A Regularized B-Spline-Heaviside Collocation Method for Cauchy Singular Integral Equations with Piecewise H\"{o}lder Solutions

Abstract

We develop a B-spline-Heaviside collocation method for Cauchy singular integral equations on a smooth closed $C^2$ contour when the exact solution is piecewise H\"{o}lder continuous with finitely many prescribed jumps. Since the Cauchy singular integral of a discontinuous function generally has logarithmic terms at the jump points, we study $M=cI+dS+K:X_\alpha\to Y_\alpha$, $X_\alpha=PH^\alpha(\Gamma,D)$, where $Y_\alpha$ is a logarithmically enlarged piecewise H\"{o}lder space. The discontinuous component is represented by a nonredundant system of normalized relative Heaviside functions adapted to the closed contour. Collocation uses point evaluations at spline nodes separated from the jump set together with logarithmic-coefficient functionals at the jumps. Assuming continuous stability of $M:X_\beta\to Y_\beta$, mesh-uniform scaled discrete stability of the regularized collocation operators, and a scaled consistency estimate for exact-jump approximants, we prove existence and uniqueness for sufficiently fine meshes and the error bound \[ \|\varphi-\varphi^H_{n_B}\|_{X_\beta} \le C h_B^{\alpha-\beta}\|\varphi\|_{X_\alpha}, \qquad 0<\beta<\alpha<1. \] We give a matrix realization of the regularized scheme, including the logarithmic and point-collocation blocks, singularity-subtracted evaluation of the Cauchy action on splines, principal-value-safe arc formulas for the Heaviside terms, and an implementation algorithm. An abstract perturbation result shows that the same rate is preserved under sufficiently accurate quadrature. Numerical experiments with arcwise errors and finite-dimensional stability and consistency indicators support the theoretical assumptions over the tested meshes.