The Method of Matched Asymptotic Expansions

Abstract

In cases where a small parameter multiplies the highest derivative in a differential equation, there occurs a sharp change in the dependent variable in a certain region of the domain of the independent variable. In constructing a solution to the differential equation through uniformly-valid expansions, one characterizes the sharp changes by a magnified scale that is different from the scale characterizing the behavior of the dependent variable outside the “boundary-layer” regions. In other words, one represents the solution by two different asymptotic expansions using the independent variables x and x/ε say. Since they are different asymptotic representations of the same function, they should be related to each other in a rational manner in an overlapping region where both are valid (Friedrichs, 1955); this leads to the asymptotic matching principle (the latter makes the two representations completely determinate).