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A B-spline-Heaviside collocation method for solving Fredholm integral equations with piecewise Holder-continuous right-hand sides

Maria Capcelea, Titu Capcelea

TL;DR

This work addresses solving linear Fredholm integral equations of the second kind on closed contours in the complex plane where the right-hand side is piecewise Hölder with finite jumps. It introduces a collocation method that combines B-spline bases with Heaviside enrichment, yielding φ^H_n(t) = ∑ α_k B_{m,k}(t) + ∑ β_r H_{t^d_r}(t) and enforcing the collocation conditions at a set of quasi-uniform nodes that include the jump points. The authors prove convergence in the piecewise Hölder norm PH_α(Γ, D) with an error bound of the form O(h^α + τ(h)) and show that quadrature errors τ(h) can decay exponentially for analytic kernels, resulting in overall rates governed by α and s'. A numerical example on an astroid-shaped contour demonstrates rapid convergence and stable behavior, with errors decreasing as the B-spline node count grows and jumps being reproduced exactly by the enrichment. The approach provides a robust framework for Fredholm equations with discontinuous data on complex contours, with potential extensions to systems, weakly singular kernels, and adaptive schemes.

Abstract

This work presents a collocation method for solving linear Fredholm integral equations of the second kind defined on a closed contour in the complex plane. The right-hand side of the equation is a piecewise continuous function that may have a finite number of jump discontinuities and is known numerically at discrete points on the contour. The proposed approach employs a combination of B-spline functions and Heaviside step functions to ensure accurate approximation near discontinuity points and smooth behavior elsewhere on the contour. Convergence in the norm of piecewise Holder spaces is established, together with explicit error estimates. Numerical results illustrate the effectiveness and convergence rate of the method.

A B-spline-Heaviside collocation method for solving Fredholm integral equations with piecewise Holder-continuous right-hand sides

TL;DR

This work addresses solving linear Fredholm integral equations of the second kind on closed contours in the complex plane where the right-hand side is piecewise Hölder with finite jumps. It introduces a collocation method that combines B-spline bases with Heaviside enrichment, yielding φ^H_n(t) = ∑ α_k B_{m,k}(t) + ∑ β_r H_{t^d_r}(t) and enforcing the collocation conditions at a set of quasi-uniform nodes that include the jump points. The authors prove convergence in the piecewise Hölder norm PH_α(Γ, D) with an error bound of the form O(h^α + τ(h)) and show that quadrature errors τ(h) can decay exponentially for analytic kernels, resulting in overall rates governed by α and s'. A numerical example on an astroid-shaped contour demonstrates rapid convergence and stable behavior, with errors decreasing as the B-spline node count grows and jumps being reproduced exactly by the enrichment. The approach provides a robust framework for Fredholm equations with discontinuous data on complex contours, with potential extensions to systems, weakly singular kernels, and adaptive schemes.

Abstract

This work presents a collocation method for solving linear Fredholm integral equations of the second kind defined on a closed contour in the complex plane. The right-hand side of the equation is a piecewise continuous function that may have a finite number of jump discontinuities and is known numerically at discrete points on the contour. The proposed approach employs a combination of B-spline functions and Heaviside step functions to ensure accurate approximation near discontinuity points and smooth behavior elsewhere on the contour. Convergence in the norm of piecewise Holder spaces is established, together with explicit error estimates. Numerical results illustrate the effectiveness and convergence rate of the method.

Paper Structure

This paper contains 6 sections, 6 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction and problem formulation
  2. B-spline--Heaviside approximation and the computational scheme for the integral equation solution
  3. Operator representation of the approximation method
  4. Convergence analysis and rate of convergence
  5. A numerical example
  6. Conclusions

Key Result

Lemma 2.1

Any function $f\in PC\left(\Gamma,{\mathcal{D}}\right)$ can be represented uniquely in the form $f=f_C+f_H$, where $f_C$ is continuous on $\Gamma$, and $f_H$ is a finite linear combination of Heaviside step functions located at the discontinuity points. The coefficients of $f_H$ are given by $\beta_

Figures (5)

  • Figure 1: The contour and the notation used
  • Figure 1: The contour and discontinuity points
  • Figure 2: Graph of the function $\varphi(t)$
  • Figure 3: Approximation error for $n_B=160$
  • Figure 4: Approximation error for $n_B=320$

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • Proof 1