Bibliothèque de L'IMIST/CNRST

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 Ḡding'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.

Includes bibliographical references (p. [331]-338) and index.

Description based on print version record.

Cover -- Contents -- Preface -- Chapter 1. General Introduction -- Chapter 2. Asymptotic Expansions -- 1 Order Symbols -- 2 Gauge Functions and Asymptotic Sequences -- Asymptotic Series -- 4 Convergence versus Asymptotic Convergence -- 5 Elementary Operations on Asymptotic Expansions -- 6 Other Types of Estimates -- 7 Generalized Asymptotic Expansions -- Exercises -- Chapter 3. Regular Perturbations -- 1 Regular Perturbations -- 2 A Nonlinear Initial Value Problem Containing a Small Parameter -- 3 Applications -- Chapter 4. The Method of the Strained Coordinate -- 1 Introduction -- 2 Applications of the Method of the Strained Coordinate -- 3 The Method of the Strained Parameter -- 4 Lighthill's Method -- 5 Temple's Method -- 6 Limitations of the Lindstedt-Poincaré Method -- Exercises -- Chapter 5. The Method of Averaging -- 1 Introduction -- 2 The Krilov-Bogoliubov-Mitropolski Theorem -- 3 Weakly Nonlinear Free Oscillations -- 4 Weakly Forced Nonlinear Oscillations -- 5 A Linear Oscillator with Increasing Damping -- Exercises -- Chapter 6. The Method of Multiple Scales -- 1 Introduction -- 2 Weakly Nonlinear Free Oscillations -- 3 The Linear Oscillator with Damping -- 4 The Equation of Mathieu -- 5 The General Case and the Error Estimate -- 6 Averaging and Multiple Scales for Perturbed Wave Equations -- Exercises -- Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations -- 1 The initial Value Problem -- 2 The Boundary Value Problem -- 3 Boundary Value Problems with Turning Points -- Exercises -- Chapter 8. Singular Perturbations of Second Order Elliptic Type. Linear Theory -- 1 Introduction -- 2 The Maximum Principle for Elliptic Operators -- 3 The Formal Approximation -- 4 Estimation of the Remainder Term and Final Results -- 5 Domains with Characteristic Boundaries -- 6 Elliptic Boundary Value Problems with Turning Points -- Exercises -- Chapter 9. Singular Perturbations of Second Order Hyperbolic Type. -- 1 Introduction -- 2 Characteristics and Subcharacteristics -- 3 The Formal Approximation -- 4 A Priori Estimates of Solutions of Initial Value Problems for Partial Differential Equations with a Singular Perturbation of Hyperbolic Type -- 5 The Estimate of the Remainder Term and Final Results -- Exercises -- Chapter 10. Singular Perturbations in Nonlinear Initial Value Problems of Second Order -- 1 Introduction -- 2 A Fixed Point Theorem -- 3 The Quasilinear Initial Value Problem -- 4 A General Nonlinear Initial Value Problem -- 5 Quasilinear Initial Value Problems with a Singular Perturbation of Second Order Hyperbolic Type -- Exercises -- Chapter 11. Singular Perturbations in Nonlinear Boundary Value Problems of Second Order -- 1 Introduction -- 2 Boundary Value Problems for Quasilinear Ordinary Differential Equations -- 3 Transition Layers -- 4 Autonomous Conservative Equations -- 5 A More General Case -- 6 Boundary Value Problems for Quasilinear Partial Differential Equations of Elliptic Type -- Exercises -- Chapter 12. Perturbations of Higher Order -- 1 Introduction -- 2 Perturbations of Higher Order in Ordinary Differential Equations -- 3 Elliptic Perturbations of Elliptic Equations -- 4 Ellipt.