Combined Fractional Variational Problems of Variable Order and Some Computational Aspects

Abstract

We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun.

This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 15-Feb-2017; Revised 17-Apr-2017; Accepted 21-Apr-2017

References in corpus (6)

• The Variable-Order Fractional Calculus of Variations
• Caputo derivatives of fractional variable order: numerical approximations
• Calculus of variations with fractional derivatives and fractional integrals
• Duality for the left and right fractional derivatives
• Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable Order
• Constrained fractional variational problems of variable order

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• The Variable-Order Fractional Calculus of Variations
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• Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels