Fractional variational calculus in terms of a combined Caputo derivative

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.