Fractional Calculus of Variations of Several Independent Variables

TL;DR

This paper develops multidimensional fractional calculus tools, including integration by parts, to derive optimality conditions, a Noether's theorem, and uniqueness results for fractional variational problems involving multiple variables.

Contribution

It introduces new multidimensional fractional integration by parts formulas and applies them to establish optimality conditions and fundamental theorems in fractional calculus of variations.

Findings

Derived multidimensional fractional integration by parts formulas.

Established optimality conditions for fractional variational problems.

Proved a generalized fractional Noether's theorem and uniqueness results.

Abstract

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given.