PRACTICAL MATHEMATICS

By DANIEL CLARK

Part I Modelling techniques 1 The basics of modelling 3 11 Introduction 3 12 What do we mean by a model? 4 13 Principles of modelling: physical laws and constitutive relations 6 14 Conservation laws 11 15 General remarks 12 16 Exercises 12 2 Units, dimensions and dimensional analysis 15 21 Introduction 15 22 Units and dimensions 16 23 Electric fields and electrostatics 18 24 Sources and further reading 20 25 Exercises 20 3 Nondimensionalisation 28 31 Nondimensionalisation and dimensional parameters 28 32 The Navier–Stokes equations and Reynolds numbers 36 33 Buckingham’s Pi-theorem 40 34 Sources and further reading 42 35 Exercises 42 4 Case studies: hair modelling and cable laying 50 41 The Euler–Bernoulli model for a beam 50 42 Hair modelling 52 43 Undersea cable laying 53 44 Modelling and analysis 54 45 Sources and further reading 58 46 Exercises 58 5 Case study: the thermistor (1) 63 51 Heat and current flow in thermistors 63 52 Nondimensionalisation 66 53 A thermistor in a circuit 67 54 Sources and further reading 69 55 Exercises 69 6 Case study: electrostatic painting 72 61 Electrostatic painting 72 62 Field equations 73 63 Boundary conditions 75 64 Nondimensionalisation 76 65 Sources and further reading 77 66 Exercises 77 Part II Analytical techniques 7 Partial differential equations 81 71 First-order quasilinear partial differential equations: theory 81 72 Example: Poisson processes 85 73 Shocks 87 74 Fully nonlinear equations: Charpitt’s method 90 75 Second-order linear equations in two variables 94 76 Further reading 97 77 Exercises 97 8 Case study: traffic modelling 104 81 Simple models for traffic flow 104 82 Traffic jams and other discontinuous solutions 107 83 More sophisticated models 110 84 Sources and further reading 111 85 Exercises 111 9 The delta function and other distributions 114 91 Introduction 114 92 A point force on a stretched string; impulses 115 93 Informal definition of the delta and Heaviside functions 117 94 Examples 120 95 Balancing singularities 122 96 Green’s functions 125 97 Sources and further reading 134 98 Exercises 134 10 Theory of distributions 140 101 Test functions 140 102 The action of a test function 141 103 Definition of a distribution 142 104 Further properties of distributions 143 105 The derivative of a distribution 143 106 Extensions of the theory of distributions 145 107 Sources and further reading 148 108 Exercises 148 11 Case study: the pantograph 157 111 What is a pantograph? 157 112 The model 158 113 Impulsive attachment for an undamped pantograph 160 114 Solution near a support 162 115 Solution for a whole span 164 116 Sources and further reading 167 117 Exercises 167 Part III Asymptotic techniques 12 Asymptotic expansions 173 121 Introduction 173 122 Order notation 175 123 Convergence and divergence 178 124 Sources and further reading 180 125 Exercises 180 13 Regular perturbation expansions 183 131 Introduction 183 132 Example: stability of a spacecraft in orbit 184 133 Linear stability 185 134 Example: the pendulum 188 135 Small perturbations of a boundary 190 136 Caveat expandator 194 137 Exercises 195 14 Case study: electrostatic painting (2) 200 141 Small parameters in the electropaint model 200 142 Exercises 202 15 Case study: piano tuning 205 151 The notes of a piano: the tonal system of Western music 205 152 Tuning an ideal piano 208 153 A real piano 209 154 Sources and further reading 211 155 Exercises 211 16 Boundary layers 216 161 Introduction 216 162 Functions with boundary layers; matching 216 163 Examples from ordinary differential equations 221 164 Case study: cable laying 224 165 Examples for partial differential equations 225 166 Exercises 230 17 Case study: the thermistor (2) 235 171 Strongly temperature-dependent conductivity 235 172 Exercises 238 18 ‘Lubrication theory’ analysis in long thin domains 240 181 ‘Lubrication theory’ approximations: slender geometries 240 182 Heat flow in a bar of variable cross-section 241 183 Heat flow in a long thin domain with cooling 244 184 Advection–diffusion in a long thin domain 246 185 Exercises 249 19 Case study: continuous casting of steel 255 191 Continuous casting of steel 255 192 Exercises 260 20 Lubrication theory for fluids 263 201 Thin fluid layers: classical lubrication theory 263 202 Thin viscous fluid sheets on solid substrates 265 203 Thin fluid sheets and fibres 271 204 Further reading 275 205 Exercises 275 21 Case study: turning of eggs during incubation 285 211 Incubating eggs 285 212 Modelling 286 213 Exercises 290 22 Multiple scales and other methods for nonlinear oscillators 292 221 The Poincar´e–Linstedt method 292 222 The method of multiple scales 294 223 Relaxation oscillations 297 224 Exercises 299 23 Ray theory and the WKB method 303 231 Introduction 303 232 Classical WKB theory 304 233 Geometric optics and ray theory: why do we say light travels in straight lines? 306 234 Kelvin’s ship waves 311 235 Exercises 314 References 318 Index 321