The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo

TL;DR

This paper introduces a novel approach to fractional calculus of variations, deriving Euler-Lagrange equations that involve only Caputo derivatives, which are more suitable for physical applications with regular boundary conditions.

Contribution

It generalizes the DuBois-Reymond lemma to fractional calculus, enabling the derivation of Euler-Lagrange equations with solely Caputo derivatives, filling a gap in current formulations.

Findings

Derived Euler-Lagrange equations with only Caputo derivatives

Generalized the DuBois-Reymond lemma for fractional calculus

Facilitated applications in physics with regular boundary conditions

Abstract

Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler-Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois-Reymond lemma and showing how Euler-Lagrange equations involving only Caputo derivatives can be obtained.