Singular Perturbation Theory Math and Analyt Technique w. appl to Engineering

309 Pages·2004·6.69 MB·English

by  R.S. Johnson

SINGULAR PERTURBATION THEORY
MATHEMATICAL AND ANALYTICAL TECHNIQUES
WITH APPLICATIONS TO ENGINEERING
MATHEMATICAL AND ANALYTICAL TECHNIQUES
WITH APPLICATIONS TO ENGINEERING
Alan Jeffrey, Consulting Editor
Published:
Inverse Problems
A. G. Ramm
Singular Perturbation Theory
R. S. Johnson
Forthcoming:
Methods for Constructing Exact Solutions of Partial Differential Equations
with Applications
S. V. Meleshko
The Fast Solution of Boundary Integral Equations
S. Rjasanow and O. Steinbach
Stochastic Differential Equations with Applications
R. Situ
SINGULAR PERTURBATION THEORY
MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH
APPLICATIONS TO ENGINEERING
R. S. JOHNSON
Springer
eBookISBN: 0-387-23217-6
Print ISBN: 0-387-23200-1
©2005 Springer Science + Business Media, Inc.
Print ©2005 Springer Science + Business Media, Inc.
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To Ros, who still, after nearly 40 years,
sometimes listens when I extol the wonders of
singular perturbation theory, fluid mechanics or water waves
—usually on a long trek in the mountains.
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CONTENTS
Foreword xi
Preface xiii
1.Mathematicalpreliminaries 1
1.1 Some introductory examples 2
1.2 Notation 10
1.3Asymptotic sequences and asymptotic expansions 13
1.4 Convergent series versus divergent series 16
1.5 Asymptotic expansions with a parameter 20
1.6 Uniformity or breakdown 22
1.7 Intermediate variables and the overlap region 26
1.8 The matching principle 28
1.9Matching with logarithmic terms 32
1.10 Composite expansions 35
Further Reading 40
Exercises 41
viii Contents
2.Introductory applications 47
2.1 Roots of equations 47
2.2 Integration of functions represented by asymptotic
expansions 55
2.3Ordinary differential equations: regular problems 59
2.4 Ordinary differential equations: simple singular problems 66
2.5 Scaling of differential equations 75
2.6 Equations which exhibit a boundary-layer behaviour 80
2.7 Where is the boundary layer? 86
2.8 Boundary layers and transition layers 90
Further Reading 103
Exercises 104
3. Further applications 115
3.1 A regular problem 116
3.2 Singular problems I 118
3.3 Singular problems II 128
3.4 Further applications to ordinary differential equations 139
Further Reading 147
Exercises 148
4.The method of multiple scales 157
4.1 Nearly linear oscillations 157
4.2 Nonlinear oscillators 165
4.3 Applications to classical ordinary differential equations 168
4.4 Applications to partial differential equations 176
4.5 A limitation on the use of the method of multiple scales 183
4.6 Boundary-layer problems 184
Further Reading 188
Exercises 188
5.Some worked examples arising from physical problems 197
5.1 Mechanical & electrical systems 198
5.2 Celestial mechanics 219
5.3 Physics of particles and of light 226
5.4 Semi- and superconductors 235
5.5 Fluid mechanics 242
ix
5.6 Extreme thermal processes 255
5.7 Chemical and biochemical reactions 262
Appendix: The Jacobian Elliptic Functions 269
Answers and Hints 271
References 283
Subject Index 287