General Fractional Derivatives with Applications in Viscoelasticity

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