The aim of this paper is to investigate, from the numerical point of view, the Jacobi polynomials to solve fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). A direct numerical method for solving a general class of FVPs and FOCPs is presented. The fractional derivative in FVPs is in the Caputo sense and in FOCPs is in the Riemann–Liouville sense. The Rayleigh–Ritz method is introduced for the numerical solution of FVPs containing left or right Caputo fractional derivatives. Rayleigh–Ritz method is one of the well-known direct methods used for the solution of variational problems. In this technique, at first, we expand the unknown function in terms of the modified Jacobi polynomials and then we derive a compact form of fractional derivative of the unknown function in terms of the Jacobi polynomials. Examples indicate that the new technique has high accuracy and is very efficient to implement.

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We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By some examples, we show the convergence of such procedure, comparing the exact solution with numerical approximations.

The aim of this paper is to generalize the Euler-Lagrange equation obtained by Almeida et al., where fractional variational problems for Lagrangians, depending on fractional operators and depending on indefinite integrals, were studied. The new problem that we address here is for cost functionals, where the interval of integration is not the whole domain of the admissible functions, but a proper subset of it. Furthermore, we present a numerical method, based on Jacobi polynomials for solving this problem.