Practical Applied Mathematics: Modelling, Analysis, Approximation

Cover Page......Page 1
Cambridge Texts in Applied Mathematics......Page 2
Title Page......Page 4
© Cambridge University Press 2005......Page 5
4 Case studies: hair modelling and cable laying......Page 6
9 The delta function and other distributions......Page 7
15 Case study: piano tuning......Page 8
22 Multiple scales and other methods for nonlinear oscillators......Page 9
23 Ray theory and the WKB method......Page 10
Who should read this book?......Page 12
Conventions......Page 13
Colemanballs......Page 14
Part I Modelling techniques......Page 16
1.1 Introduction......Page 18
1.2 What do we mean by a model?......Page 19
1.3 Principles of modelling: physical laws and constitutive relations......Page 21
1.3.1 Example: inviscid fluid mechanics......Page 22
1.3.2 Example: viscous fluids......Page 23
1.4 Conservation laws......Page 26
1.6 Exercises......Page 27
2.1 Introduction......Page 30
2.2 Units and dimensions......Page 31
2.2.1 Example: heat flow......Page 32
2.3 Electric fields and electrostatics......Page 33
2.5 Exercises......Page 35
Electromagnetism......Page 36
Other exercises......Page 40
3.1.1 Example: advection–diffusion......Page 43
3.1.2 Example: the damped pendulum......Page 47
3.1.3 Example: beams and strings......Page 49
3.2 The Navier–Stokes equations and Reynolds numbers......Page 51
3.2.1 Water in the bathtub......Page 54
Example: the drag on a cylinder......Page 55
3.5 Exercises......Page 57
4.1 The Euler–Bernoulli model for a beam......Page 65
4.2 Hair modelling......Page 67
4.3 Undersea cable laying......Page 68
4.4 Modelling and analysis......Page 69
4.4.2 Effective forces and nondimensionalisation......Page 71
4.6 Exercises......Page 73
5.1 Heat and current flow in thermistors......Page 78
5.1.2 A simple model for the heat flow......Page 79
5.2 Nondimensionalisation......Page 81
5.3 A thermistor in a circuit......Page 82
5.5 Exercises......Page 84
6.1 Electrostatic painting......Page 87
6.2 Field equations......Page 88
6.3 Boundary conditions......Page 90
6.4 Nondimensionalisation......Page 91
6.6 Exercises......Page 92
Part II Analytical techniques......Page 94
7.1 First-order quasilinear partial differential equations: theory......Page 96
7.2 Example: Poisson processes......Page 100
7.3 Shocks......Page 102
7.3.1 The Rankine–Hugoniot conditions......Page 104
7.4 Fully nonlinear equations: Charpit’s method......Page 105
7.5 Second-order linear equations in two variables......Page 109
Two real roots: hyperbolic equations......Page 110
Double roots: parabolic equations......Page 111
7.7 Exercises......Page 112
8.1 Simple models for traffic flow......Page 119
Blinkered drivers......Page 120
Local speed–density laws......Page 121
Red lights and shocks......Page 122
Green lights and expansion fans......Page 124
8.3 More sophisticated models......Page 125
8.5 Exercises......Page 126
9.1 Introduction......Page 129
9.2 A point force on a stretched string; impulses......Page 130
9.3 Informal definition of the delta and Heaviside functions......Page 132
9.4.2 Continuous and discrete probabilities......Page 135
9.4.3 The fundamental solution of the heat equation......Page 136
9.5 Balancing singularities......Page 137
9.5.1 The Rankine–Hugoniot conditions......Page 138
9.5.2 Case study: cable-laying......Page 139
9.6.1 Ordinary differential equations......Page 140
9.6.2 Partial differential equations......Page 145
9.8 Exercises......Page 149
10.1 Test functions......Page 155
10.2 The action of a test function......Page 156
10.3 Definition of a distribution......Page 157
10.5 The derivative of a distribution......Page 158
10.6.2 Fourier transforms......Page 160
10.8 Exercises......Page 163
11.1 What is a pantograph?......Page 172
11.2 The model......Page 173
11.3 Impulsive attachment for an undamped pantograph......Page 175
11.4 Solution near a support......Page 177
11.5 Solution for a whole span......Page 179
11.7 Exercises......Page 182
Part III Asymptotic techniques......Page 186
12.1 Introduction......Page 188
12.2 Order notation......Page 190
12.2.1 Asymptotic sequences and expansions......Page 192
12.3 Convergence and divergence......Page 193
12.5 Exercises......Page 195
13.1 Introduction......Page 198
13.2 Example: stability of a spacecraft in orbit......Page 199
13.3 Linear stability......Page 200
13.3.1 Stability of critical points in a phase plane......Page 201
13.3.2 Side track: example of a system that is linearly neutrally stable but nonlinearly stable or unstable......Page 202
13.4 Example: the pendulum......Page 203
13.5.1 Example: flow past a nearly circular cylinder......Page 205
13.5.2 Example: water waves......Page 207
13.6 Caveat expandator......Page 209
13.7 Exercises......Page 210
14.1 Small parameters in the electropaint model......Page 215
14.2 Exercises......Page 217
15.1 The notes of a piano: the tonal system of Western music......Page 220
15.2 Tuning an ideal piano......Page 223
15.3 A real piano......Page 224
15.5 Exercises......Page 226
16.2 Functions with boundary layers; matching......Page 231
16.2.1 Matching......Page 233
16.3 Examples from ordinary differential equations......Page 236
16.4 Case study: cable laying......Page 239
16.5.1 Large-Peclet-number advection–diffusion past an infinite flat plate......Page 240
16.5.2 Traffic flow with small anticipation......Page 242
16.5.3 A thin elliptical conductor in a uniform electric field......Page 243
16.6 Exercises......Page 245
17.1 Strongly temperature-dependent conductivity......Page 250
17.2 Exercises......Page 253
18.1 ‘Lubrication theory’ approximations: slender geometries......Page 255
18.2 Heat flow in a bar of variable cross-section......Page 256
18.3 Heat flow in a long thin domain with cooling......Page 259
18.4 Advection–diffusion in a long thin domain......Page 261
18.5 Exercises......Page 264
19.1 Continuous casting of steel......Page 270
19.2 Exercises......Page 275
20.1 Thin fluid layers: classical lubrication theory......Page 278
20.2 Thin viscous fluid sheets on solid substrates......Page 280
20.2.1 Viscous fluid spreading horizontally under gravity: intuitive argument......Page 281
20.2.2 Viscous fluid spreading under gravity: systematic argument......Page 283
20.2.3 A viscous fluid layer on a vertical wall......Page 285
20.3 Thin fluid sheets and fibres......Page 286
20.3.1 The viscous sheet equations by a systematic argument......Page 287
20.5 Exercises......Page 290
21.1 Incubating eggs......Page 300
Heat flow......Page 301
Fluid flow in the egg......Page 302
Rotation of the yolk......Page 303
Diffusion......Page 304
21.3 Exercises......Page 305
22.1 The Poincar´e–Linstedt method......Page 307
22.2 The method of multiple scales......Page 309
22.3 Relaxation oscillations......Page 312
22.4 Exercises......Page 314
23.1 Introduction......Page 318
23.2 Classical WKB theory......Page 319
23.3 Geometric optics and ray theory: why do we say light travels in straight lines?......Page 321
23.4 Kelvin’s ship waves......Page 326
23.5 Exercises......Page 329
References......Page 333
A,B,C......Page 336
D,E......Page 337
F,G,H,I,K,L,M......Page 338
N,O,P,R,S......Page 339
T,U......Page 340
V,W,Y......Page 341
Back Page......Page 342