An efficient and stable Lagrangian matrix approach to Abel integral and integro-differential equations

Abstract

This article studies Abel integral equations (AIEs) and singular integro-differential equations (SIDEs) and aims to develop two numerical schemes for them. It also emphasises on the comparative analysis of both AIEs & SIDEs which is based on mainly two process namely Gauss-Legendre roots as collocation node points and random node points over the domain [0,1]. For generating interpolating basis functions (IBF), we used Lagrangian interpolating polynomial and for orthonormal Lagrangian basis functions (OLBF), we used Gram-Schmidt orthogonalization algorithm, respectively. Firstly, we introduced the function approximation by using generated IBF and OLBF, then established the error bounds of these approximations. The constructed approximations by both the schemes convert the AIEs and SIDEs into the system of algebraic equations. We have also established error bounds, stability and convergence analysis of the proposed schemes by considering several mild mathematical conditions. Moreover, the stability of schemes is also established numerically. Finally, the test functions with the support of graphs clearly show the reliability and computational efficiency of the proposed methods.