Fractional calculus of variations for a combined Caputo derivative

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

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