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Practical Applied Mathematics Modelling, Analysis, Approximation PDF

299 Pages·2004·1.51 MB·English
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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 3 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 6 CONTENTS 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 8 CONTENTS 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. 9 10 CONTENTS

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Practical Applied Mathematics. Modelling, Analysis, Approximation. Sam Howison. OCIAM. Mathematical Institute. Oxford University. May 31, 2004
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