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viii Contents

2. Introductory applications 47
2.1 Roots of equations 47
2.2 Integration of functions represented by asymptotic
expansions 55
2.3 Ordinary 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