A note on the equivalence of fractional relaxation equations to differential equations with varying coefficients

TL;DR

This paper demonstrates that fractional relaxation equations with constant coefficients can be reformulated as first-order differential equations with variable coefficients, highlighting the role of Mittag-Leffler functions in fractional calculus and suggesting broader applications.

Contribution

It establishes an equivalence between fractional relaxation equations and variable coefficient differential equations, providing a new perspective for analyzing fractional processes.

Findings

Fractional relaxation equations can be transformed into first-order equations with varying coefficients.

Mittag-Leffler functions are central to understanding fractional relaxation processes.

The approach can be extended to more complex fractional evolution equations.

Abstract

In this note we show how a initial value problem for a relaxation process governed by a differential equation of non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag-Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a more deep knowledge of certain phenomena, both deterministic and stochastic ones, nowadays investigated with the techniques of the fractional calculus.