Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

TL;DR

This paper extends fractional variational calculus by allowing different bounds for derivatives and integrals, deriving new Euler-Lagrange equations, and providing approximation methods for these equations.

Contribution

It introduces a generalized framework for fractional variational problems with non-coinciding bounds and derives new Euler-Lagrange equations for this setting.

Findings

Derived new Euler-Lagrange equations for fractional variational problems.

Proposed approximation methods for fractional derivatives in the Lagrangian.

Validated the approximations in a weak sense.

Abstract

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense.