On the fractional relaxation equation with Scarpi derivative

TL;DR

This paper addresses the solution of the fractional relaxation equation with Scarpi derivatives within variable order fractional calculus, providing a general framework and explicit solutions for specific transition functions.

Contribution

It introduces the concepts of Scarpi derivatives and transition functions, solving the relaxation equation in the most general variable order fractional setting.

Findings

Complete solution for arbitrary transition functions

Explicit solutions for exponential-type transition functions

Explicit solutions for Mittag-Leffler transition functions

Abstract

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. Thus, we solve the relaxation equation in, what seems to be, the most general case. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition functions which make essentials of our problem. Next, we completely solve our initial value problem for an arbitrary transition function, and we calculate the solution in the case of an exponential-type transition function, as well as in the case of a Mittag-Leffler transition.