The Laguerre Collocation Method for Third Kind Integral Equations on Unbounded Domains

Abstract

AbstractThe aim of this paper is to approximate the solution of a class of integral equations of the third kind on an unbounded domain. For computing such approximation, the collocation method based on the generalized Laguerre abscissas is considered. In this method, the unknown function is interpolated at the nodal points ${\lbrace t_i\rbrace _{i=1}^{n+1}}$, where ${\lbrace t_i\rbrace _{i=1}^{n}}$ are the zeros of generalized Laguerre polynomials and ${t_{n+1}=4n}$. Then, the given equation is transformed to the Fredholm integral equation of the second kind. In the sequel, according to the integration interval, we apply the Gauss–Laguerre collocation method on the interval ${[0,\infty )}$ by using the given nodal points. Therefore, the solution of the third kind integral equation is reduced to the solution of a system of linear equations. Convergence analysis of the method in some Sobolev-type space is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique.