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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

 

 

Contents

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

 

 14.1Introduction

 

 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

 

 18.1Introduction

 

 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

 

 Introduction

 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

 

CHAPTER 1.

INTRODUCTION

 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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 [2] Acheson, DJ, Elementary Fluid Dynamics, OUP (1990)

 [3] Addison, J, title, D. Phil. Thesis, Oxford University

 [4] Barenblatt, GI, Scaling, Self- Similarity and Intermediate Asymptotics, CUP (1996)

 [5] Bender, CM & Orszag, SA, Advanced Mathematical Methods for Scientists and Engineers, McGraw–hill (1978)

 [6] Carrier, GF, Krook, M & Pearson, CE, Functions of a Complex Variable, Hod Books (?? ??)

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 [8] Dewynne, JN, Ockendon, JR & Wilmott, P, On a mathematicalmodel for fiber tapering, SIAM J. Appl. Math. 49,983–990 (1989)

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 [16] Goldstein, AA, Optimal temperament, in [24], pp 242–251 283284 BIBLIOGRAPHY

 [17] Hildebrand, F, Methods of Applied Mathematics

 [18] Hinch, EJ, Perturbation Methods, CUP (1991)

 [19] Howell, PD, Models for thin viscous sheets, Europ J Appl Math 7,  321–343 (1996)

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 [25] Lighthill, MJ, Introduction to Fourier Analysis and Generalised Functions, CUP (1958)

 [26] McMahon, TA, Rowing: a similarity analysis, Science, 173,349– 351 (1971)

 [27] Ockendon, JR, Howison, SD, Lacey, AA & Movchan, AB, Applied Partial Differential Equations, OUP (revised edition 2003)

 [28] Ockendon, H & Ockendon, JR Viscous Flow, CUP (1995)

 [29] Ockendon, H & Ockendon, JR Waves and Compressible Flow,  Springer (2003)

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 [34] Richards, JI & Youn, HK, Theory of Distributions, CUP (1990)

 [35] Robinson, FNH, Electromagnetism, OUP (1973)

 [36] Rodeman, R, Longcope, DB & Shampine, LF, Response of a string to an accelerating mass, J. Appl. Mech. 98,675–680 (1976)

 [37] Schwarz, L, Th´ eorie des Distributions, vols 1 & II, Hermann et Cie, Paris (1951,1952)

 [38] Stakgold, I, BIBLIOGRAPHY 285

 [39] Tayler, AB, Mathematical Models in Applied Mechanics, OUP (1986, reissued 2003)

 [40] Taylor, GI, The formation of a blast wave by a very intense explosion. I, Theoretical discussion; II, The atomic explosion of 1945, Proc. Roy. Soc. A201,159–174 & 175–186 (1950)

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