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General Fractional Derivatives: Theory, Methods and Applications provides knowledge of the special functions with respect to another function, and the integro-differential operators where the integrals are of the convolution type and exist the singular, weakly singular and nonsingular kernels, which exhibit the fractional derivatives, fractional integrals, general fractional derivatives, and general fractional integrals of the constant and variable order without and with respect to another function due to the appearance of the power-law and complex herbivores to figure out the modern developments in theoretical and applied science. Features: Give some new results for fractional calculus of constant and variable orders. Discuss some new definitions for fractional calculus with respect to another function. Provide definitions for general fractional calculus of constant and variable orders. Report new results of general fractional calculus with respect to another function. Propose news special functions with respect to another function and their applications. Present new models for the anomalous relaxation and rheological behaviors. This book serves as a reference book and textbook for scientists and engineers in the fields of mathematics, physics, chemistry and engineering, senior undergraduate and graduate students. Dr. Xiao-Jun Yang is a full professor of Applied Mathematics and Mechanics, at China University of Mining and Technology, China. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Modelling and Analysis, International Journal of Numerical Methods for Heat & Fluid Flow, and Thermal Science.
Cover; Half Title; Title Page; Copyright Page; Dedication; Contents; Preface; Author; 1. Introduction; 1.1 History of fractional calculus; 1.1.1 The contribution for fractional calculus and applications; 1.1.2 The contribution for generalized fractional calculus and applications; 1.2 History of special functions; 1.3 Special functions with respect to another function; 2. Fractional Derivatives of Constant Order and Applications; 2.1 Fractional derivatives within power-law kernel; 2.2 Riemann-Liouville fractional calculus; 2.2.1 Riemann-Liouville fractional integrals
2.2.2 Riemann-Liouville fractional derivatives2.2.3 Riemann-Liouville fractional derivatives of a purely imaginary order; 2.3 Liouville-Sonine-Caputo fractional derivatives; 2.3.1 Motivations; 2.3.2 Liouville-Sonine-Caputo fractional derivatives; 2.4 Liouville-Grüunwald-Letnikov fractional derivatives; 2.4.1 Motivations; 2.4.2 Liouville-Grüunwald-Letnikov fractional derivatives; 2.4.3 Kilbas-Srivastava-Trujillo fractional derivatives; 2.5 Tarasov type fractional derivatives; 2.5.1 Tarasov type fractional derivatives; 2.5.2 Extended Tarasov type fractional derivatives
2.6 Riesz fractional calculus2.7 Feller fractional calculus; 2.8 Richard fractional calculus; 2.9 Erdélyi-Kober type fractional calculus; 2.9.1 Erdélyi-Kober type operators of fractional integration and fractional derivative; 2.9.2 Fractional integrals and fractional derivatives of the Erdélyi-Kober-Riesz, Erdélyi-Kober-Feller and Erdélyi-Kober-Rich; 2.10 Katugampola fractional calculus; 2.10.1 Katugampola fractional integrals and Katugampola fractional derivatives; 2.10.2 Katugampola type fractional integrals and Katugampola type fractional derivatives involving the exponential function
2.11 Hadamard fractional calculus2.11.1 Hadamard fractional integrals and fractional derivatives; 2.11.2 Hadamard type fractional integrals and fractional derivatives; 2.12 Marchaud fractional derivatives; 2.13 Tempered fractional calculus; 2.13.1 Motivations; 2.13.2 Tempered fractional derivatives; 2.13.3 Tempered fractional derivatives with respect to another function; 2.13.4 Tempered fractional derivatives of a purely imaginary order; 2.13.5 Tempered fractional integrals; 2.13.6 Tempered fractional integrals of a purely imaginary order
2.13.7 Tempered fractional derivatives in the sense of Liouville-Sonine and Liouville-Sonine-Caputo types2.13.8 Tempered fractional derivatives involving power-sine and power-cosine functions; 2.13.9 Tempered fractional calculus involving power-Kohlrausch-Williams-Watts function; 2.13.9.1 Tempered fractional derivative in the Liouville-SonineCaputo type involving the kernel of the power-Kohlrausch-Williams-Watts function; 2.13.9.2 Tempered fractional integral involving the kernel of the power-Kohlrausch-Williams-Watts function; 2.13.10 Sabzikar-Meerschaert-Chen tempered fractional calculus
Includes bibliographical references and index.
Boca Raton : CRC Press, Taylor & Francis Group, 2019.