Operational Calculus for the 1st Level General Fractional Derivatives and its Applications

TL;DR

This paper develops an operational calculus for the 1st level General Fractional Derivatives, unifying existing types, and applies it to derive explicit solutions for fractional differential equations.

Contribution

It introduces a Mikusiński type operational calculus for 1st level GFDs, encompassing Riemann-Liouville and Caputo types, and applies it to solve related initial-value problems.

Findings

Unified operational calculus for 1st level GFDs

Closed-form solutions for fractional differential equations

Includes Riemann-Liouville and Caputo derivatives as special cases

Abstract

The 1st level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann-Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the Fractional Calculus literature. In this paper, we first construct an operational calculus of Mikusi\'nski type for the 1st level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann-Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for derivation of the closed form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st level GFDs.