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

Citation

Bender, C. M., & Orszag, S. A. (1999). Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer. (Reprint of the 1978 McGraw-Hill edition.)

Chapter Summary

The book has eleven chapters and an appendix, in about 590 pages. The publisher describes it as a clear, practical and self-contained presentation of asymptotics and perturbation theory, showing how to obtain approximate analytical solutions to differential and difference equations when exact solutions are unavailable and brute-force numerical methods may not give useful answers. The presentation begins with a review of differential and difference equations, develops local asymptotic methods, explains perturbation and summation theory, and concludes with global methods: boundary-layer theory, WKB theory and multiple-scale analysis. The publisher adds that the discussion emphasizes applications and care rather than rigor, with many examples, 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying difficulty and an appendix summarizing the properties of special functions. The Springer edition is a reprint of the original McGraw-Hill book, with a roman numeral and a subtitle added to the title.

Chapters 1–2: Ordinary Differential Equations; Difference Equations

The opening chapters review the two kinds of equations the rest of the book approximates. Chapter 1 covers ordinary differential equations (pp. 3–35) and Chapter 2 difference equations (pp. 36–57). Together they supply the exact theory against which later approximations are judged.

Chapters 3–5: Approximate Solution of Linear Differential Equations; Approximate Solution of Nonlinear Differential Equations; Approximate Solution of Difference Equations

These three chapters make up the local-analysis core, at roughly 85, 60 and 40 pages. Linear differential equations come first, then nonlinear ones, then difference equations. This is the publisher's "local asymptotic methods for differential and difference equations".

Chapter 6: Asymptotic Expansion of Integrals

The sixth chapter, of about 70 pages, moves from equations to integrals and treats their asymptotic expansion.

Chapters 7–8: Perturbation Series; Summation of Series

The next two chapters concern perturbation series and the summation of series. The publisher presents these as the "perturbation and summation theory" part of the book.

Chapters 9–11: Boundary-Layer Theory; WKB Theory; Multiple-Scale Analysis

The last three chapters, about 65, 60 and 33 pages, develop the global methods: boundary-layer theory (Chapter 9), WKB theory (Chapter 10) and multiple-scale analysis (Chapter 11). They draw on the local methods and the perturbation techniques that precede them. An appendix summarizes properties of special functions, and the book ends with references and an index.

Key Concepts

Chapters 1–2: Differential and difference equations

• Differential and difference equations: the two types of equation reviewed first and approximated throughout.

Chapters 3–6: Local analysis and integrals

• Local asymptotic analysis: determining the behaviour of solutions of linear and nonlinear differential equations and of difference equations in a limit.
• Asymptotic expansion of integrals: approximating integrals when a parameter becomes large.
• Approximation against exact results: the comparisons of approximate and exact solutions that the publisher highlights.

Chapters 7–8: Perturbation and summation

• Perturbation series: expansions in a small parameter.
• Summation of series: methods for obtaining useful values from a series.

Chapters 9–11: Global methods

• Boundary-layer theory: matching solutions across regions of rapid change.
• WKB theory: approximate solutions of differential equations with a large or small parameter.
• Multiple-scale analysis: methods for problems with more than one characteristic scale.

Critical Analysis

Strengths

• A method book built around approximation checks. The publisher describes 190 computer-generated plots and tables that compare approximate and exact solutions, so readers see how good each approximation is, not just how to derive it.
• A deliberate progression. The chapter order follows the publisher's outline: review, local methods, perturbation and summation, then global methods. The global chapters can therefore reuse the tools introduced earlier.
• Many problems. The publisher counts over 600 problems with varying levels of difficulty, and the book has an appendix on special functions.
• Endorsed by a specialist reviewer. A. D. Wood, reviewing the 1999 reprint in Mathematical Reviews (quoted by the publisher), called it an excellent introduction to asymptotic and perturbation methods for master's students or beginning research students, written in a fresh and lively style, with graphs and tables comparing exact and approximate results that were in advance of their time.

Limitations

• Not an undergraduate teaching book despite the original title. Wood notes that the original title suggested an undergraduate text for scientists and engineers, which the book is not. It is aimed at master's and beginning research students, although parts could serve a final-year undergraduate course in applied mathematics or mathematical physics.
• Care rather than rigor. By the publisher's own description, the discussion stresses care rather than rigor. Readers who need formal proofs of asymptotic statements will need a more rigorous source.
• Volume I only. The title carries a roman numeral I and the subtitle names asymptotic methods and perturbation theory, so topics outside that scope are not in this volume.
• A reprint of a 1978 text. The Springer edition changes only the title (Wood notes the added roman numeral and subtitle), so it does not include later developments in the field or in numerical computing.

Where It Fits in the Course

This is the advanced, specialist book of the engineering-mathematics list, and it assumes the material of the others. Erwin Kreyszig's Advanced Engineering Mathematics is a broad, reference-style textbook covering differential equations, linear algebra, vector calculus, Fourier and complex analysis and numerical methods, and Glyn James's Advanced Modern Engineering Mathematics is a second-level British text. Anthony Croft and Robert Davison's Mathematics for Engineers and K. A. Stroud's Engineering Mathematics are first-year and programmed texts that build from arithmetic and algebra. Riley, Hobson and Bence's Mathematical Methods for Physics and Engineering is a large single-volume reference that extends to Green's functions, variational calculus, tensors, group theory and special functions.

Read Bender and Orszag after a complete engineering-mathematics sequence, with ordinary differential equations and complex variables in hand. Use Riley, Hobson and Bence for the wider toolbox, and turn to Bender and Orszag when the problem has a small or large parameter and an approximate analytical answer is needed.

How to Use This Book

The publisher intends the book for graduate students and advanced undergraduates and says it assumes only limited familiarity with differential equations and complex variables. The reviewer quoted by the publisher places it at master's or beginning-research level, so "limited familiarity" is best read as a minimum: confident work with series, ordinary differential equations and complex variables will make the book much easier.

A practical plan:

• Core: Chapters 1 to 3 as a base, then Chapter 6 on integrals and Chapters 7 and 8 on perturbation and summation.
• Choose by need: Chapters 9 to 11 are the global methods for boundary-layer, WKB and multiple-scale problems. Chapters 4 and 5 can be taken as needed for nonlinear equations and difference equations.
• Work the problems and the comparisons. Try problems at several difficulty levels, and check each approximation against the exact or numerical solution, as the book does.
• Use the appendix on special functions as a reference when working through examples.