The B-spline-Heaviside collocation method for solving Cauchy singular integral equations with piecewise Holder continuous coefficients
Maria Capcelea, Titu Capcelea
TL;DR
This work addresses the numerical solution of Cauchy singular integral equations on a closed smooth contour Γ when the data (coefficients and right-hand side) are piecewise Hölder continuous with finite jump discontinuities. It introduces a B-spline–Heaviside collocation method that augments periodic B-splines with explicit jump components to model discontinuities, enabling accurate representation of both smooth and non-smooth parts of the solution. The authors prove existence, uniqueness, and convergence of the scheme in the piecewise Hölder space $PH_\alpha(Γ,\mathcal{D})$, with explicit rates $O((2\pi/n_B)^{\alpha-\beta})$ (up to a logarithmic factor) for approximations in $PH_\beta(Γ,\mathcal{D})$, under standard invertibility assumptions on the operator. This approach reduces Gibbs phenomena, preserves Hölder regularity, and provides a rigorous, high-fidelity tool for singular integral equations with discontinuous data, with potential impact in physics and engineering applications.
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 Hölder 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 Hölder 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 Hölder spaces and derive explicit estimates for the convergence rate of the method.