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Discrete direct methods in the fractional calculus of variations. (English) Zbl 1350.49033
Summary: 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 Grünwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on the 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 the mesh points.

##### MSC:
 49M25 Discrete approximations in optimal control 26A33 Fractional derivatives and integrals 34A08 Fractional ordinary differential equations and fractional differential inclusions
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##### References:
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