Fractional Euler-Lagrange differential equations via Caputo derivatives

TL;DR

This paper reviews recent advances in fractional variational calculus, deriving Euler-Lagrange equations involving Caputo derivatives for various boundary conditions and constraints, expanding the theoretical framework of fractional calculus.

Contribution

It introduces necessary optimality conditions for functionals with Caputo derivatives, covering fixed/free boundaries and integral constraints, advancing fractional variational calculus.

Findings

Derived Euler-Lagrange equations with Caputo derivatives

Addressed boundary conditions and integral constraints

Enhanced theoretical understanding of fractional variational problems

Abstract

We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.