On fractional calculus with general analytic kernels

Abstract

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.

23 pages. Accepted for publication in Applied Mathematics and Computation

References in corpus (3)

• A Caputo fractional derivative of a function with respect to another function
• On some new properties of fractional derivatives with Mittag-Leffler kernel
• Incomplete Riemann-Liouville fractional derivative operators and incomplete hypergeometric functions

Cited by in corpus (5)

• On some analytic properties of tempered fractional calculus
• On tempered fractional calculus with respect to functions and the associated fractional differential equations
• A complex analysis approach to Atangana-Baleanu fractional calculus
• Analysis of Impulsive $\phi$--Hilfer Fractional Differential Equations
• Weak Pontryagin's Maximum Principle for Optimal Control Problems Involving a General Analytic Kernel