Fractional calculus of variations for a combined Caputo derivative

We generalize the fractional Caputo derivative to the fractional derivative CDγα,β, 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 CDγα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

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