Variational Approach for Fractional Partial Differential Equations

TL;DR

This paper introduces a new fractional variational derivative based on a modified Riemann-Liouville derivative, enabling the derivation of Euler-Lagrange equations for fractional PDEs, thus advancing fractional variational methods.

Contribution

It proposes a novel fractional variational derivative and establishes a fractional Euler-Lagrange principle applicable to fractional partial differential equations.

Findings

New fractional variational derivative defined using modified Riemann-Liouville derivative

Established fractional Euler-Lagrange equations for PDEs

Facilitates application of variational methods to fractional models

Abstract

Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational principles are proposed. However, it becomes difficult to apply the existing fractional variational theories to fractional differential models, due to the definitions of fractional variational derivatives which not only contain the left fractional derivatives but also appear right ones. In this paper, a new definition of fractional variational derivative is introduced by using a modified Riemann-Liouville derivative and the fractional Euler-Lagrange principle is established for fractional partial differential equations.