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High Accuracy Method for Integral Equations with Discontinuous Kernels

Sheon-Young Kang, Israel Koltracht, George Rawitscher

TL;DR

This work develops a high-accuracy discretization for Fredholm integral equations of the second kind with kernels that are discontinuous or non-smooth along the main diagonal. By introducing (p1,p2)-semismooth kernels and employing a Gauss-type Clenshaw-Curtis quadrature based on Chebyshev polynomials, the authors obtain a discretization that achieves spectral accuracy when the kernel is infinitely differentiable away from the diagonal and remains robust to diagonal singularities. The method extends to a composite rule for long intervals and to kernels with diagonal singularities, with numerical results showing superior accuracy compared to standard Nyström-type methods. Applications to non-local Schrödinger equations demonstrate the approach’s practicality for integro-differential problems, including those with non-separable kernels, and indicate potential for fast, structured solvers when kernel properties (low-rank, Toeplitz-like) are present.

Abstract

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis Quadrature for Fredholm integral equations of the second kind, whose kernel is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when the kernel is infinitely differentiable away from the main diagonal, and is also applicable when the kernel is singular along the boundary, and at isolated points on the main diagonal. The corresponding composite rule is described. Application to integro-differential Schroedinger equations with non-local potentials is given.

High Accuracy Method for Integral Equations with Discontinuous Kernels

TL;DR

This work develops a high-accuracy discretization for Fredholm integral equations of the second kind with kernels that are discontinuous or non-smooth along the main diagonal. By introducing (p1,p2)-semismooth kernels and employing a Gauss-type Clenshaw-Curtis quadrature based on Chebyshev polynomials, the authors obtain a discretization that achieves spectral accuracy when the kernel is infinitely differentiable away from the diagonal and remains robust to diagonal singularities. The method extends to a composite rule for long intervals and to kernels with diagonal singularities, with numerical results showing superior accuracy compared to standard Nyström-type methods. Applications to non-local Schrödinger equations demonstrate the approach’s practicality for integro-differential problems, including those with non-separable kernels, and indicate potential for fast, structured solvers when kernel properties (low-rank, Toeplitz-like) are present.

Abstract

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis Quadrature for Fredholm integral equations of the second kind, whose kernel is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when the kernel is infinitely differentiable away from the main diagonal, and is also applicable when the kernel is singular along the boundary, and at isolated points on the main diagonal. The corresponding composite rule is described. Application to integro-differential Schroedinger equations with non-local potentials is given.

Paper Structure

This paper contains 8 sections, 6 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discretization of a Smooth Kernel
  3. Gauss Type Quadrature for a Semismooth Kernel
  4. The Composite Rule
  5. Kernels with Singularities on the Main Diagonal
  6. Numerical Examples
  7. Application to Non-Local Schroedinger Equations
  8. Summary and Conclusions

Key Result

lemma 1

Let ${\bf{A}}$ and ${\bf{B}}$ be $n \times n$ matrices and ${\bf{c}}=[c_{1},...,c_{n}]^{T}.$ Then $({\bf{A}} \circ {\bf{B}}) {\bf{c}}=diag({\bf{A}}diag(c_{1},...,c_{n}) {\bf{B}}^{T}),$ where ${\bf{A}} \circ {\bf{B}}$ denotes the Schur product of ${\bf{A}}$ and ${\bf{B}},$$({\bf{A}} \circ {\bf{B}})

Figures (4)

  • Figure 1: Comparison of numerical solutions of Example 1, $\lambda=0.1$.
  • Figure 2: Comparison of numerical solutions of Example 2.
  • Figure 3: Comparison of numerical solutions of Example 3.
  • Figure 4: Comparison of numerical solutions of Example 4.

Theorems & Definitions (8)

  • Definition 1
  • lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • theorem 1