I have read several books on perturbation methods, and this book absolutely the best. Perturbation methods are used in problems where the equations (algebraic, ODEs, or PDEs) involve a small parameter, or that evolve on multiple time scales that differ by orders of magnitude. The basic idea is to build an approximate solution to the equation based on a series expansion involving the small parameter. Usually two or more series expansions are built and then combined to yield a solution to the equation that incorporates dynamics on both the slow and fast time scales (for example).The first two chapters introduce the basic concepts of asymptotic expansions. Several informative, though relatively simple, examples are developed in detail to illustrate the techniques. The third chapter introduces techniques for solving problems involving multiple scales. Chapters four and five discuss the WKB method and homogenization techniques. The final chapter shows how perturbation methods can be used in conjunction with techniques from dynamical systems analysis, especially stability analysis and bifurcation theory.
The contents of the book are extremely well-chosen and pertinent. Many of the problems and techniques are based on realistic physical systems, and represent the types of problems an applied mathematician is likely to encounter in practice. The book's prose is clear and informative. Only occasionally did I feel that the author had explained something poorly or left out an important piece of information.
There are of course a couple of problems with the book. First, most of the exercises are significantly more difficult than the examples, and no hints or solutions are given. This can be a draw-back if you are using the book for self-study. There are a number of typos and errors in the text, though the author's website has a list of errata. Finally, solving perturbation problems in practice often involves subtle tricks, and unfortunately, this book does not discuss or give examples of many of those tricks.
Overall, this is definitely the best introductory book on perturbation methods. You should definitely consider this book if you work in applied math or with differential equations.
