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The linear two-point boundary-value problem on an infinite interval

A numerical method, using finite-difference approximations to the second-order differential equation, is given which tests the suitability of the finite point chosen to represent infinity before computing the numerical solution. The theory is illustrated with examples and suggestions for further applications of the method are presented. 1. Introduction. Consider the linear two-point boundary-value problem on an infinite interval (1) Ly = -y" + p(x)y' + q(x)y= f(x) with y(a) a, y(Acm) = 0. More general boundary conditions will be considered later. A typical method is to apply the second boundary condition at a finite point or at several finite points and observe the variation of the solutions (Fox (2)). The basic objective is to obtain an accurate numerical solution in a relatively small finite interval starting at a. The proposed method, involving finite-difference approxima- tions, finds a solution over such an interval and also produces values b and ,3 such that the solution of (1) on (a, b) is given by the solution of