On some new properties of fractional derivatives with Mittag-Leffler kernel

TL;DR

This paper introduces a new series-based formula for fractional derivatives with Mittag-Leffler kernels, clarifying their non-locality, and proves fundamental properties and applications in fractional differential equations and mechanics.

Contribution

It presents a novel series representation of fractional derivatives with Mittag-Leffler kernels and explores their properties, including existence, uniqueness, and rules like product and chain, with applications.

Findings

New series formula for fractional derivatives with Mittag-Leffler kernel

Proved existence and uniqueness for related fractional ODEs

Extended product and chain rules for these derivatives

Abstract

We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann-Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics.