CIX. A Technique for rendering approximate solutions to physical problems uniformly valid

Abstract

Summary A method is described for treating some of the characteristically non-linear problems of physics, in particular those involving a non-linear partial differential equation for which an approximate linearization is permissible everywhere except in a limited region, such as the neighbourhood of (§ 5) a singular characteristic of the approximate solution, or of (§ 6) the point at infinity, where the approximation is valueless. The method involves a transformation of an independent variable, which is determined progressively with successive approximations to the solution : only one step being necessary if a first approximation valid uniformly (even in the critical region) is to be obtained. The method is most easily understood in its application to simple first order ordinary differential equations, which are studied in detail in §§2 and 3 as a preparation for the extension to more complicated problems in §§4, 5 and 6. Physically, the longest section, §6, concerns the “spread” of a progressive wave at infinity, an important and essentially non-linear process.