Extended Mittag-Leffler Function and truncated $\nu$-fractional derivatives

TL;DR

This paper introduces a new class of fractional derivatives based on an extended Mittag-Leffler function, exploring their fundamental properties and relationships with fractional integrals.

Contribution

It presents the definition and analysis of $ u$-fractional derivatives using a 4-parameter extended Mittag-Leffler function, including key calculus properties and integral relationships.

Findings

The $ u$-fractional derivative satisfies chain, product, Rolle's, and mean-value theorems.

A generalized inverse property and fundamental theorem of calculus are established.

Connections between $ u$-fractional derivatives and fractional integrals are demonstrated.

Abstract

The main objective of this article is to present $\nu$-fractional derivative $\mu$-differentiable functions by considering 4-parameters extended Mittag-Leffler function (MLF). We investigate that the new $\nu$-fractional derivative satisfies various properties of order calculus such as chain rule, product rule, Rolle's and mean-value theorems for $\mu$-differentiable function and its extension. Moreover, we define the generalized form of inverse property and the fundamental theorem of calculus and the mean-value theorem for integrals. Also, we establish a relationship with fractional integral through truncated $\nu$-fractional integral.