On the fractional relaxation equation with Scarpi derivative

Abstract

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition functions which represent the essentials of our problem. Next, we provide an integral representation for the solution of our initial value problem where the transition function is chosen arbitrary. After that, we find an integral representation in several special cases in which we choose the transition function concretely. Finally, we give some numerical insights which prove our theoretical results.