Discrete Direct Methods in the Fractional Calculus of Variations

TL;DR

This paper extends finite difference methods to fractional calculus of variations, discretizing fractional derivatives to transform the problem into a standard optimization task for numerical solutions.

Contribution

It generalizes direct methods to fractional derivatives using Grunwald-Letnikov approximation, enabling numerical solutions for fractional variational problems.

Findings

Transforms fractional variational problems into static optimization problems.

Provides a discretization approach for fractional derivatives in variational calculus.

Facilitates numerical approximation of solutions to fractional problems.

Abstract

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.