Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory

by Carl M. Bender, Steven A. Orszag

A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.

Genres: Mathematics, Physics, Textbooks, Reference, Science

607 pages, Paperback

First published January 1, 1978

Reviews

[Bessel · Rating 5 out of 5]

I was introduced to this book when I first took a course on Perturbation and Asymptotic Methods. The chapters on Boundary Layer theory, and WKB theory are well written and fun to read. In addition, I also found Summation of Series chapter quite interesting in regards to accelerated convergence (a numerical technique that is sadly ignored in beginning numerical methods courses). My only gripe with this book is the difficulty of some of the problems. Not all problems are difficult, but even some of the "Intermediate" exercises are quite daunting. Regardless, I think this is a MUST own for any scientist/engineer in a research/teaching field. As a supplement to this book, I also recommend reading Chapter 6 (titled Perturbation Theory) in Landau and Lifshitz vol. 3 (Quantum Mechanics Non-Relativistic Theory).

[Tomáš Ševček · Rating 5 out of 5]

A classic on asymptotics and perturbation methods. The book starts with a recap of ODEs and difference equations (although one should keep in mind the book was originally published in the '70s, so a lot of the things commonly covered in ODE courses back then are no longer covered, so some of the material may be new even to people with experience with ODEs). The following chapters then cover the most basic ideas of asymptotics and perturbation methods - approximate solutions to ODEs and difference equations near ordinary and (ir)regular singular points, asymptotic expansions of integrals (Watson's lemma, Laplace's method, stationary phase and steepest descent), boundary-layer theory, WKB theory and multiple-scale analysis. In order to follow the text, knowledge of calculus, complex analysis and ODEs/difference equations is advisable. The exposition is attentive to detail, but not rigorous. The methods are explained using examples. Personally, I don't like this approach very much. I would appreciate a little more rigor. However, this branch of mathematics really seems to hinge on one's intuition and experience a lot and that is, in my opinion, the only way one can truly understand how to apply these methods (and maybe even why they work so inexplicably well). Therefore, I recommend using a symbolic software which can do asymptotics (such as Mathematica) to experiment with the methods, see how well they work, compare them with numerical methods, etc. While not perfect, this book is still better than a lot of the other books on the topic - a topic which seems to be in dire need of a clearly written textbook which makes use of the modern computational tools.

[albin james · Rating 4 out of 5]

This book has a very nice presentation style. It covers quite a lot of topics but the authors try not to overwhelm the student with results. It's also written in a very friendly and unassuming tone which is usually a plus for scared students. I also really liked that they were consistent with their choice of clever opening quotes for each chapter: always some reasonable long and relaxing quote from Sherlock Holmes =) =)

[Kyle · Rating 5 out of 5]

This is considered a classic in the field for good reason. Bender and Orszag cover a wide range of methods to solve differential equations using clear language, understandable examples, and a bit of personality that makes it an excellent volume. If you want an overview of the typical methods that can be used to solve single dependent variable differential equations, this book probably has you covered.

The book assumes facility with upper college level mathematics, so you should certainly note the word "Advanced," but I think the authors do an excellent job of explaining the approach for each method.