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From the Publisher
The motivation for creating the book has been to help restore asymptotic and perturbation analysis to their proper place in an engineer's arsenal of mathematical tools, especially those involved in computer oriented studies.
Several excellent text books on perturbation theory have been written during the past three decades, and most have a common denominator of establishing as much mathematical rigor as possible. This has been done to provide a solid foundation for the approximate methods and "common sense mathematics" used by scientists and engineers for over 100 years. This is certainly a desirable goal that is just as important an issue for the applied mathematician as it is for the engineer.
Since the foundations are now well established, study of these methods by engineers needn't focus on details the mathematician is more appropriately concerned with. Courses on perturbation methods are often referred to by engineers as being "too theoretical." The perceived focus on theory clouds the utility of asymptotic methods.
This book is intended for use by practicing engineers and graduate students who want to master the techniques of asymptotic analysis while leaving formal justification to a more advanced treatise. That is not to say the text is simply a recipe book for asymptotic methods. On the contrary, careful attention focuses on justification of error estimates and explanations are given for why the methods work as well as they do. However, formal proofs are at a minimum. This is similar to mastering integral calculus without learning that there are types of integrals other than the one defined by Riemann.
Thus, the primary goal of this book is to provide a practical and understandable introduction to the intriguing field of asymptotic methods. The layout of the text is as follows.
- Chapter 1 gives an overview and history, followed by introduction of the order relations and operations with asymptotic expansions.
- Chapter 2 focuses on asymptotic expansion of integrals, including Laplace's method, stationary phase and steepest descent.
- Chapter 3 introduces the general principles of singular perturbation theory, including examples for both ODE's and PDE's.
- Chapter 4 covers the branch of perturbation theory known as multiple-scale analysis, including the Method of Averaging and the WKB method.
- Appendix A summarizes salient properties of the Gamma function.
- Appendix B presents useful concepts for doing contour integrals.
- Appendix C includes useful trigonometric relationships.
Practical examples illustrating how useful asymptotics can be when used in conjunction with computational methods are scattered throughout the text. The text includes an excellent Bibliography, a detailed Index and a large selection of carefully prepared homework problems.
For university people...The material presented in this book is appropriate for a one-semester, graduate or advanced undergraduate level course. Successful study of this material requires an understanding of basic calculus and proficiency in solving differential equations. The material in Chapter 2 requires a knowledge of complex variable theory, including familiarity with contour integration. If the reader's focus is strictly upon perturbation theory, Chapter 2 can be skipped.
For practicing engineers...The book has been written by an engineer for other engineers. The style is intended to clearly explain the powerful concepts that have guided engineering analysis for more than a century. Upon discovering these methods, the computational analyst will gain an outstanding method for understanding complex problems and for determining and controlling numerical error.
About the Author
David C. Wilcox was born in Wilmington, Delaware. He was educated as an aeronautical engineer at the Massachusetts Institute of Technology (BS 1966) and the California Institute of Technology (PhD 1970). After spending the early part of his career with several Southern California aerospace companies, in 1973, he founded DCW Industries, Inc., for which he is currently the President. He has taught several aerospace and mechanical engineering courses at both USC and UCLA.
In the early 1990s, Dr. Wilcox shifted the focus of his corporation from defense contracting to book publishing. Building on an international reputation based on numerous scientific-journal publications, he has written and published two undergraduate fluid-mechanics texts entitled Basic Fluid Mechanics (1997, 2000, 2007) and Elements of Fluid Mechanics (2005). He has also written a graduate-level text entitled Turbulence Modeling for CFD (1993, 1998, 2006). These books are currently in use at universities throughout the world.
Dr. Wilcox has also written three nontechnical books, the most recent of which is a partial autobiography entitled An Improbable Life (2007). The other two books are a fictional novel about a political compaign, . . . And the Donkey They Rode in On (2001), and collection of essays about politics entitled Cliches of Liberalism (1999).
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