Asymptotic Treatment of Differential Equations

Contents

Preface VII

1. The basics of asymptotics 1
1.1. Introduction 1
1.2. Order relations 5
1.3. Asymptotic sequences 10
1.4. Asymptotic expansions 12
1.5. Asymptotic series 31
1.6. Asymptotic behaviour of functions dependent on asymptotic parameters 39
1.7. Singular linear o.d.e.s in the complex field 41
1.8. Linear o.d.e.s with large parameters, uniformly valid asymptotic expansions, and the asymptotic resonance phenomenon 45

2. Perturbation theory 54
2.1. Regular and singular perturbation problems for non-linear differential equations with a small parameter 54
2.2. Regular perturbation problems for systems of o.d.e.s 67
2.3. Classification of singular perturbation problems 70
2.4. Methods in singular perturbation theory and the matching of asymptotic expansions 78
2.5. Optimum meshes found by asymptotics: numerical methods specific to boundary layer type problems 115
2.6. Rigorous results in the theory of boundary layer problems; the method of boundary layer type functions 116
2.7. The multiple scale method 144
2.8. Averaging methods 164

3. Model examples 176
3.1. Prandtl\'s example 176
3.2. Friedrichs\' example 183
3.3. Principles of intermediate matching; Lagerstrom model example 189

4. Models of asymptotic approximation of the Navier-Stokes model 198
4.1. Models of asymptotic approximation of the Navier-Stokes model at large Reynolds numbers 198
4.2. Models of asymptotic approximation of the Navier-Stokes model at small Reynolds numbers 213

5. Asymptotic approximation of the Boltzmann model for small and large mean free path 233
5.1. The linearized Boltzmann equation at small Knudsen numbers 233
5.2. Boltzmann\'s equation at small Knudsen numbers 239
5.3. Boltzmann\'s equation at large Knudsen numbers 242

6 Other models of asymptotic approximation 244
6.1 Synoptic flows 244
6.2 Elastic behaviour 249
6.3 Abstract and concrete applications of asymptotic analysis 251

References 253

Index 264