General Fractional Derivatives with Applications in Viscoelasticity

By Xiao-Jun Yang, Feng Gao and Yang Ju

General Fractional Derivatives with Applications in Viscoelasticity introduces the newly established fractional-order calculus operators involving singular and non-singular kernels with applications to fractional-order viscoelastic models from the calculus operator viewpoint. Fractional calculus and its applications have gained considerable popularity and importance because of their applicability to many seemingly diverse and widespread fields in science and engineering. Many operations in physics and engineering can be defined accurately by using fractional derivatives to model complex phenomena. Viscoelasticity is chief among them, as the general fractional calculus approach to viscoelasticity has evolved as an empirical method of describing the properties of viscoelastic materials. General Fractional Derivatives with Applications in Viscoelasticity makes a concise presentation of general fractional calculus.

• Presents a comprehensive overview of the fractional derivatives and their applications in viscoelasticity
• Provides help in handling the power-law functions
• Introduces and explores the questions about general fractional derivatives and its applications

• Language: English
• Publisher: Academic Press
• Release date: Apr 3, 2020
• ISBN: 9780128172094

Xiao-Jun Yang

Dr. Xiao-Jun Yang is a full professor of China University of Mining and Technology, China. He was awarded the 2019 Obada-Prize, the Young Scientist Prize (Turkey), and Springer's Distinguished Researcher Award. His scientific interests include: Viscoelasticity, Mathematical Physics, Fractional Calculus and Applications, Fractals, Analytic Number Theory, and Special Functions. He has published over 160 journal articles and 4 monographs, 1 edited volume, and 10 chapters. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Methods in the Applied Sciences, Mathematical Modelling and Analysis, Journal of Thermal Stresses, and Thermal Science, and an associate editor of Journal of Thermal Analysis and Calorimetry, Alexandria Engineering Journal, and IEEE Access.

Book preview

General Fractional Derivatives With Applications in Viscoelasticity

First edition

Xiao-Jun Yang

Feng Gao

Yang Ju

Table of Contents

Cover image

Title page

Copyright

Preface

1: Special functions

Abstract

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

References

2: Fractional derivatives with singular kernels

Abstract

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

2.11. Kilbas–Srivastava–Trujillo fractional difference derivatives

2.12. Riesz fractional calculus

2.13. Feller fractional calculus

2.14. Herrmann fractional calculus

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

2.22. Hadamard fractional calculus

2.23. Marchaud fractional derivatives

2.24. Riemann–Liouville-type tempered fractional calculus

2.25. Liouville–Weyl-type tempered fractional calculus

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

2.27. Hilfer derivatives

2.28. Mixed fractional derivatives

References

3: Fractional derivatives with nonsingular kernels

Abstract

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

References

4: Variable-order fractional derivatives with singular kernels

Abstract

4.1. Riemann–Liouville-type variable-order fractional calculus with singular kernel

4.2. Variable-order Hilfer-type fractional derivatives with singular kernel

4.3. Liouville–Weyl-type variable-order fractional calculus

4.4. Riesz-, Feller-, and Herrmann-type variable-order fractional derivatives with singular kernel

4.5. Variable-order tempered fractional derivatives with weakly singular kernel

References

5: Variable-order general fractional derivatives with nonsingular kernels

Abstract

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

References

6: General derivatives

Abstract

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

References

7: Applications of fractional-order viscoelastic models

Abstract

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

7.5. Mathematical models with fractional derivatives with respect to another function

References

References

References

Index

Copyright

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Preface

Xiao-Jun Yang; Feng Gao; Yang Ju     

The main purpose of this monograph is to provide an introduction to the newly-established 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, variable-order 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, Hadamard-type 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, Liouville–Weyl–Riesz-type tempered fractional calculus, Riemann–Liouville–Riesz-type 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–Liouville-type 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, Feller-type 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 variable-order general fractional derivatives with the nonsingular kernels, e.g., Mittag-Leffler, Wiman, Prabhakar, Lorenzo–Hartely, and Miller–Ross.

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 Bo-Ming 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.

1

Special functions

Abstract

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 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.

Keywords

Euler gamma function; Euler beta function; Mittag-Leffler function; Miller–Ross function; Rabotnov function; Lorenzo–Hartley function; Prabhakar function; Kilbas–Saigo–Saxena function; Wiman function; Gorenflo–Mainardi function; subsine function; subcosine function; hyperbolic subsine function; hyperbolic subcosine function; Euler gamma function with respect to another function; Euler beta function with respect to another function; Mittag-Leffler function with respect to another function; Miller–Ross function with respect to another function; Rabotnov function with respect to another function; Lorenzo–Hartley function with respect to another function; Prabhakar function with respect to another function; Kilbas–Saigo–Saxena function with respect to another function; Wiman function with respect to another function; Gorenflo–Mainardi function with respect to another function; subsine function with respect to another function; subcosine function with respect to another function; hyperbolic subsine function with respect to another function; hyperbolic subcosine function with respect to another function

Chapter Outline

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

In this chapter, we introduce the Euler gamma and beta