Operational Calculus for the general fractional derivatives with the Sonine kernels

TL;DR

This paper develops an operational calculus for general fractional derivatives with Sonine kernels, enabling analytical solutions to fractional differential equations with new convolution series generalizing exponential and Mittag-Leffler functions.

Contribution

It introduces a Mikusiński-type operational calculus for fractional derivatives with Sonine kernels, expanding analytical tools for solving related differential equations.

Findings

Constructed an operational calculus for these derivatives.

Derived solutions in convolution series form.

Generalized exponential and Mittag-Leffler functions.

Abstract

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusi\'nski type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.