$ hp $-version collocation method for a class of nonlinear Volterra integral equations of the first kind

TL;DR

This paper introduces an $hp$-version collocation method using Legendre polynomials for solving nonlinear Volterra integral equations of the first kind, with proven convergence and verified by numerical experiments.

Contribution

It develops a new $hp$-version collocation approach with rigorous convergence analysis for nonlinear Volterra equations of the first kind.

Findings

Convergence of the method is theoretically established.

Error estimates under the $L^2$-norm are provided.

Numerical experiments confirm the theoretical results.

Abstract

In this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of $hp$-version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the $L^2$-norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results.