Global polynomial approximation of nonlinear functional Volterra-Fredholm equations using the Legendre-Galerkin scheme

In this work, we propose a numerical method for solving nonlinear functional mixed Volterra-Fredholm integral equations. We have developed a method using the Galerkin approach, where Legendre polynomials are chosen as basis functions. This framework provides global polynomial approximations and achieves spectral accuracy for sufficiently smooth solutions. To further enhance accuracy, we apply an iterated Galerkin method that achieves superconvergence without increasing the dimension of the polynomial subspace. Convergence rates in both L 2 and uniform norms are provided under suitable regularity assumptions. Finally, numerical examples demonstrate the efficiency and superior performance of the proposed methods and results are compared with the existing scheme performance.