B-spline collocation method for solving Fredholm integral equations with discontinuous right-hand side

In this paper, we propose a method for approximating the solution of the linear Fredholm integral equation of the second kind which is defined on a closed contour $\Gamma $ in the complex plane. The right-hand side of the equation is a piecewise continuous function that is numerically defined on a finite set of points on $\Gamma $. To approximate the solution, we use a linear combination of B-spline functions and Heaviside step functions defined on $\Gamma $. We discuss both theoretical and practical aspects of the pointwise convergence of the method, including its performance in the vicinity of the points where discontinuities occur.