculus with singular kernel

4.1.1 History of the variable-order fractional calculus with singular kernel In 1952, Feller introduced the extended version of Marcel Riesz’ potentials with the aid of the variable-order Riesz fractional derivative with singular kernel [141]. In 1993, Samko and Ross proposed the variable-order integration and differentiation with singular kernel [43,142]. The fractional random fields with the variable-order General Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00009-1 Copyright © 2020 Elsevier Inc. All rights reserved.

312

General Fractional Derivatives With Applications in Viscoelasticity

Riesz fractional derivative were proposed in [143]. The more general versions of the variable-order integration and differentiation with singular kernel were reported in [144–157]. The variable-order Caputo differentiation was considered in [147]. First, we present the Riemann–Liouville-type variable-order fractional integrals.

4.1.2 Riemann–Liouville-type variable-order fractional integrals In this section, we introduce the variable-order fractional integrals and derivatives with singular kernel, which are called the variable-order fractional calculus with singular kernel. Definition 4.1 ([145]). Let α (τ ) > 0 and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable order fractional integral with singular kernel is defined as α(·) RL Ia+ f

t (t) = a

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

(4.1)

and the right-sided Riemann–Liouville-type variable-order fractional integral with singular kernel is

α(·) RL Ib− f

b (t) = t

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

(4.2)

where α (·) is a function of τ . Definition 4.2. Let α (t) > 0 and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable-order fractional integral with singular kernel is defined as α(·) R RL Ia+ f

1 (t) =  (α (t))

t a

f (τ ) (t − τ )1−α(τ )

dτ ,

(4.3)

and the right-sided Riemann–Liouville-type variable-order fractional integral with singular kernel is

α(·) R RL Ib− f

1 (t) =  (α (t))

b t

where α (·) is a function of t.

f (τ ) (τ − t)1−α(τ )

dτ ,

(4.4)

Variable-order fractional derivatives with singular kernels

313

4.1.3 Riemann–Liouville-type variable-order fractional derivatives Definition 4.3 ([148]). Let κ < α (t) < 1 + κ and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as

α(·) RL Da+ f

dκ (t) = κ dt

t a

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

(4.5)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is

α(·) RL Db− f

  b 1 d κ f (τ ) dτ , (t) = − dt  (κ − α (τ )) (τ − t)α(τ )−κ+1

(4.6)

t

where α (·) is a function of τ . Definition 4.4 ([148]). Let κ < α (t) < 1 + κ and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as

CGS α(·) RL Da+ f

1 dκ (t) =  (κ − α (t)) dt κ

t a

f (τ ) (t − τ )α(τ )−κ+1

dτ ,

(4.7)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is

CGS α(·) RL Db− f

  b f (τ ) 1 d κ dτ , − (t) =  (κ − α (t)) dt (τ − t)α(τ )−κ+1

(4.8)

t

where α (·) is a function of t. Definition 4.5 ([25]). Let κ < α (t) < 1 + κ and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as

CGS α(·) RL Da+ f

⎤ ⎡ t 1 f (τ ) dκ ⎣ dτ ⎦ , (t) = κ dt  (κ − α (t)) (t − τ )α(τ )−κ+1 a

(4.9)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is ⎤ ⎡  κ b f (τ ) d 1 CGS α(·) ⎣ dτ ⎦ , (4.10) RL Db− f (t) = − dt  (κ − α (t)) (τ − t)α(τ )−κ+1 t

where α (·) is a function of t. Definition 4.6 ([148]). Let 0 < α (t) < 1 and −∞ < a < b < ∞. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) RL Da+ f

= =

d dt d dt



(t)

1−α(·) f RL Ia+

t a

(t)

(4.11)

f (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ ,

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is α(·) RL Db− f d = − dt d = − dt

(t)

α(·) Ib− f (t) b t

(4.12)

f (τ ) 1 (1−α(τ )) (τ −t)α(τ ) dτ .

Definition 4.7 ([145]). Let α (t) ≥ 0 and κ = [α (t)] + 1. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) LSC Da+ f

(t)

α(·) = Ia+ f (κ) (t) t f (κ) (τ ) 1 = (κ−α(τ )) (t−τ )α(τ )−κ+1 dτ , a

(4.13)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is α(·) LSC Db− f

(x)

α(·) = Ib− f (κ) (x) b f (κ) (τ ) 1 = (−1)κ (κ−α(τ )) (τ −t)α(τ )−κ+1 dτ . t

(4.14)

Variable-order fractional derivatives with singular kernels

315

Definition 4.8 ([145]). Let 0 < α (t) < 1. The left-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is defined as α(·) LS Da+ f

(t)

α(·) 

 = Ia+ f (1) (t)

=

t

(4.15)

f (1) (τ )

1 (1−α(τ )) (t−τ )α(τ ) dτ ,

a

and the right-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is α(·) LS Db− f

(t)

α(·) 

 = Ib− f (1) (t)

=−

b

(4.16) f (1) (τ )

1 (1−α(τ )) (τ −t)α(τ ) dτ .

t

Definition 4.9 ([144]; also see [149]). Let 0 < α (τ, t) < 1. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as H L α(·) RLq Ia+ f

t (t) = a

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

(4.17)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is H L α(·) RLq Ib− f

b (t) = t

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

(4.18)

Definition 4.10 ([149]). Let 0 < α (τ, t) < 1. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) RLq Da+ f

(t)

=

1−α(·)

=

d dt d dt



RLq Ia+

t a

f (t)

f (τ ) 1 (1−α(τ,t)) (t−τ )α(τ,t) dτ ,

(4.19)

316

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is α(·) RLq Db− f d = − dt d = − dt



(t)

α(·) RLq Ib− f

b t

(t)

(4.20)

f (τ ) 1 (1−α(τ,t)) (t−τ )α(τ,t) dτ .

Definition 4.11 ([150]). Let 0 < α (τ, t) < 1. The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) RLq Da+ f

1 d (t) =  (1 − α (τ, t)) dt

t a

f (τ ) (t − τ )α(τ,t)

dτ ,

(4.21)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is α(·) RLq Db− f

  b f (τ ) 1 d dτ . − (t) =  (1 − α (τ, t)) dt (τ − t)α(τ,t)

(4.22)

t

Definition 4.12 ([150]). Let 0 < α (τ, t) < 1. The left-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is defined as α(·) LSq Da+ f

(t)  α(·)  = RLq Ia+ f (1) (t)

=

t a

(4.23)

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

and the right-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is α(·) LSq Db− f

(t)  α(·)  = RLq Ib− f (1) (t)

=−

b t

f (1) (τ )

1 (1−α(τ,t)) (τ −t)α(τ,t) dτ .

(4.24)

Variable-order fractional derivatives with singular kernels

317

Definition 4.13 ([142]). Let α (τ, t) = α (t). The left-sided Riemann–Liouville-type variable-order fractional integral with singular kernel is defined as

α(·) RLqs Ia+ f

1 (t) =  (α (t))

t a

f (τ ) (t − τ )1−α(t)

(4.25)

dτ ,

and the right-sided Riemann–Liouville-type variable-order fractional integral with singular kernel is

α(·) RLqs Ib− f

1 (t) =  (α (t))

b t

f (τ ) (τ − t)1−α(t)

(4.26)

dτ .

Definition 4.14 ([142]). Let α (τ, t) = α (t). The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as

α(·) SR RLqs Da+ f

1 d (t) =  (1 − α (t)) dt

t

f (τ ) (t − τ )α(t)

a

dτ ,

(4.27)

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is

α(·) RLqs Db− f

1 d (t) = −  (1 − α (t)) dt

b t

f (τ ) (τ − t)α(t)

dτ .

(4.28)

Definition 4.15 ([25]). Let α (τ, t) = α (t). The left-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) RLqs Da+ f

= =

d dt d dt



(t)

1−α(·) f RLqs Ia+



1 (1−α(t))

t a

(t)

f (τ ) dτ (t−τ )α(t)

(4.29)

 ,

and the right-sided Riemann–Liouville-type variable-order fractional derivative with singular kernel is

318

General Fractional Derivatives With Applications in Viscoelasticity

α(·) RLqs Db− f d = − dt d = − dt d = − dt

(t)

α(·) I f (t) RLqs b−



b t 

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

b t

f (τ ) dτ (τ −t)α(t)

(4.30)  .

Definition 4.16 ([151]; for the complement function, see [145]). Let α (τ, t) = α (t). The left-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is defined as α(·) LSqs Da+ f

(t)  α(·)  = RLqs Ia+ f (1) (t)

=

t a

=

f (1) (τ ) 1 (1−α(t)) (τ −t)α(t) dτ

1 (1−α(t))

t a

(4.31)

f (1) (τ ) dτ , (t−τ )α(t)

and the right-sided Liouville–Sonine-type variable-order fractional derivative with singular kernel is α(·) LSqs Db− f

(t)  α(·)  = RLqs Ib− f (1) (t)

=−

b t

f (1) (τ ) 1 (1−α(t)) (τ −t)α(t) dτ

1 = − (1−α(t))

b t

(4.32)

f (1) (τ ) dτ . (τ −t)α(t)

For more information of the theory and applications of the variable-order fractional calculus with singular kernel, see [152,155].

4.2

Variable-order Hilfer-type fractional derivatives with singular kernel

In this section, we present the variable-order Hilfer-type fractional derivatives with singular kernel [25]. Definition 4.17 ([25]). Let 0 < α (t) < 1, 0 ≤ β (t) ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞.

Variable-order fractional derivatives with singular kernels

319

The left-sided variable-order Hilfer-type fractional derivative with singular kernel is defined as 

 α(·),β(·) HL H L β(t)(1−α(t)) d H L (1−β(t))(1−α(t)) f (t) = RLq Ia+ I f (t) , (4.33) RLq Da+ dt RLq a+ where H L I β(t)(1−α(t)) f (t) RLq a+ t f (τ ) 1 = (β(τ,t)(1−α(τ,t))) dτ (t−τ )1−β(τ,t)(1−α(τ,t)) a

(4.34)

H L I (1−β(t))(1−α(t)) f (t) RLq a+ t f (τ ) 1 = ((1−β(τ,t))(1−α(τ,t))) dτ , (t−τ )1−(1−β(τ,t))(1−α(τ,t)) a

(4.35)

and

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β(·) HL H L β(t)(1−α(t)) d H L (1−β(t))(1−α(t)) f (t) = −RLq Ib− I f (t) , (4.36) RLq Db− dt RLq b− where H L I β(t)(1−α(t)) f (t) RLq b− ∞ f (τ ) 1 dτ = (β(τ,t)(1−α(τ,t))) (τ −t)1−β(τ,t)(1−α(τ,t)) t

(4.37)

H L I (1−β(t))(1−α(t)) f (t) RLq b− ∞ f (τ ) 1 = ((1−β(τ,t))(1−α(τ,t))) dτ . (τ −t)1−(1−β(τ,t))(1−α(τ,t)) t

(4.38)

and

Definition 4.18 ([25]). Let 0 < α (t) < 1, 0 ≤ β (t) ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivatives with singular kernel is defined as 

 α(·),β(·) β(t)(1−α(t)) d (1−β(t))(1−α(t)) D f = I I f (4.39) (t) (t) , RL a+ RL a+ RL a+ dt

320

General Fractional Derivatives With Applications in Viscoelasticity

where β(t)(1−α(t)) f (t) RL Ia+ t f (τ ) 1 = (β(τ )(1−α(τ ))) (t−τ )1−β(τ )(1−α(τ )) dτ a

(4.40)

(1−β(t))(1−α(t)) f (t) RL Ia+ t f (τ ) 1 = ((1−β(τ ))(1−α(τ ))) (t−τ )1−(1−β(τ ))(1−α(τ )) dτ , a

(4.41)

and

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β(·) β(t)(1−α(t)) d (1−β(t))(1−α(t)) D f = − I I f (t) (t) , (4.42) RL b− RL b− RL b− dt where β(t)(1−α(t)) f RL Ib−

=

b t

(t)

f (τ ) 1 (β(τ )(1−α(τ ))) (τ −t)1−β(τ )(1−α(τ )) dτ

(4.43)

and (1−β(t))(1−α(t)) f RL Ib−

=

b t

(t)

f (τ ) 1 ((1−β(τ ))(1−α(τ ))) (τ −t)1−(1−β(τ ))(1−α(τ )) dτ .

(4.44)

Definition 4.19 ([25]). Let 0 < α (t) < 1, 0 ≤ β (t) ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with singular kernel is defined as 

 α(·),β(·) β(t)(1−α(t)) d (1−β(t))(1−α(t)) R D f = I I f (4.45) (t) (t) , RL RL a+ a+ RL a+ dt where R I β(t)(1−α(t)) f RL a+

=

1 (β(t)(1−α(t)))

(t) t a

f (τ ) dτ (t−τ )1−β(τ )(1−α(τ ))

(4.46)

Variable-order fractional derivatives with singular kernels

321

and R I (1−β(t))(1−α(t)) f RL a+

=

1 ((1−β(t))(1−α(t)))

(t) t

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

a

(4.47)

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β(·) β(t)(1−α(t)) d (1−β(t))(1−α(t)) R D f = − I I f (t) (t) , (4.48) RL b− RL b− RL b− dt where R I β(t)(1−α(t)) f RL b−

=

1 (β(t)(1−α(t)))

(t) b t

(4.49)

f (τ )

(τ −t)1−β(τ )(1−α(τ ))

dτ

and R I (1−β(t))(1−α(t)) f RL b−

=

1 ((1−β(t))(1−α(t)))

(t) b

f (τ )

(τ −t)1−(1−β(τ ))(1−α(τ ))

t

(4.50) dτ .

Definition 4.20. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with singular kernel is defined as 

 α(·),β HL H L β(1−α(t)) d H L (1−β)(1−α(t)) I f (4.51) (t) , RLq Da+ f (t) = RLq Ia+ dt RLq a+ where H L β(1−α(t)) f RLq Ia+

t (t) = a

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

(4.52)

and H L (1−β)(1−α(t)) f RLq Ia+

t (t) = a

f (τ ) 1 dτ , 1−(1−β)(1−α(τ,t))  ((1 − β) (1 − α (τ, t))) (t − τ ) (4.53)

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β HL H L β(1−α(t)) d H L (1−β)(1−α(t)) D f = − I I f (4.54) (t) (t) , RLq b− RLq b− dt RLq b−

322

General Fractional Derivatives With Applications in Viscoelasticity

where H L β(1−α(t)) f RLq Ib−

∞ (t) = t

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

(4.55)

and H L (1−β)(1−α(t)) f RLq Ib−

∞ (t) = t

f (τ ) 1 dτ .  ((1 − β) (1 − α (τ, t))) (τ − t)1−(1−β)(1−α(τ,t)) (4.56)

Definition 4.21. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with singular kernel is defined as 

 α(·),β β(1−α(t)) d (1−β)(1−α(t)) D f = I I f (4.57) (t) (t) , RL a+ RL a+ RL a+ dt where β(1−α(t)) f RL Ia+

t (t) = a

f (τ ) 1 dτ  (β (1 − α (τ ))) (t − τ )1−β(1−α(τ ))

(4.58)

and (1−β)(1−α(t)) f RL Ia+

t

f (τ ) 1 dτ , (4.59) 1−(1−β)(1−α(τ ))  ((1 − β) (1 − α (τ ))) (t − τ )

(t) = a

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β β(1−α(t)) d (1−β)(1−α(t)) D f = − I I f (4.60) (t) (t) , RL b− RL b− RL b− dt where β(1−α(t)) f RL Ib−

b (t) = t

f (τ ) 1 dτ  (β (1 − α (τ ))) (τ − t)1−β(1−α(τ ))

(4.61)

and (1−β)(1−α(t))

RL Ib−

b f (t) = t

f (τ ) 1 dτ . (4.62)  ((1 − β) (1 − α (τ ))) (τ − t)1−(1−β)(1−α(τ ))

Variable-order fractional derivatives with singular kernels

323

Definition 4.22. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with singular kernel is defined as 

 α(·),β β(1−α(t)) d (1−β)(1−α(t)) R D f = I I f (4.63) (t) (t) , RL RL a+ a+ RL a+ dt where β(1−α(t)) R f RL Ia+

1 (t) =  (β (1 − α (t)))

t a

f (τ ) (t − τ )1−β(1−α(τ ))

(4.64)

dτ

and (1−β)(1−α(t)) R f RL Ia+

1 (t) =  ((1 − β) (1 − α (t)))

t

f (τ ) (t − τ )

1−(1−β)(1−α(τ ))

a

dτ , (4.65)

and the right-sided variable-order Hilfer-type fractional derivative with singular kernel is 

 α(·),β β(1−α(t)) d (1−β)(1−α(t)) R D f = − I I f (4.66) (t) (t) , RL b− RL b− RL b− dt where β(1−α(t)) R f RL Ib−

1 (t) =  (β (1 − α (t)))

b t

f (τ ) (τ − t)1−β(1−α(τ ))

(4.67)

dτ

and R (1−β)(1−α(t)) f RL Ib−

4.3

1 (t) =  ((1 − β) (1 − α (t)))

b t

f (τ ) (τ − t)

1−(1−β)(1−α(τ ))

dτ . (4.68)

Liouville–Weyl-type variable-order fractional calculus

In this section, we present the Liouville–Weyl-type variable-order fractional calculus including the Liouville–Weyl-type variable-order fractional integrals and derivatives.

4.3.1 Liouville–Weyl-type variable-order fractional integrals Definition 4.23. Let α (τ ) > 0. The left-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is defined as

324

General Fractional Derivatives With Applications in Viscoelasticity

α(·) RL I+ f

t (t) = −∞

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

(4.69)

and the right-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is α(·) RL I− f

∞ (t) = t

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

(4.70)

where α (·) is a function of τ . Definition 4.24. Let α (t) > 0. The left-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is defined as α(·) R RL I+ f

1 (t) =  (α (t))

t

f (τ ) (t − τ )1−α(τ )

−∞

dτ ,

(4.71)

and the right-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is α(·) R RL I− f

1 (t) =  (α (t))

∞

f (τ ) (τ − t)1−α(τ )

t

dτ ,

(4.72)

where α (·) is a function of t.

4.3.2 Liouville–Weyl-type variable-order fractional derivatives Definition 4.25. Let κ < α (t) < 1 + κ. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) RL D+ f

dκ (t) = κ dt

t −∞

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

(4.73)

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is 

α(·) RL D− f

d (t) = − dt

κ ∞ t

where α (·) is a function of τ .

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

(4.74)

Variable-order fractional derivatives with singular kernels

325

Definition 4.26. Let κ < α (t) < 1 + κ. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as CGS α(·) RL D+ f

dκ 1 (t) =  (κ − α (t)) dt κ

t −∞

f (τ ) (t − τ )α(τ )−κ+1

(4.75)

dτ ,

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is CGS α(·) RL D− f

  ∞ f (τ ) 1 d κ dτ , − (t) =  (κ − α (t)) dt (τ − t)α(τ )−κ+1

(4.76)

t

where α (·) is a function of t. Definition 4.27. Let κ < α (t) < 1 + κ. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as CGS α(·) RL D+ f

⎤ ⎡ t 1 f (τ ) dκ ⎣ dτ ⎦ , (t) = κ dt  (κ − α (t)) (t − τ )α(τ )−κ+1

(4.77)

−∞

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is 

CGS α(·) RL D− f

d (t) = − dt

κ

⎡ 1 ⎣  (κ − α (t))

∞ t

⎤ f (τ ) (τ − t)α(τ )−κ+1

dτ ⎦ ,

(4.78)

where α (·) is a function of t. Definition 4.28. Let 0 < α (t) < 1. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) RL D+ f

= =

d dt d dt



(t)

1−α(·) f RL I+

t −∞

(t)

f (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ ,

(4.79)

326

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is α(·) RL D− f d = − dt d = − dt

(t)

α(·) I− f (t) ∞ t

(4.80)

f (τ ) 1 (1−α(τ )) (τ −t)α(τ ) dτ .

Definition 4.29. Let α (t) ≥ 0 and κ = [α (t)] + 1. The left-sided Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) LSC D+ f

(t)

α(·) = I+ f (κ) (t)

=

t −∞

(4.81) f (κ) (τ )

1 (κ−α(τ )) (t−τ )α(τ )−κ+1 dτ ,

and the right-sided Liouville-type variable-order fractional derivative with singular kernel is α(·) LSC D− f

(x)

α(·) = I− f (κ) (x) ∞ f (κ) (τ ) 1 = (−1)κ (κ−α(τ )) (τ −t)α(τ )−κ+1 dτ . t

(4.82)

Definition 4.30. Let 0 < α (t) < 1. The left-sided Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) LS D+ f

(t)

 α(·)  = I+ f (1) (t) t f (1) (τ ) 1 = (1−α(τ )) (t−τ )α(τ ) dτ , −∞

(4.83)

and the right-sided Liouville-type variable-order fractional derivative with singular kernel is α(·) LS D− f

(t)

 α(·)  = I− f (1) (t) ∞ f (1) (τ ) 1 = − (1−α(τ )) (t−τ )α(τ ) dτ . t

(4.84)

Variable-order fractional derivatives with singular kernels

327

Definition 4.31. Let 0 < α (τ, t) < 1. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as H L α(·) RLq I+ f

t

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

(t) = −∞

(4.85)

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is H L α(·) RLq I− f

∞ (t) = t

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

(4.86)

Definition 4.32. Let 0 < α (τ, t) < 1. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) RLq D+ f

= =

d dt d dt



(t)

1−α(·) f RLq I+

t −∞

(t)

(4.87)

f (τ ) 1 (1−α(τ,t)) (t−τ )α(τ,t) dτ ,

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is α(·) RLq D− f d = − dt d = − dt



α(·) RLq I− f

∞ t

(t)

(t)

(4.88)

f (τ ) 1 (1−α(τ,t)) (τ −t)α(τ,t) dτ .

Definition 4.33. Let 0 < α (τ, t) < 1. The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) RLq D+ f

d 1 (t) =  (1 − α (τ, t)) dt

t −∞

f (τ ) (t − τ )α(τ,t)

dτ ,

(4.89)

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is α(·) RLq D− f

  ∞ d f (τ ) 1 − dτ . (t) =  (1 − α (τ, t)) dt (τ − t)α(τ,t) t

(4.90)

328

General Fractional Derivatives With Applications in Viscoelasticity

Definition 4.34. Let 0 < α (τ, t) < 1. The left-sided Liouville-type variable-order fractional derivative with singular kernel as is defined as α(·) LSq D+ f

(t)

 α(·)  = RLq I+ f (1) (t) t f (1) (τ ) 1 = (1−α(τ,t)) (t−τ )α(τ,t) dt, −∞

(4.91)

and the right-sided Liouville-type variable-order fractional derivative with singular kernel is α(·) LSq D− f

(t)

 α(·)  = RLq I− f (1) (t) ∞ f (1) (τ ) 1 = − (1−α(τ,t)) dτ . (τ −t)α(τ,t) t

(4.92)

Definition 4.35. Let α (τ, t) = α (t). The left-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is defined as α(·) RLqs I+ f

1 (t) =  (α (t))

t −∞

f (τ ) (t − τ )1−α(t)

(4.93)

dτ ,

and the right-sided Liouville–Weyl-type variable-order fractional integral with singular kernel is α(·) RLqs I− f

1 (t) =  (α (t))

∞ t

f (τ ) (τ − t)1−α(t)

(4.94)

dτ .

Definition 4.36. Let α (τ, t) = α (t). The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) SR RLqs D+ f

d 1 (t) =  (1 − α (t)) dt

t −∞

f (τ ) (t − τ )α(t)

dτ ,

(4.95)

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is α(·)

RLqs D− f (t) = −

d 1  (1 − α (t)) dt

∞ t

f (τ ) (τ − t)α(t)

dτ .

(4.96)

Variable-order fractional derivatives with singular kernels

329

Definition 4.37. Let α (τ, t) = α (t). The left-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is defined as α(·) RLqs D+ f

(t)

=

1−α(·)

=

d dt



RLqs I+ 

t

1 (1−α(t))

d dt

f (t)

−∞

(4.97)



f (τ ) dτ (t−τ )α(t)

,

and the right-sided Liouville–Weyl-type variable-order fractional derivative with singular kernel is α(·) RLqs D− f



d = − dt

α(·) RLqs I− f

∞

d = − dt

t



d = − dt

(t)

(t)

f (τ ) 1 (1−α(t)) (τ −t)α(t) dτ

1 (1−α(t))

∞ t

f (τ ) dτ (τ −t)α(t)

(4.98)  .

Definition 4.38. Let α (τ, t) = α (t). The left-sided Liouville-type variable-order fractional derivative with singular kernel is defined as α(·) LSqs D+ f

(t)  α(·)  = RLqs I+ f (1) (t)

= =

t −∞

f (1) (τ ) 1 (1−α(t)) (t−τ )α(t) dτ

1 (1−α(t))

t −∞

(4.99)

f (1) (τ ) dτ , (t−τ )α(t)

and the right-sided Liouville-type variable-order fractional derivative with singular kernel as α(·) LSqs D− f

(t)  α(·)  = RLqs I− f (1) (t)

=−

∞ t

f (1) (τ ) 1 (1−α(t)) (τ −t)α(t) dτ

1 = − (1−α(t))

∞ t

f (1) (τ ) dτ . (τ −t)α(t)

(4.100)

330

General Fractional Derivatives With Applications in Viscoelasticity

4.4 Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel In 1952, Feller considered the extended version of the Marcel Riesz’ potentials with the aid of the variable-order Riesz fractional derivative with singular kernel [141]. In 2015, the Riesz-type variable-order fractional derivative with singular kernel (see [150]) was proposed by Zayernouri and Karniadakis. The Feller- and Herrmanntype variable-order fractional derivatives with singular kernel based on the Riemann– Liouville fractional derivatives were proposed in the book [25]. Meanwhile, the Riesz-, Feller-, and Richard-type fractional derivatives of variable-order based on the Liouville–Sonine fractional derivatives were presented in [25]. Here we introduce the Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel.

4.4.1 Riesz-type variable-order fractional derivative with singular kernel Definition 4.39 ([25,150]). Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as Ri D α(·) f ZK [a,b]

(t)



(t)

α(·) α(·) RLqs Da+ f (t) + RLqs Db− f t f (τ ) d 1 1 × (1−α(t)) dτ = − 2 cos(πα(t)/2) dt (t−τ )α(t) a b f (τ ) d 1 1 × (1−α(t)) dτ , + 2 cos(πα(t)/2) dt (τ −t)α(t) t 1 = − 2 cos(πα(t)/2)

×

(4.101)

where α(·) RLqs Da+ f

1 d (t) =  (1 − α (t)) dt

t

f (τ ) (t − τ )α(t)

a

dτ

(4.102)

and α(·) RLqs Db− f

1 d (t) = −  (1 − α (t)) dt

b t

f (τ ) (τ − t)α(t)

dτ .

(4.103)

Definition 4.40 ([25]). Let 0 < α (t) < 1 and 0 < α (τ, t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as

Variable-order fractional derivatives with singular kernels

Ri D α(·) f GZK [a,b]

(t)



1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 × + 2 cos(πα(t)/2)

α(·) RLq Da+ f

t

d dt

d dt

α(·) (t) + RLq Db− f (t)

f (τ ) 1 (1−α(τ,t)) (t−τ )α(τ,t) dτ

a

b

331

(4.104)

f (τ ) 1 (1−α(τ,t)) (τ −t)α(τ,t) dτ ,

t

where α(·) RLq Da+ f

t

d (t) = dt

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

a

(4.105)

and α(·) RLq Db− f

b

d (t) = − dt

t

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

(4.106)

Definition 4.41 ([25]). Let 0 < α (t) < 1 and 0 < α (τ, t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as Ri D α(·) f SZK [a,b]

(t)

1 × = − 2 cos(πα(t)/2) 1 = − 2 cos(πα(t)/2) × 1 × + 2 cos(πα(t)/2)

d dt



α(·) RL Da+ f

t

d dt

a

b

α(·) (t) + RL Db− f (t)

f (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ

(4.107)

f (τ ) 1 (1−α(τ )) (τ −t)α(τ ) dτ ,

t

where α(·) RL Da+ f

d (t) = dt

t

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

a

(4.108)

and α(·) RL Db− f

d (t) = − dt

b t

1 f (τ ) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.109)

332

General Fractional Derivatives With Applications in Viscoelasticity

Definition 4.42 ([25]). Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as Ri D α(·) f LSqs [a,b]

(t)



α(·) α(·) LSqs Da+ f (t) + LSqs Db− f t f (1) (τ ) 1 1 × (1−α(t)) dτ = − 2 cos(πα(t)/2) (t−τ )α(t) a b f (1) (τ ) 1 1 × (1−α(t)) dτ , + 2 cos(πα(t)/2) (τ −t)α(t) t 1 = − 2 cos(πα(t)/2) ×

(t) (4.110)

where α(·) LSqs Da+ f

1 (t) =  (1 − α (t))

t

f (1) (τ ) (t − τ )α(t)

a

dτ

(4.111)

and α(·) LSqs Db− f

1 (t) = −  (1 − α (t))

b t

f (1) (τ ) (τ − t)α(t)

dτ .

(4.112)

Definition 4.43 ([25]). Let 0 < α (t) < 1 and 0 < α (τ, t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as α(·) Ri GLSq D[a,b] f

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2)

×

b t



α(·) LSq Da+ f

t a

α(·) (t) + LSq Db− f (t)

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

(4.113)

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

where α(·) LSq Da+ f

t (t) = a

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

(4.114)

and α(·)

b

RLq Db− f (t) = − t

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

(4.115)

Variable-order fractional derivatives with singular kernels

333

Definition 4.44 ([25]). Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel is defined as Ri D α(·) f GLC [a,b]

(t)

1 = − 2 cos(πα(t)/2)



×

t

1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2)

×

b t

a

α(·) GLC Da+ f

α(·) (t) + GLC Db− f (t) (4.116)

f (1) (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ

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

where t

α(·) GLC Da+ f

(t) = a

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

(4.117)

and b

α(·) GLC Db− f

(t) = − t

1 f (1) (τ ) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.118)

4.4.2 Feller-type variable-order fractional derivative with singular kernel Definition 4.45 ([25]). Let 0 < α (t) < 1. The Feller-type variable-order fractional derivative with singular kernel is defined as F e D α(·) f ZK [a,b]



(t)

α(·) α(·) = − H+ (ϑ, α (t)) RLqs Da+ f (t) + H− (ϑ, α (t)) RLqs Db− f (t) t f (τ ) d 1 × (1−α(t)) = − sin((α(t)+ϑ)π/2) α(t) dτ sin(πϑ) dt

× + sin((α(t)−ϑ)π/2) sin(πϑ)

d 1 (1−α(t)) dt

b t

a

(4.119)

(t−τ )

f (τ ) dτ , (τ −t)α(t)

where α(·) RLqs Da+ f

1 d (t) =  (1 − α (t)) dt

t a

f (τ ) (t − τ )α(t)

dτ

(4.120)

334

General Fractional Derivatives With Applications in Viscoelasticity

and α(·) RLqs Db− f

1 d (t) = −  (1 − α (t)) dt

b

f (τ ) (τ − t)α(t)

t

(4.121)

dτ .

Definition 4.46 ([25]). Let 0 < α (t) < 1. The Feller-type variable-order fractional derivative with singular kernel 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) t f (1) (τ ) 1 × = − sin((α(t)+ϑ)π/2) α(t) dτ sin(πϑ) (1−α(t)) + sin((α(t)−ϑ)π/2) sin(πϑ)

×

1 (1−α(t))

b t

(4.122)

(t−τ )

a

f (1) (τ ) dτ , (τ −t)α(t)

where α(·) LSqs Da+ f

1 (t) =  (1 − α (t))

t

f (1) (τ ) (t − τ )α(t)

a

(4.123)

dτ

and α(·) LSqs Db− f

1 (t) = −  (1 − α (t))

b t

f (1) (τ ) (τ − t)α(t)

(4.124)

dτ .

4.4.3 Herrmann-type variable-order fractional derivative with singular kernel Definition 4.47 ([25]). Let 0 < α (t) < 1. The Herrmann-type variable-order fractional derivative with singular kernel is defined as Rc D α(·) f ZK [a,b]

(t)



α(·) α(·) RLqs Da+ f (t) + RLqs Db− f t f (τ ) d 1 1 × (1−α(t)) dτ = − 2 cos(πα(t)/2) dt (t−τ )α(t) a b f (τ ) d 1 1 × (1−α(t)) dτ , + 2 cos(πα(t)/2) dt (τ −t)α(t) t 1 = − 2 cos(πα(t)/2) ×

(t) (4.125)

Variable-order fractional derivatives with singular kernels

335

where α(·) RLqs Da+ f

t

1 d (t) =  (1 − α (t)) dt

f (τ ) (t − τ )α(t)

a

dτ

(4.126)

and α(·) RLqs Db− f

1 d (t) = −  (1 − α (t)) dt

b t

f (τ ) (τ − t)α(t)

dτ .

(4.127)

Definition 4.48 ([25]). Let 0 < α (t) < 1. The Herrmann-type variable-order fractional derivative with singular kernel is defined as α(·) Rc ZKLC D[a,b] f

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 × + 2 cos(πα(t)/2)



α(·) RLqs Da+ f

1 (1−α(t))

1 (1−α(t))

b t

t a

α(·) (t) + RLqs Db− f (t) (4.128)

f (1) (τ ) dτ (t−τ )α(t)

f (1) (τ ) dτ , (τ −t)α(t)

where α(·) LSqs Da+ f

1 (t) =  (1 − α (t))

t

f (1) (τ ) (t − τ )α(t)

a

dτ

(4.129)

and α(·) LSqs Db− f

1 (t) = −  (1 − α (t))

b t

f (1) (τ ) (τ − t)α(t)

dτ .

(4.130)

Definition 4.49. Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as Ri D α(·) f ZK R

(t)

1 = − 2 cos(πα(t)/2) × 1 × = − 2 cos(πα(t)/2) 1 + 2 cos(πα(t)/2) ×



α(·) (t) + RLqs D− f (t) t f (τ ) α(t) dτ

α(·) RLqs D+ f

d 1 (1−α(t)) dt

d 1 (1−α(t)) dt

∞ t

−∞

(t−τ )

f (τ ) dτ , (τ −t)α(t)

(4.131)

336

General Fractional Derivatives With Applications in Viscoelasticity

where α(·) RLqs D+ f

d 1 (t) =  (1 − α (t)) dt

t −∞

f (τ ) (t − τ )α(t)

dτ

(4.132)

and α(·) RLqs D− f

d 1 (t) = −  (1 − α (t)) dt

∞ t

f (τ ) (τ − t)α(t)

dτ .

(4.133)

Definition 4.50. Let 0 < α (t) < 1 and 0 < α (τ, t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as Ri D α(·) f GZK R

(t)

1 = − 2 cos(πα(t)/2) × 1 × = − 2 cos(πα(t)/2) 1 + 2 cos(πα(t)/2) ×

d dt



α(·) RLq D+ f

d dt

∞ t

t −∞

α(·) (t) + RLq D− f (t)

f (τ ) 1 (1−α(τ,t)) (t−τ )α(τ,t) dτ

(4.134)

f (τ ) 1 (1−α(τ,t)) (τ −t)α(τ,t) dτ ,

where α(·) RLq D+ f

d (t) = dt

t −∞

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

(4.135)

and α(·) RLq D− f

d (t) = − dt

∞ t

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

(4.136)

Definition 4.51. Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as Ri D α(·) f SZK R

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2) ×

d dt



α(·) α(·) D f + D f (t) (t) RL + RL −

d dt

∞ t

t −∞

f (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ

f (τ ) 1 (1−α(τ )) (τ −t)α(τ ) dτ ,

(4.137)

Variable-order fractional derivatives with singular kernels

337

where α(·) RL D+ f

d (t) = dt

t −∞

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

(4.138)

and α(·) RL D− f

∞

d (t) = − dt

t

f (τ ) 1 dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.139)

Definition 4.52. Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as Ri D α(·) f LSqs R

(t)



α(·) α(·) LSqs D+ f (t) + LSqs D− f t f (1) (τ ) 1 1 = − 2 cos(πα(t)/2) × (1−α(t)) dτ (t−τ )α(t) −∞ ∞ f (1) (τ ) 1 1 + 2 cos(πα(t)/2) × (1−α(t)) dτ , (τ −t)α(t) t 1 = − 2 cos(πα(t)/2) ×

(t) (4.140)

where α(·) LSqs D+ f

1 (t) =  (1 − α (t))

t −∞

f (1) (τ ) (t − τ )α(t)

dτ

(4.141)

and α(·) LSqs D− f

1 (t) = −  (1 − α (t))

∞ t

f (1) (τ ) (τ − t)α(t)

dτ .

(4.142)

Definition 4.53. Let 0 < α (t) < 1 and 0 < α (τ, t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as α(·) Ri GLSq DR f

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2) ×

∞ t



α(·) α(·) D f + D f (t) (t) LSq + LSq −

t −∞

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

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

(4.143)

338

General Fractional Derivatives With Applications in Viscoelasticity

where t

α(·) LSq D+ f

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

(t) = −∞

(4.144)

and ∞

α(·) RLq D− f

(t) = − t

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

(4.145)

Definition 4.54. Let 0 < α (t) < 1. The Riesz-type variable-order fractional derivative with singular kernel on the real line is defined as Ri D α(·) f GLC R

(t)



1 = − 2 cos(πα(t)/2) × 1 × = − 2 cos(πα(t)/2)

∞

1 + 2 cos(πα(t)/2) ×

α(·) GLC D+ f

t −∞

t

α(·)

(t) + GLC D− f (t)

(4.146)

f (1) (τ ) 1 (1−α(τ )) (t−τ )α(τ ) dτ

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

where α(·) GLC Da+ f

t (t) = −∞

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

(4.147)

and α(·) GLC Db− f

∞ (t) = − t

f (1) (τ ) 1 dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.148)

Definition 4.55. Let 0 < α (t) < 1. The Feller-type variable-order fractional derivative with singular kernel on the real line is defined as F e D α(·) f ZK R



(t) α(·)

α(·)

= − H+ (ϑ, α (t)) RLqs D+ f (t) + H− (ϑ, α (t)) RLqs D− f (t) = − sin((α(t)+ϑ)π/2) × sin(πϑ) + sin((α(t)−ϑ)π/2) × sin(πϑ)

d 1 (1−α(t)) dt

d 1 (1−α(t)) dt

∞ t

t −∞

f (τ ) dτ (t−τ )α(t)

f (τ ) dτ , (τ −t)α(t)

(4.149)

Variable-order fractional derivatives with singular kernels

339

where α(·) RLqs D+ f

d 1 (t) =  (1 − α (t)) dt

t −∞

f (τ ) (t − τ )α(t)

dτ

(4.150)

and α(·) RLqs D− f

d 1 (t) = −  (1 − α (t)) dt

∞ t

f (τ ) (τ − t)α(t)

dτ .

(4.151)

Definition 4.56. Let 0 < α (t) < 1. The Feller-type variable-order fractional derivative with singular kernel on the real line is defined as α(·) Fe ZKLC DR f

(t)

α(·) α(·) = − H+ (ϑ, α (t)) RLqs D+ f (t) + H− (ϑ, α (t)) RLqs D− f (t)

= − sin((α(t)+ϑ)π/2) × sin(πϑ) + sin((α(t)−ϑ)π/2) × sin(πϑ)

t

1 (1−α(t))

1 (1−α(t))

∞ t

−∞

(4.152)

f (1) (τ ) dτ (t−τ )α(t)

f (1) (τ ) dτ , (τ −t)α(t)

where α(·) LSqs D+ f

1 (t) =  (1 − α (t))

t −∞

f (1) (τ ) (t − τ )α(t)

dτ

(4.153)

and α(·) LSqs D− f

1 (t) = −  (1 − α (t))

∞ t

f (1) (τ ) (τ − t)α(t)

dτ .

(4.154)

Definition 4.57. Let 0 < α (t) < 1. The Herrmann-type variable-order fractional derivative with singular kernel on the real line is defined as Rc D α(·) f ZK R

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2) ×



α(·) (t) + RLqs D− f (t) t f (τ ) α(t) dτ

α(·) RLqs D+ f

d 1 (1−α(t)) dt

d 1 (1−α(t)) dt

∞ t

−∞

(t−τ )

f (τ ) dτ , (τ −t)α(t)

(4.155)

340

General Fractional Derivatives With Applications in Viscoelasticity

where α(·) RLqs D+ f

1 d (t) =  (1 − α (t)) dt

t

f (τ ) (t − τ )α(t)

−∞

dτ

(4.156)

and α(·) RLqs D− f

1 d (t) = −  (1 − α (t)) dt

−∞ t

f (τ ) (τ − t)α(t)

dτ .

(4.157)

Definition 4.58. Let 0 < α (t) < 1. The Herrmann-type variable-order fractional derivative with singular kernel on the real line is defined as α(·) Rc ZKLC DR f

(t)

1 = − 2 cos(πα(t)/2) × 1 = − 2 cos(πα(t)/2) × 1 + 2 cos(πα(t)/2) ×



α(·) RLqs D+ f

1 (1−α(t))

1 (1−α(t))

∞ t

t −∞

α(·) (t) + RLqs D− f (t) (4.158)

f (1) (τ ) dτ (t−τ )α(t)

f (1) (τ ) dτ , (τ −t)α(t)

where α(·) LSqs D+ f

1 (t) =  (1 − α (t))

t −∞

f (1) (τ ) (t − τ )α(t)

dτ

(4.159)

and α(·) LSqs D− f

1 (t) = −  (1 − α (t))

∞ t

f (1) (τ ) (τ − t)α(t)

dτ .

(4.160)

For the numerical methods of the above operators, see [156].

4.5

Variable-order tempered fractional derivatives with weakly singular kernel

In this section, we present the variable-order tempered fractional derivatives with weakly singular kernel. Definition 4.59 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is defined as

Variable-order fractional derivatives with singular kernels

RL α(·),λ Cp Da+ f

t

d (t) = dt

1 f (τ ) e−λ(t−τ ) dτ ,  (1 − α (τ )) (t − τ )α(τ )

a

341

(4.161)

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is RL α(·),λ Cp Db− f

d (t) = − dt

b t

1 f (τ ) e−λ(τ −t) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.162)

Definition 4.60 ([25]). Let κ < α (t) < κ + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is defined as RL α(·),λ Cp Da+ f

t

dκ (t) = κ dt

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

a

(4.163)

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is RL α(·),λ Cp Db− f

dκ (t) = (−1) dt κ

b

κ

t

1 f (τ ) e−λ(τ −t) dτ . (4.164)  (κ − α (τ )) (τ − t)α(τ )−κ+1

Definition 4.61 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is defined as LS α(·),λ Cp Da+ f

t

1 e−λ(t−τ ) (1) f (τ ) dτ ,  (1 − α (τ )) (t − τ )α(τ )

(t) = a

(4.165)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is LS α(·),λ Cp Db− f

b (t) = − t

1 e−λ(τ −t) (1) f (τ ) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.166)

Definition 4.62 ([25]). Let α (t) > 0, κ = [α (t)] + 1, − ∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type variable-order tempered fractional derivative with weakly singular kernel is defined as LSC α(·),λ Cp Da+ f

t (t) = a

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

(4.167)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Sonine–Caputo-type variable-order tempered fractional derivative with weakly singular kernel is

LSC α(·),λ Cp Db− f

b (t) = (−1)

κ t

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

(4.168)

Definition 4.63 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel is defined as α(·),λ Cp Ia+ f

t (t) = a

1

α(τ )−1

(t − τ )1−α(τ )

E1,α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(4.169)

and the right-sided variable-order tempered fractional integral with weakly singular kernel is α(·),λ Cp Ib− f

b (t) = t

1

α(τ )−1

(τ − t)1−α(τ )

E1,α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.170)

Definition 4.64 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel is defined as RL α(·),λ Cp Ia+ f

t (t) =

1 (t − τ )

1−α(τ )

a

α(τ )−κ

E1,α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(4.171)

and the right-sided variable-order tempered fractional integral with weakly singular kernel is RL α(·),λ Cp Ib− f

b (t) = t

1 (τ − t)1−α(τ )

α(τ )−κ

E1,α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.172)

Definition 4.65 ([25]). Let 1 > α (τ, t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is defined as

α(·),λ RL Cpq Da+ f

d (t) = dt

t a

1 f (τ ) e−λ(t−τ ) dτ ,  (1 − α (τ, t)) (t − τ )α(τ,t)

(4.173)

Variable-order fractional derivatives with singular kernels

343

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is

α(·),λ RL Cpq Db− f

d (t) = − dt

b t

1 f (τ ) e−λ(τ −t) dτ .  (1 − α (τ, t)) (τ − t)α(τ,t)

(4.174)

Definition 4.66 ([25]). Let 1 > α (τ, t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is defined as α(·),λ LS Cpq Da+ f

t (t) = a

1 e−λ(t−τ ) f (1) (τ ) dτ ,  (1 − α (τ, t)) (t − τ )α(τ,t)

(4.175)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is

α(·),λ LS Cpq Db− f

b (t) = − t

1 e−λ(τ −t) f (1) (τ ) dτ .  (1 − α (τ, t)) (τ − t)α(τ,t)

(4.176)

Definition 4.67. Let 1 > α (τ, t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel is defined as α(·),λ Cpq Ia+ f

t (t) = a

1 (t − τ )1−α(τ,t)

α(τ,t)−1

E1,α(τ,t) (−λ (t − τ )) f (τ ) dτ ,

(4.177)

and the right-sided variable-order tempered fractional integral with weakly singular kernel is α(·),λ Cpq Ib− f

b (t) = t

1 (τ − t)1−α(τ,t)

α(τ,t)−1

E1,α(τ,t) (−λ (τ − t)) f (τ ) dτ .

(4.178)

Definition 4.68. Let 1 > a (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is defined as α(·),λ RL Cpqs Da+ f

d (t) = dt

t a

1 f (τ ) e−λ(t−τ ) dτ ,  (1 − α (t)) (t − τ )α(t)

(4.179)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel is

α(·),λ RL Cpqs Db− f

d (t) = − dt

b t

f (τ ) 1 e−λ(τ −t) dτ .  (1 − α (t)) (τ − t)α(t)

(4.180)

Definition 4.69. Let 1 > a (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is defined as α(·),λ LS Cpqs Da+ f

t (t) = a

e−λ(t−τ ) (1) 1 f (τ ) dτ ,  (1 − α (t)) (t − τ )α(t)

(4.181)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel is

α(·),λ LS Cpqs Db− f

b (t) = − t

e−λ(τ −t) (1) 1 f (τ ) dτ .  (1 − α (t)) (τ − t)α(t)

(4.182)

Definition 4.70. Let 1 > a (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel is defined as α(·),λ Cpqs Ia+ f

t (t) =

1 (t − τ )1−α(t)

a

α(t)−1

E1,α(t) (−λ (t − τ )) f (τ ) dτ ,

(4.183)

and the right-sided variable-order tempered fractional integral with weakly singular kernel is α(·),λ Cpqs Ib− f

b (t) = t

1 (τ − t)1−α(t)

α(t)−1

E1,α(t) (−λ (τ − t)) f (τ ) dτ .

(4.184)

Definition 4.71. Let 1 > α (t) > 0 and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as RL α(·),λ f Cp D+

d (t) = dt

t −∞

f (τ ) 1 e−λ(t−τ ) dτ ,  (1 − α (τ )) (t − τ )α(τ )

(4.185)

Variable-order fractional derivatives with singular kernels

345

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is RL α(·),λ f Cp D−

d (t) = − dt

∞ t

f (τ ) 1 e−λ(τ −t) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.186)

Definition 4.72. Let κ < α (t) < κ + 1 and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as RL α(·),λ f Cp D+

dκ (t) = κ dt

t −∞

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

(4.187)

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is RL α(·),λ f Cp D−

dκ (t) = (−1) dt κ

∞

κ

t

f (τ ) 1 e−λ(τ −t) dτ . (4.188)  (κ − α (τ )) (τ − t)α(τ )−κ+1

Definition 4.73. Let 1 > α (t) > 0 and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as LS α(·),λ f Cp D+

t (t) = −∞

e−λ(t−τ ) (1) 1 f (τ ) dτ ,  (1 − α (τ )) (t − τ )α(τ )

(4.189)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is LS α(·),λ f Cp D−

∞ (t) = − t

e−λ(τ −t) (1) 1 f (τ ) dτ .  (1 − α (τ )) (τ − t)α(τ )

(4.190)

Definition 4.74. Let α (t) > 0, κ = [α (t)] + 1 and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as LSC α(·),λ f Cp D+

t (t) = −∞

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

(4.191)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Sonine–Caputo-type variable-order tempered fractional derivative with weakly singular kernel on the real line is LSC α(·),λ f Cp D−

∞ (t) = (−1)

e−λ(τ −t) 1 f (κ) (τ ) dτ . (4.192)  (κ − α (τ )) (τ − t)α(τ )−κ+1

κ t

Definition 4.75. Let 1 > α (t) > 0 and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel on the real line is defined as α(·),λ f Cp I+

t (t) =

1

α(τ )−1

(t − τ )

1−α(τ )

−∞

E1,α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(4.193)

and the right-sided variable-order tempered fractional integral with weakly singular kernel on the real line is α(·),λ f Cp I−

∞ (t) = t

1

α(τ )−1

(τ − t)1−α(τ )

E1,α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.194)

Definition 4.76. Let κ + 1 > α (t) > κ and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel on the real line is defined as RL α(·),λ f Cp I+

t

1

(t) =

α(τ

(t − τ )1−α(τ )

−∞

E1,α(τ)−κ ) (−λ (t − τ )) f (τ ) dτ ,

(4.195)

and the right-sided variable-order tempered fractional integral with weakly singular kernel is RL α(·),λ f Cp I−

∞ (t) =

1 (τ − t)

1−α(τ )

t

α(τ )−κ

E1,α(τ ) (−λ (τ − t)) f (τ ) dτ .

(4.196)

Definition 4.77. Let 1 > α (τ, t) > 0 and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as

α(·),λ RL f Cpq D+

d (t) = dt

t −∞

f (τ ) 1 e−λ(t−τ ) dτ ,  (1 − α (τ, t)) (t − τ )α(τ,t)

(4.197)

Variable-order fractional derivatives with singular kernels

347

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is α(·),λ RL f Cpq D−

∞

d (t) = − dt

t

f (τ ) 1 e−λ(τ −t) dτ .  (1 − α (τ, t)) (τ − t)α(τ,t)

(4.198)

Definition 4.78. Let 1 > α (τ, t) > 0 and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as

α(·),λ LS f Cpq D+

t (t) = −∞

e−λ(t−τ ) 1 f (1) (τ ) dτ ,  (1 − α (τ, t)) (t − τ )α(τ,t)

(4.199)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is α(·),λ LS f Cpq D−

∞ (t) = − t

e−λ(τ −t) 1 f (1) (τ ) dτ .  (1 − α (τ, t)) (τ − t)α(τ,t)

(4.200)

Definition 4.79. Let 1 > α (τ, t) > 0 and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel on the real line is defined as α(·),λ f Cpq I+

t

1

(t) =

(t − τ )

α(τ,t)−1

1−α(τ,t)

−∞

E1,α(τ,t) (−λ (t − τ )) f (τ ) dτ ,

(4.201)

and the right-sided variable-order tempered fractional integral with weakly singular kernel on the real line is α(·),λ f Cpq I−

∞ (t) = t

1 (τ − t)1−α(τ,t)

α(τ,t)−1

E1,α(τ,t) (−λ (τ − t)) f (τ ) dτ .

(4.202)

Definition 4.80. Let 1 > a (t) > 0 and λ ∈ R. The left-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as

α(·),λ RL f Cpqs D+

d (t) = dt

t −∞

f (τ ) 1 e−λ(t−τ ) dτ ,  (1 − α (t)) (t − τ )α(t)

(4.203)

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General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Riemann–Liouville-type variable-order tempered fractional derivative with weakly singular kernel on the real line is α(·),λ RL f Cpqs D−

d (t) = − dt

∞ t

1 f (τ ) e−λ(τ −t) dτ .  (1 − α (t)) (τ − t)α(t)

(4.204)

Definition 4.81. Let 1 > a (t) > 0 and λ ∈ R. The left-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is defined as α(·),λ LS f Cpqs D+

t (t) = −∞

1 e−λ(t−τ ) (1) f (τ ) dτ ,  (1 − α (t)) (t − τ )α(t)

(4.205)

and the right-sided Liouville–Sonine-type variable-order tempered fractional derivative with weakly singular kernel on the real line is α(·),λ LS f Cpqs D−

∞ (t) = − t

1 e−λ(τ −t) (1) f (τ ) dτ .  (1 − α (t)) (τ − t)α(t)

(4.206)

Definition 4.82. Let 1 > a (t) > 0 and λ ∈ R. The left-sided variable-order tempered fractional integral with weakly singular kernel on the real line is defined as α(·),λ f Cpqs I+

t (t) =

1

α(t)−1

(t − τ )

1−α(t)

−∞

E1,α(t) (−λ (t − τ )) f (τ ) dτ ,

(4.207)

and the right-sided variable-order tempered fractional integral with weakly singular kernel on the real line is α(·),λ f Cpqs I−

∞ (t) = t

1 (τ − t)1−α(t)

α(t)−1

E1,α(t) (−λ (τ − t)) f (τ ) dτ .

(4.208)

For more information on the applications of the variable-order fractional calculus and variable-order tempered fractional calculus, see [25].

Variable-order general fractional derivatives with nonsingular kernels

5

Contents 5.1

Riemann–Liouville-type variable-order general fractional derivatives with Mittag-Leffler–Gauss-like nonsingular kernel 350 5.1.1

Liouville–Sonine–Caputo-type variable-order general fractional derivatives with Mittag-Leffler nonsingular kernel 353

5.2

Hilfer-type variable-order fractional derivatives with Mittag-Leffler nonsingular kernel 357 5.3 Variable-order general fractional derivatives with Gorenflo–Mainardi nonsingular kernel 359 5.4 Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel 364 5.5 Variable-order general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel 365 5.6 Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel 370 5.7 Variable-order general fractional derivative with Miller–Ross nonsingular kernel 371 5.8 Variable-order Hilfer-type fractional derivatives with Miller–Ross nonsingular kernel 376 5.9 Variable-order general fractional derivative with Prabhakar nonsingular kernel 377 5.10 Variable-order Hilfer-type fractional derivatives with Prabhakar nonsingular kernel 382

In this chapter, we present the variable-order fractional derivatives with nonsingular kernels, which are called the variable-order general fractional derivatives. We introduce the variable-order general fractional derivatives with Mittag-Leffler–Gauss-like, Mittag-Leffler, Wiman, Prabhakar, one-parameter Lorenzo–Hartely, and Miller–Ross nonsingular kernels. We also propose the Hilfer-type variable-order general fractional derivatives with nonsingular kernels, e.g., Mittag-Leffler, Wiman, Prabhakar, oneparameter Lorenzo–Hartely, and Miller–Ross. In 2017, Yang first presented the variable-order general fractional derivatives with Mittag-Leffler nonsingular kernel [142]. In 2019, he suggested the variable-order general fractional derivatives with nonsingular kernels [25], e.g., Mittag-Leffler–GaussGeneral Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00010-8 Copyright © 2020 Elsevier Inc. All rights reserved.

350

General Fractional Derivatives With Applications in Viscoelasticity

like, Mittag-Leffler, Wiman, Prabhakar, one-parameter Lorenzo–Hartely, and Miller– Ross. First, we introduce the variable-order fractional derivatives with nonsingular kernels.

5.1

Riemann–Liouville-type variable-order general fractional derivatives with Mittag-Leffler–Gauss-like nonsingular kernel

  Let E2α −λt 2α be the Gauss-like function involving the structure of the MittagLeffler function, which is called the Mittag-Leffler–Gauss-like function. Definition 5.1 ([11]). 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 Mittag-Leffler–Gauss-like nonsingular kernel is defined as RL α(·),κ,λ f Mlt Da+

dκ (t) = κ dt

t

  E2α(τ ) −λ (t − τ )2α(τ ) f (τ ) dτ ,

(5.1)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is RL α(·),κ,λ f Mlt Db−

dκ (t) = (−1) dt κ

b

κ

  E2α(τ ) −λ (τ − t)2α(τ ) f (τ ) dτ .

(5.2)

t

Definition 5.2 ([25]). 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 Mittag-Leffler–Gauss-like nonsingular kernel is defined as LSC α(·),κ,λ f Mlt Da+

t (t) =

  E2α(τ ) −λ (t − τ )2α(τ ) f (κ) (τ ) dτ ,

(5.3)

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is LSC α(·),κ,λ f Mlt Db−

b (t) = (−1)

κ

  E2α(τ ) −λ (τ − t)2α(τ ) f (κ) (τ ) dτ .

(5.4)

t

Definition 5.3 ([25]). Let α (t) > 0, 1 + κ > α (t) > κ, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R.

Variable-order general fractional derivatives with nonsingular kernels

351

The left-sided variable-order Prabhakar–Gauss-like fractional integral is defined as

RL α(·),κ,λ f Mlt Ia+

t (t) = a

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

(5.5) and the right-sided variable-order Prabhakar–Gauss-like fractional integral as RL I α(·),κ,λ f Mlt b−

= (−1)κ

(t)

b

1

t

(τ −t)2α(τ )+1−κ

  −1 2α(τ ) f (τ ) dτ . E2α(τ ),κ−2α(τ ) −λ (τ − t)

(5.6)

Definition 5.4 ([25]). Let 1 > α (t) > 0. The left-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is defined as LSC α(·),λ Mlt Da+ f

t (t) =

  E2α(τ ) −λ (t − τ )2α(τ ) f (1) (τ ) dτ ,

(5.7)

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is

LSC α(·),λ Mlt Db− f

b (t) = −

  E2α(τ ) −λ (τ − t)2α(τ ) f (1) (τ ) dτ .

(5.8)

t

Definition 5.5 ([25]). Let 1 > α (t) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is defined as RL α(·),λ Mlt Da+ f

d (t) = dt

t

  E2α(τ ) −λ (t − τ )2α(τ ) f (τ ) dτ ,

(5.9)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel is

RL α(·),κ,λ f Mlt Db−

d (t) = − dt

b t

  E2α(τ ) −λ (τ − t)2α(τ ) f (τ ) dτ .

(5.10)

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General Fractional Derivatives With Applications in Viscoelasticity

Definition 5.6 ([25]). Let 1 > α (t) > 0. The left-sided variable-order Prabhakar–Gauss-like fractional integral is defined as RL α(·),λ Mlt Ia+ f

t (t) = a

  −1 2α(τ ) E −λ − τ f (τ ) dτ , (5.11) (t ) 2α(τ ),1−2α(τ ) (t − τ )2α(τ ) 1

and the right-sided variable-order Prabhakar–Gauss-like fractional integral is RL α(·),λ Mlt Ib− f

b

1

(t) = −

(τ − t)

t

  −1 2α(τ ) E −λ − t) f (τ ) dτ . (5.12) (τ 2α(τ ) 2α(τ ),1−2α(τ )

Definition 5.7 ([25]). Let 1 > α (t) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel on the real line is defined as d (t) = dt

RL α(·),λ f Mlt D+

t

  E2α(τ ) −λ (t − τ )2α(τ ) f (τ ) dτ ,

(5.13)

−∞

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler–Gauss-like nonsingular kernel on the real line is RL α(·),κ,λ f Mlt D−

d (t) = − dt

∞

  E2α(τ ) −λ (τ − t)2α(τ ) f (τ ) dτ .

(5.14)

t

Definition 5.8 ([25]). Let 1 > α (t) > 0. The left-sided variable-order Prabhakar–Gauss-like fractional integral on the real line is defined as RL α(·),λ f Mlt I+

t (t) = −∞

  −1 2α(τ ) E −λ − τ f (τ ) dτ , (5.15) (t ) 2α(τ ),1−2α(τ ) (t − τ )2α(τ ) 1

and the right-sided variable-order Prabhakar–Gauss-like fractional integral on the real line is RL α(·),λ f Mlt I−

∞ (t) = − t

1 (τ − t)

  −1 2α(τ ) E −λ − t) f (τ ) dτ . (5.16) (τ 2α(τ ) 2α(τ ),1−2α(τ )

Motivated by of the general fractional derivatives and integrals of variable-order involving the Mittag-Leffler kernel with the normalization function [135], the variableorder general fractional calculus with Mittag-Leffler nonsingular kernel was presented in [25].

Variable-order general fractional derivatives with nonsingular kernels

353

5.1.1 Liouville–Sonine–Caputo-type variable-order general fractional derivatives with Mittag-Leffler nonsingular kernel Definition 5.9 ([25]). 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 Mittag-Leffler nonsingular kernel is defined as

LSC α(·),κ,λ f Ml Da+

t

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

(t) =

(5.17)

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Mittag-Leffler nonsingular kernel is

LSC α(·),κ,λ f Ml Db−

b (t) = (−1)

κ

  Eα(τ ) −λ (τ − t)α(τ ) f (κ) (τ ) dτ .

(5.18)

t

Definition 5.10 ([25]). 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 Mittag-Leffler nonsingular kernel is defined as

RL α(·),κ,λ f Ml Da+

dκ (t) = κ dt

t

  Eα(τ ) −λ (t − τ )α(τ ) f (τ ) dτ ,

(5.19)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Mittag-Leffler nonsingular kernel is

RL α(·),κ,λ f Ml Db−

dκ (t) = (−1) dt κ

b

κ

  Eα(τ ) −λ (τ − t)α(τ ) f (τ ) dτ .

(5.20)

t

Definition 5.11 ([25]). Let α (t) > 0, 1 + κ > α (t) > κ, κ = [α (t)] +1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order fractional integral with Mittag-Leffler nonsingular kernel is defined as

RL α(·),κ,λ f Ml Ia+

t (t) = a

  −1 α(τ ) E −λ − τ f (τ ) dτ , (5.21) (t ) (t − τ )α(τ )+1−κ α(τ ),κ−α(τ ) 1

354

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Riemann–Liouville-type variable-order fractional integral in Mittag-Leffler nonsingular kernel is

RL α(·),κ,λ f Ml Ib−

b (t) = (−1)

(τ − t)

t

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

κ

(5.22) Definition 5.12. Let 1 > α (t) > 0. The left-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as LSC α(·),λ Ml Da+ f

t (t) =

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

(5.23)

a

and the right-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is LSC α(·),λ Ml Db− f

b (t) = −

  Eα(τ ) −λ (τ − t)α(τ ) f (1) (τ ) dτ .

(5.24)

t

Definition 5.13 ([25]). Let 1 > α (t) > 0. The left-sided Riemann–Liouville-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as RL α(·),λ Ml Da+ f

d (t) = dt

t

  Eα(τ ) −λ (t − τ )α(τ ) f (τ ) dτ ,

(5.25)

a

and the right-sided Riemann–Liouville-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is RL α(·),κ,λ f Ml Db−

d (t) = − dt

b

  Eα(τ ) −λ (τ − t)α(τ ) f (τ ) dτ .

(5.26)

t

Definition 5.14 ([25]). Let 1 > α (t) > 0. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as RL α(·),λ Ml Ia+ f

t (t) = a

  −1 α(τ ) E −λ − τ f (τ ) dτ , (5.27) (t ) (t − τ )α(τ ) α(τ ),1−α(τ ) 1

Variable-order general fractional derivatives with nonsingular kernels

355

the right-sided variable-order fractional integral with Prabhakar nonsingular kernel as

RL α(·),λ Ml Ib− f

b (t) = − t

1 (τ − t)

  −1 α(τ ) E −λ − t) f (τ ) dτ . (τ α(τ ) α(τ ),1−α(τ )

(5.28)

Definition 5.15 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as t

LSC α(·),λ Mlq Da+ f

(t) =

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

(5.29)

a

and the right-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is b

LSC α(·),λ Mlq Db− f

(t) = −

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

(5.30)

t

Definition 5.16 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Riemann–Liouville-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as

α(·),λ RL Mlq Da+ f

d (t) = dt

t

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

(5.31)

a

and the right-sided Riemann–Liouville-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is

α(·),κ,λ RL f Mlq Db−

d (t) = − dt

b

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

(5.32)

t

Definition 5.17 ([25]). Let 1 > α (t, τ ) > 0. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as RL α(·),λ Mlq Ia+ f

t (t) = a

1 (t − τ )

  −1 α(t,τ ) E −λ − τ f (τ ) dτ , (5.33) (t ) α(t,τ ) α(t,τ ),1−α(t,τ )

356

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided variable-order fractional integral with Prabhakar nonsingular kernel is

RL α(·),λ Mlq Ib− f

b (t) = −

1 (τ − t)

t

  −1 α(t,τ ) E −λ − t) f (τ ) dτ . (τ α(t,τ ),1−α(t,τ ) α(t,τ ) (5.34)

Definition 5.18 ([25]). Let 0 < α (t) < 1. The left-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as LSC α(·),λ Mlq Da+ f

t (t) =

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

(5.35)

a

and the right-sided Liouville–Sonine-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is α(·),λ LSC Mlqs Db− f

b (t) = −

  Eα(t) −λ (τ − t)α(t) f (1) (τ ) dτ .

(5.36)

t

Definition 5.19 ([25]). Let 0 < α (t) < 1. The left-sided Riemann–Liouville-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as α(·),λ RL Mlqs Da+ f

d (t) = dt

t

  Eα(t) −λ (t − τ )α(t) f (τ ) dτ ,

(5.37)

a

and the right-sided variable-order fractional derivative with Mittag-Leffler nonsingular kernel is α(·),κ,λ RL f Mlqs Db−

d (t) = − dt

b

  Eα(t) −λ (τ − t)α(t) f (τ ) dτ .

(5.38)

t

Definition 5.20 ([25]). Let 0 < α (t) < 1. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as α(·),λ RL Mlqs Ia+ f

t (t) = a

1 (t − τ )

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

(5.39)

Variable-order general fractional derivatives with nonsingular kernels

357

and the right-sided variable-order fractional integral with Prabhakar nonsingular kernel is

α(·),λ RL Mlqs Ib− f

b (t) = −

(τ − t)

t

5.2

1

  −1 α(t) −λ − t) f (τ ) dτ . E (τ α(t),1−α(t) α(t)

(5.40)

Hilfer-type variable-order fractional derivatives with Mittag-Leffler nonsingular kernel

Definition 5.21. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided Hilfer-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as  α(·),β(·),λ RL f Mlq Da+

(t) =

RL β(t)(1−α(t)),λ Mlq Ia+

d RL (1−β(t))(1−α(t)),λ  I f (t) , (5.41) dt Mlq a+

where RL β(t)(1−α(t)),λ f Mlq Ia+

t (t) =

  Eβ(t)(1−α(t)) −λ (t − τ )β(t)(1−α(t)) f (τ ) dτ

(5.42)

a

and

RL (1−β(t))(1−α(t)),λ f Mlq Ia+

t (t) =

  E(1−β(t))(1−α(t)) −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ ,

a

(5.43) and the right-sided Hilfer-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is  α(·),β(·),λ RL f Mlq Db−

(t) =

RL β(t)(1−α(t)),λ Mlq Ib−

d RL (1−β(t))(1−α(t)),λ  I f (t) , (5.44) dt Mlq −b

where RL β(t)(1−α(t)),λ f Mlq Ib−

b (t) = t

  Eβ(t)(1−α(t)) −λ (t − τ )β(t)(1−α(t)) f (τ ) dτ

(5.45)

358

General Fractional Derivatives With Applications in Viscoelasticity

and RL I (1−β(t))(1−α(t)),λ f Mlq b− b 

(t)

 = E(1−β(t))(1−α(t)) −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ .

(5.46)

t

Definition 5.22. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, f ∈ L1 (a, b) and −∞ < a < b < ∞. The left-sided Hilfer-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is defined as  α(·),β(·),λ RL f Mlqs Da+

(t) =

β(t)(1−α(t)),λ RL Mlqs Ia+

d RL (1−β(t))(1−α(t)),λ  I f (t) , dt Mlqs a+ (5.47)

where β(1−α(t)),λ RL f Mlqs Ia+

t (t) =

  Eβ(1−α(τ )) −λ (t − τ )β(1−α(τ )) f (τ ) dτ

(5.48)

a

and RL I (1−β(t))(1−α(t)),λ f Mlqs a+ t 

(t)

 = E(1−β(t))(1−α(t)) −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ ,

(5.49)

a

and the right-sided Hilfer-type variable-order fractional derivative with Mittag-Leffler nonsingular kernel is α(·),β R RL Db− f

   β(t)(1−α(t)) d (1−β(t))(1−α(t)) f (t) = −RL Ib− (t) , RL Ib− dt

(5.50)

where β(1−α(t)),λ RL f Mlqs Ib−

b (t) =

  Eβ(1−α(τ )) −λ (τ − t)β(1−α(τ )) f (τ ) dτ

(5.51)

t

and RL I (1−β(t))(1−α(t)),λ f Mlqs b− b 

(t)

 = E(1−β(t))(1−α(t)) −λ (τ − t)(1−β(t))(1−α(t)) f (τ ) dτ . t

(5.52)

Variable-order general fractional derivatives with nonsingular kernels

5.3

359

Variable-order general fractional derivatives with Gorenflo–Mainardi nonsingular kernel

In this section, we present the so-called variable-order fractional calculus with Gorenflo–Mainardi nonsingular kernel. Definition 5.23 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as RL D α(·),γ ,λ f GM a+

= =

d dt d dt



(t)

α(·),γ ,λ

GM Ia+

t

Gα(τ ),γ

 f (t)

(5.53)



 −λ (t − τ )α(τ ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is RL D α(·),γ ,λ f GM b−



 d = − dt

d = − dt

(t)

α(·),γ ,λ f GM Ib−

b

 (t)

(5.54)

  Gα(τ ),γ −λ (τ − t)α(τ ) f (τ ) dτ ,

t

where α(·),γ ,λ f GM Ia+

t (t) =

  Gα(τ ),γ −λ (t − τ )α(τ ) f (τ ) dτ

(5.55)

  Gα(τ ),γ −λ (τ − t)α(τ ) f (τ ) dτ .

(5.56)

a

and α(·),γ ,λ f GM Ib−

b (t) = t

Definition 5.24 ([25]). 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 with Gorenflo–Mainardi nonsingular kernel is defined as RL D α(·),γ ,λ f GM a+

=

dκ dt κ

=

dκ dt κ

 t a

(t)

α(·),γ ,λ f GM Ia+



 (t)

Gα(τ ),γ −λ (t − τ )

(5.57)  α(τ )

f (τ ) dτ ,

360

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided variable-order general fractional derivative with Gorenflo– Mainardi nonsingular kernel is RL D α(·),γ ,λ f (t) GM b−   d κ  α(·),γ ,λ = − dt f (t) GM Ib−  d κ b   = − dt Gα(τ ),γ −λ (τ − t)α(τ ) f t

(5.58) (τ ) dτ .

Definition 5.25 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as LS D α(·),γ ,λ f (t) GM a+  α(·),γ ,λ  (1) = GM Ia+ f (t) t 

= Gα(τ ),γ

 −λ (t − τ )α(τ ) f (1) (τ ) dτ ,

(5.59)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is LS D α(·),γ ,λ f (t) GM b−  α(·),γ ,λ  = GM Ib− −f (1) (t) b 

= − Gα(τ ),γ −λ (τ − t)

(5.60)  α(τ )

f (1) (τ ) dτ .

t

Definition 5.26 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as LSC D α(·),γ ,λ f (t) GM a+  α(·),γ ,λ  (κ) = GM Ia+ f (t) t 

= Gα(τ ),γ

 −λ (t − τ )α(τ ) f (κ) (τ ) dτ,

(5.61)

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is LSC D α(·),γ ,λ f (t) GM b−  α(·),γ ,λ  = GM Ib− (−1)κ f (κ) (t) b   = (−1)κ Gα(τ ),γ −λ (τ − t)α(τ ) f (κ) (τ ) dτ . t

(5.62)

Variable-order general fractional derivatives with nonsingular kernels

361

Definition 5.27 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as α(·),γ ,λ R f GM Ia+

t (t) =

  −1 α(τ ) −λ − τ f (τ ) dτ, (t ) (t − τ )κ−γ −1 Eα(τ ),κ−γ

(5.63)

a

and the right-sided variable-order fractional integral with Prabhakar nonsingular kernel is

α(·),γ ,λ R f GM Ib−

b (t) = (−1)

κ

  −1 α(τ ) −λ − t) f (τ ) dτ . (τ (τ − t)κ−γ −1 Eα(τ ),κ−γ

t

(5.64) Definition 5.28 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as α(·),γ ,λ R f GM Ia+

t (t) =

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

(5.65)

a

and the right-sided variable-order fractional integral with Prabhakar nonsingular kernel is α(·),γ ,λ R f GM Ib−

b (t) = −

  −1 α(τ ) −λ − t) f (τ ) dτ . (τ (τ − t)−γ Eα(τ ),1−γ

(5.66)

t

Definition 5.29. Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as RL D α(·),γ ,λ f GMq a+

= =

d dt d dt



(t)

α(·),γ ,λ f GMq Ia+

t

 (t)

(5.67)

  Gα(t,τ ),γ −λ (t − τ )α(t,τ ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is

362

General Fractional Derivatives With Applications in Viscoelasticity

RL D α(·),γ ,λ f GMq b−



d = − dt d = − dt

(t)



α(·),γ ,λ

GMq Ib−

b

 f (t)



Gα(t,τ ),γ −λ (τ − t)

(5.68)  α(t,τ )

f (τ ) dτ ,

t

where α(·),γ ,λ f GMq Ia+

t (t) =

  Gα(t,τ ),γ −λ (t − τ )α(t,τ ) f (τ ) dτ

(5.69)

  Gα(t,τ ),γ −λ (τ − t)α(t,τ ) f (τ ) dτ .

(5.70)

a

and α(·),γ ,λ f GMq Ib−

b (t) = t

Definition 5.30 ([25]). Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as LS D α(·),γ ,λ f GMq a+

(t)   α(·),γ ,λ = GMq Ia+ f (1) (t) t



= Gα(t,τ ),γ −λ (t − τ )

(5.71)  α(t,τ )

f (1) (τ ) dτ ,

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is LS D α(·),γ ,λ f GMq b−

(t)   α(·),γ ,λ = GMq Ib− −f (1) (t) b

= − Gα(t,τ ),γ

(5.72)



 −λ (τ − t)α(t,τ ) f (1) (τ ) dτ .

t

Definition 5.31 ([25]). Let 1 > α (t, τ ) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order fractional integral with Prabhakar nonsingular kernel is defined as α(·),γ ,λ R f GMq Ia+

t (t) = a

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

(5.73)

Variable-order general fractional derivatives with nonsingular kernels

363

and the right-sided variable-order fractional integral with Prabhakar nonsingular kernel is

α(·),γ ,λ R f GMq Ib−

b (t) = −

  −1 α(t,τ ) f (τ ) dτ . (5.74) (τ − t)−γ Eα(t,τ ),1−γ −λ (τ − t)

t

Definition 5.32 ([25]). Let 0 < α (t) < 1. Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as RL D α(·),γ ,λ f GMqs a+

= =

d dt d dt

(t)



α(·),γ ,λ f GMqs Ia+

t

Gα(t),γ

 (t)

(5.75)



 −λ (t − τ )α(t) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is RL D α(·),γ ,λ f GMqs b−



d = − dt d = − dt



(t) α(·),γ ,λ

GMqs Ib−

b

 f (t)

(5.76)

  Gα(t),γ −λ (τ − t)α(t) f (τ ) dτ ,

t

where α(·),γ ,λ f GMqs Ia+

t (t) =

  Gα(t),γ −λ (t − τ )α(t) f (τ ) dτ

(5.77)

  Gα(t),γ −λ (τ − t)α(t) f (τ ) dτ .

(5.78)

a

and α(·),γ ,λ f GMqs Ib−

b (t) = t

Definition 5.33 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is defined as

364

General Fractional Derivatives With Applications in Viscoelasticity

α(·),γ ,λ LS f (t) GMqs Da+  α(·),γ ,λ  (1) = GMqs Ia+ f (t) t   = Gα(t),γ −λ (t − τ )α(t) f (1) (τ ) dτ , a

(5.79)

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Gorenflo–Mainardi nonsingular kernel is α(·),γ ,λ LS f (t) GMqs Db−  α(·),γ ,λ  = GMqs Ib− −f (1) (t) b   = − Gα(t),γ −λ (τ − t)α(t) f (1) (τ ) dτ . t

(5.80)

Definition 5.34 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, γ ∈ R and λ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),γ ,λ R f GMqs Ia+

t (t) =

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

(5.81)

a

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is α(·),γ ,λ R f GMqs Ib−

b =−

  −1 −λ (τ − t)α(t) f (τ ) dτ . (τ − t)−γ Eα(t),1−γ

(5.82)

t

5.4

Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel

Definition 5.35. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, γ ∈ R, λ ∈ R, and f ∈ L1 (a, b), −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with Gorenflo– Mainardi nonsingular kernel is defined as    α(·),β(·),γ ,λ RL RL β(t)(1−α(t)),λ d RL (1−β(t))(1−α(t)),λ f (t) = Mlq Ia+ I f (t) , (5.83) Mlq Da+ dt Mlq a+ where RL β(t)(1−α(t)),γ ,λ f Mlq Ia+

t (t) = a

  Gβ(t)(1−α(t)),γ −λ (t − τ )β(t)(1−α(t)) f (τ ) dτ (5.84)

Variable-order general fractional derivatives with nonsingular kernels

365

and RL I (1−β(t))(1−α(t)),γ ,λ f Mlq a+ t 

(t)

 = G(1−β(t))(1−α(t)),γ −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ ,

(5.85)

a

and the right-sided variable-order Hilfer-type fractional derivative with Gorenflo– Mainardi nonsingular kernel is    α(·),β(·),γ ,λ RL RL β(t)(1−α(t)),λ d RL (1−β(t))(1−α(t)),λ f (t) = Mlq Ib− I f (t) , (5.86) Mlq Db− dt Mlq b− where b

RL β(t)(1−α(t)),γ ,λ f Mlq Ib−

(t) =

  Gβ(t)(1−α(t)),γ −λ (τ − t)β(t)(1−α(t)) f (τ ) dτ (5.87)

t

and RL I (1−β(t))(1−α(t)),γ ,λ f Mlq b− b 

(t)

 = G(1−β(t))(1−α(t)),γ −λ (τ − t)(1−β(t))(1−α(t)) f (τ ) dτ .

(5.88)

t

5.5

Variable-order general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel

In this section, we present the so-called variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel. Definition 5.36 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and γ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as RL D α(·),γ f a+ A

= =

d dt d dt



(t)

α(·),γ

A Ia+

t





f (t)

Fα(τ ) −γ (t − τ )

(5.89)  α(τ )

f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is

366

General Fractional Derivatives With Applications in Viscoelasticity

RL D α(·),γ f A b−



d = − dt d = − dt



(t)

α(·),γ f A Ib−

b

 (t)



Fα(τ ) −γ (τ − t)

(5.90)  α(τ )

f (τ ) dτ ,

t

where α(·),γ f A Ia+

t (t) =

  Fα(τ ) −γ (t − τ )α(τ ) f (τ ) dτ

(5.91)

  Fα(τ ) −γ (τ − t)α(τ ) f (τ ) dτ .

(5.92)

a

and α(·),γ f A Ib−

b (t) = t

Definition 5.37 ([25]). Let α (t) > 0, κ = [α (t)] +1, −∞ < a < b < ∞ and γ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as RL D α(·),γ f (t) a+ A   α(·),γ dκ = dt κ A Ia+ f (t) t   dκ = dt Fα(τ ) −γ (t − τ )α(τ ) f κ a

(5.93) (τ ) dτ ,

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is RL D α(·),γ f (t) A b−  d κ  α(·),γ = − dt f A Ib−

 d κ = − dt

b

 (t)

(5.94)

  Fα(τ ) −γ (τ − t)α(τ ) f (τ ) dτ .

t

Definition 5.38 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and γ ∈ C. The left-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as LS D α(·),γ f (t) a+ A  α(·),γ  (1) = A Ia+ f (t) t 

= Fα(τ ) −γ (t − τ ) a

(5.95)  α(τ )

f (1) (τ ) dτ ,

Variable-order general fractional derivatives with nonsingular kernels

367

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is LS D α(·),λ f (t) A b−   α(·),γ = A Ib− −f (1) (t) b 

(5.96)

 −γ (τ − t)α(τ ) f (1) (τ ) dτ .

= − Fα(τ ) t

Definition 5.39 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and γ ∈ R. The left-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as LSC D α(·),γ f (t) a+ A  α(·),γ  (κ) = A Ia+ f (t) t 

= Fα(τ ) −γ (t − τ )

 α(τ )

(5.97) f (κ) (τ ) dτ ,

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is LSC D α(·),λ f (t) A b−  α(·),γ  = A Ib− (−1)κ f (κ) (t) b   = (−1)κ Fα(τ ) −γ (τ − t)α(τ ) f (κ) (τ ) dτ . t

(5.98)

Definition 5.40 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and γ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as R α(·),γ f A Ia+

t (t) =

  −1 α(τ ) −γ − τ f (τ ) dτ , (5.99) (t ) (t − τ )κ−α(τ )−1 Eα(τ ),κ−α(τ )

a

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is R α(·),γ f A Ib−

b (t) = (−1)

κ

  −1 α(τ ) −γ − t) f (τ ) dτ . (τ (τ − t)κ−α(τ )−1 Eα(τ ),κ−α(τ )

t

(5.100) Definition 5.41. Let 1 > α (t) > 0, −∞ < a < b < ∞ and γ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as

368

General Fractional Derivatives With Applications in Viscoelasticity

R α(·),γ f A Ia+

(t) =

   t E −1 α(τ ) α(τ ),1−α(τ ) −γ (t − τ ) (t − τ )α(τ )

a

f (τ ) dτ

(5.101)

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is R α(·),γ f A Ib−

(t) = −

  b E −1 α(τ ) α(τ ),1−α(τ ) −γ (τ − t) (τ − t)α(τ )

t

f (τ ) dτ .

(5.102)

Definition 5.42 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as RL α(·),γ Aq Da+ f

t

d (t) = dt

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

(5.103)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is RL α(·),γ Aq Db− f

d (t) = − dt

b

  Fα(t,τ ) −γ (τ − t)α(t,τ ) f (τ ) dτ .

(5.104)

t

Definition 5.43 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as LS α(·),γ Aq Da+ f

t (t) =

  Fα(t,τ ) −γ (t − τ )α(t,τ ) f (1) (τ ) dτ ,

(5.105)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is LS α(·),λ Aq Db− f

b (t) = −

  Fα(t,τ ) −γ (τ − t)α(t,τ ) f (1) (τ ) dτ .

(5.106)

t

Definition 5.44 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as LS α(·),γ Aq Da+ f

t (t) = a

  Fα(t,τ ) −γ (t − τ )α(t,τ ) f (1) (τ ) dτ ,

(5.107)

Variable-order general fractional derivatives with nonsingular kernels

369

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is

LS α(·),λ Aq Db− f

b (t) = −

  Fα(t,τ ) −γ (τ − t)α(t,τ ) f (1) (τ ) dτ .

(5.108)

t

Definition 5.45 ([25]). Let 1 > α (t, τ ) > 0. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as R α(·),γ f Aq Ia+

(t) =

   t E −1 α(t,τ ) α(t,τ ),1−α(t,τ ) −γ (t − τ ) (t − τ )α(t,τ )

a

f (τ ) dτ,

(5.109)

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is

R α(·),γ f Aq Ib−

(t) = −

  b E −1 α(t,τ ) α(t,τ ),1−α(t,τ ) −γ (τ − t) (τ − t)α(t,τ )

t

f (τ ) dτ .

(5.110)

Definition 5.46 ([25]). Let 0 < α (t) < 1. The left-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as RL α(·),γ Aqs Da+ f

d (t) = dt

t

  Fα(t) −γ (t − τ )α(t) f (τ ) dτ ,

(5.111)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is

RL α(·),γ Aqs Db− f

d (t) = − dt

b

  Fα(t) −γ (τ − t)α(t) f (τ ) dτ .

(5.112)

t

Definition 5.47 ([25]). Let 0 < α (t) < 1. The left-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as α(·),γ LS Aqs Da+ f

t (t) = a

  Fα(t) −γ (t − τ )α(t) f (1) (τ ) dτ ,

(5.113)

370

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is α(·),λ LS Aqs Db− f

b (t) = −

  Fα(t) −γ (τ − t)α(t) f (1) (τ ) dτ .

(5.114)

t

Definition 5.48 ([25]). Let 0 < α (t) < 1. The left-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as α(·),γ LS Aqs Da+ f

t

  Fα(t) −γ (t − τ )α(t) f (1) (τ ) dτ ,

(t) =

(5.115)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is α(·),λ LS Aqs Db− f

b (t) = −

  Fα(t) −γ (τ − t)α(t) f (1) (τ ) dτ .

(5.116)

t

Definition 5.49 ([25]). Let 0 < α (t) < 1. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),γ R f Aqs Ia+

(t) =

   t E −1 α(t) α(t),1−α(t) −γ (t − τ ) (t − τ )α(t)

a

f (τ ) dτ

(5.117)

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel as α(·),γ R f Aqs Ib−

(t) = −

  b E −1 α(t) α(t),1−α(t) −γ (τ − t) t

5.6

(τ − t)α(t)

f (τ ) dτ .

(5.118)

Variable-order Hilfer-type fractional derivatives with Gorenflo–Mainardi nonsingular kernel

Definition 5.50. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, γ ∈ R, λ ∈ R, and f ∈ L1 (a, b), −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is defined as

Variable-order general fractional derivatives with nonsingular kernels

 RL α(·),β(·),γ f Aqs Da+

(t) =

RL β(t)(1−α(t)),γ Aqs Ia+

371

d RL (1−β(t))(1−α(t)),γ  I f (t) , (5.119) dt Aqs a+

where RL β(t)(1−α(t)),γ f Aqs Ia+

t (t) =

  Fβ(t)(1−α(t)) −γ (t − τ )β(t)(1−α(t)) f (τ ) dτ

(5.120)

a

and RL I (1−β(t))(1−α(t)),γ f Mlq a+ t 

(t)

 = F(1−β(t))(1−α(t)) −γ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ ,

(5.121)

a

and the right-sided variable-order Hilfer-type fractional derivative with one-parameter Lorenzo–Hartley nonsingular kernel is  RL α(·),β(·),γ f Aqs Db−

(t) =

RL β(t)(1−α(t)),γ Aqs Ib−

d RL (1−β(t))(1−α(t)),γ  I f (t) , (5.122) dt Aqs b−

where RL β(t)(1−α(t)),γ f Aqs Ib−

b (t) =

  Fβ(t)(1−α(t)) −γ (τ − t)β(t)(1−α(t)) f (τ ) dτ

(5.123)

t

and RL I (1−β(t))(1−α(t)),γ f Mlq b−

b

(t)

  = F(1−β(t))(1−α(t)) −γ (τ − t)(1−β(t))(1−α(t)) f (τ ) dτ .

(5.124)

t

5.7

Variable-order general fractional derivative with Miller–Ross nonsingular kernel

In this section, we present the variable-order general fractional derivative with Miller– Ross nonsingular kernel [25]. Definition 5.51 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as

372

General Fractional Derivatives With Applications in Viscoelasticity

RL D α(·),λ f (t) MR a+   α(·),λ d = dt MR Ia+ f (t) t   d = dt Mα(τ ) −λ (t − τ )α(τ ) f a

(5.125) (τ ) dτ ,

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is RL D α(·),λ f MR b−

α(·),λ MR Ib− f

d = − dt d = − dt

(t)





b

Mα(τ )

 (t)

(5.126)



 −λ (τ − t)α(τ ) f (τ ) dτ ,

t

where α(·),λ MR Ia+ f

t (t) =

  Mα(τ ) −λ (t − τ )α(τ ) f (τ ) dτ

(5.127)

  Mα(τ ) −λ (τ − t)α(τ ) f (τ ) dτ .

(5.128)

a

and α(·),λ MR Ib− f

b (t) = t

Definition 5.52 ([25]). Let α (t) > 0, κ = [α (t)] + 1, −∞ < a < b < ∞ and λ ∈ R. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as RL D α(·),λ f MR a+

= =

dκ dt κ dκ dt κ



(t) α(·),λ

MR Ia+

t

Mα(τ )

 f (t)

(5.129)



 −λ (t − τ )α(τ ) f (τ ) dτ ,

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is RL D α(·),λ f (t) MR b−  d κ  α(·),λ = − dt MR Ib− f

 d κ = − dt

b t

 (t)

  Mα(τ ) −λ (τ − t)α(τ ) f (τ ) dτ .

(5.130)

Variable-order general fractional derivatives with nonsingular kernels

373

Definition 5.53 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as LS D α(·),λ f (t) MR a+  α(·),λ  (1) = MR Ia+ f (t) t 

= Mα(τ )

 −λ (t − τ )α(τ ) f (1) (τ ) dτ ,

(5.131)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is LS D α(·),λ f (t) MR b−  α(·),λ  = MR Ib− −f (1) (t) b 

= − Mα(τ ) −λ (τ − t)

(5.132)  α(τ )

f (1) (τ ) dτ .

t

Definition 5.54 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞ and λ ∈ R. The left-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as LSC D α(·),λ f (t) MR a+   α(·),λ = MR Ia+ f (κ) (t) t 

= Mα(τ )

 −λ (t − τ )α(τ ) f (κ) (τ ) dτ ,

(5.133)

a

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is LSC D α(·),λ f (t) MR b−   α(·),λ = MR Ib− (−1)κ f (κ) (t) b   = (−1)κ Mα(τ ) −λ (τ − t)α(τ ) f (κ) (τ ) dτ . t

(5.134)

Definition 5.55 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),λ R MR Ia+ f

t (t) = a

1 (t − τ )α(τ )+1

E1,−α(τ ) (−λ (t − τ )) f (τ ) dτ ,

(5.135)

374

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is

α(·),γ ,λ R f MR Ib−

b (t) = t

1 (τ − t)α(τ )+1

E1,−α(τ ) (−λ (τ − t)) f (τ ) dτ .

(5.136)

Definition 5.56 ([25]). Let κ + 1 > α > κ, −∞ < a < b < ∞ and λ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),λ R MR Ia+ f

t (t) = a

1 (t − τ )

α(τ )+2−κ

E1,κ−α(τ )−1 (−λ (t − τ )) f (τ ) dτ ,

(5.137)

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is

α(·),γ ,λ R f MR Ib−

b (t) = t

1 (t − τ )

α(τ )+2−κ

E1,κ−α(τ )−1 (−λ (τ − t)) f (τ ) dτ .

(5.138)

Definition 5.57 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as α(·),λ RL MRq Da+ f

d (t) = dt

t

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

(5.139)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is

α(·),λ RL MRq Db− f

d (t) = − dt

b

  Mα(τ,t) −λ (τ − t)α(τ,t) f (τ ) dτ .

(5.140)

t

Definition 5.58 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as α(·),λ LS MRq Da+ f

t (t) = a

  Mα(τ,t) −λ (t − τ )α(τ,t) f (1) (τ ) dτ ,

(5.141)

Variable-order general fractional derivatives with nonsingular kernels

375

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is

α(·),λ LS MRq Db− f

b (t) = −

  Mα(τ,t) −λ (τ − t)α(τ,t) f (1) (τ ) dτ .

(5.142)

t

Definition 5.59 ([25]). Let 1 > α (t, τ ) > 0. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),λ R MRq Ia+ f

t

1

(t) =

(t − τ )α(τ,t)+1

a

E1,−α(τ,t) (−λ (t − τ )) f (τ ) dτ ,

(5.143)

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is

α(·),γ ,λ R f MRq Ib−

b (t) = t

1 (τ − t)α(τ,t)+1

E1,−α(τ,t) (−λ (τ − t)) f (τ ) dτ .

(5.144)

Definition 5.60 ([25]). Let 1 > α (t) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as α(·),λ RL MRqs Da+ f

d (t) = dt

t

  Mα(t) −λ (t − τ )α(t) f (τ ) dτ ,

(5.145)

a

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is

α(·),λ RL MRqs Db− f

d (t) = − dt

b

  Mα(t) −λ (τ − t)α(t) f (τ ) dτ .

(5.146)

t

Definition 5.61 ([25]). Let 1 > α (t) > 0. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is defined as α(·),λ LS MRqs Da+ f

t (t) = a

  Mα(t) −λ (t − τ )α(t) f (1) (τ ) dτ ,

(5.147)

376

General Fractional Derivatives With Applications in Viscoelasticity

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Miller–Ross nonsingular kernel is α(·),λ LS MRqs Db− f

b (t) = −

  Mα(t) −λ (τ − t)α(t) f (1) (τ ) dτ .

(5.148)

t

Definition 5.62 ([25]). Let 1 > α (t) > 0. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),λ R MRq Ia+ f

t

1

(t) =

(t − τ )α(t)+1

a

E1,−α(t) (−λ (t − τ )) f (τ ) dτ ,

(5.149)

and the variable-order general fractional integral with Prabhakar nonsingular kernel is α(·),γ ,λ R f MRq Ib−

b (t) =

(τ − t)α(t)+1

t

5.8

1

E1,−α(t) (−λ (τ − t)) f (τ ) dτ .

(5.150)

Variable-order Hilfer-type fractional derivatives with Miller–Ross nonsingular kernel

Definition 5.63. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, γ ∈ R, λ ∈ R, and f ∈ L1 (a, b), −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with Miller–Ross nonsingular kernel is defined as  α(·),β(·),λ RL f MRqs Da+

(t) =

β(t)(1−α(t)),λ RL MRqs Ia+

 d RL (1−β(t))(1−α(t)),λ I f (t) , dt MRqs a+ (5.151)

where β(t)(1−α(t)),λ RL f MRqs Ia+

t (t) =

  Mβ(t)(1−α(t)) −λ (t − τ )β(t)(1−α(t)) f (τ ) dτ

(5.152)

a

and RL I (1−β(t))(1−α(t)),λ f MRqs a+ t 

(t)

 = M(1−β(t))(1−α(t)) −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ , a

(5.153)

Variable-order general fractional derivatives with nonsingular kernels

377

and the right-sided variable-order Hilfer-type fractional derivative with Miller–Ross nonsingular kernel is  α(·),β(·),γ RL f MRqs Db−

β(t)(1−α(t)),γ RL MRqs Ib−

(t) =

 d RL (1−β(t))(1−α(t)),λ I f (t) , dt MRqs b− (5.154)

where β(t)(1−α(t)),λ RL f MRqs Ib−

b (t) =

  Mβ(t)(1−α(t)) −λ (τ − t)β(t)(1−α(t)) f (τ ) dτ (5.155)

t

and RL I (1−β(t))(1−α(t)),γ f MRqs b−

(t)

b   = M(1−β(t))(1−α(t)) −λ (τ − t)(1−β(t))(1−α(t)) f (τ ) dτ .

(5.156)

t

5.9

Variable-order general fractional derivative with Prabhakar nonsingular kernel

In this section, we present the variable-order general fractional derivative with Prabhakar nonsingular kernel [25]. Definition 5.64 ([25]). 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 with Prabhakar nonsingular kernel is defined as RL D α(·),υ,ϕ,λ f (t) GP a+  α(·),υ,ϕ,λ d = dt f GP Ia+

=

d dt

t a

 (t)

(5.157)

  ϕ (t − τ )υ−1 Eα(τ ),υ −λ (t − τ )α(τ ) f (τ ) dτ ,

and the right-sided Riemann–Liouville-type variable order general fractional derivative with Prabhakar nonsingular kernel is RL D α(·),υ,ϕ,λ f (t) GP b−  α(·),υ,ϕ,λ d = − dt f GP Ib− d = − dt

b t

ϕ

 (t)

(5.158) 

(τ − t)υ−1 Eα(τ ),υ −λ (τ − t)

 α(τ )

f (τ ) dτ ,

378

General Fractional Derivatives With Applications in Viscoelasticity

where α(·),υ,ϕ,λ f GP Ia+

t (t) =

  ϕ (t − τ )υ−1 Eα(τ ),υ −λ (t − τ )α(τ ) f (τ ) dτ

(5.159)

  ϕ (τ − t)υ−1 Eα(τ ),υ −λ (τ − t)α(τ ) f (τ ) dτ .

(5.160)

a

and α(·),υ,ϕ,λ f GP Ib−

b (t) = t

Definition 5.65 ([25]). Let 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 with Prabhakar nonsingular kernel is defined as RL D α(·),υ,ϕ,λ f (t) GP a+  α(·),υ,ϕ,λ dκ = dt f κ GP Ia+

=

dκ dt κ

t a

 (t)

(5.161)

  ϕ (t − τ )υ−1 Eα(τ ),υ −λ (t − τ )α(τ ) f (τ ) dτ ,

and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Prabhakar nonsingular kernel is RL D α(·),υ,ϕ,λ f (t) GP b−  d κ  α(·),υ,ϕ,λ = − dt f GP Ib−



d = − dt

κ b t

 (t)

ϕ

(τ − t)υ−1 Eα(τ ),υ

(5.162) 

 −λ (τ − t)α(τ ) f (τ ) dτ .

Definition 5.66 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is defined as LS D α(·),ϕ,υ,λ f (t) GP a+  α(·),ϕ,υ,λ  (1) = GP Ia+ f (t) t  ϕ = (t − τ )υ−1 Eα(τ ),υ −λ (t a

− τ)

 α(τ )

(5.163) f (1) (τ ) dτ ,

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is

Variable-order general fractional derivatives with nonsingular kernels

LS D α(·),ϕ,υ,λ f (t) GP b−  α(·),ϕ,υ,λ  = GP Ib− −f (1) (t) b  ϕ = − (τ − t)υ−1 Eα(τ ),υ −λ (τ t

379

(5.164)  − t)α(τ ) f (1) (τ ) dτ .

Definition 5.67 ([25]). Let κ + 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 with Prabhakar nonsingular kernel is defined as LSC D α(·),ϕ,υ,λ f (t) a+ GP  α(·),ϕ,υ,λ  (κ) = GP Ia+ f (t) t  ϕ = (t − τ )υ−1 Eα(τ ),υ −λ (t a

− τ)

 α(τ )

(5.165) f (κ) (τ ) dτ ,

and the right-sided Liouville–Sonine–Caputo-type variable-order general fractional derivative with Prabhakar nonsingular kernel is LSC D α(·),ϕ,υ,λ f (t) GP b−  α(·),ϕ,υ,λ  = GP Ib− (−1)κ f (κ) (t) b  ϕ = (−1)κ (τ − t)υ−1 Eα(τ ),υ −λ (τ t

(5.166)  − t)α(τ ) f (κ) (τ ) dτ .

Definition 5.68 ([25]). Let κ + 1 > α (t) > κ, −∞ < a < b < ∞, ϕ ∈ R, and υ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as α(·),ϕ,υ,λ R f GP Ia+

t (t) =

  −ϕ (t − τ )κ−υ−1 Eα(τ ),κ−υ −λ (t − τ )α(τ ) f (τ ) dτ , (5.167)

a

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is α(·),ϕ,υ,λ R f GP Ib−

b (t) =

  −ϕ (τ − t)κ−υ−1 Eα(τ ),κ−υ −λ (τ − t)α(τ ) f (τ ) dτ . (5.168)

t

Definition 5.69 ([25]). Let 1 > α (t) > 0, −∞ < a < b < ∞, ϕ ∈ R, υ ∈ R and λ ∈ R. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as

380

General Fractional Derivatives With Applications in Viscoelasticity

α(·),ϕ,υ,λ R f GP Ia+

t (t) = a

  1 −ϕ α(τ ) E −λ − τ f (τ ) dτ , (5.169) (t ) (t − τ )υ α(τ ),κ−υ

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is α(·),ϕ,υ,λ R f GP Ib−

b (t) = t

  1 −ϕ α(τ ) E −λ − t) f (τ ) dτ . (5.170) (τ (τ − t)υ α(τ ),κ−υ

Definition 5.70. Let 1 > α (t, τ ) > 0. The left-sided Riemann–Liouville-type variable-order general fractional derivative with Prabhakar nonsingular kernel is defined as α(·),υ,ϕ,λ RL f GP q Da+

t

d (t) = dt

  ϕ (t − τ )υ−1 Eα(t,τ ),υ −λ (t − τ )α(t,τ ) f (τ ) dτ ,

a

(5.171) and the right-sided Riemann–Liouville-type variable-order general fractional derivative with Prabhakar nonsingular kernel is α(·),υ,ϕ,λ RL f GP q Db−

d (t) = − dt

b

  ϕ (τ − t)υ−1 Eα(t,τ ),υ −λ (τ − t)α(t,τ ) f (τ ) dτ .

t

(5.172) Definition 5.71 ([25]). Let 1 > α (t, τ ) > 0. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is defined as α(·),ϕ,υ,λ LS f GP q Da+

t (t) =

  ϕ (t − τ )υ−1 Eα(t,τ ),υ −λ (t − τ )α(t,τ ) f (1) (τ ) dτ , (5.173)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is α(·),ϕ,υ,λ LS f GP q Db−

b (t) = −

  ϕ (τ − t)υ−1 Eα(t,τ ),υ −λ (τ − t)α(t,τ ) f (1) (τ ) dτ .

t

(5.174) Definition 5.72 ([25]). Let 1 > α (t, τ ) > 0. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as

Variable-order general fractional derivatives with nonsingular kernels

α(·),ϕ,υ,λ R f GP q Ia+

t (t) := a

381

  1 −ϕ α(t,τ ) E −λ − τ f (τ ) dτ , (5.175) (t ) (t − τ )υ α(t,τ ),κ−υ

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is b

α(·),ϕ,υ,λ R f GP q Ib−

(t) = t

  1 −ϕ α(t,τ ) f (τ ) dτ . (5.176) υ Eα(t,τ ),κ−υ −λ (τ − t) (τ − t)

Definition 5.73 ([25]). Let 0 < α (t) < 1. The left-sided Riemann–Liouville-type variable order general fractional derivative with Prabhakar nonsingular kernel is defined as α(·),υ,ϕ,λ RL f GP qs Da+

d (t) = dt

t

  ϕ (t − τ )υ−1 Eα(t),υ −λ (t − τ )α(t) f (τ ) dτ , (5.177)

a

and the right-sided Riemann–Liouville-type variable order general fractional derivative with Prabhakar nonsingular kernel is α(·),υ,ϕ,λ RL f GP qs Db−

d (t) = − dt

b

  ϕ (τ − t)υ−1 Eα(t),υ −λ (τ − t)α(t) f (τ ) dτ . (5.178)

t

Definition 5.74 ([25]). Let 0 < α (t) < 1. The left-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is defined as α(·),ϕ,υ,λ LS f GP qs Da+

t (t) =

  ϕ (t − τ )υ−1 Eα(t),υ −λ (t − τ )α(t) f (1) (τ ) dτ , (5.179)

a

and the right-sided Liouville–Sonine-type variable-order general fractional derivative with Prabhakar nonsingular kernel is

α(·),ϕ,υ,λ LS f GP qs Db−

b (t) := −

  ϕ (τ − t)υ−1 Eα(t),υ −λ (τ − t)α(t) f (1) (τ ) dτ .

t

(5.180) Definition 5.75 ([25]). Let 0 < α (t) < 1. The left-sided variable-order general fractional integral with Prabhakar nonsingular kernel is defined as

382

General Fractional Derivatives With Applications in Viscoelasticity

α(·),ϕ,υ,λ R f GP qs Ia+

t

  1 −ϕ α(t) −λ − τ f (τ ) dτ , (5.181) E (t ) (t − τ )υ α(t),κ−υ

(t) = a

and the right-sided variable-order general fractional integral with Prabhakar nonsingular kernel is α(·),ϕ,υ,λ R f GP qs Ib−

b

  1 −ϕ α(t) f (τ ) dτ . (5.182) υ Eα(t),κ−υ −λ (τ − t) (τ − t)

(t) = t

5.10

Variable-order Hilfer-type fractional derivatives with Prabhakar nonsingular kernel

Definition 5.76. Let 0 < α (t) < 1, 0 ≤ β ≤ 1, γ ∈ R, λ ∈ R, and f ∈ L1 (a, b), −∞ < a < b < ∞. The left-sided variable-order Hilfer-type fractional derivative with Prabhakar nonsingular kernel is defined as  α(·),β(·),λ RL f GP qs Da+

(t) =

β(t)(1−α(t)),λ RL GP qs Ia+

d RL (1−β(t))(1−α(t)),λ  I f (t) , dt GP qs a+ (5.183)

where RL I α(·),υ,ϕ,λ f (t) GP qs a+ t ϕ = (t − τ )υ−1 Eβ(t)(1−α(t)),υ a

  −λ (t − τ )β(t)(1−α(t)) f (τ ) dτ

(5.184)

and RL I (1−β(t))(1−α(t)),υ,ϕ,λ f (t) GP qs a+ t ϕ = (t − τ )υ−1 E(1−β(t))(1−α(t)),υ a

(5.185)   −λ (t − τ )(1−β(t))(1−α(t)) f (τ ) dτ ,

and the right-sided variable-order Hilfer-type fractional derivative with Prabhakar nonsingular kernel is  α(·),β(·),λ RL f GP qs Db−

(t) =

β(t)(1−α(t)),λ RL GP qs Ib−

d RL (1−β(t))(1−α(t)),λ  I f (t) , dt GP qs b− (5.186)

Variable-order general fractional derivatives with nonsingular kernels

383

where RL I α(·),υ,ϕ,λ f (t) GP qs b− b ϕ = (τ − t)υ−1 Eβ(t)(1−α(t)),υ t



 −λ (τ − t)β(t)(1−α(t)) f (τ ) dτ

(5.187)

and RL I GP qs b

(1−β(t))(1−α(t)),υ,ϕ,λ

−b

f (t)

(5.188)   ϕ = (τ − t)υ−1 E(1−β(t))(1−α(t)),υ −λ (τ − t)(1−β(t))(1−α(t)) f (τ ) dτ . t

For more details on the applications of the above operators, the readers are referred to [25].

General derivatives

6

Contents 6.1 6.2 6.3 6.4 6.5 6.6

Classical derivatives 385 Derivatives with respect to another function 386 General derivatives with respect to power-law function 390 General derivatives with respect to exponential function 392 General derivatives with respect to logarithmic function 394 Other general derivatives 395 6.6.1 6.6.2 6.6.3

General derivative with respect to negative power-law function 395 General derivative with respect to logarithmic function with parameter General derivative with respect to exponential function with parameter

395 396

In this chapter, we present the concepts of the general derivatives with singular and nonsingular kernels, and other general derivatives. In 2019, based on the Newton–Leibniz derivatives and integrals, Yang proposed new, general derivatives and integrals with respect to another function.

6.1

Classical derivatives

Definition 6.1 ([157]). The Newton–Leibniz derivative of a function f with respect to the variable t is defined as D (1) f (t) =

f (t + t) − f (t) df (t) = lim . t→0 dt t

(6.1)

Definition 6.2 ([157]). The Newton–Leibniz integral of a function f with respect to the variable t is defined as (1) 0 It f (t) =

t f (τ ) dτ .

(6.2)

0

The relations between (6.1) and (6.2), the Newton–Leibniz formulas, are given as [157] d f (t) = dt

t f (τ ) dτ 0

General Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00011-X Copyright © 2020 Elsevier Inc. All rights reserved.

(6.3)

386

General Fractional Derivatives With Applications in Viscoelasticity

and t f (t) =

d f (τ ) dτ + f (0) . dτ

(6.4)

0

Definition 6.3 ([157]). The Newton–Leibniz integral of a function f with respect to the variable t is defined as (1) a It f

t (t) =

f (τ ) dτ .

(6.5)

a

The relations between (6.4) and (6.5), the Newton–Leibniz formulas, are given as [157] d f (t) = dt

t f (τ ) dτ

(6.6)

a

and t f (t) =

d f (τ ) dτ + f (a) . dτ

(6.7)

a

The Newton–Leibniz integral of a function f with respect to the variable t is defined as [157] (1)

t

It f (t) =

f (τ ) dτ .

(6.8)

−∞

6.2

Derivatives with respect to another function

The general derivatives and integrals with respect to another function are presented as follows. Definition 6.4 ([158]). Let h(1) (τ ) > 0. The general derivative of a function  with respect to another function is defined as     1 dτ d d (1)  (τ ) =  (τ ) Dτ,h  (τ ) = dh(τ ) dτ dh (τ ) dτ dτ (6.9) d 1 d (τ ) =  (τ ) = (1) . dh (τ ) h (τ ) dτ

General derivatives

387

Definition 6.5 ([158]). Let h(1) (τ ) > 0. The general integral of a function  with respect to another function is defined as τ (1)  (t) h(1) (t) dt. (6.10) 0 Iτ,h  (τ ) = 0

Definition 6.6 ([158]). The general derivative of a function  of higher order with respect to another function is defined as   1 d n (n) Dτ,h  (τ ) =  (τ ) . (6.11) h(1) (τ ) dτ Their relationships between (6.9) and (6.10) are presented as [158] τ 1 d  (t) h(1) (t) dt  (τ ) = (1) h (τ ) dτ

(6.12)

0

and τ   (τ ) =

1

d h(1) (t) dt



  (t) h(1) (t) dt +  (0) .

(6.13)

0

It is clear that its physical meaning is the slope of the tangent line with respect to another function h (·). More generally, we have the following results: Definition 6.7 ([158]). Let h(1) (τ ) > 0. The general integral of a function  with respect to another function is defined as τ (1)  (t) h(1) (t) dt. (6.14) a Iτ,h  (τ ) = a

Thus, for h(1) (τ ) > 0, 1 d  (τ ) = (1) h (τ ) dτ

τ  (t) h(1) (t) dt

(6.15)

a

and  (τ )

τ  1 d =  h(1) (t) dt +  (a) (t) h(1) (t) dt =

a τ a

d(t) dt dt

(6.16)

+  (a) .

Observe that, for h (τ ) = τ , (6.9) and (6.10) become (6.1) and (6.2), respectively.

388

General Fractional Derivatives With Applications in Viscoelasticity

The general integral with respect to another function is defined as (1) Iτ,h  (τ ) =

τ  (t) h(1) (t) dt.

(6.17)

−∞

The properties of the general derivatives and integrals with respect to another function are presented as follows. Property 6.1. Let h(1) (τ ) > 0 and  (τ ) = (h (τ ) − h (a))n for n ∈ N. Then we have (1)

Dτ,h  (τ ) = n (h (τ ) − h (a))n−1 .

(6.18)

Proof. Using the definition of the general derivative with respect to another function, we have (1)

Dτ,h  (τ )   d n = dh(τ ) (h (τ ) − h (a))

(6.19)

= n (h (τ ) − h (a))n−1 , completing the proof. Property 6.2. Let h(1) (τ ) > 0 and  (τ ) = n (h (τ ) − h (a))n−1 for n ∈ N. Then we have (1) n a Iτ,h  (τ ) = (h (τ ) − h (a)) .

(6.20)

Proof. Using definition, (1) a Iτ,h  (τ ) τ 

 n (h (t) − h (a))n−1 h(1) (t) dt

=

a

τ   = n (h (t) − h (a))n−1 (dh (t))

(6.21)

a

τ   = n (h (t) − h (a))n−1 (d (h (t) − h (a))), a

which leads to (1) a Iτ,h  (τ )

= (h (τ ) − h (a))n − (h (a) − h (a))n = (h (τ ) − h (a))n . Thus, we finish the proof.

(6.22)

General derivatives

389

Property 6.3. Let h(1) (τ ) > 0 and  (τ ) = eλ(h(τ )−h(a)) for n ∈ N and λ ∈ R. Then we have (1)

Dτ,h  (τ ) = λ (τ ) .

(6.23)

Proof. Using the definitions of the general derivative with respect to another function, we have (1)  (τ ) Dτ,h  λ(h(τ )−h(a))  d = dh(τ ) e

(6.24)

= λeλ(h(τ )−h(a)) . Thus, we complete the proof. Property 6.4. Let h(1) (τ ) > 0 and  (τ ) = λeλ(h(τ )−h(a)) for n ∈ N. Then we have (1) λ(h(τ )−h(a)) . a Iτ,h  (τ ) = e

(6.25)

Proof. By definition and direct computation, (1) a Iτ,h  (τ )

τ = λeλ(h(t)−h(a)) h(1) (t) dt a

τ

= λeλ(h(t)−h(a)) (dh (t))

(6.26)

a τ

= λeλ(h(t)−h(a)) (d (h (t) − h (a))) a

= eλ(h(τ )−h(a)) , finishing the proof. Property 6.5. Let h(1) (τ ) > 0, h (0) = 0, h (∞) = ∞ and x ∈ N. Then ∞  (x) = 0

e−h(τ ) h (τ )x−1 h(1) (τ ) dτ .

(6.27)

390

General Fractional Derivatives With Applications in Viscoelasticity

Proof. Direct calculation gives ∞ 0

= =

e−h(τ ) h (τ )x−1 h(1) (τ ) dτ ∞ 0 ∞

e−h(τ ) h (τ )x−1 d (h (τ )) (6.28) e−τ τ x−1 dτ

0

=  (x) , completing the proof. Property 6.6. Let h(1) (τ ) > 0, h (0) = 0, h (1) = 1, x ∈ R+ and y ∈ R+ . Then 1 B (x, y) =

e−h(τ ) (h (τ ))x−1 (1 − h (τ ))y−1 h(1) (τ ) dτ .

(6.29)

0

Proof. Again, by direct calculation, 1

e−h(τ ) (h (τ ))x−1 (1 − h (τ ))y−1 h(1) (τ ) dτ

0

1 = e−h(τ ) (h (τ ))x−1 (1 − h (τ ))y−1 (dh (τ )) 0

1

(6.30)

= e−τ τ x−1 (1 − τ )y−1 dτ 0

= B (x, y) , which completes the proof.

6.3

General derivatives with respect to power-law function

In fact, the general derivatives with respect to power-law function are called the general derivatives with singular kernels. Definition 6.8 ([158]). Let h (τ ) = τ α (τ = 0). The general derivative of a function  with respect to power-law function is defined as (1) Dτ,τ α  (τ ) =

where α ∈ R.

1 d (τ ) , ατ α−1 dτ

(6.31)

General derivatives

391

Definition 6.9 ([158]). Let h (τ ) = τ α (τ = 0). The general integral of a function  with respect to power-law function is defined as (1) 0 Iτ,τ α  (τ ) = α

τ  (t) t α−1 dt.

(6.32)

0

Definition 6.10 ([158]). The general derivative of a function  of higher order with respect to power-law function is defined as  (n) Dτ,τ α  (τ ) =

1

d α−1 dτ ατ

n (6.33)

 (τ ) .

Their relationships between (6.31) and (6.32) are presented as [158] 1

d  (τ ) = α−1 dτ τ

τ  (t) t α−1 dt

(6.34)

0

and τ  (τ ) =

d (t) dt +  (0) . dt

(6.35)

0

More generally, 1

d  (τ ) = α−1 dτ τ

τ  (t) t α−1 dt

(6.36)

a

and τ   (τ ) =

1

d t α−1 dt



 τ d (t) α−1 dt +  (a) =  (t) t dt +  (a) . (6.37) dt

a

a

The properties of the general derivatives and integrals with respect to another function are presented as follows. Property 6.7. Let  (τ ) = (τ α − a α )n for n ∈ N. Then n−1  (1) Dτ,τ α  (τ ) = n τ α − a α .

(6.38)

392

General Fractional Derivatives With Applications in Viscoelasticity

Proof. Using the definition of the general derivative with respect to another function, we have (1)

Dτ,h  (τ ) = =



1 d (τ α − a α )n ατ α−1 dτ  α  d α n d(τ α ) (τ − a )

 (6.39)

= n (τ α − a α )n−1 . Thus, we complete the proof. Property 6.8. Let  (τ ) = n (τ α − a α )n−1 for n ∈ N. Then  α (1) a Iτ,τ α  (τ ) = τ

− aα

n

.

(6.40)

Proof. By definition, (1) a Iτ,τ α  (τ ) τ  α =α n (t

− a α )n−1 t α−1 dt a τ  α n (t − a α )n−1 (dt α ) =

(6.41)

a

= (τ α − a α )n , yielding the claim. We remark that Chen presented the power-law function as the Hausdorff measure with the aid of the hypotheses of fractal invariance and fractal equivalence [13].

6.4

General derivatives with respect to exponential function

In fact, the general derivatives with respect to exponential function are called the general derivatives with nonsingular kernels. Definition 6.11 ([158]). Let h (τ ) = eλτ and λ ∈ R. The general derivative of a function  with respect to exponential function is defined as (1) Dτ,e λτ  (τ ) =

1 d (τ ) . λeλτ dτ

(6.42)

General derivatives

393

Definition 6.12 ([158]). Let h (τ ) = eλτ and λ ∈ R. The general integral with of a function  with respect to exponential function is defined as (1) 0 Iτ,eλτ  (τ ) =

τ  (t) λeλt dt.

(6.43)

0

The general derivative of higher order with respect to another function is defined as [158]   1 d n (n)  (τ ) . (6.44) Dτ,eλτ  (τ ) = λeλτ dτ Their relationships between (6.42) and (6.43) are given as follows [158]:  (τ ) =

τ

1 d eλτ dτ

 (t) eλt dt

(6.45)

0

and τ  (τ ) =

d  (t) dt +  (0) . dt

(6.46)

0

Generally, we have the following results. Definition 6.13 ([158]). Let h (τ ) = eλτ and λ ∈ R. The general integral of a function  with respect to exponential function is defined as (1) a Iτ,eλτ  (τ ) =

τ  (t) λeλt dt.

(6.47)

a

Thus, we have 1 d  (τ ) = λτ e dτ

τ  (t) eλt dt

(6.48)

 (t) eλt dt +  (a)

(6.49)

a

and  (τ ) τ  1 = eλt =

a τ a

d dt



d dτ  (t) dt

+  (a) .

394

6.5

General Fractional Derivatives With Applications in Viscoelasticity

General derivatives with respect to logarithmic function

Definition 6.14 ([158]). The general derivative of a function  with respect to logarithmic function is defined as (1)

Dτ,ln τ  (τ ) = τ

d (τ ) . dτ

(6.50)

Definition 6.15 ([158]). The general integral of a function  with respect to logarithmic function is defined as (1) 0 Iτ,ln τ  (τ ) =

τ

1  (t) dt. t

(6.51)

0

The general derivative of a function  of higher order with respect to logarithmic function is defined as [158]   d n (n)  (τ ) . (6.52) Dτ,ln τ  (τ ) = τ dτ Their relationships between (6.50) and (6.51) can be written as follows [158]:  (τ )  d  τ 1 = τ dτ t  (t) dt d = τ dτ

τ 0

0

(6.53)

1 t  (t) dt

and  (τ ) τ  d  = τ dt  (t) 1t dt +  (0) =

0 τ 0

d dt  (t) dt

(6.54)

+  (0) .

Definition 6.16 ([158]). The general integral of a function  with respect to logarithmic function is defined as (1)

τ

a Iτ,ln τ  (τ ) = a

1  (t) dt. t

(6.55)

General derivatives

395

More generally,  (τ )  d  τ 1 = τ dτ t  (t) dt d = τ dτ

τ a

(6.56)

a

1 t  (t) dt

and  (τ ) τ  d   (t) 1t dt +  (a) = τ dt =

a τ a

6.6

d dt  (t) dt

(6.57)

+  (a) .

Other general derivatives

6.6.1 General derivative with respect to negative power-law function Definition 6.17 ([159]). Let h (τ ) = −τ −α and 0 < α. The general derivative of a function with respect to negative power-law function is defined as (1)

D−t −α (t) =

t α+1 d (t) . a dt

(6.58)

Definition 6.18 ([159]). Let h (τ ) = −τ −α and 0 < α. The general integral of a function M with respect to negative power-law function is defined as t dτ (1) M (τ ) α+1 . (6.59) a It,−t −α M (t) = a τ a

6.6.2 General derivative with respect to logarithmic function with parameter Definition 6.19 ([159]). Let g (t) = ln (t − 1). The general derivative of a function with respect to logarithmic function with parameter is defined as (1)

Dln(t−1) (t) = (t − 1)

d (t) . dt

(6.60)

396

General Fractional Derivatives With Applications in Viscoelasticity

Definition 6.20 ([159]). Let g (t) = ln (t − 1). The general integral of a function M with respect to logarithmic function with parameter is defined as τ

(1)

a Iτ,ln(τ −1) M (τ ) = a

M (t) dt. t −1

(6.61)

6.6.3 General derivative with respect to exponential function with parameter Definition 6.21 ([159]). Let h (τ ) = eλτ − 1. The general derivative of a function with respect to exponential function with parameter is defined as Deλt −1 (t) = λe−λt (1)

d (t) . dt

(6.62)

Definition 6.22 ([159]). Let h (τ ) = eλτ − 1. The general integral of a function M with respect to exponential function with parameter is defined as (1) a Iτ,eλτ −1 M (τ ) = λ

τ M (t) eλt dt.

(6.63)

a

For a = 0, one has [159] (1) 0 Iτ,−τ −α M (τ ) = a

τ M (t)

dt t α+1

,

(6.64)

0 (1) 0 Iτ,ln(τ −1) M (τ ) =

τ

M (t) dt, t −1

(6.65)

M (t) eλt dt.

(6.66)

0

and (1) 0 Iτ,eλτ M (τ ) = λ

τ 0

For a = 0, one also gets [159] t (1) −∞ It,−t −α M (t) = a −∞

M (τ )

dτ , τ α+1

(6.67)

General derivatives

397

t

(1) −∞ It,ln(t−1) M (t) =

−∞

M (τ ) dτ , τ −1

(6.68)

M (τ ) eλτ dτ .

(6.69)

and t (1) −∞ It,eλt −1 M (t) = λ −∞

For h (τ ) = eλτ − 1 and λ ∈ R+ , one has [159] (1)

D1−e−λt (t) = λeλt (1) a It,1−e−λt M (t) = λ

d (t) , dt

t

(6.70)

M (τ ) e−λτ dτ ,

(6.71)

M (τ ) e−λτ dτ ,

(6.72)

M (τ ) e−λτ dτ .

(6.73)

a (1) 0 It,1−e−λt M (t) = λ

t a

and (1) 0 It,1−e−λt M (t) = λ

t a

The above operators were applied in anomalous viscoelasticity and discussed in [158, 159].

Applications of fractional-order viscoelastic models

7

Contents 7.1 7.2 7.3 7.4

Mathematical models with classical derivatives 399 Mathematical models with general derivatives 400 Mathematical models with fractional derivatives 409 Mathematical models with fractional derivatives with nonsingular kernels 7.4.1 7.4.2 7.4.3

411

Mathematical models with fractional derivatives with Mittag-Leffler nonsingular kernel 411 Mathematical models with fractional derivatives with Wiman nonsingular kernel 413 Mathematical models with fractional derivatives with Prabhakar nonsingular kernel 415

7.5 Mathematical models with fractional derivatives with respect to another function 417

Linear viscoelasticity is a subject which stretches in its influence and importance from the early days of rheology to the present day [160]. In 1678, Robert Hooke first reported the quantitative concept of elasticity [161]. In 1687, Isaac Newton proposed the hypothesis that “the resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another” [162]. In 1867 and 1868, James Clerk Maxwell put forward the idea for the concept of viscosity of solids [163,164]. Kelvin (1875) [165] and Voigt (1889) [166] respectively proposed the theoretical work, which is now known as the Kelvin–Voigt model (also called the Kelvin model or Voigt model). In 1878, Ludwig Boltzmann structured the integral form of linear viscoelasticity, which is now known as the Boltzmann’s superposition principle [167]. Thomson (1888) [168] and Wiechert (1893) [169] independently introduced the concept of the relaxation times. In 1929, Jeffreys [170] reported the theoretical work with the crust of the Earth. In 1935, Burgers proposed the mechanical model, which is now known as the Burgers model [171]. In 1948, Zener coined the theoretical work, which is now known as Zener model [172]. For the historical investigation of the rheology, the reader is referred to the references [160,173–177]. We now investigate and research the fractional-order viscoelastic models with the different derivatives.

7.1

Mathematical models with classical derivatives

The Hookean spring element is given as follows [161]: σ (τ ) = ζ ε (τ ) , General Fractional Derivatives With Applications in Viscoelasticity. https://doi.org/10.1016/B978-0-12-817208-7.00012-1 Copyright © 2020 Elsevier Inc. All rights reserved.

(7.1)

400

General Fractional Derivatives With Applications in Viscoelasticity

where ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The Newtonian dashpot element reads [162]: σ (τ ) = γ D (1) ε (τ ) ,

(7.2)

where γ is the viscosity of the material, D (1) is the Newton–Leibniz derivative, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the Maxwell model can be written as follows [163, 164]: D (1) ε (τ ) =

σ (τ ) D (1) σ (τ ) + , γ ζ

(7.3)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model can be given as follows [165, 166]: σ (τ ) = γ ε (τ ) + ζ D (1) ε (τ ) ,

(7.4)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Jeffreys model can be reported as follows [170]:     (7.5) 1 + aD (1) σ (τ ) = b cD (1) + dD (1) ε (τ ) , where a, b, c, and d are the material constants. The constitutive equation for the Burgers model becomes [171]:     1 + aD (1) + bD (2) σ (τ ) = cD (1) + dD (2) ε (τ ) ,

(7.6)

where a, b, c, and d are the material constants. The constitutive equation for the Zener model can be given as follows [172]:     (7.7) 1 + aD (1) σ (τ ) = b + cD (1) ε (τ ) , where a, b, and c are the material constants.

7.2

Mathematical models with general derivatives

The Newtonian-like dashpot element containing the general derivative with respect to another function is as follows [158]: σ (τ ) =

γ dε (τ ) (1) = γ Dτ,h ε (τ ) , h(1) (τ ) dτ

(7.8)

Applications of fractional-order viscoelastic models

401

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.8) can be presented as follows [158]: (1)

σ0 = γ Dτ,h ε (τ )

(7.9)

with the solution τ ε (τ ) =

σ0 (1) h (t) dt. γ

(7.10)

0

The constitutive equation for the Maxwell-like model containing the general derivative with respect to another function becomes (1)

(1) Dτ,h ε (τ ) =

σ (τ ) Dτ,h σ (τ ) + , γ ζ

(7.11)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the Kelvin–Voigt-like model containing the general derivative with respect to another function can be presented as follows: (1)

σ (τ ) = γ ε (τ ) + ζ Dτ,h ε (τ ) ,

(7.12)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the Burgers-like model containing the general derivative with respect to another function can be represented in the form:     (1) (2) (1) (2) + bDτ,h + dDτ,h σ (τ ) = cDτ,h ε (τ ) , (7.13) 1 + aDτ,h where a, b, c, and d are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the Zener-like model containing the general derivative with respect to another function can be presented as follows:     (1) (1) (7.14) 1 + aDτ,h σ (τ ) = b + cDτ,h ε (τ ) , where a, b, and c are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The Newtonian-like dashpot element containing the general derivative with respect to power-law function is given as [14] (1)

σ (τ ) = γ Dτ,τ α ε (τ ) ,

(7.15)

402

General Fractional Derivatives With Applications in Viscoelasticity

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.15) can be written as follows [158]: (1)

σ0 = γ Dτ,τ α ε (τ ) with the solution σ0 ε (τ ) = τ α . γ

(7.16)

(7.17)

The strain for the Newtonian-like dashpot element containing the general derivative with respect to power-law function is given in Fig. 7.1.

Figure 7.1 The strain distribution for the Newtonian-like dashpot element containing the general derivative with respect to power-law function with σ0 /γ = 0.6.

The constitutive equation for the anomalous Maxwell-like model containing the general derivative with respect to another function can be given as follows: (1)

Dτ,τ α ε (τ ) =

σ (τ ) 1 (1) + Dτ,τ α σ (τ ) , γ ζ

(7.18)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Kelvin–Voigt-like model containing the general derivative with respect to power-law function can be presented as follows: (1) σ (τ ) = γ ε (τ ) + ζ Dτ,τ α σ (τ ) ,

(7.19)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time.

Applications of fractional-order viscoelastic models

403

The constitutive equation for the anomalous Burgers-like model containing the general derivative with respect to power-law function can be represented in the form:     (1) (2) (1) (2) 1 + aDτ,τ (7.20) α + bDτ,τ α σ (τ ) = cDτ,τ α + dDτ,τ α ε (τ ) , where a, b, c, and d are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Zener-like model containing the general derivative with respect to power-law function is as follows:     (1) (1) 1 + aDτ,τ (7.21) α σ (τ ) = b + cDτ,τ α ε (τ ) , where a, b, and c are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The Newtonian-like dashpot element containing the general derivative with respect to exponential function is given as [158] (1)

σ (τ ) = γ Dτ,eλτ ε (τ ) ,

(7.22)

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.22) can be written as follows [158]: (1)

σ0 = γ Dτ,eλτ ε (τ ) with the solution σ0 ε (τ ) = eλτ . γ

(7.23)

(7.24)

The strain for the Newtonian-like dashpot element containing the general derivative with respect to exponential function is presented in Fig. 7.2. The constitutive equation for the anomalous Maxwell-like model containing the general derivative with respect to exponential function can be given as follows: (1)

Dτ,eλτ ε (τ ) =

σ (τ ) 1 (1) + Dτ,eλτ σ (τ ) , γ ζ

(7.25)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Kelvin–Voigt-like model containing the general derivative with respect to exponential function can be presented as follows: (1) σ (τ ) = γ ε (τ ) + ζ Dτ,e λτ σ (τ ) ,

(7.26)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time.

404

General Fractional Derivatives With Applications in Viscoelasticity

Figure 7.2 The strain distribution for the Newtonian-like dashpot element containing the general derivative with respect to exponential function with σ0 /γ = 0.6 and λ = 0.5.

The constitutive equation for the anomalous Burgers-like model containing the general derivative with respect to exponential function can be represented in the form: 

   (2) (2) (1) (2) 1 + aDτ,e λτ + bDτ,eλτ σ (τ ) = cDτ,eλτ + dDτ,eλτ ε (τ ) ,

(7.27)

where a, b, c, and d are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Zener-like model containing the general derivative with respect to exponential function reads as follows: 

   (1) (1) 1 + aDτ,e λτ σ (τ ) = b + cDτ,eλτ ε (τ ) ,

(7.28)

where a, b, and c are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The Newtonian-like dashpot element containing the general derivative with respect to logarithmic function is considered as follows [158]: (1) σ (τ ) = γ Dτ,ln τ ε (τ ) ,

(7.29)

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.29) can be presented as follows [158]: (1)

σ0 = γ Dτ,ln τ ε (τ )

(7.30)

Applications of fractional-order viscoelastic models

405

with the solution ε (τ ) =

σ0 ln τ. γ

(7.31)

The strain for the Newtonian-like dashpot element containing the general derivative with respect to logarithmic function is displayed in Fig. 7.3.

Figure 7.3 The strain distribution for the Newtonian-like dashpot element containing the general derivative with respect to logarithmic function with σ0 /γ = 0.6.

The constitutive equation for the anomalous Maxwell-like model containing the general derivative with respect to logarithmic function can be given as follows: (1) Dτ,ln τ ε (τ ) =

σ (τ ) 1 (1) + Dτ,ln τ σ (τ ) , γ ζ

(7.32)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Kelvin–Voigt-like model containing the general derivative with respect to logarithmic function is as follows: (1) σ (τ ) = γ ε (τ ) + ζ Dτ,ln τ σ (τ ) ,

(7.33)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time.

406

General Fractional Derivatives With Applications in Viscoelasticity

The constitutive equation for the anomalous Burgers-like model containing the general derivative with respect to logarithmic function can be represented in the form:     (1) (2) (1) (2) (7.34) 1 + aDτ,ln τ + bDτ,ln τ σ (τ ) = cDτ,ln τ + dDτ,ln τ ε (τ ) , where a, b, c, and d are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Zener-like model containing the general derivative with respect to logarithmic function becomes:     (1) (1) (7.35) 1 + aDτ,ln τ σ (τ ) = b + cDτ,ln τ ε (τ ) , where a, b, and c are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The Newtonian-like dashpot element involving the general derivative with respect to negative power-law function can be written as follows [159]: (1)

σ (τ ) = γ Dτ,τ −α ε (τ ) ,

(7.36)

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.36) can be given as follows: (1)

σ0 = γ Dτ,τ −α ε (τ ) with the solution ε (τ ) =

σ0 −α τ . γ

The strain for the Newtonian-like dashpot element involving the general derivative with respect to negative power-law function is depicted in Fig. 7.4. The constitutive equation for the anomalous Maxwell-like model involving the general derivative with respect to negative power-law function can be suggested as follows [159]: (1)

Dτ,τ −α ε (τ ) =

σ (τ ) 1 (1) + Dτ,τ −α σ (τ ) , γ ζ

(7.37)

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Kelvin–Voigt-like model involving the general derivative with respect to negative power-law function can be given as follows [159]: (1)

σ (τ ) = γ ε (τ ) + ζ Dτ,τ −α σ (τ ) ,

(7.38)

Applications of fractional-order viscoelastic models

407

Figure 7.4 The strain distribution for the Newtonian-like dashpot element involving the general derivative with respect to negative power-law function with σ0 /γ = 0.6.

where γ is the viscosity of the material, ζ is the elastic modulus of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Burgers-like model involving the general derivative with respect to negative power-law function can be represented in the form [159]: (1)

(2)

(1)

(2)

σ (τ ) + aDτ,τ −α σ (τ ) + bDτ,τ −α σ (τ ) = cDτ,τ −α ε (τ ) + dDτ,τ −α ε (τ ) , (7.39) where a, b, c, and d are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the anomalous Zener-like model involving the general derivative with respect to negative power-law function can be presented as follows [159]: (1) (1) σ (τ ) + aDτ,τ −α σ (τ ) = bε (τ ) + cDτ,τ −α ε (τ ) ,

(7.40)

where a, b, and c are the material constants, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. In fact, one can present the general case as follows: The Newtonian-like dashpot element involving the general derivative with respect to scaling law can be written as follows: (1)

σ (τ ) = γ Dτ,φτ −α ε (τ ) ,

(7.41)

408

General Fractional Derivatives With Applications in Viscoelasticity

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time, and the scaling-law is φτ −α with the constant φ ∈ R. When σ (0) = σ0 , (7.36) reads as follows: (1)

σ0 = γ Dτ,φτ −α ε (τ ) with the solution ε (τ ) =

φσ0 −α τ . γ

The strain for the Newtonian-like dashpot element involving the general derivative with respect to scaling law is depicted in Fig. 7.5.

Figure 7.5 The strain distribution for the Newtonian-like dashpot element involving the general derivative with respect to scaling law with φσ0 /γ = 0.8.

The constitutive equation for the anomalous Maxwell-like model involving the general derivative with respect to scaling law is (1) Dτ,φτ −α ε (τ ) =

σ (τ ) 1 (1) + Dτ,φτ −α σ (τ ) , γ ζ

(7.42)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the anomalous Kelvin–Voigt-like model involving the general derivative with respect to scaling law can be given as follows: (1)

σ (τ ) = γ ε (τ ) + ζ Dτ,φτ −α σ (τ ) ,

(7.43)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

409

The constitutive equation for the anomalous Burgers-like model involving the general derivative with respect to scaling law can be represented in the form: (1) (2) (2) (2) σ (τ ) + aDτ,φτ (7.44) −α σ (τ ) + bDτ,φτ −α σ (τ ) = cDτ,φτ −α ε (τ ) + dDτ,φτ −α ε (τ ) ,

where a, b, c, and d are the material constants. The constitutive equation for the anomalous Zener-like model involving the general derivative with respect to scaling law becomes: (1) (1) σ (τ ) + aDτ,φτ −α σ (τ ) = bε (τ ) + cDτ,φτ −α ε (τ ) ,

(7.45)

where a, b, and c are the material constants.

7.3

Mathematical models with fractional derivatives

The Scott–Blair dashpot element with Riemann–Liouville fractional derivative is considered as follows [178]: α σ (τ ) = γ D0+ ε (τ ) ,

(7.46)

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.46) reads [14]: α σ0 = γ D0+ ε (τ )

with the solution [14] ε (τ ) =

σ0 τ α . γ (1+α)

The strain for the Scott–Blair dashpot element with Riemann–Liouville fractional derivative is presented in Fig. 7.6. The Newtonian-like dashpot element with Liouville–Sonine–Caputo fractional derivative becomes [14]: α σ (τ ) = γ LS D0+ ε (τ ) ,

(7.47)

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.47) can be written as follows [14] α σ0 = γ LS D0+ ε (τ )

with the solution [14] ε (τ ) =

σ0 τ α . γ (1+α)

410

General Fractional Derivatives With Applications in Viscoelasticity

Figure 7.6 Strain distribution for the Scott–Blair dashpot element with Riemann–Liouville fractional derivative with σ0 /γ = 0.6.

The strain for the dashpot element with Liouville–Sonine–Caputo fractional derivative is in agreement with the result in Fig. 7.6 (see [14]). The Gerasimov dashpot element with Liouville–Sonine–Caputo fractional derivative is expressed as follows [78,179]: α ε (τ ) , σ (τ ) = γ LS D+

(7.48)

where γ is the material parameter, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. The constitutive equation for the Maxwell-like model with Riemann–Liouville fractional derivative can be written as follows [180]: β

α ε (τ ) = D0+

σ (τ ) D0+ σ (τ ) + , γ ζ

(7.49)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Maxwell-like model with Riemann–Liouville fractional derivative is as follows [181]: β

α ε (τ ) = D0+

σ (τ ) D0+ σ (τ ) + , γ ζ

(7.50)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

411

The constitutive equation for the Maxwell-like model with Liouville–Sonine– Caputo fractional derivative can be written as follows [14]: α LS D0+ ε (τ ) =

σ (τ ) + γ

α LS D0+ σ

ζ

(τ )

,

(7.51)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The fractional Kelvin–Voigt-like model with Liouville–Sonine–Caputo fractional derivative is given as follows [75]: α σ (τ ) = γ ε (τ ) + ζ LS D0+ σ (τ ) ,

(7.52)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The fractional Zener-like model with Riemann–Liouville fractional derivative is as follows [182]: α α σ (τ ) = bε (τ ) + cD0+ ε (τ ) . σ (τ ) + aD0+

(7.53)

The fractional Zener-like model with Liouville–Sonine–Caputo fractional derivative is presented as follows [183,184]: β

α ε (τ ) , σ (τ ) + a LS D0+ σ (τ ) = bε (τ ) + cLS D0+

(7.54)

where a, b, and c are the material constants. The fractional Zener-like model with Liouville–Sonine–Caputo fractional derivative reads [14] α α σ (τ ) = bε (τ ) + cLS D0+ ε (τ ) , σ (τ ) + a LS D0+

(7.55)

where a, b, and c are the material constants. For more models with Riemann–Liouville fractional derivatives, the readers are referred to the references [185–190].

7.4

Mathematical models with fractional derivatives with nonsingular kernels

7.4.1 Mathematical models with fractional derivatives with Mittag-Leffler nonsingular kernel The Newtonian-like dashpot element with Hille–Tamarkin general fractional derivative is given as follows ([25]; for λ = ±1, see [134,135]): RL α,λ D0+ ε (τ ) , σ (τ ) = γ Ml

(7.56)

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time.

412

General Fractional Derivatives With Applications in Viscoelasticity

When σ (0) = σ0 , (7.56) can be given as follows: α,λ σ0 = γ RL Ml D0+ ε (τ )

with the solution   τα σ0 1+ . ε (τ ) = γ

(1 + α) The strain for the Newtonian-like dashpot element with Hille–Tamarkin general fractional derivative is given in Fig. 7.7.

Figure 7.7 The strain distribution for the Newtonian-like dashpot element with Hille–Tamarkin general fractional derivative with σ0 /γ = 0.6.

The constitutive equation for the Maxwell model with Hille–Tamarkin general fractional derivative can be written as follows ([133]; for λ = ±1, see [134,135]): RL α,λ Ml D0+ ε (τ ) =

σ (τ ) + γ

RL D α,λ σ Ml 0+

ζ

(τ )

,

(7.57)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with Hille–Tamarkin general fractional derivative is as follows ([133]; for λ = ±1, see [134,135]): α,λ σ (τ ) = γ ε (τ ) + ζ RL Ml D0+ ε (τ ) ,

(7.58)

where γ is the viscosity of the material, ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

413

The Newtonian-like dashpot element with Liouville–Sonine–Caputo type general fractional derivative via Mittag-Leffler function is given as follows ([133]; for λ = ±1, see [134,135]): LSC α,λ σ (τ ) = γ Ml D0+ ε (τ ) ,

(7.59)

where γ is the viscosity of the material, ε (t) is the strain, σ (t) is the stress, and τ is the time. When σ (0) = σ0 , (7.59) can be given as follows: α,λ σ0 = γ LSC Ml D0+ ε (τ )

with the solution ε (τ ) =

  τα σ0 1+ . γ

(1 + α)

The strain for the Newtonian-like dashpot element with Liouville–Sonine–Caputo type general fractional derivative via Mittag-Leffler function is in agreement with the result in Fig. 7.7 (see [134,135]). The constitutive equation for the Maxwell model with Liouville–Sonine–Caputo type general fractional derivative via Mittag-Leffler function can be written as follows ([133]; for λ = ±1, see [134,135]): LSC α,λ Ml D0+ ε (τ ) =

σ (τ ) + γ

LSC D α,λ σ Ml 0+

ζ

(τ )

,

(7.60)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with Liouville–Sonine– Caputo type general fractional derivative via Mittag-Leffler function becomes ([133]; for λ = ±1, see [134,135]): α,λ σ (τ ) = γ ε (τ ) + ζ LSC Ml D0+ ε (τ ) ,

(7.61)

where γ is the viscosity of the material, ζ is the elastic modulus of the material.

7.4.2 Mathematical models with fractional derivatives with Wiman nonsingular kernel The Newtonian-like dashpot element with general fractional derivative with Wiman nonsingular kernel is represented as follows ([25]; for λ = ±1, see [133–135]): α,υ,λ RL σ (τ ) = γ GW D0+ ε (τ ) ,

(7.62)

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time.

414

General Fractional Derivatives With Applications in Viscoelasticity

When σ (0) = σ0 , (7.62) can be given as follows: α,υ,λ σ0 = γ RL GW D0+ ε (τ )

with the solution ε (τ ) =

σ0 γ



 τ 1−υ λτ 1+α−υ − .

(2 − υ) (2+α − υ)

The strain for the Newtonian-like dashpot element with general fractional derivative with Wiman nonsingular kernel is illustrated in Fig. 7.8.

Figure 7.8 The strain distribution for the Newtonian-like dashpot element with general fractional derivative with Wiman nonsingular kernel with σ0 /γ = 0.6 and υ = 0.8.

The constitutive equation for the Maxwell model with general fractional derivative with Wiman nonsingular kernel can be written as follows (for λ = ±1, see [133–135]): α,υ,λ RL GW D0+ ε (τ ) =

σ (τ ) + γ

RL D α,υ,λ σ GW 0+

ζ

(τ )

,

(7.63)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with general fractional derivative with Wiman nonsingular kernel is as follows (for λ = ±1, see [133–135]): α,υ,λ σ (τ ) = γ ε (τ ) + ζ RL GW D0+ ε (τ ) ,

(7.64)

where γ is the viscosity of the material, ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

415

The Newtonian-like dashpot element with general fractional derivative with Wiman nonsingular kernel is given as follows ([25]; for λ = ±1, see [133–135]): LSC α,υ,λ σ (τ ) = γ GW D0+ ε (τ ) ,

(7.65)

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.65) can be given as follows: α,υ,λ σ0 = γ LSC GW D0+ ε (τ )

with the solution ε (τ ) =

σ0 γ



 τ 1−υ λτ 1+α−υ − .

(2 − υ) (2+α − υ)

The strain for the Newtonian-like dashpot element with general fractional derivative with Wiman nonsingular kernel is in agreement with the result in Fig. 7.8 (see [134,135]). The constitutive equation for the Maxwell model with general fractional derivative with Wiman nonsingular kernel can be written as follows (for λ = ±1, see [133–135]): LSC α,υ,λ GW D0+ ε (τ ) =

σ (τ ) + γ

LSC D α,υ,λ σ GW 0+

ζ

(τ )

,

(7.66)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with general fractional derivative with Wiman nonsingular kernel can be given as follows (for λ = ±1, see [133–135]): α,λ σ (τ ) = γ ε (τ ) + ζ LSC Ml D0+ ε (τ ) ,

(7.67)

where γ is the viscosity of the material, ζ is the elastic modulus of the material.

7.4.3 Mathematical models with fractional derivatives with Prabhakar nonsingular kernel The Newtonian-like dashpot element with Kilbas–Saigo–Saxena-type general fractional derivative in Prabhakar nonsingular kernel is given as follows ([25]; for λ = ±1, see [133–135]): α,υ,ϕ,λ

RL D0+ σ (τ ) = γ GP

ε (τ ) ,

(7.68)

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.68) can be given as follows: α,υ,ϕ,λ

σ0 = γ RL GP D0+

ε (τ )

416

General Fractional Derivatives With Applications in Viscoelasticity

with the solution   σ0 −ϕ ε (τ ) = τ −υ Eα,1−υ −λτ α . γ The constitutive equation for the Maxwell model with Kilbas–Saigo–Saxena type general fractional derivative in Prabhakar nonsingular kernel is as follows (for λ = ±1, see [133–135]): RL α,υ,ϕ,λ ε (τ ) = GP D0+

σ (τ ) + γ

RL D α,υ,ϕ,λ σ GP 0+

(τ )

ζ

,

(7.69)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with Kilbas–Saigo–Saxenatype general fractional derivative in Prabhakar nonsingular kernel can be given as follows (for λ = ±1, see [133–135]): α,υ,ϕ,λ

σ (τ ) = γ ε (τ ) + ζ RL GP D0+

(7.70)

ε (τ ) ,

where γ is the viscosity of the material, ζ is the elastic modulus of the material. The Newtonian-like dashpot element with Garra–Gorenflo–Polito–Tomovski type general fractional derivative in Prabhakar nonsingular kernel is as follows ([25]; for λ = ±1, see [133–135]): α,υ,ϕ,λ

σ (τ ) = γ LS GP D0+

(7.71)

ε (τ ) ,

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.68) can be given as follows: α,υ,ϕ,λ

σ0 = γ LS GP D0+

ε (τ )

with the solution   σ0 −ϕ ε (τ ) = τ −υ Eα,1−υ −λτ α . γ The constitutive equation for the Maxwell model with Garra–Gorenflo–Polito– Tomovski-type general fractional derivative in Prabhakar nonsingular kernel can be written as follows (for λ = ±1, see [133–135]): α,υ,ϕ,λ LS ε (τ ) = GP D0+

σ (τ ) + γ

LS D α,υ,ϕ,λ σ GP 0+

ζ

(τ )

,

(7.72)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt model with Garra–Gorenflo–Polito– Tomovski-type general fractional derivative in Prabhakar nonsingular kernel reads (for λ = ±1, see [133–135]): α,υ,ϕ,λ

σ (τ ) = γ ε (τ ) + ζ LS GP D0+

ε (τ ) ,

(7.73)

where γ is the viscosity of the material, ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

7.5

417

Mathematical models with fractional derivatives with respect to another function

The Newtonian-like dashpot element involving Osler fractional derivative with respect to another function is given as follows: σ (τ ) α = γ D0+,h ε (τ )   τ γ d 1 = (1−α) (1) h (τ ) dτ τ

(7.74) f (t) (1) (h(τ )−h(t))α h (t) dt,

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.68) can be given as follows: α ε (τ ) σ0 = γ D0+,h

with the solution ε (τ ) =

σ0 (h (τ ) − h (0))α . γ

(1+α)

The Boltzmann-like type superposition integrals based on the Osler fractional integrals with respect to another function can be written as follows [192]:  τ α J (h (τ ) − h (t)) h(1) (t)D0+,h σ (t) dt (7.75) ε (τ ) = σ (0) J (h (τ )) + 0

and 

τ

σ (τ ) = ε (0) G (h (τ )) + 0

α G (h (τ ) − h (t)) h(1) (τ ) D0+,h ε (t)dt,

(7.76)

where α σ D0+,h

d 1 1 (τ ) = (1)

(1 − α) h (τ ) dτ

τ 0

h(1) (t) σ (t) dt (h (τ ) − h (t))α

(7.77)

h(1) (t) ε (t) dt, (h (τ ) − h (t))α

(7.78)

and α ε (τ ) = D0+,h

1 1 d

(1 − α) h(1) (τ ) dt

τ 0

with the creep compliance function J (τ ) and the relaxation modulus function G (τ ).

418

General Fractional Derivatives With Applications in Viscoelasticity

The constitutive equation for the Maxwell-like model involving Osler fractional derivative with respect to another function becomes [192] α ε (τ ) = D0+,h

α σ (τ ) D0+,h σ (τ ) + , γ ζ

(7.79)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt-like model involving Osler fractional derivative with respect to another function can be given as follows [192]: α σ (τ ) = γ ε (τ ) + ζ D0+,h ε (τ ) ,

(7.80)

where γ is the viscosity of the material, ζ is the elastic modulus of the material. The constitutive equation for the Burgers-like model involving Osler fractional derivative with respect to another function can be represented in the form [193] β

β

α α σ (τ ) + bD0+,h σ (τ ) = cD0+,h ε (τ ) + dD0+,h ε (τ ) , σ (τ ) + aD0+,h

(7.81)

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Osler fractional derivative with respect to another function can be presented as follows [194]: α α σ (τ ) = bε (τ ) + cD0+,h ε (τ ) , σ (τ ) + aD0+,h

(7.82)

where a, b, and c are the material constants. The Newtonian-like dashpot element involving Osler fractional derivative with respect to power function is given as follows [191]: σ (τ ) α = γ D0+,τ α ε (τ )   τ α2 γ d = (1−α) τ 1−α dτ 0

(7.83) f (t) dt, t 1−α (τ α −t α )α

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.83) is as follows: α σ0 = γ D0+,τ α ε (τ )

with the solution 2

σ0 τ α . ε (τ ) = γ (1+α) The strain for the Newtonian-like dashpot element involving Osler fractional derivative with respect to power function is depicted in Fig. 7.9.

Applications of fractional-order viscoelastic models

419

Figure 7.9 The strain distribution for the Newtonian-like dashpot element involving Osler fractional derivative with respect to power function with σ0 /γ = 0.6.

The constitutive equation for the Maxwell-like model involving Osler fractional derivative with respect to power function can be written as follows: α D0+,τ α ε (τ ) =

α σ (τ ) D0+,τ α σ (τ ) + , γ ζ

(7.84)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt-like model involving Osler fractional derivative with respect to power function can be given as follows: α σ (τ ) = γ ε (τ ) + ζ D0+,τ α ε (τ ) ,

(7.85)

where γ is the viscosity of the material, ζ is the elastic modulus of the material. The constitutive equation for the Burgers-like model involving Osler fractional derivative with respect to power function can be represented in the form: β

β

α α σ (τ ) + aD0+,τ (7.86) α σ (τ ) + bD0+,τ α σ (τ ) = cD0+,τ α ε (τ ) + dD0+,τ α ε (τ ) ,

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Osler fractional derivative with respect to power function is as follows: α α σ (τ ) + aD0+,τ α σ (τ ) = bε (τ ) + cD0+,τ α ε (τ ) ,

where a, b, and c are the material constants.

(7.87)

420

General Fractional Derivatives With Applications in Viscoelasticity

The Newtonian-like dashpot element involving Osler fractional derivative with respect to scaling law becomes σ (τ ) α = γ D0+,φt −α ε (τ )

=

α 2 γ τ 1+α φ α (1−α)

d dτ

τ 0

(7.88)

f (t) dt, t 1+α (τ α −t α )α

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, τ is the time, and the scaling law is φt −α with the constant φ ∈ R. When σ (0) = σ0 , (7.88) can be given as follows: α σ0 = γ D0+,φt −α ε (τ )

with the solution φ α σ0 t −α ε (τ ) = . γ (1+α) 2

The strain for the Newtonian-like dashpot element involving Osler fractional derivative with respect to scaling law is given in Fig. 7.10.

Figure 7.10 The strain distribution for the Newtonian-like dashpot element involving Osler fractional derivative with respect to scaling law with σ0 /γ = 0.6 and φ = 0.5.

Applications of fractional-order viscoelastic models

421

The constitutive equation for the Maxwell-like model involving Osler fractional derivative with respect to scaling law can be written as follows: α D0+,φt −α ε (τ ) =

α σ (τ ) D0+,φt −α σ (τ ) + , γ ζ

(7.89)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt-like model involving Osler fractional derivative with respect to scaling law reads: α σ (τ ) = γ ε (τ ) + ζ D0+,φt −α ε (τ ) ,

(7.90)

where γ is the viscosity of the material, ζ is the elastic modulus of the material. The constitutive equation for the Burgers-like model involving Osler fractional derivative with respect to scaling law can be represented in the form: β

ρ

α α σ (τ ) + aD0+,φt −α σ (τ ) + bD0+,φt −α σ (τ ) = cD0+,φt −α ε (τ ) + dD0+,φt −α ε (τ ) ,

(7.91) where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Osler fractional derivative with respect to scaling law can be presented as follows: α α σ (τ ) + aD0+,φt −α σ (τ ) = bε (τ ) + cD0+,φt −α ε (τ ) ,

(7.92)

where a, b and c are the material constants. The Colombaro–Garra–Giusti–Mainardi dashpot element involving Almeida fractional derivatives with respect to another function can be reported as follows [191]: σ (τ ) α = γ LS D0+,h ε (τ )   τ γ h(1) (t) d 1 = (1−α) ε (t) dt, α (1) (h(t)−h(τ )) h (t) dt

(7.93)

0

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.93) can be given as follows: α σ0 = γ LS D0+,h ε (τ )

with the solution ε (τ ) =

σ0 (h (τ ) − h (0))α . γ

(1+α)

The constitutive equation for the Maxwell-like model involving Almeida fractional derivatives with respect to another function can be written as follows [195]:

422

General Fractional Derivatives With Applications in Viscoelasticity

α LS D0+,h ε (τ ) =

σ (τ ) + γ

α LS D0+,h σ

(τ )

ζ

(7.94)

,

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt-like model involving Almeida fractional derivatives with respect to another function can be represented as follows [193]: α ε (τ ) . σ (τ ) = γ ε (τ ) + ζ LS D0+,h

(7.95)

The constitutive equation for the Burgers-like model involving Almeida fractional derivatives with respect to another function can be presented in the form [193]: β

β

α α σ (τ )+bLS D0+,h σ (τ ) = cLS D0+,h ε (τ )+d LS D0+,h ε (τ ) , (7.96) σ (τ )+a LS D0+,h

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Almeida fractional derivatives with respect to another function can be written as follows [194]: β

β

σ (τ ) + a LS D0+,h σ (τ ) = bε (τ ) + cLS D0+,h ε (τ ) ,

(7.97)

where a, b, and c are the material constants. The Newtonian-like dashpot element involving Almeida fractional derivatives with respect to scaling law is as follows: σ (τ ) α = γ LS D0+,φτ −α ε (τ )

=

γ φ α (1−α)

τ 0



(7.98)

1  dε(t) α dt dt, t −α −τ −α

where γ is the viscosity of the material, ε (t) is the strain, σ (t) is the stress, and τ is the time. When σ (0) = σ0 , (7.98) can be given as follows: α σ0 = γ LS D0+,φτ −α ε (τ )

with the solution φ α σ0 τ −α . γ (1+α) 2

ε (τ ) =

The constitutive equation for the Maxwell-like model involving Almeida fractional derivatives with respect to scaling law can be written as follows: α LS D0+,φτ −α ε (τ ) =

σ (τ ) + γ

α LS D0+,φτ −α σ

ζ

(τ ) ,

(7.99)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

423

The constitutive equation for the Kelvin–Voigt-like model involving Almeida fractional derivatives with respect to scaling law can be given as follows: α σ (τ ) = γ ε (τ ) + ζ LS D0+,φτ −α ε (τ ) .

(7.100)

The constitutive equation for the Burgers-like model involving Almeida fractional derivatives with respect to scaling law can be represented as: β

α σ (τ ) + a LS D0+,φτ −α σ (τ ) + b LS D0+,φτ −α σ (τ ) β

α = cLS D0+,φτ −α ε (τ ) + d LS D0+,φτ −α ε (τ ) ,

(7.101)

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Almeida fractional derivatives with respect to scaling law can be presented as follows: α α σ (τ ) + a LS D0+,φτ −α σ (τ ) = bε (τ ) + LS D0+,φτ −α ε (τ ) ,

(7.102)

where a, b, and c are the material constants. The Newtonian-like dashpot element involving Almeida fractional derivatives with respect to scaling law is given as follows: σ (τ ) α = γ LS D0+,τ −α ε (τ ) τ γ 1   = (1−α) α t −α −τ −α 0

(7.103) dε(t) dt dt,

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.103) reads as follows: α σ0 = γ LS D0+,τ −α ε (τ )

with the solution σ0 t −α . γ (1+α) 2

ε (τ ) =

The strain for the Newtonian-like dashpot element involving Almeida fractional derivatives with respect to scaling law is in agreement with the result in Fig. 7.10. The constitutive equation for the Maxwell-like model involving Almeida fractional derivatives with respect to power function can be written as follows: α LS D0+,τ −α ε (τ ) =

σ (τ ) + γ

α LS D0+,τ −α σ

ζ

(τ )

,

(7.104)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material.

424

General Fractional Derivatives With Applications in Viscoelasticity

The constitutive equation for the Kelvin–Voigt-like model involving Almeida fractional derivatives with respect to scaling law can be given as follows: α σ (τ ) = γ ε (τ ) + ζ LS D0+,τ −α ε (τ ) .

(7.105)

The constitutive equation for the Burgers-like model involving Almeida fractional derivatives with respect to power function can be represented as: β

α σ (τ ) + a LS D0+,τ −α σ (τ ) + b LS D0+,τ −α σ (τ ) β

α = cLS D0+,τ −α ε (τ ) + d LS D0+,τ −α ε (τ ) ,

(7.106)

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Almeida fractional derivatives with respect to power function can be presented as follows: α α σ (τ ) + a LS D0+,τ −α σ (τ ) = bε (τ ) + LS D0+,τ −α ε (τ ) ,

(7.107)

where a, b, and c are the material constants. The Newtonian-like dashpot element involving Almeida fractional derivatives with respect to exponential function is given as follows: σ (τ ) α = γ LS D0+,e λτ ε (τ )

=

γ

(1−α)

τ 0



(7.108)

1  dε(t) α dt dt, eλt −eλτ

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.108) can be given as follows: α σ0 = γ LS D0+,e λτ ε (τ )

with the solution

 α σ0 eλτ − 1 ε (τ ) = . γ (1+α)

The strain for the Newtonian-like dashpot element involving Almeida fractional derivatives with respect to exponential function is demonstrated in Fig. 7.11. The constitutive equation for the Maxwell-like model involving Almeida fractional derivatives with respect to exponential function can be written as follows: α LS D0+,eλτ ε (τ ) =

σ (τ ) + γ

α LS D0+,eλτ σ

ζ

(τ ) ,

(7.109)

where γ is the viscosity of the material, and ζ is the elastic modulus of the material.

Applications of fractional-order viscoelastic models

425

Figure 7.11 The strain distribution for the Newtonian-like dashpot element involving Almeida fractional derivatives with respect to exponential function with σ0 /γ = 0.6 and λ = 0.5.

The constitutive equation for the Kelvin–Voigt-like model involving Almeida fractional derivatives with respect to exponential function can be given as follows: α σ (τ ) = γ ε (τ ) + ζ LS D0+,e λτ ε (τ ) .

(7.110)

The constitutive equation for the Burgers-like model involving Almeida fractional derivatives with respect to exponential function can be represented as: β

α σ (τ ) + a LS D0+,e λτ σ (τ ) + b LS D0+,eλτ σ (τ ) β

(7.111)

α = cLS D0+,e λτ ε (τ ) + d LS D0+,eλτ ε (τ ) ,

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Almeida fractional derivatives with respect to exponential function can be presented as follows: α α σ (τ ) + a LS D0+,e λτ σ (τ ) = bε (τ ) + LS D0+,eλτ ε (τ ) ,

where a, b, and c are the material constants.

(7.112)

426

General Fractional Derivatives With Applications in Viscoelasticity

The Newtonian-like dashpot element involving Almeida fractional derivatives with respect to logarithmic function is given as follows: σ (τ ) α = γ LS D1+,ln τ ε (τ )

=

γ

(1−α)

τ 1

(7.113)

dε(t) 1 (ln t−ln τ )α dt dt,

where γ is the viscosity of the material, ε (τ ) is the strain, σ (τ ) is the stress, and τ is the time. When σ (0) = σ0 , (7.113) can be given as follows: α σ0 = γ LS D1+,ln τ ε (τ )

with the solution ε (τ ) =

σ0 (ln (τ ))α . γ (1+α)

The strain for the Newtonian-like dashpot element involving Almeida fractional derivatives with respect to logarithmic function is demonstrated in Fig. 7.12.

Figure 7.12 The strain distribution for the Newtonian-like dashpot element involving Almeida fractional derivatives with respect to logarithmic function with σ0 /γ = 0.6.

Applications of fractional-order viscoelastic models

427

The constitutive equation for the Maxwell-like model involving Almeida fractional derivatives with respect to logarithmic function can be written as follows: α LS D1+,ln τ ε (τ ) =

σ (τ ) + γ

α LS D1+,ln τ σ

ζ

(τ )

(7.114)

,

where γ is the viscosity of the material, and ζ is the elastic modulus of the material. The constitutive equation for the Kelvin–Voigt-like model involving Almeida fractional derivatives with respect to logarithmic function is as follows: α σ (τ ) = γ ε (τ ) + ζ LS D1+,ln τ ε (τ ) .

(7.115)

The constitutive equation for the Burgers-like model involving Almeida fractional derivatives with respect to logarithmic function can be represented as: β

β

α α σ (τ )+a LS D1+,ln τ σ (τ )+bLS D1+,ln τ σ (τ ) = cLS D1+,ln τ ε (τ )+d LS D1+,ln τ ε (τ ) , (7.116)

where a, b, c, and d are the material constants. The constitutive equation for the Zener-like model involving Almeida fractional derivatives with respect to logarithmic function can be presented as follows: α α σ (τ ) + a LS D1+,ln τ σ (τ ) = bε (τ ) + LS D1+,ln τ ε (τ ) ,

(7.117)

where a, b, and c are the material constants. For more viscoelastic models with the different derivatives, the readers are referred to the reference [25].

References

[1] L. Euler, De progressionibus transcendentibus sen quaroum termini generales algebrare dari nequeunt, Commentarii Academiae Scientiarum Petropolitanae, vol. 5, 1738, pp. 36–57, presented to the St. Petersburg Academy on November 28, 1729. [2] A.M. Legendre, Recherches sur diverses sortes d’integrales definies, Mem. de l’Institut, vol. 10, 1809, pp. 416–509. [3] G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71, Cambridge University Press, Cambridge, UK, 1999. [4] A.M. Legendre, Essai sur la Theorie des Nombres, Courcier, Paris, 1808. [5] B.C. Carlson, Special Functions of Applied Mathematics, Academic, New York, 1977. [6] I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, Edinburgh, 1961. [7] L. Euler, Evolutio Formula Integralis, etc., Commentarii Academiae Scientiarum Petropolitanae, vol. 16, 1771, pp. 91–139 (first published in 1772). [8] T.H. Gronwall, An elementary exposition of the theory of the gamma function, Annals of Mathematics 17 (3) (1916) 124–166. [9] O. Schlomilch, Analytische Studien, Engelmann, Leipzig, 1848. [10] F.W. Newman, On  (a) especially when a is negative, The Cambridge and Dublin Mathematical Journal 3 (1848) 57–60. [11] W. Freeden, M. Gutting, Special Functions of Mathematical (Geo-)Physics, Springer Science & Business Media, 2013. [12] W.W. Bell, Special Functions and Their Applications, Prentice Hall, London, UK, 1965, revised English edition, translated and edited by Richard A. Silverman. [13] W.W. Bell, Special Functions for Scientists and Engineers, Van Nostrand, London, 1968. [14] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. [15] V.S. Kiryakova, Generalized Fractional Calculus and Applications, CRC Press, 1993. [16] C.S. Meijer, Uber Whittakersche bzw. Besselsche Funktionen und deren Produkte, Nieuw Archief Voor Wiskunde (2) 18 (4) (1936) 10–39 (in German). [17] H. Hankel, Die Euler’schen integrale bei unbeschrankter variabilitat der arguments, Leipzig, 1863. [18] N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, John Wiley & Sons, 2011. [19] G. Gasper, M. Rahman, G. George, Basic Hypergeometric Series, vol. 96, Cambridge University Press, 2004. [20] F. Oberhettinger, Tables of Mellin Transforms, Springer Science & Business Media, 2012.  [21] L. Euler, Evolutio formulae integralis x f −1 (log x)m/n dx integratione a valore x = 0 ad x = 1 extensa, Novi Commentarii academiae scientiarum Petropolitanae 16 (1772) 91–139, Opera Omnia, Ser. I 17 (1771) 316–357. [22] F.E. Prym, Zur Theorie der Gammafunktion, Journal für die Reine und Angewandte Mathematik 82 (1877) 165–172. [23] F. Oberhettinger, L. Badii, Tables of Laplace Transforms, Springer-Verlag, 1973.

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Index

A Almeida fractional derivative, 120 Anomalous viscoelasticity, 397 B Burgers model, 399, 400 C Caputo fractional derivative, 120 Constitutive equation, 400–416, 418, 419, 421–425, 427 Continuous derivative, 100 D Dashpot element, 410 Derivative Feller fractional, 132, 133 fractional, 97, 98, 104, 105, 108, 210, 223–226, 315, 316, 318, 340, 341, 349–353, 409, 411–415 Herrmann fractional, 138 Hilfer, 98, 195 Riesz fractional, 129 E Euler beta function, 1–5, 59, 60 Euler gamma function, 1, 3, 59, 60 Exponential function, 98, 166–170, 173, 392, 393, 396, 403, 404, 424, 425 F Feller derivative, 204, 206 fractional calculus, 97, 132, 134, 139 derivative, 132, 133 integral, 132 Feller type fractional integral, 134

Finite interval, 151, 153 Fractal equivalence, 392 Fractal invariance, 392 Fractional calculus, 98, 111, 128, 137, 173, 191, 207, 224, 227, 228, 232, 241, 248, 254, 263, 267, 287, 293, 295, 348, 352 derivative, 97, 98, 104, 105, 108, 210, 223–226, 315, 316, 318, 340, 341, 349–353, 409, 411–415 difference, 123, 124 derivative, 97, 98, 123, 124, 127, 143, 144, 146–148, 150 derivative formula, 123 operator, 124 integral, 98, 100, 102, 107, 108, 132, 160, 165, 173, 174, 191, 192, 223–225, 227, 228, 232, 234, 240, 241, 243, 247, 342–344, 346–348, 364, 367–370, 373–376, 379–381 integration, 107, 150 order, 124 Function integral, 275 Prabhakar, 23, 30, 41, 70, 263 Prabhakar type, 1, 2, 74, 81 Rabotnov type, 2 spaces, 98 Wiman, 2, 47, 50, 252, 258 Wiman type, 2 Functional spaces, 97 G Gauss hypergeometric function, 223 Gerasimov dashpot element, 410 H Hadamard fractional calculus, 98, 160, 164 integrals, 160, 164

440

Hadamard type fractional calculus, 98 integrals, 166 Hankel contour, 4 Herrmann derivative, 205, 207 fractional calculus, 97, 137, 139 derivative, 138 integral, 137 Hilfer derivative, 98, 195 fractional derivative, 197–207 Hilfer type general fractional derivative, 256, 289 Hookean spring element, 399 Hyperbolic subcosine function, 1, 2, 7, 13, 20, 54, 57, 62, 65, 69, 90, 92 subsine function, 1, 2, 7, 13, 20, 54, 57, 62, 64, 68, 90, 92 supercosine function, 38, 80 supersine function, 37, 79 I Integral Feller fractional, 132 fractional, 98, 100, 102, 107, 108, 132, 160, 165, 173, 174, 191, 192, 223–225, 227, 228, 232, 234, 240, 241, 243, 247, 342–344, 346–348, 364, 367–370, 373–376, 379–381 function, 275 Herrmann fractional, 137 Riesz fractional, 128 Sonine, 211 Inverse operator, 240, 243, 247, 249, 254, 287, 295, 300, 306 J Jeffreys model, 400 K Kelvin model, 399 L Laplace transform, 5, 119, 214, 215, 220–222, 226, 227, 254, 267

Index

inverse, 5 properties, 5 Lebesgue measurable functions, 98, 99 Lebesgue summable functions, 100 Linear viscoelasticity, 399 Liouville fractional derivatives, 97, 120 Logarithmic function, 394–396, 404–406, 426, 427 M Marchaud fractional derivatives, 98, 171 Marchaud type fractional derivatives, 172 Maxwell model, 400, 412–416 N Nonsingular kernel, 210, 211, 224–229, 232–235, 243, 349, 385, 392, 411 Prabhakar, 263–274, 354–357, 361–364, 367–370, 373–382, 415 Sonine, 224, 226–229 Wiman, 252–254, 256–263, 413–415 Normalization function, 352 O Osler fractional calculus, 107, 108 derivatives, 108 integrals, 417 P Power function, 97, 418, 419, 423, 424 Power weight, 99 Prabhakar function, 23, 30, 41, 70, 263 general fractional integrals, 254, 255 nonsingular kernel, 263–274, 354–357, 361–364, 367–370, 373–382, 415, 416 Prabhakar type function, 1, 2, 74, 81 general fractional integrals, 255, 271, 282 hyperbolic subcosine function, 1, 2 subsine function, 1, 2 supercosine function, 25, 32, 44, 72, 76, 83 supersine function, 24, 31, 43, 71, 75, 83 subcosine function, 1, 2 subsine function, 1, 2

Index

supercosine function, 24, 31, 42, 71, 75, 82 supersine function, 23, 30, 42, 70, 74, 82 Prym decomposition, 4 R Rabotnov function, 1, 16 Rabotnov type function, 2 hyperbolic subcosine function, 1, 2, 17, 67 hyperbolic subsine function, 1, 2, 16, 66 subcosine function, 1, 2, 16, 66 subsine function, 1, 2, 16, 66 Rheology, 399 Riemann fractional derivative, 102 Riesz derivative, 203, 205 fractional calculus, 97, 128, 130, 132 derivative, 129 integral, 128 S Scaling law, 407–409, 420–424 Slipperiness, 399 Sonine condition, 211 general fractional calculus, 224, 232 derivative, 225, 229 integral, 224, 226 integral, 211 nonsingular kernel, 224, 226–229 Subcosine function, 1, 2, 7, 13, 20, 54, 57, 61, 64, 68, 90, 92 hyperbolic, 1, 2, 7, 13, 20, 54, 57, 62, 65, 69, 90, 92 Prabhakar type, 1, 2 Prabhakar type hyperbolic, 1, 2 Rabotnov type, 1, 2, 16, 66 Wiman type, 2 Wiman type hyperbolic, 2 Subsine function, 1, 2, 7, 12, 20, 53, 56, 61, 64, 68, 89, 91 hyperbolic, 1, 2, 7, 13, 20, 54, 57, 62, 64, 68, 90, 92 Prabhakar type, 1, 2 Prabhakar type hyperbolic, 1, 2 Rabotnov type, 1, 2, 16, 66 Wiman type, 2 Wiman type hyperbolic, 2

441

Supercosine function, 36, 78 hyperbolic, 38, 80 Prabhakar type, 24, 31, 42, 71, 75, 82 Prabhakar type hyperbolic, 25, 32, 44, 72, 76, 83 Wiman type, 48, 51, 85, 88 Wiman type hyperbolic, 48, 51, 86, 88 Superposition integrals, 417 Supersine function, 36, 78 hyperbolic, 37, 79 Prabhakar type, 23, 30, 42, 70, 74, 82 Prabhakar type hyperbolic, 24, 31, 43, 71, 75, 83 Wiman type, 48, 51, 85, 87 Wiman type hyperbolic, 48, 51, 86, 88 Symmetric fractional difference derivative, 143, 144, 147, 148 T Tempered fractional calculus, 178, 182, 186, 220 V Viscoelastic models, 427 Viscosity, 400–403, 405–408, 410–421 Viscosity concept, 399 Voigt model, 399 W Wiman function, 2, 47, 50, 252, 258 nonsingular kernel, 252–254, 256–263, 413–415 Wiman type function, 2 hyperbolic subcosine function, 2 subsine function, 2 supercosine function, 48, 51, 86, 88 supersine function, 48, 51, 86, 88 subcosine function, 2 subsine function, 2 supercosine function, 48, 51, 85, 88 supersine function, 48, 51, 85, 87 Z Zener model, 399, 400