Perturbations: theory and methods

James A. Murdock9780471612940, 9780898714432, 0471612944, 0898714435

Perturbations: Theory and Methods gives a thorough introduction to both regular and singular perturbation methods for algebraic and differential equations. Unlike most introductory books on the subject, this one distinguishes between formal and rigorous asymptotic validity, which are commonly confused in books that treat perturbation theory as a bag of heuristic tricks with no foundation. The meaning of ‘uniformity’ is carefully explained in a variety of contexts. All standard methods, such as rescaling, multiple scales, averaging, matching, and the WKB method are covered, and the asymptotic validity (in the rigorous sense) of each method is carefully proved.

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Table of contents :
Perturbations Theory and Methods……Page 1
CONTENTS……Page 10
PREFACE TO THE CLASSICS EDITION……Page 14
PREFACE……Page 16
PART I INTRODUCTION TO PERTURBATION THEORY……Page 22
CHAPTER 1 ROOT FINDING……Page 24
CHAPTER 2 REGULAR PERTURBATIONS……Page 104
CHAPTER 3 DIRECT ERROR ESTIMATION……Page 156
PART II OSCILLATORY PHENOMENA……Page 176
CHAPTER 4 PERIODIC SOLUTIONS AND LINDSTEDT SERIES……Page 178
CHAPTER 5 MULTIPLE SCALES……Page 248
CHAPTER 6 AVERAGING……Page 296
PART III TRANSITION LAYERS……Page 362
CHAPTER 7 INITIAL LAYERS……Page 364
CHAPTER 8 BOUNDARY LAYERS……Page 438
CHAPTER 9 METHODS OF THE WKB TYPE……Page 454
APPENDIX A TAYLOR’S THEOREM……Page 484
THE IMPLICIT FUNCTION THEOREM APPENDIX B……Page 490
APPENDIX C SECOND ORDER DIFFERENTIAL EQUATIONS……Page 498
APPENDIX D SYSTEMS OF DIFFERENTIAL EQUATIONS……Page 506
APPENDIX E FOURIER SERIES……Page 512
APPENDIX F LIPSCHITZ CONSTANTS AND VECTOR NORMS……Page 516
APPENDIX G LOGICAL QUANTIFIERS AND UNIFORMITY……Page 518
SYMBOL INDEX……Page 524
INDEX……Page 526