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A global approximation method for second-kind nonlinear integral equations

Luisa Fermo, Anna Lucia Laguardia, Concetta Laurita, Maria Grazia Russo

TL;DR

This work introduces a Nyström-type framework for nonlinear second-kind integral equations with smooth or weakly singular kernels, combining Gauss-Legendre quadrature and Legendre-node product rules to discretize the integral operators. It establishes stability and convergence in the uniform norm, linking error decay to best polynomial approximation in the relevant function spaces, and provides precise rates tied to the smoothness of data. A key contribution is the rigorous treatment of nonlinearities via fixed-point arguments, including solvability, regularity, and error estimates, plus a practical boundary-integral application to the interior Neumann problem for Laplace’s equation with nonlinear boundary conditions. The method is validated through extensive numerical experiments on Hammerstein-type problems and boundary-integral reformulations, demonstrating fast convergence, high accuracy, and applicability to nonlinear BIEs in two-dimensional domains.

Abstract

A global approximation method of Nyström type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered.

A global approximation method for second-kind nonlinear integral equations

TL;DR

This work introduces a Nyström-type framework for nonlinear second-kind integral equations with smooth or weakly singular kernels, combining Gauss-Legendre quadrature and Legendre-node product rules to discretize the integral operators. It establishes stability and convergence in the uniform norm, linking error decay to best polynomial approximation in the relevant function spaces, and provides precise rates tied to the smoothness of data. A key contribution is the rigorous treatment of nonlinearities via fixed-point arguments, including solvability, regularity, and error estimates, plus a practical boundary-integral application to the interior Neumann problem for Laplace’s equation with nonlinear boundary conditions. The method is validated through extensive numerical experiments on Hammerstein-type problems and boundary-integral reformulations, demonstrating fast convergence, high accuracy, and applicability to nonlinear BIEs in two-dimensional domains.

Abstract

A global approximation method of Nyström type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered.
Paper Structure (11 sections, 10 theorems, 149 equations, 1 figure, 6 tables)

This paper contains 11 sections, 10 theorems, 149 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Function spaces and best polynomial approximation
  3. The solvability of equation (\ref{['EqInFr']})
  4. The Nyström method for the case of continuous kernels
  5. Numerical tests
  6. The Nyström method for the case of weakly singular kernels
  7. Numerical experiments
  8. An application to Boundary Integral Equations
  9. Numerical experiments
  10. Proofs
  11. Declarations

Key Result

Theorem 2.1

\newlabelteo:BestApprx Let $h: \Omega \rightarrow \mathbb{R},$ with $\Omega$ open subset of $\mathbb{R}^2$ and $\sigma: (-1,1) \rightarrow \mathbb{R}^2$ such that $Im( \sigma) \subseteq \Omega.$ Assume that $h \in \mathbf{W}^r( \Omega)$ and $\sigma(x)= (x, f(x))$ with $f \in W^r,$ then where $\mathcal{C}=\mathcal{C}(r)$ is a positive constant independent of $h$ and $f$, $B_r$ is the $r$-th Bell n

Figures (1)

  • Figure 6.1: The boundary $\Gamma$ of the domain $D$ in Example \ref{['ExampleBIE3']} \newlabelFig:amoeba

Theorems & Definitions (31)

  • Theorem 2.1
  • Remark 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 2
  • Theorem 4.1
  • Remark 3
  • Theorem 4.2
  • ...and 21 more