Generalized Fractional Derivative, Fractional differential ring

TL;DR

This paper introduces a generalized fractional derivative framework that encompasses existing definitions, establishing a differential ring structure and demonstrating its advantages through applications.

Contribution

It proposes a new generalized fractional derivative that unifies various existing definitions within a differential ring framework.

Findings

The generalized derivative includes previous definitions as special cases.

Applications illustrate the advantages of the generalized fractional derivative.

The framework extends the algebraic structure of fractional derivatives.

Abstract

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of which must be properly defined in its algebra. We introduce a generalized version of the fractional derivative that extends the existing ones in the literature. To those extensions, it is associated with a differentiable operator and a differential ring and applications that show the advantages of the generalization. We also review the different definitions of fractional derivatives proposed by Michele Caputo in \cite{GJI:GJI529}, Khalil, Al Horani, Yousef, Sababheh in \cite{khalil2014new}, Anderson and Ulness in \cite{anderson2015newly}, Guebbai and Ghiat in \cite{guebbai2016new}, Udita N. Katugampola in \cite{katugampola2014new}, Camrud in \cite{camrud2016conformable} and it is shown how the generalized version contains the previous ones as a particular case.