A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

TL;DR

This paper introduces a generalized derivative concept based on arbitrary kernels, unifying various fractional derivatives, and explores its properties, applications, and an existence theorem for related boundary value problems.

Contribution

It proposes a new derivative operator with respect to arbitrary kernels, encompassing many existing fractional derivatives and enabling the development of new fractional operators.

Findings

The new derivative operator admits a right inverse via conjugate kernels.

It includes Riemann-Liouville and Hadamard fractional derivatives as special cases.

An existence theorem for boundary value problems involving this derivative is established.

Abstract

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.