The Fractional Calculus of Variations

Abstract

In this chapter, we consider general fractional problems of the calculus of variations, where the Lagrangian depends on a combined Caputo fractional derivative of variable fractional order \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}\) given as a combination of the left and the right Caputo fractional derivatives of orders, respectively, \({\alpha (\cdot ,\cdot )}\) and \({\beta (\cdot ,\cdot )}\). More specifically, here we study some problems of the calculus of variations with integrands depending on the independent variable t, an arbitrary function x and a fractional derivative \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}x\).