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Multiple scale and singular perturbation methods. (English) Zbl 0846.34001
Applied Mathematical Sciences. 114. Berlin: Springer-Verlag. viii, 632 p. (1996).
This book is a revised and updated version, including a substantial portion of new material, of the authors’ textbook [PM] [Perturbation methods in applied mathematics (1981; Zbl 0456.34001)] – which, in turn, was a revised and updated version of J. D. Cole’s famous text from 1968 (Zbl 0162.12602). As stated in the preface, the authors’ aim 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”.
As compared to [PM], the arrangement of the material has changed; the organization of the book is now as follows: Regular perturbation problems (for ODE and PDE) are considered in the first chapter. Singular perturbation problems of boundary layer type and the corresponding limit process expansions are discussed in Chapter 2 for ODE and in Chapter 3 for PDE, including physical examples from boundary layer theory in fluid mechanics.
Chapters 4-6 are devoted to perturbation problems with a “cumulative effect”, i.e., to problems where the effects of a small perturbation accumulate over a long time. Chapter 4 (which, essentially, is Chapter 3 of [PM]) deals with multiple-variable expansions (now called “multiple scale expansions”) for ODE. Chapter 5, entitled “Near-identity averaging transformations: transient and sustained resonance”, is new; it is an expanded and updated account of many of the results from the first author’s expository paper [SIAM Rev. 29, 391-461 (1987; Zbl 0645.34031)]. Chapter 6 deals with multiple scale expansions for PDE, including interesting physical examples, among others from shallow water flow and gas dynamics.
Each chapter is complemented by a separate collection of references, each subsection by a number of problems to be solved by the reader. The book can be recommended as a text in either an advanced undergraduate course or a graduate-level course on the subject.
Reviewer: W.Müller (Berlin)

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35C15 Integral representations of solutions to PDEs
34C29 Averaging method for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
35B20 Perturbations in context of PDEs
35B25 Singular perturbations in context of PDEs