General Fractional Derivatives with Applications in Viscoelasticity 1st Edition by Xiao Jun Yang, Feng Gao, Yang Ju ISBN 0128172088 9780128172087

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ISBN 10: 0128172088 
ISBN 13: 9780128172087
Author: Xiao Jun Yang, Feng Gao, 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.

General Fractional Derivatives with Applications in Viscoelasticity 1st Table of contents:

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

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Tags: Xiao Jun Yang, Feng Gao, Yang Ju, General Fractional, Derivatives, Viscoelasticity