Duality for the left and right fractional derivatives

Abstract

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators.

This is a preprint of a paper whose final and definite form will appear in the international journal Signal Processing, ISSN 0165-1684. Paper submitted Dec/2013; revised Apr, July and Sept 2014; accepted for publication 18/Sept/2014

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• A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations
• Discrete-Time Fractional Variational Problems
• Fractional conservation laws in optimal control theory
• Calculus of variations with fractional derivatives and fractional integrals
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Cited by in corpus (2)

• The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only
• Pulse solutions of the fractional effective models of the Fermi-Pasta-Ulam lattice with long-range interactions