Practical Applied Math. Modelling, Analysis, Approximation

285 Pages·2003·1.28 MB·English

by  S. Howison

Practical Applied Mathematics
Modelling, Analysis, Approximation
Sam Howison
OCIAM
Mathematical Institute
Oxford University
October 10, 2003
2
Contents
1 Introduction 9
1.1 What is modelling/why model? . . . . . . . . . . . . . . . . . 9
1.2 How to use this book . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 9
I Modelling techniques 11
2 The basics of modelling 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 What do we mean by a model? . . . . . . . . . . . . . . . . . 14
2.3 Principles of modelling . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Example: inviscid fluid mechanics . . . . . . . . . . . . 17
2.3.2 Example: viscous fluids . . . . . . . . . . . . . . . . . . 18
2.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Units and dimensions 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Units and dimensions . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Example: heat flow . . . . . . . . . . . . . . . . . . . . 27
3.3 Electric fields and electrostatics . . . . . . . . . . . . . . . . . 28
4 Dimensional analysis 39
4.1 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Example: advection-diffusion . . . . . . . . . . . . . . 39
4.1.2 Example: the damped pendulum . . . . . . . . . . . . 43
4.1.3 Example: beams and strings . . . . . . . . . . . . . . . 45
4.2 The Navier–Stokes equations . . . . . . . . . . . . . . . . . . . 47
4.2.1 Water in the bathtub . . . . . . . . . . . . . . . . . . . 50
4.3 Buckingham’s Pi-theorem . . . . . . . . . . . . . . . . . . . . 51
3
4 CONTENTS
4.4 Onwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Case study: hair modelling and cable laying 61
5.1 The Euler–Bernoulli model for a beam . . . . . . . . . . . . . 61
5.2 Hair modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Cable-laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Modelling and analysis . . . . . . . . . . . . . . . . . . . . . . 65
5.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . 67
5.4.2 Effective forces and nondimensionalisation . . . . . . . 67
6 Case study: the thermistor 1 73
6.1 Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 A simple model . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 75
6.3 A thermistor in a circuit . . . . . . . . . . . . . . . . . . . . . 77
6.3.1 The one-dimensional model . . . . . . . . . . . . . . . 78
7 Case study: electrostatic painting 83
7.1 Electrostatic painting . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 86
7.4 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 87
II Mathematical techniques 91
8 Partial differential equations 93
8.1 First-order equations . . . . . . . . . . . . . . . . . . . . . . . 93
8.2 Example: Poisson processes . . . . . . . . . . . . . . . . . . . 97
8.3 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . 101
8.4 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . 102
8.4.1 Example: spray forming . . . . . . . . . . . . . . . . . 102
9 Case study: traffic modelling 105
9.1 Case study: traffic modelling . . . . . . . . . . . . . . . . . . . 105
9.1.1 Local speed-density laws . . . . . . . . . . . . . . . . . 107
9.2 Solutionswithdiscontinuities: shocksandtheRankine–Hugoniot
relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.2.1 Traffic jams . . . . . . . . . . . . . . . . . . . . . . . . 109
9.2.2 Traffic lights . . . . . . . . . . . . . . . . . . . . . . . . 109
CONTENTS 5
10 The delta function and other distributions 111
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.2 A point force on a stretched string; impulses . . . . . . . . . . 112
10.3 Informal definition of the delta and Heaviside functions . . . . 114
10.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.4.1 A point force on a wire revisited . . . . . . . . . . . . . 117
10.4.2 Continuous and discrete probability. . . . . . . . . . . 117
10.4.3 The fundamental solution of the heat equation . . . . . 119
10.5 Balancing singularities . . . . . . . . . . . . . . . . . . . . . . 120
10.5.1 The Rankine–Hugoniot conditions . . . . . . . . . . . . 120
10.5.2 Case study: cable-laying . . . . . . . . . . . . . . . . . 121
10.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.6.1 Ordinary differential equations . . . . . . . . . . . . . . 122
10.6.2 Partial differential equations . . . . . . . . . . . . . . . 125
11 Theory of distributions 137
11.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.2 The action of a test function . . . . . . . . . . . . . . . . . . . 138
11.3 Definition of a distribution . . . . . . . . . . . . . . . . . . . . 139
11.4 Further properties of distributions . . . . . . . . . . . . . . . . 140
11.5 The derivative of a distribution . . . . . . . . . . . . . . . . . 141
11.6 Extensions of the theory of distributions . . . . . . . . . . . . 142
11.6.1 More variables . . . . . . . . . . . . . . . . . . . . . . . 142
11.6.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . 142
12 Case study: the pantograph 155
12.1 What is a pantograph? . . . . . . . . . . . . . . . . . . . . . . 155
12.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.2.1 What happens at the contact point? . . . . . . . . . . 158
12.3 Impulsive attachment . . . . . . . . . . . . . . . . . . . . . . . 159
12.4 Solution near a support . . . . . . . . . . . . . . . . . . . . . . 160
12.5 Solution for a whole span . . . . . . . . . . . . . . . . . . . . . 162
III Asymptotic techniques 171
13 Asymptotic expansions 173
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
13.2 Order notation . . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.2.1 Asymptotic sequences and expansions . . . . . . . . . . 177
13.3 Convergence and divergence . . . . . . . . . . . . . . . . . . . 178
6 CONTENTS
14 Regular perturbations/expansions 183
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
14.2 Example: stability of a spacecraft in orbit . . . . . . . . . . . 184
14.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14.3.1 Stability of critical points in a phase plane . . . . . . . 186
14.3.2 Example (side track): a system which is neutrally sta-
ble but nonlinearly stable (or unstable) . . . . . . . . . 187
14.4 Example: the pendulum . . . . . . . . . . . . . . . . . . . . . 188
14.5 Small perturbations of a boundary . . . . . . . . . . . . . . . 189
14.5.1 Example: flow past a nearly circular cylinder . . . . . . 189
14.5.2 Example: water waves . . . . . . . . . . . . . . . . . . 192
14.6 Caveat expandator . . . . . . . . . . . . . . . . . . . . . . . . 193
15 Case study: electrostatic painting 2 201
15.1 Small parameters in the electropaint model . . . . . . . . . . . 201
16 Case study: piano tuning 207
16.1 The notes of a piano . . . . . . . . . . . . . . . . . . . . . . . 207
16.2 Tuning an ideal piano . . . . . . . . . . . . . . . . . . . . . . . 209
16.3 A real piano . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
17 Methods for oscillators 219
17.0.1 Poincar´e–Linstedt for the pendulum . . . . . . . . . . . 219
18 Boundary layers 223
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
18.2 Functions with boundary layers; matching . . . . . . . . . . . 224
18.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 225
18.3 Cable laying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
19 ‘Lubrication theory’ analysis: 231
19.1 ‘Lubrication theory’ approximations: slender geometries . . . . 231
19.2 Heat flow in a bar of variable cross-section . . . . . . . . . . . 232
19.3 Heat flow in a long thin domain with cooling . . . . . . . . . . 235
19.4 Advection-diffusion in a long thin domain . . . . . . . . . . . 237
20 Case study: continuous casting of steel 247
20.1 Continuous casting of steel . . . . . . . . . . . . . . . . . . . . 247
21 Lubrication theory for fluids 253
21.1 Thin fluid layers: classical lubrication theory . . . . . . . . . . 253
21.2 Thin viscous fluid sheets on solid substrates . . . . . . . . . . 256
CONTENTS 7
21.2.1 Viscous fluid spreading horizontally under gravity: in-
tuitive argument . . . . . . . . . . . . . . . . . . . . . 256
21.2.2 Viscous fluid spreading under gravity: systematic ar-
gument . . . . . . . . . . . . . . . . . . . . . . . . . . 258
21.2.3 A viscous fluid layer on a vertical wall . . . . . . . . . 261
21.3 Thin fluid sheets and fibres . . . . . . . . . . . . . . . . . . . . 261
21.3.1 The viscous sheet equations by a systematic argument 263
21.4 The beam equation (?) . . . . . . . . . . . . . . . . . . . . . . 266
22 Ray theory and other ‘exponential’ approaches 277
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
23 Case study: the thermistor 2 281
8 CONTENTS
Chapter 1
Introduction
Book born out of fascination with applied math as meeting place of physical
world and mathematical structures.
have to be generalists, anything and everything potentially interesting to
an applied mathematician
1.1 What is modelling/why model?
1.2 How to use this book
case studies as strands
must do exercises
1.3 acknowledgements
Have taken examples from many sources, old examples often the best. If you
teach a course using other peoples’ books and then write your own this is
inevitable.
errors all my own
ACF, Fowkes/Mahoney, O2, green book, Hinch, ABT, study groups
Conventions. Let me introduce a couple of conventions that I use in this
book. I use ‘we’, as in ‘we can solve this by a Laplace transform’, to signal
the usualpolite fictionthatyou, thereader, andI,theauthor, areengagedon
a joint voyage of discovery. ‘You’ is mostly used to suggest that you should
get your pen out and work though some of the ‘we’ stuff, a good idea in view
9
10 CHAPTER 1. INTRODUCTION
of my fallible arithmetic. ‘I’ is associated with authorial opinions and can
mostly be ignored if you like.
I have tried to draw together a lot of threads in this book, and in writing
itIhaveconstantlyfelttheneedtosidestepinordertopointoutaconnection
with something else. On the other hand, I don’t want you to lose track of
Marginal notes are the argument. As a compromise, I have used marginal notes and footnotes1
usually directly rel- with slightly different purposes.
evant to the current
discussion, often be-
ingusedtofillinde-
tails or point out a
feature of a calcula-
tion.
1Footnotes are more digressional and can, in principle, be ignored.