A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties

TL;DR

This paper introduces a new truncated M-fractional derivative that unifies several existing fractional derivatives, satisfying classical properties and enabling solutions to fractional heat equations.

Contribution

The paper proposes a novel truncated M-fractional derivative unifying multiple fractional derivatives with classical properties and provides analytical solutions to fractional heat equations.

Findings

Unified fractional derivative framework with classical properties

Derived analytical solution for fractional heat equation

Graphical analysis illustrating the derivative's behavior

Abstract

We introduce a truncated $M$-fractional derivative type for $\alpha$-differentiable functions that generalizes four other fractional derivatives types recently introduced by Khalil et al., Katugampola and Sousa et al., the so-called conformable fractional derivative, alternative fractional derivative, generalized alternative fractional derivative and $M$-fractional derivative, respectively. We denote this new differential operator by $_{i}\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order of the derivative is such that $ 0 <\alpha<1 $, $\beta>0$ and $ M $ is the notation to designate that the function to be derived involves the truncated Mittag-Leffler function with one parameter. The definition of this truncated $M$-fractional derivative type satisfies the properties of the integer-order calculus. We also present, the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the $M$-fractional heat equation and present a graphical analysis.