Special Functions of Fractional Calculus in Form of Convolution Series and their Applications

TL;DR

This paper introduces a new class of special functions in Fractional Calculus based on convolution series generated by Sonine kernels, providing analytic solutions to complex fractional differential equations.

Contribution

It develops a novel class of special functions related to fractional integrals and derivatives, and derives analytic solutions for multi-term fractional differential equations.

Findings

Convolution series generated by Sonine kernels form a new class of special functions.

Analytic solutions are derived for complex fractional differential equations.

Mittag-Leffler functions are specific cases within this new class.

Abstract

In this paper, we first discuss the convolution series that are generated by the Sonine kernels from a class of functions continuous on the real positive semi-axis that have an integrable singularity of power function type at the point zero. These convolution series are closely related to the general fractional integrals and derivatives with the Sonine kernels and represent a new class of the special functions of Fractional Calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel $\kappa(t) = t^{\alpha -1}/\Gamma(\alpha),\ 0<\alpha <1$. The main result of the paper is derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that were not yet considered in the Fractional Calculus publications.