Practical Applied Mathematics Modelling, Analysis, Approximation

299 Pages·2004·1.51 MB·English

Practical Applied Mathematics
Modelling, Analysis, Approximation
Sam Howison
OCIAM
Mathematical Institute
Oxford University
May 31, 2004
2
Contents
I Modelling techniques 15
1 The basics of modelling 17
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 What do we mean by a model? . . . . . . . . . . . . . . . . . . . 18
1.3 Principles of modelling . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Example: inviscid fluid mechanics . . . . . . . . . . . . . 20
1.3.2 Example: viscous fluids . . . . . . . . . . . . . . . . . . . 21
1.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Units, dimensions and dimensional analysis 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Units and dimensions . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Example: heat flow. . . . . . . . . . . . . . . . . . . . . . 28
2.3 Electric fields and electrostatics . . . . . . . . . . . . . . . . . . . 29
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Non-dimensionalisation 39
3.1 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Example: advection-diffusion . . . . . . . . . . . . . . . . 39
3.1.2 Example: the damped pendulum . . . . . . . . . . . . . . 42
3.1.3 Example: beams and strings . . . . . . . . . . . . . . . . 44
3.2 The Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Water in the bathtub . . . . . . . . . . . . . . . . . . . . 48
3.3 Buckingham’s Pi-theorem . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Case studies: hair modelling and cable laying 59
4.1 The Euler–Bernoulli model for a beam . . . . . . . . . . . . . . . 59
4.2 Hair modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Cable-laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Modelling and analysis . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 64
4.4.2 Effective forces and nondimensionalisation . . . . . . . . . 64
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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4 CONTENTS
5 Case study: the thermistor (1) 71
5.1 Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 A black box model . . . . . . . . . . . . . . . . . . . . . . 71
5.1.2 A simple model for heat flow . . . . . . . . . . . . . . . . 72
5.2 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 A thermistor in a circuit . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 The one-dimensional model . . . . . . . . . . . . . . . . . 76
5.4 Sources and further reading . . . . . . . . . . . . . . . . . . . . . 76
6 Case study: electrostatic painting 79
6.1 Electrostatic painting . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
II Mathematical techniques 85
7 Partial differential equations 87
7.1 First-order equations . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Example: Poisson processes . . . . . . . . . . . . . . . . . . . . . 90
7.3 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . . 94
7.4 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5 Second-order linear equations in two variables . . . . . . . . . . . 99
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8 Case study: traffic modelling 107
8.1 Simple models for traffic flow . . . . . . . . . . . . . . . . . . . . 107
8.2 Traffic jams and other discontinuous solutions . . . . . . . . . . . 109
8.3 More sophisticated models . . . . . . . . . . . . . . . . . . . . . . 111
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9 Distributions 117
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.2 A point force on a stretched string; impulses. . . . . . . . . . . . 118
9.3 Informal definition of the delta and Heaviside functions . . . . . 120
9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.4.1 A point force on a wire revisited . . . . . . . . . . . . . . 122
9.4.2 Continuous and discrete probability. . . . . . . . . . . . . 122
9.4.3 The fundamental solution of the heat equation . . . . . . 123
9.5 Balancing singularities . . . . . . . . . . . . . . . . . . . . . . . . 124
9.5.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . . 125
9.5.2 Case study: cable-laying . . . . . . . . . . . . . . . . . . . 125
9.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.6.1 Ordinary differential equations . . . . . . . . . . . . . . . 126
9.6.2 Partial differential equations . . . . . . . . . . . . . . . . 130
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
CONTENTS 5
10 Theory of distributions 139
10.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.2 The action of a test function . . . . . . . . . . . . . . . . . . . . 140
10.3 Definition of a distribution. . . . . . . . . . . . . . . . . . . . . . 140
10.4 Further properties of distributions . . . . . . . . . . . . . . . . . 141
10.5 The derivative of a distribution . . . . . . . . . . . . . . . . . . . 142
10.6 Extensions of the theory of distributions . . . . . . . . . . . . . . 143
10.6.1 More variables . . . . . . . . . . . . . . . . . . . . . . . . 143
10.6.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . 143
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
11 Case study: the pantograph 153
11.1 What is a pantograph? . . . . . . . . . . . . . . . . . . . . . . . . 153
11.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.2.1 What happens at the contact point? . . . . . . . . . . . . 155
11.3 Impulsive attachment . . . . . . . . . . . . . . . . . . . . . . . . 156
11.4 Solution near a support . . . . . . . . . . . . . . . . . . . . . . . 157
11.5 Solution for a whole span . . . . . . . . . . . . . . . . . . . . . . 159
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
III Asymptotic techniques 167
12 Asymptotic expansions 169
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
12.2 Order notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2.1 Asymptotic sequences and expansions . . . . . . . . . . . 172
12.3 Convergence and divergence . . . . . . . . . . . . . . . . . . . . . 173
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13 Regular perturbation expansions 177
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.2 Example: stability of a spacecraft in orbit . . . . . . . . . . . . . 178
13.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.3.1 Stability of critical points in a phase plane. . . . . . . . . 179
13.3.2 Example (side track): a system which is neutrally stable
but nonlinearly stable (or unstable) . . . . . . . . . . . . 180
13.4 Example: the pendulum . . . . . . . . . . . . . . . . . . . . . . . 181
13.5 Small perturbations of a boundary . . . . . . . . . . . . . . . . . 182
13.5.1 Example: flow past a nearly circular cylinder . . . . . . . 182
13.5.2 Example: water waves . . . . . . . . . . . . . . . . . . . . 185
13.6 Caveat expandator . . . . . . . . . . . . . . . . . . . . . . . . . . 186
13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
14 Case study: electrostatic painting (2) 191
14.1 Small parameters in the electropaint model . . . . . . . . . . . . 191
14.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
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15 Case study: piano tuning 195
15.1 The notes of a piano . . . . . . . . . . . . . . . . . . . . . . . . . 195
15.2 Tuning an ideal piano . . . . . . . . . . . . . . . . . . . . . . . . 197
15.3 A real piano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
15.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
16 Boundary layers 205
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
16.2 Functions with boundary layers; matching . . . . . . . . . . . . . 205
16.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
16.3 Examples from ordinary differential equations . . . . . . . . . . . 209
16.4 Cable laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
16.5 Examples for partial differential equations . . . . . . . . . . . . . 213
16.5.1 Large Peclet number heat flow . . . . . . . . . . . . . . . 213
16.5.2 Traffic flow with small anticipation . . . . . . . . . . . . . 214
16.5.3 A thin elliptical conductor in a uniform electric field . . . 216
16.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
17 Case study: the thermistor (2) 221
17.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
18 ‘Lubrication theory’ analysis: 225
18.1 ‘Lubrication theory’ approximations: slender geometries . . . . . 225
18.2 Heat flow in a bar of variable cross-section . . . . . . . . . . . . . 226
18.3 Heat flow in a long thin domain with cooling . . . . . . . . . . . 228
18.4 Advection-diffusion in a long thin domain . . . . . . . . . . . . . 230
18.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
19 Case study: continuous casting of steel 239
19.1 Continuous casting of steel . . . . . . . . . . . . . . . . . . . . . 239
20 Lubrication theory for fluids 247
20.1 Thin fluid layers: classical lubrication theory . . . . . . . . . . . 247
20.2 Thin viscous fluid sheets on solid substrates . . . . . . . . . . . . 249
20.2.1 Viscous fluid spreading horizontally under gravity: intu-
itive argument . . . . . . . . . . . . . . . . . . . . . . . . 249
20.2.2 Viscous fluid spreading under gravity: systematic argument251
20.2.3 A viscous fluid layer on a vertical wall . . . . . . . . . . . 253
20.3 Thin fluid sheets and fibres . . . . . . . . . . . . . . . . . . . . . 254
20.3.1 The viscous sheet equations by a systematic argument . . 255
20.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
21 Case Study: eggs 267
21.1 Incubating eggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
21.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
21.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
CONTENTS 7
22 Methods for oscillators 273
22.1 The Poincar´e–Linstedt method . . . . . . . . . . . . . . . . . . . 273
22.2 The method of multiple scales . . . . . . . . . . . . . . . . . . . . 275
22.3 Relaxation oscillations . . . . . . . . . . . . . . . . . . . . . . . . 277
22.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
23 Ray theory and other ‘exponential’ approaches 283
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
23.2 Classical WKB theory . . . . . . . . . . . . . . . . . . . . . . . . 284
23.3 Geometric optics and ray theory . . . . . . . . . . . . . . . . . . 285
23.4 Kelvin’s ship waves . . . . . . . . . . . . . . . . . . . . . . . . . . 290
23.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
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Notes to typeset-
ter/copyeditor/publisher
1. There is some trouble with bold greek fonts. My macro
\BGx
for example, does not seem to produce bold ξ (nor does
\mathbf{\xi}
which gives ξ, the same as the ordinary ξ). There is a smiliar problem
with σ.
2. I need a nice curly D for the space of distributions on page 141.
3. I also need bold calligraphic font for vector distributions on page 149.
4. I would appreciate help with Fig 15.1.
5. Need to think about separating the ‘colemanballs’ from the end of the
preceding exercises, maybe a line or a bit of graphics?
6. The table in exercise 1 of ‘Other exercises’ in Ch 2 has a missing vertical
line, I do not know why.
7. I try to be sparing with commas.
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