Numerical solution of fractional delay Volterra integro-differential equations by Bernstein polynomials

We apply spectral collocation and Galerkin methods with shifted orthonormal Bernstein polynomials (SOBPs) to a class of fractional delay Volterra integro-differential equations (FDVIDEs). To this end, we first obtain the SOBPs operational matrix for fractional derivatives in the Caputo sense and convert the original equation to a system of algebraic equations. In addition, the convergence analysis of the method is presented. Some examples are provided to investigate the efficiency of the proposed methods. In each example, the Galerkin method and the collocation method are compared with other methods in terms of accuracy and CPU time. The numerical results show the efficiency and validity of the method as well as the suitability of the error bound. They also show that spectral methods yield acceptable approximate solutions even on long intervals.