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

References in corpus (13)

• 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
• Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
• Fractional Variational Calculus with Classical and Combined Caputo Derivatives
• A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String
• Fractional Order Optimal Control Problems with Free Terminal Time
• Nonconservative Lagrangian Mechanics: A generalized function approach
• Extending Bauer's corollary to fractional derivatives
• Time scales: from Nabla calculus to Delta calculus and vice versa via duality
• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
• Stationarity-conservation laws for certain linear fractional differential equations

Cited by in corpus (9)

• Existence and Uniqueness of Solution for a Fractional Riemann-Liouville Initial Value Problem on Time Scales
• Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales
• On a Fractional Oscillator Equation with Natural Boundary Conditions
• Combined Fractional Variational Problems of Variable Order and Some Computational Aspects
• Symmetric duality for left and right Riemann-Liouville and Caputo fractional differences
• Time-Fractional Optimal Control of Initial Value Problems on Time Scales
• A collocation method of lines for two-sided space-fractional advection-diffusion equations with variable coefficients
• 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