Efficient spectral-collocation methods for a class of linear Fredholm integro-differential equations on the half-line

Abstract

In this paper, an extension of the Legendre spectral collocation method has been proposed for the numerical solution of a class of linear Fredholm integro-differential equation on the half-line. The properties of mapped Legendre functions are first presented. These properties together with the Legendre–Gauss points are then utilized to reform the Fredholm integro-differential equation in semi-infinite interval into a singular equation in finite interval and to reduce it to the solution of a simple matrix equation. Besides, in order to show the efficiency and accuracy of the proposed method, some numerical examples are considered and solved through a survey of three approaches, namely: Exponential, rational and logarithmic Legendre functions collocation methods. Furthermore, a comparison of the results, shows that using exponential functions, leads to more accurate results and faster convergence.