The Theory of Singular Perturbations book cover

The Theory of Singular Perturbations

The subject of this textbook is the mathematical theory of singular perturbations, which despite its respectable history is still in a state of vigorous development. Singular perturbations of cumulative and of boundary layer type are presented. Attention has been given to composite expansions of solutions of initial and boundary value problems for ordinary and partial differential equations, linear as well as quasilinear; also turning points are discussed.

The main emphasis lies on several methods of approximation for solutions of singularly perturbed differential equations and on the mathematical justification of these methods. The latter implies a priori estimates of solutions of differential equations; this involves the application of Gronwall's lemma, maximum principles, energy integrals, fixed point theorems and Gåding's theorem for general elliptic equations. These features make the book of value to mathematicians and researchers in the engineering sciences, interested in the mathematical justification of formal approximations of solutions of practical perturbation problems. The text is selfcontained and each chapter is concluded with some exercises.

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Published: November 1996

Imprint: North-holland

ISBN: 978-0-444-82170-6

Reviews

  • ... the authors have offered a broad and thoughtful mathematical introduction to singular perturbation theory, in contrast to other works which directly motivate such study through applications. The intrigue and challenge of the subject are certainly well-displayed. I recommend their book highly.
    Zeitschrift für Angewandte Mathematic und Mechanik

Contents

  • Preface. 1. General introduction. 2. Asymptotic expansions. 3. Regular perturbations. 4. The method of the strained coordinate. 5. The method of averaging. 6. The method of multiple scales. 7. Singular perturbations of linear ordinary differential equations. 8. Singular perturbations of second order elliptic type. Linear theory. 9. Singular perurbations of second order hyperbolic type. 10. Singular perturbations in nonlinear initial value problems of second order. 11. Singular perturbations in nonlinear boundary value problems of second order. 12. Perturbations of higher order. References. Subject index.

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