Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

TL;DR

This paper introduces a new nonlocal fractional derivative with Mittag-Leffler kernel, develops its integration by parts formula, and derives related Euler-Lagrange equations with an illustrative example.

Contribution

It defines a novel nonlocal fractional derivative with Mittag-Leffler kernel and establishes its integration by parts and Euler-Lagrange equations.

Findings

Derived the right fractional derivative and integral with Mittag-Leffler kernel.

Established the integration by parts formula for the new derivative.

Presented an example illustrating the application of the theory.

Abstract

In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the recently introduced nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the $Q-$operator to confirm our results. The corresponding Euler-Lagrange equations are obtained and one illustrative example is discussed.