A finite difference method for a class of two-point boundary value problems over a semi-infinite range

摘要

We examine the three-point finite difference discretization of the two-point boundary value problem: − y″ + f(x)y = g(x), 0 ⩽ x $̌∞, y(0) = y(∞) = 0. Our purpose here is to study the resulting infinite tridiagonal linear system. We first show that, under suitable assumptions, the infinite linear system possesses a unique solution provided inf f(x) = ϵ > 0. We show that the l∞-norm of the discretization error is bounded by Ch2 where h is the step-size and C is a constant independent of h. We assume that ∣ g(x) ∣ ⩽ M e−αχ, for suitable postive constants M and α. We then obtain a bound for the numerical solution yn, at xn, and the bound involves only h, ϵ, M a α. From this bound we conclude that yn → 0 as n → ∞. An interesting application of this bound is that we can obtain an a priori estimate for n so that for this n the numerical solution yn at xn is ‘almost zero’ in the se that ∣ yn ∣ ⩽ δ for a preassigned δ. In other words, this bound provides an a priori estimate for n for truncating the infinite linear system (or equivalently, for truncating the semi-infinite range at xn) so that yn is within the tolerance δ.