Convergence analysis of a Nyström-type method for a class of nonlinear integral equations with highly oscillatory kernels

Abstract

In this paper, we present a Nyström-type method for the numerical solution of a class of nonlinear highly oscillatory Volterra integral equations with a trigonometric kernel. The implementation of this method leads to a nonlinear system that involves oscillatory integrals, which is then addressed using a two-point generalized quadrature rule to construct a fully discretized scheme. The error analysis of the method, in terms of both frequency and step length, is also presented. It is demonstrated that the proposed method outperforms the one recently introduced in the literature. To validate the method, several numerical examples are provided, confirming its efficiency and accuracy.