A note on the proper choice of scales in two variable expansions

Abstract

By way of a class of weakly nonlinear differential equations it is shown that the simplified two scale procedure advocated by Greenlee and Snow in general leads to expansions of which already the second term is secular on the slow scale, while such a secularity can be avoided if the more elaborate choice of scales proposed by Kevorkian and Cole is used. The relevance of this fact for applications is pointed out by demonstrating by means of a numerical example that two terms of Kevorkian and Cole's expansion describe the long time behavior of the solution much more accurately than the same number of terms of Greenlee and Snow's expansion, while being at least as simple to derive