On the $\psi$-Hilfer fractional derivative

TL;DR

This paper introduces the $\psi$-Hilfer fractional derivative, explores its properties, and demonstrates its applications through examples involving Mittag-Leffler functions and a broad class of integrals and derivatives.

Contribution

The paper presents a novel fractional derivative with respect to another function, expanding fractional calculus with new properties, results, and applications.

Findings

Properties and important results of the $\\psi$-Hilfer fractional derivative

Examples involving Mittag-Leffler functions

A wide class of integrals and derivatives using the new derivative

Abstract

In this paper we introduce a new fractional derivative with respect to another function the so-called $\psi$-Hilfer fractional derivative. We discuss some properties and important results of the fractional calculus. In this sense, we present some uniformly convergent sequence of function results and examples involving the Mittag-Leffler function with one parameter. Finally, we present a wide class of integrals and fractional derivatives, by means of the fractional integral with respect to another function and the $\psi$-Hilfer fractional derivative.