# Nabla Euler -Lagrange equations in discrete fractional variational calculus within Riemann and Caputo

## TL;DR

This paper derives various fractional difference Euler-Lagrange equations in Riemann and Caputo frameworks using fractional difference calculus and integration by parts, with an illustrative example.

## Contribution

It introduces new fractional difference Euler-Lagrange equations in Riemann and Caputo contexts using novel integration by parts formulas.

## Key findings

- Derived fractional difference Euler-Lagrange equations in Riemann and Caputo forms.
- Presented an example illustrating the application of the derived equations.
- Extended fractional variational calculus within discrete settings.

## Abstract

Different fractional difference types of Euler-Lagrange equations are obtained within Riemann and Caputo by making use of different versions of integration by part forumlas in fractional difference calculus. An example is presented to illustrate part of the results.