A discrete time method to the first variation of fractional order variational functionals

TL;DR

This paper introduces a discrete-time method for computing the first variation of fractional order variational functionals, enabling more effective solutions to fractional calculus of variations problems.

Contribution

It generalizes the classical variational approach to fractional derivatives using a discrete method based on Grunwald-Letnikov derivatives.

Findings

Provides a new discrete method for fractional variational problems.

Utilizes first order splines as variations with known fractional derivatives.

Facilitates straightforward approximations of fractional derivatives.

Abstract

The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grunwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.