Two Numerical Approaches for Nonlinear Weakly Singular Integral Equations

TL;DR

This paper introduces two numerical methods for solving nonlinear weakly singular integral equations, extending singularity subtraction techniques and demonstrating their convergence and efficiency through numerical experiments.

Contribution

It generalizes singularity subtraction to nonlinear equations and proposes a more efficient linearization-based approach with proven convergence.

Findings

The classical approach converges under mild conditions.

The new linearization approach is more efficient.

Numerical experiments confirm the efficiency of the new method.

Abstract

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as two numerical experiments confirm.