This work presents a collocation method for solving linear Fredholm integral equations of the second kind defined on a closed contour in the complex plane. The right-hand side of the equation is a piecewise continuous function that may have a finite number of jump discontinuities and is known numerically at discrete points on the contour. The proposed approach employs a combination of B-spline functions and Heaviside step functions to ensure accurate approximation near discontinuity points and smooth behavior elsewhere on the contour. Convergence in the norm of piecewise Holder spaces is established, together with explicit error estimates. Numerical results illustrate the effectiveness and convergence rate of the method.

翻译：本文提出了一种求解复平面上闭合围道上定义的第二类线性Fredholm积分方程的配点法。该方程的右端项为分段连续函数，可能具有有限个跳跃间断点，且仅在围道上的离散点处数值已知。所提出的方法结合了B样条函数与海维赛德阶跃函数，以确保在间断点附近实现精确逼近，并在围道其他区域保持光滑特性。本文建立了该方法在分段Hölder空间范数下的收敛性，并给出了显式误差估计。数值结果验证了该方法的有效性与收敛速率。