Fractional calculus of variations for 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.

This is a preprint of a paper whose final and definite form has been published in: Fract. Calc. Appl. Anal., Vol. 14, No 4 (2011), pp. 523--537; DOI: 10.2478/s13540-011-0032-6

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Cited by in corpus (12)

• Generalized fractional calculus with applications to the calculus of variations
• Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
• Numerical solution of fractional Sturm-Liouville equation in integral form
• Towards a combined fractional mechanics and quantization
• Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable Order
• Combined Fractional Variational Problems of Variable Order and Some Computational Aspects
• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
• Constrained fractional variational problems of variable order
• Fractional Herglotz variational problems of variable order
• Fractional Calculus on Time Scales
• Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels
• Series Representation of Jointly S$\alpha$S Distribution via Symmetric Covariations