New quadrature approach based on operational matrix for solving a class of fractional variational problems

Abstract

This manuscript presents a new numerical approach to approximate the solution of a class of fractional variational problems. The presented approach is consisting of using the shifted Legendre orthonormal polynomials as basis functions of the operational matrix of fractional derivatives (described in the Caputo sense) and that of fractional integrals (described in the sense of Riemann-Liouville) with the help of the Legendre-Gauss quadrature formula together with the Lagrange multipliers method for converting such fractional variational problems into easier problems that consist of solving an algebraic system in the unknown coefficients. The convergence of the proposed method is analyzed. Finally, in order to demonstrate the accuracy of the present method, some test problems are introduced with their approximate solutions and comparisons with other numerical approaches.