A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String

TL;DR

This paper develops a fractional calculus of variations framework for multiple integrals, extending classical theorems and equations to fractional derivatives, with an application to modeling vibrating strings.

Contribution

It introduces a novel fractional calculus of variations for multiple integrals using Riemann-Liouville derivatives, including fractional Green, Gauss, Euler-Lagrange, and boundary conditions.

Findings

Derived fractional versions of Green and Gauss theorems

Formulated fractional Euler-Lagrange equations

Applied theory to fractional vibrating string model

Abstract

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.