Necessary Optimality Conditions for Fractional Difference Problems of the Calculus of Variations

TL;DR

This paper develops a discrete-time fractional calculus of variations, establishing necessary optimality conditions and demonstrating that solutions align with classical problems when the fractional order is an integer.

Contribution

It introduces a novel discrete-time fractional calculus of variations with new Euler-Lagrange and Legendre conditions, bridging fractional and classical variational problems.

Findings

Solutions match classical variational problems at integer orders

New Euler-Lagrange and Legendre conditions derived

Examples illustrate applicability of the fractional calculus

Abstract

We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.