Fractional variational calculus in terms of a combined Caputo derivative

Abstract

We generalize the fractional Caputo derivative to the fractional derivative ${}^C{D}_{\gamma }^{\alpha ,\beta }$, 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 ${}^C{D}_{\gamma }^{\alpha ,\beta }$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

Submitted 30-April-2010; accepted 23-June-2010; for presentation at the IFAC Workshop on Fractional Derivative and Applications (IFAC FDA2010) to be held in University of Extremadura, Badajoz, Spain, October 18-20, 2010

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• Discrete-Time Fractional Variational Problems
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Cited by in corpus (5)

• The Variable-Order Fractional Calculus of Variations
• Fractional Variational Calculus with Classical and Combined Caputo Derivatives
• Composition Functionals in Fractional Calculus of Variations
• Combined Delta-Nabla Sum Operator in Discrete Fractional Calculus
• Fractional Calculus on Time Scales