The B-spline-Heaviside collocation method for solving Cauchy singular integral equations with piecewise Holder continuous coefficients

TL;DR

This paper introduces a novel numerical collocation method using B-spline and Heaviside functions to solve Cauchy singular integral equations with piecewise H"{o}lder continuous coefficients, ensuring convergence and explicit error estimates.

Contribution

The paper develops an efficient B-spline-Heaviside collocation scheme for singular integral equations with piecewise H"{o}lder coefficients, including convergence analysis and error bounds.

Findings

Method converges in piecewise H"{o}lder spaces

Explicit convergence rate estimates derived

Applicable to equations with jump discontinuities

Abstract

In this paper, we propose a numerical method for approximating the solution of a Cauchy singular integral equation defined on a closed, smooth contour in the complex plane. The coefficients and the right-hand side of the equation are piecewise H\"{o}lder continuous functions that may exhibit a finite number of jump discontinuities and are given numerically at a finite set of points on the contour. We introduce an efficient approximation scheme for piecewise H\"{o}lder continuous functions based on linear combinations of B-spline basis functions and Heaviside step functions, which serves as the foundation for the proposed collocation algorithm. We establish the convergence of the resulting sequence of approximations to the exact solution in the norm of piecewise H\"{o}lder spaces and derive explicit estimates for the convergence rate of the method.