Direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method

This paper presents an adaptive Legendre–Gauss pseudospectral method for solving a type of constrained fractional variational problems (FVPs). The fractional derivative is defined in the Caputo sense. In the presented method, by dividing the domain of the problem into a uniform mesh the given FVP reduces to a nonlinear mathematical programming problem, and there is no need to solve the complicated fractional Euler–Lagrange equations. The method developed in this paper adjusts both the mesh spacing and the number of collocation points on each subinterval in order to improve the accuracy. The method is easy to implement and yields very accurate results. Some error estimates and convergence properties of the method are discussed. Numerical examples are included to confirm the efficiency and convergence of the proposed method.

Fractional variational problem Caputo derivative Adaptive pseudospectral method Shifted Legendre–Gauss points Nonlinear programming