| Preface | p. xi |
| Modelling techniques | |
| The basics of modelling | p. 3 |
| Introduction | p. 3 |
| What do we mean by a model? | p. 4 |
| Principles of modelling: physical laws and constitutive relations | p. 6 |
| Conservation laws | p. 11 |
| General remarks | p. 12 |
| Exercises | p. 12 |
| Units, dimensions and dimensional analysis | p. 15 |
| Introduction | p. 15 |
| Units and dimensions | p. 16 |
| Electric fields and electrostatics | p. 18 |
| Sources and further reading | p. 20 |
| Exercises | p. 20 |
| Nondimensionalisation | p. 28 |
| Nondimensionalisation and dimensional parameters | p. 28 |
| The Navier-Stokes equations and Reynolds numbers | p. 36 |
| Buckingham's Pi-theorem | p. 40 |
| Sources and further reading | p. 42 |
| Exercises | p. 42 |
| Case studies: hair modelling and cable laying | p. 50 |
| The Euler-Bernoulli model for a beam | p. 50 |
| Hair modelling | p. 52 |
| Undersea cable laying | p. 53 |
| Modelling and analysis | p. 54 |
| Sources and further reading | p. 58 |
| Exercises | p. 58 |
| Case study: the thermistor (1) | p. 63 |
| Heat and current flow in thermistors | p. 63 |
| Nondimensionalisation | p. 66 |
| A thermistor in a circuit | p. 67 |
| Sources and further reading | p. 69 |
| Exercise | p. 69 |
| Case study: electrostatic painting | p. 72 |
| Electrostatic painting | p. 72 |
| Field equations | p. 73 |
| Boundary conditions | p. 75 |
| Nondimensionalisation | p. 76 |
| Sources and further reading | p. 77 |
| Exercises | p. 77 |
| Analytical techniques | |
| Partial differential equations | p. 81 |
| First-order quasilinear partial differential equations: theory | p. 81 |
| Example: Poisson processes | p. 85 |
| Shocks | p. 87 |
| Fully nonlinear equations: Charpitt's method | p. 90 |
| Second-order linear equations in two variables | p. 94 |
| Further reading | p. 97 |
| Exercises | p. 97 |
| Case study: traffic modelling | p. 104 |
| Simple models for traffic flow | p. 104 |
| Traffic jams and other discontinuous solutions | p. 107 |
| More sophisticated models | p. 110 |
| Sources and further reading | p. 111 |
| Exercises | p. 111 |
| The delta function and other distributions | p. 114 |
| Introduction | p. 114 |
| A point force on a stretched string; impulses | p. 115 |
| Informal definition of the delta and Heaviside functions | p. 117 |
| Examples | p. 120 |
| Balancing singularities | p. 122 |
| Green's functions | p. 125 |
| Sources and further reading | p. 134 |
| Exercises | p. 134 |
| Theory of distributions | p. 140 |
| Test functions | p. 140 |
| The action of a test function | p. 141 |
| Definition of a distribution | p. 142 |
| Further properties of distributions | p. 143 |
| The derivative of a distribution | p. 143 |
| Extensions of the theory of distributions | p. 145 |
| Sources and further reading | p. 148 |
| Exercises | p. 148 |
| Case study: the pantograph | p. 157 |
| What is a pantograph? | p. 157 |
| The model | p. 158 |
| Impulsive attachment for an undamped pantograph | p. 160 |
| Solution near a support | p. 162 |
| Solution for a whole span | p. 164 |
| Sources and further reading | p. 167 |
| Exercises | p. 167 |
| Asymptotic techniques | |
| Asymptotic expansions | p. 173 |
| Introduction | p. 173 |
| Order notation | p. 175 |
| Convergence and divergence | p. 178 |
| Sources and further reading | p. 180 |
| Exercises | p. 180 |
| Regular perturbation expansions | p. 183 |
| Introduction | p. 183 |
| Example: stability of a spacecraft in orbit | p. 184 |
| Linear stability | p. 185 |
| Example: the pendulum | p. 188 |
| Small perturbations of a boundary | p. 190 |
| Caveat expandator | p. 194 |
| Exercises | p. 195 |
| Case study: electrostatic painting (2) | p. 200 |
| Small parameters in the electropaint model | p. 200 |
| Exercises | p. 202 |
| Case study: piano tuning | p. 205 |
| The notes of a piano: the tonal system of Western music | p. 205 |
| Tuning an ideal piano | p. 208 |
| A real piano | p. 209 |
| Sources and further reading | p. 211 |
| Exercises | p. 211 |
| Boundary layers | p. 216 |
| Introduction | p. 216 |
| Functions with boundary layers; matching | p. 216 |
| Examples from ordinary differential equations | p. 221 |
| Case study: cable laying | p. 224 |
| Examples for partial differential equations | p. 225 |
| Exercises | p. 230 |
| Case study: the thermistor (2) | p. 235 |
| Strongly temperature-dependent conductivity | p. 235 |
| Exercises | p. 238 |
| 'Lubrication theory' analysis in long thin domains | p. 240 |
| 'Lubrication theory' approximations: slender geometries | p. 240 |
| Heat flow in a bar of variable cross-section | p. 241 |
| Heat flow in a long thin domain with cooling | p. 244 |
| Advection-diffusion in a long thin domain | p. 246 |
| Exercises | p. 249 |
| Case study: continuous casting of steel | p. 255 |
| Continuous casting of steel | p. 255 |
| Exercises | p. 260 |
| Lubrication theory for fluids | p. 263 |
| Thin fluid layers: classical lubrication theory | p. 263 |
| Thin viscous fluid sheets on solid substrates | p. 265 |
| Thin fluid sheets and fibres | p. 271 |
| Further reading | p. 275 |
| Exercises | p. 275 |
| Case study: turning of eggs during incubation | p. 285 |
| Incubating eggs | p. 285 |
| Modelling | p. 286 |
| Exercises | p. 290 |
| Multiple scales and other methods for nonlinear oscillators | p. 292 |
| The Poincare-Linstedt method | p. 292 |
| The method of multiple scales | p. 294 |
| Relaxation oscillations | p. 297 |
| Exercises | p. 299 |
| Ray theory and the WKB method | p. 303 |
| Introduction | p. 303 |
| Classical WKB theory | p. 304 |
| Geometric optics and ray theory: why do we say light travels in straight lines? | p. 306 |
| Kelvin's ship waves | p. 311 |
| Exercises | p. 314 |
| References | p. 318 |
| Index | p. 321 |
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