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Practical Applied Mathematics Modelling, Analysis, Approximation

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Practical Applied Mathematics Modelling, Analysis, Approximation -Sam Howison OCIAM Mathematical Institute, Oxford University October 10, 2003. 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

Practical Applied Mathematics Modelling, Analysis, Approximation

Sam Howison OCIAM Mathematical Institute
Oxford University October 10, 2003




1 Introduction 9


 1.1 What is modelling/ Why model?


 1.2 How to use this book


 1.3 acknowledgements


 I Modelling techniques 11


 2 The basics of modelling 13


 2.1 Introduction


 2.2 What do we mean by amodel?


 2.3 Principles of modelling


 2.3.1 Example: inviscidfluidmechanics


 2.3.2 Example: viscousfluids


 2.4 Conservationlaws


 2.5 Conclusion


 3 Unit sanddimensions 25


 3.1 Introduction


 3.2 Unit sanddimensions


 3.2.1 Example: heatflow


 3.3 Electric fields and electrostatics


 4 Dimensional analysis 39


 4.1 Nondimensionalisation


 4.1.1 Example: advection- Diffusion


 4.1.2 Example: the damped pendulum


 4.1.3 Example: beamsandstrings


 4.2 The Navier–Stoke sequations


 4.2.1 Water in the bathtub


 4.3 Buckingham’sPi- Theorem


 4.4 Onwards


 5 Case study: hair modelling and cable laying 61


 5.1 The Euler–Bernoulli model for a beam


 5.2 Hair modelling


 5.3 Cable- Laying


 5.4 Modelling and analysis


 5.4.1 Boundary conditions


 5.4.2 Effective force sand nondimensionalisation


 6 Case study: the thermistor 1 73


 6.1 Thermistors


 6.1.1 A simple model


 6.2 Nondimensionalisation


 6.3 Athermistor in a circuit


 6.3.1 Theone- Dimensional model


 7 Case study: electrostatic painting 83


 7.1 Electrostatic painting


 7.2 Fieldequations


 7.3 Boundary conditions


 7.4 Nondimensionalisation


 II Mathematical techniques 91


 8 Partial differential equations 93


 8.1 First- Order equations


 8.2 Example: Poisson processes


 8.3 Shocks


 8.3.1 The Rankine–Hugoniot conditions


 8.4 Nonlinear equations


 8.4.1 Example: spray forming


 9 Case study: traffic modelling 105


 9.1 Case study: traffic modelling


 9.1.1 Localspeed- Densitylaws


 9.2 Solutions with discontinuities: shocks and the Rankine Hugoniot relations


 9.2.1 Trafficjams


 9.2.2 Trafficlights


 10 The delta function and other distributions 111


 10.1 Introduction


 10.2 Apointforceonastretchedstring; impulses


 10.3 Informal definition of the delta and Heaviside functions


 10.4 Examples


 10.4.1 A point force on a wirere visited


 10.4.2 Continuous and discrete probability


 10.4.3 The fundamental solution of the heat equation


 10.5 Balancingsingularities


 10.5.1 The Rankine–Hugoniot conditions


 10.5.2 Case study: cable- Laying


 10.6 Green’s functions


 10.6.1 Ordinary differential equations


 10.6.2 Partial differential equations


 11 Theory of distributions 137


 11.1 Test functions


 11.2 The action of atest function


 11.3 Definition of adistribution


 11.4 Further properties of distributions


 11.5 The derivative of adistribution


 11.6 Extensions of the theory of distributions


 11.6.1 Morevariables


 11.6.2 Fourier transforms


 12 Case study: the pantograph 155


 12.1 What is a pantograph?


 12.2 The model


 12.2.1What happens at the contactpoint?


 12.3 Impulsive at tachment


 12.4 Solution near a support


 12.5 Solution for a whole span


 III Asymptotic techniques 171


 13 A symptotic expansions 173


 13.1 Introduction


 13.2 Order notation


 13.2.1 A symptotics equence sand expansions


 13.3Convergence anddivergence


 14 Regular perturbations/ Expansions 183




 14.2 Example: stability of a spacecraft in orbit


 14.3 Linear stability


 14.3.1 Stability of critical points in a phase plane


 14.3.2 Example (side track): a system which is neutrally stable butnon linearly stable (orunstable)


 14.4 Example: the pendulum


 14.5 Small perturbations of a boundary


 14.5.1 Example: flow pastanearly circular cylinder


 14.5.2 Example: waterwaves


 14.6 Caveatex pandator


 15 Case study: electrostatic painting 2 201


 15.1 Small parameters in the electropaint model


 16 Case study: piano tuning 207


 16.1 The notes of apiano


 16.2 Tun in ganideal piano


 16.3 Areal piano


 17 Methods for oscillators 219


 17.0.1 Poincar´ eLinstedt for the pendulum


 18 Boundary layers 223




 18.2 Functions with boundary layers; matching


 18.2.1 Matching


 18.3 Cablelaying


 19 ‘Lubrication theory’ analysis: 231


 19.1 ‘Lubrication theory’a pproximations: slender geometries


 19.2 Heat flow in abarof variable cross- Section


 19.3 Heat flow in a long thin domain with cooling


 19.4 Advection- Diffusionin a long thin domain


 20 Case study: continuous casting of steel 247


 20.1 Continuous casting of steel


 21 Lubrication theory for fluids 253


 21.1 Thin fluid layers: classical lubrication theory


 21.2 Thin viscous fluid sheets on solid substrates


 21.2.1 Viscous fluid spreading horizontally under gravity: intuitive argument


 21.2.2 Viscous fluid spreading under gravity: systematic argument


 21.2.3 A viscous fluid layer on a vertical wall


 21.3 Thin fluid sheet sand fibres


 21.3.1 The viscous sheet equations by a systematic argument 263


 21.4 The beam equation (?)


 22 Ray theory and other ‘exponential’ approaches 277


 22.1 Introduction


 23 Case study: the thermistor 2 281



 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 usual polite fiction that you, the reader, and I, the author, are engaged on 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




 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 it I have constantly felt the need to sidestep in order to point out a connection  with something else. On the other hand, I don’t want you to lose track of the argument. As a compromise, I have used marginal notes and footnotes Marginal notes are usually directly relevant to the current discussion, often being used to fill in details or point out a feature of a calculation With slightly different purposes




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