On fractional derivatives with exponential kernel and their discrete versions

TL;DR

This paper introduces new fractional derivatives and integrals with exponential kernels, derives related calculus formulas, and explores their discrete versions, providing theoretical foundations and an example.

Contribution

It presents novel definitions of fractional derivatives with exponential kernels and their discrete analogs, along with integration by parts and Euler-Lagrange equations.

Findings

Derived integration by parts formula for exponential kernel derivatives

Established Euler-Lagrange equations for the new fractional derivatives

Formulated discrete counterparts of the continuous fractional calculus results

Abstract

In this manuscript we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. Then, we provide the integration by parts formula and we use $Q-$operator to confirm our results. The related Euler-Lagrange equations were obtained and one example is analyzed. Moreover, we formulate and discus the discrete counterparts of our results.