Introduction

Why study perturbation techniques? The answer lies in the fact that we can simplify the problem significantly if we study the structure of the equation. In some cases, analytical solutions may be obtained, which eliminates any need for computational work. More commonly, the application of these techniques results in a problem that may be solved numerically much easier than the original problem.

Perturbation methods had been known since the days of the ancient astronomers to predict the trajectories of planets subject to small disturbances. The modern concept of regular and singular perturbation problems was developed in the 1960s and refined over the 1970s in fluid mechanics to investigate the high Reynolds number flow past a flat plate, the boundary layer. The function of a boundary layer is to bring the velocity to zero on a solid body. Because the boundary layer is very thin, of O(Re −1/2), the velocity will vary rapidly across the layer, making numerical solutions very difficult, if not impossible. In the intervening years, boundary layer–type behavior, the rapid variation of any quantity over a short distance or period of time, has been used in a wide variety of disciplines to describe such behavior.

There are a number of books on this subject, and these appear in the bibliography; they include Kevorkian and Cole (1981, 1996), VanDyke (1975), Nayfeh (1973), Bellman (1964), Bender and Orszag (1999), Holmes (1995), Hinchcliffe (2003), and Howison (2005). The book by VanDyke (1975) is specifically oriented toward fluid mechanics, whereas the others are more general treatments.