Singular-Perturbation Theory: An Introduction with Applications

by Donald R. Smith

• ISBN13: 9780521300421
• ISBN10: 0521300428
• Format: Hardcover
• Copyright: 1985-08-30
• Publisher: Cambridge University Press

Summary

This book presents an introduction to singular-perturbation problems, problems which depend on a parameter in such a way that solutions behave nonuniformly as the parameter tends toward some limiting value of interest. The nature of the nonuniformity can vary from case to case, and so the author considers and solves a variety of problems, mostly for ordinary differential equations. He constructs (approximate) solutions for oscillation problems, using the methods of averaging and of multiple scales. For problems of the nonoscillatory type, where solutions exhibit 'fast dynamics' in a thin initial layer, he derives solutions using the O'Malley/Hoppensteadt method and the method of matched expansions. He obtains solutions for boundary-value problems, where solutions exhibit rapid variation in thin layers, using a multivariable method. After a suitable approximate solution is constructed for each problem, the author linearizes the problem about the proposed approximate solution, and, emphasizing the use of the Banach/Picard fixed-point theorem, presents a study of the linearization. This book will be a welcome text and reference to students at the graduate and senior undergraduate levels studying perturbation theory for differential equations, and to pure and applied mathematicians, engineers, and scientists who use differential equations in the modeling of natural phenomena.

Table of Contents

Preface

Acknowledgments

Preliminary results

Part I. Initial-Value Problems of Oscillatory Type: 1. Precession of the planet Mercury

2. Krylov/Bogoliubov averaging

3. The multiscale technique

4. Error estimates for perturbed-oscillation problems

Part II. Initial-Value Problems of Overdamped Type: 5. Linear overdamped initial-value problems

6. Nonlinear overdamped initial-value problems

7. Conditionally stable problems

Part III. Boundary-Value Problems: 8. Linear scalar problems

9. Linear first-order systems

10. Nonlinear problems

References

Name index

Subject index.