Method of Multiple Scales

Abstract

In the method of matched asymptotic expansions (Chapter 5), the solution is constructed in different regions that are then patched together to form a composite expansion. The method of multiple scales¹, on the other hand, starts with a generalized version of a composite expansion. This involves separate coordinates for each region, which are considered to be independent of one another. Consequently, the given equation is transformed into a partial differential equation even if it was an ordinary differential equation to begin with. On the other hand, the method of multiple scales may also be viewed as a generalization of the method of strained parameters in that the relevant scales are given implicitly rather than explicitly in terms of the original variables (Kevorkian, 1966).

This name is a bit awkward because the method of matched asymptotic expansions also uses multiple scales though each scale is effectively confined to a certain region in the latter method.