Analysis and spectral element solution of nonlinear integral equations of hammerstein type

Abstract

We employ the spectral element method with Gauss-Lobatto-Legendre collocation points to approximate nonlinear integral equations of Hammerstein type. Using the Banach Fixed Point Theorem, we establish sufficient conditions for the existence and uniqueness of solutions in the L2 norm, as well as the convergence of the proposed method, under a different aspect of the existing works in the literature, indicating that the numerical error decays exponentially provided that the kernel function be smooth enough. The iterative Picard process was used to approximate the nonlinear problem. Numerical experiments involving one- and two-dimensional nonlinear equations illustrate the effectiveness of this approach.