Download 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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[12] Fowkes, ND & Mahoney, JJ, An Introduction to Mathematical Modelling, Wiley UK (1994)
[13]
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current- Limiting circuit devices, SIAM J Appl Math 52,998–1011 (1992)
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[15] Gershenfeld
[16] Goldstein, AA, Optimal temperament, in [24], pp 242–251 283284 BIBLIOGRAPHY
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[22] Kevorkian, J & Cole, JD Perturbation Methods in Applied Math- ematics, Springer–Verlag (1981)
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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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system for an electric locomotive, Proc. Roy. Soc. A 322,447–468 (1971)
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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)
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Taylor, GI, The formation of a blast wave by a very intense explosion.
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Soc. A201,159–174 & 175–186 (1950)
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