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

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.