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The Boundary Function Method for Singular Perturbed Problems

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Adelaida B. Vasil'eva, Valentin F. Butuzov, and Leonid V. Kalachev


1995 / Softcover / ISBN: 978-0-898713-33-6 / List Price $112.00 / SIAM Member Price $78.40 / Order Code AM14

This is the first book published in English devoted solely to the boundary function method, which is one of the asymptotic methods. This method provides an effective and simple way to obtain asymptotic approximations for the solutions of certain ordinary and partial differential equations containing small parameters in front of the highest derivatives. These equations, called singularly perturbed equations, are often used in modeling. In addition to numerous examples, the book includes discussions on singularly perturbed problems from chemical kinetics and heat conduction, semiconductor device modeling, and mathematical biology.

The book also contains a variety of original ideas and explicit calculations previously available only in journal literature, as well as many concrete applied problems illustrating the boundary function method algorithms. Quite general asymptotic results described in the book are rigorous in the sense that, along with the asymptotic algorithms, in most cases the theorems on estimation of the remainder terms are presented. A survey of results of Russian mathematicians on the subject is provided; many of these results are not well known in the West.

Based on the Russian edition of the textbook by Vasil'eva and Butuzov, this American edition, prepared by Kalachev, differs in many aspects. The text of the book has been revised substantially, some new material has been added to every chapter, and more examples, exercises, and new references on asymptotic methods and their applications have been included.

This book is intended for specialists in perturbation theory and for those who use perturbation theory in their work, including scientists in national laboratories and commercial enterprises; mechanical, electrical, and chemical engineers; university professors and graduate and undergraduate students in applied mathematics, engineering, or mathematical biology; and specialists in ordinary and partial differential equations. It also serves as an excellent introduction to perturbation ideas for those who are not familiar with asymptotic methods but want to start using them in their work.

Chapter 1: Basic Ideas. Regular and Singular Perturbations; Asymptotic Approximations. Asymptotic and Convergent Series; Examples of Asymptotic Expansions for Solutions of Regularly and Singularly Perturbed Problems; Chapter 2: Singularly Perturbed Ordinary Differential Equations.
Initial Value Problem; The Critical Case; Boundary Value Problems; Spike-type Solutions and Other Contrast (Dissipative) Structures; Chapter 3: Singularly Perturbed Partial Differential Equations. The Method of Vishik-Lyusternik; Corner Boundary Functions; The Smoothing Procedure; Systems of Equations in Critical Cases; Periodic Solutions; Hyperbolic Systems; Chapter 4: Applied Problems. Mathematical Model of Combustion Process in the Case of Autocatalytic Reaction; Heat Conduction in Thin Bodies; Application of the Boundary Function Method in The Theory of Semiconductor Devices; Relaxation Waves in the FitzHugh-Nagumo System; On Some Other Applied Problems; Index.


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