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Introduction to Perturbation Methods (Texts in Applied Mathematics), by Mark H. Holmes
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This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas.
- Sales Rank: #8243763 in Books
- Published on: 1995-01-01
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .81" w x 6.10" l, 1.10 pounds
- Binding: Paperback
- 369 pages
Most helpful customer reviews
26 of 26 people found the following review helpful.
Best available introduction to perturbation methods
By David Elder
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.
4 of 4 people found the following review helpful.
Online Book Review
By D. Subbaram Naidu
Mark Holmes: Introduction to Perturbation Methods, Second Edition, Springer, New York, NY, 2013
Reviewed by D. Subbaram Naidu, Idaho State University
The book is composed of 6 chapters with the topics of Introduction to Asymptotic Approximations, Matched Asymptotic Expansions, Multiple Scales, The WKB and Related Methods, The Method of Homogenization, and Introduction to Bifurcation and Stability and appendices on Taylor Series, Solution and Properties of Transition Layer Equations, Asymptotic Approximations of Integrals, Second-Order Difference Equations, and Delay Equations.
This reviewer, with background in singular perturbations and time scales in automatic control theory and applications, enjoyed reading this book that has a refreshing flavor to perturbation methods. Some of the interesting and useful features of this book are the coverage of both ordinary and partial differential equations related to applications to nonlinear wave and diffusion problems and others, generous sprinkling of a number of examples and both section-wise and chapter-wise exercises throughout the book paving the best way of "learning mathematics", solutions to some of the exercises available from the author's home page and the MATLAB� files, that were used to generate the figures, also available from author's web page.
The book can be used for both undergraduate and graduate courses in mathematics and physics and also in aerospace, electrical and mechanical engineering areas. Those working in industry will find this book useful in addressing some of the nonlinear problems in real-world situations.
7 of 8 people found the following review helpful.
nice introduction
By G. Rogers
I found this book explains the ideas very well and also provides examples that help to understand the methods. I also liked all the exercies at the end of each chapter. This is one book that anyone who studies applied mathematics should have on their bookshelf.
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