Fractional Variational Calculus with Classical and Combined Caputo Derivatives

TL;DR

This paper extends classical calculus of variations to fractional derivatives, deriving Euler-Lagrange equations and conditions for problems involving combined Caputo fractional derivatives.

Contribution

It introduces a fractional variational calculus framework with combined Caputo derivatives, including derivation of Euler-Lagrange equations and boundary conditions.

Findings

Derived Euler-Lagrange equations for fractional variational problems

Established transversality conditions for boundary problems

Extended classical calculus of variations to fractional derivatives

Abstract

We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange equations to the basic and isoperimetric problems are proved, as well as transversality conditions.