Discrete Direct Methods in the Fractional Calculus of Variations

Abstract

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.

This work was partially presented 16-May-2012 by Shakoor Pooseh, at FDA'2012, who received a 'Best Oral Presentation Award'. Submitted 26-Aug-2012; revised 25-Jan-2013; accepted 29-Jan-2013; for publication in Computers and Mathematics with Applications

References in corpus (3)

• Fractional Noether's theorem in the Riesz-Caputo sense
• Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics
• Leitmann's direct method for fractional optimization problems

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• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
• Uniform approximation of fractional derivatives and integrals with application to fractional differential equations
• Generalized Fractional Order Derivatives, Its Properties and Applications
• A Discretization Method to Solve Fractional Variational Problems with Dependence on Hadamard Derivatives
• Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control