This book is a revised and updated version, including a substantial portion of new material, of the authors' text Perturbation Methods in Applied Mathematics (Springer-Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject.

Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is ``close'' to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, , in the governing system (consisting of differential equations and boundary conditions) so that for the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of .

In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.

In a singular perturbation problem, also called a layer-type problem, there are one or more thin layers at the boundary or in the interior of the domain where the above procedure fails. Often, this failure is due to the fact that multiplies the highest derivative in the differential equation; therefore the leading approximation obeys a lower-order equation that cannot satisfy all the prescribed boundary conditions. Layer-type problems for ordinary differential equations are discussed in Chapter 2 and for partial differential equations in Chapter 3.

Regular perturbations also fail if the govening system is to be solved over an infinite domain and contains small terms with a cumulative effect. The two principal techniques for deriving asymptotic solutions that remain valid in the far field are multiple scale expansions and the method of averaging. These techniques are discussed in Chapters 4 and 5 for systems of ordinary differential equations. Applications of multiple scale methods to problems in partial differential equations appear in Chapter 6.

The aim of this book is to survey perturbation methods as currently used in various application areas. We introduce a particular topic by means of a simple illustrative example and then build up to more challenging problems. Whenever possible (and practical), we give the general theory for a procedure that applies to a broad class of problems. However, we do not consider rigorous proofs for the validity of our results; to do so would take us far afield from our stated aim. Also, in spite of the progress in this regard in recent years, rigorous justification of asymptotic validity remains generally out of reach except for simple, well-understood problems.

The basic ideas discussed in this book are, as is usual in scientific work, the contributions of many people. We have made some attempt to cite original sources, but we do not claim perfect historical accuracy, nor do we give a complete list of references. Rather, we have tried to present the state of the art in a systematic and unified manner. There are several excellent references that cover various aspects of layer-type problems in some detail; fewer are available on multiple scale methods. We present a comprehensive treatment of both types of problems, including recent developments in multiple scale and averaging methods not available in other reference work.