An efficient method based on hybrid functions for Fredholm integral equation of the first kind with convergence analysis

In this article, a numerical direct method based on hybrid Block-Pulse functions and Legendre polynomials is proposed to solve Fredholm integral equation of the first kind. These hybrid basic functions are orthogonal and have compact support on [0, 1]. The properties of the hybrid functions are utilized to convert the integral equations into a system of linear algebraic equations. The main purpose of this paper is to obtain an error estimate and to show the convergence analysis for the numerical direct method based on hybrid functions under the L²-norm. The main characteristic of the method is low cost of setting up the equations without using any projection method. Finally, some numerical examples are provided to illustrate the validity and the efficiency of the proposed scheme especially for problems with non-sufficiently smooth solutions belonging to class C¹ or C².