TABLE OF CONTENTS Perturbation methods are used to provide approximate solutions of nonlinear problems characterized by the presence of a small parameter. These approximations represent an asymptotic series expansion of the true solution truncated typically after a few terms. A condition for the successful application of these methods is that the reduced problem, obtained by the leading order analysis, is amenable to analytic solution.
We first discuss the simple case of regular perturbation expansions and then proceed to analyze the more complex singular perturbation expansions. For the latter, there is no unique procedure and various methods can be applied depending on the specific problem at hand. We describe a few of these, including the method of strained coordinates, matched asymptotic expansions, the WKB method, and the multiple scale method. Several examples are discussed to highlight the potential and limitations of each technique. However, let us first examine the main features of perturbation methods in general and introduce some concepts related to asymptotic expansions.
Equations deriving from mathematical models of physical phenomena usually contain various parameters arising from elementary processes which together constitute the specific phenomenon being studied, such as mass or heat transport, chemical reactions, convective motion, and so on. In most cases such equations are nonlinear and can only be solved numerically. However, it is sometimes possible to obtain analytical approximations of the solution, valid in the entire domain of interest or only in some portion of it, which can be useful in investigating the model at...
