Numerical Solution of Nonlinear Abel Integral Equations: An $ hp $-Version Collocation Approach

TL;DR

This paper introduces an $hp$-version collocation method using Jacobi polynomials for solving nonlinear Abel integral equations, providing error analysis and demonstrating efficiency through numerical experiments.

Contribution

It presents a novel $hp$-collocation approach with error estimates for nonlinear Abel integral equations, advancing numerical solution techniques.

Findings

Error estimates in $L^2$-norm are established.

Numerical experiments confirm the method's efficiency.

Existence and uniqueness of solutions are thoroughly analyzed.

Abstract

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of the nonlinear Abel integral equations is studied completely. An $hp$-version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in $L^2$-norm. The efficiency of the method is illustrated by some numerical experiments.