Boundary value problems in infinite intervals

Abstract

The treatment of differential equations on infinite intervals (assuming that solutions exist on the entire interval) depends crucially on the asymptotic behaviour of the solutions at infinity. Often, power series with logarithmic coefficients solve the differential equations formally and may be proven to be asymptotic to true solutions.

In this paper it is suggested to use numerical integration over a finite portion of the interval, and to switch to the asymptotic approximation when the latter is sufficiently accurate. In order to integrate numerically over large intervals a step roughly proportional to the value of the independent variable may often be used. One-step methods with step control are well suited for this purpose.

The shooting method is applied in order to solve boundary value problems. If numerical integration is done in the stable direction, the secant method is an efficient tool for determining missing initial conditions. Bisection is preferrably used when integrating in an instable direction.

These methods are illustrated with a practical example.