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.