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 Holder continuous functions that may have 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 Holder continuous functions based on linear combinations of B-spline functions and Heaviside step functions, which serves as the foundation for the proposed collocation algorithm. We then establish the convergence of the sequence of the constructed approximations to the exact solution of the equation in the norm of piecewise Holder spaces and derive estimates for the convergence rate of the method.

翻译：本文提出了一种数值方法，用于逼近定义在复平面上一条封闭光滑曲线上的Cauchy奇异积分方程的解。该方程的系数和右端项为分段Hölder连续函数，可能包含有限个跳跃间断点，且这些函数在曲线上有限个点处以数值形式给出。我们引入了一种基于B样条函数与Heaviside阶跃函数线性组合的高效分段Hölder连续函数逼近方案，该方案构成了所提配置算法的基础。随后，我们在分段Hölder空间的范数下，证明了所构造的逼近序列收敛于方程的精确解，并推导了该方法的收敛速率估计。