Fractional calculus of variations for a combined Caputo derivative

TL;DR

This paper introduces a generalized fractional Caputo derivative combining left and right derivatives, formulates variational problems using it, and derives the corresponding Euler-Lagrange equations and conditions.

Contribution

It extends fractional calculus of variations by defining a new combined Caputo derivative and establishing fundamental variational principles for it.

Findings

Derived Euler-Lagrange equations for the new derivative

Formulated variational problems using the combined Caputo derivative

Established transversality conditions for the generalized fractional variational calculus

Abstract

We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional derivative of order $\beta$. The fractional variational problems under our consideration are formulated in terms of ${{^CD}^{\alpha,\beta}_{\gamma}}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.