A B-spline-Heaviside collocation method for solving Fredholm integral equations with piecewise Holder-continuous right-hand sides

TL;DR

This paper introduces a novel collocation method combining B-splines and Heaviside functions to solve Fredholm integral equations with piecewise discontinuous right-hand sides on complex contours, ensuring accurate approximation near discontinuities.

Contribution

The paper develops a new B-spline-Heaviside collocation approach with proven convergence and explicit error estimates for solving Fredholm equations with discontinuous data.

Findings

Method achieves high accuracy near discontinuities.

Convergence proven in piecewise Holder spaces.

Numerical results confirm effectiveness and convergence rate.

Abstract

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.