On the local $M$-derivative

TL;DR

This paper introduces a new fractional derivative involving Mittag-Leffler functions that generalizes existing derivatives, preserves key calculus properties, and extends classical theorems like Rolle's and the mean value theorem.

Contribution

The paper proposes a novel fractional derivative operator that generalizes the Katugampola derivative using Mittag-Leffler functions, maintaining core calculus properties and extending classical theorems.

Findings

The new derivative satisfies linearity, product, quotient, and chain rules.

It extends classical theorems such as Rolle's and the mean value theorem.

The associated fractional integral generalizes the fundamental theorem of calculus.

Abstract

We introduce a new fractional derivative that generalizes the so-called alternative fractional derivative recently proposed by Katugampola. We denote this new differential operator by $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order, is such that $0<\alpha<1$, $\beta>0$ and $M$ is used to denote that the function to be derived involves a Mittag-Leffler function with one parameter. This new derivative satisfies some properties of integer-order calculus, e.g.\ linearity, product rule, quotient rule, function composition and the chain rule. Besides as in the case of the Caputo derivative, the derivative of a constant is zero. Because Mittag-Leffler function is a natural generalization of the exponential function, we can extend some of the classical results of integer-order calculus, namely: Rolle's theorem, the mean value theorem and its extension. Further, when the order of the derivative is $\alpha=1$ and the parameter of the Mittag-Leffler function is also unitary, our definition is equivalent to the definition of the ordinary derivative of order one. Finally, we present the corresponding fractional integral from which, as a natural consequence, new results emerge which can be interpreted as applications. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts.