Practical Applied Math. - Modelling, Analysis, Approximation - S. Howison (2003) WW.pdf

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Practical Applied Mathematics
Modelling, Analysis, Approximation
Sam Howison
OCIAM
Mathematical Institute
Oxford University
October 10, 2003
2
Contents
1 Introduction 9
1.1 What is modelling/why model? ................. 9
1.2 Howtousethisbook....................... 9
1.3 acknowledgements......................... 9
I Modelling techniques
11
2 The basics of modelling 13
2.1 Introduction ............................ 13
2.2 Whatdowemeanbyamodel? ................. 14
2.3 Principles of modelling . . . ................... 16
2.3.1 Example:inviscidfluidmechanics............ 17
2.3.2 Example:viscousfluids.................. 18
2.4 Conservationlaws......................... 21
2.5 Conclusion ............................. 22
3Un sanddimen ions 25
3.1 Introduction ............................ 25
3.2 Unitsanddimensions....................... 25
3.2.1 Example:heatflow.................... 27
3.3 Electricfieldsandelectrostatics ................. 28
4 Dimensional analysis 39
4.1 Nondimensionalisation ...................... 39
4.1.1 Example:advection-diffusion .............. 39
4.1.2 Example: the damped pendulum ............ 43
4.1.3 Example:beamsandstrings............... 45
4.2 TheNavier–Stokesequations................... 47
4.2.1 Waterinthebathtub................... 50
4.3 Buckingham’sPi-theorem .................... 51
3
4
CONTENTS
4.4 Onwards.............................. 53
5 Case study: hair modelling and cable laying 61
5.1 TheEuler–Bernoullimodelforabeam ............. 61
5.2 Hair modelling .......................... 63
5.3 Cable-laying............................ 64
5.4 Modelling and analysis ...................... 65
5.4.1 Boundary conditions . . ................. 67
5.4.2 Effectiveforcesandnondimensionalisation ....... 67
6 Case study: the thermistor 1 73
6.1 Thermistors............................ 73
6.1.1 Asimplemodel...................... 73
6.2 Nondimensionalisation ...................... 75
6.3 Athermistorinacircuit ..................... 77
6.3.1 Theone-dimensionalmodel ............... 78
7 Case study: electrostatic painting 83
7.1 Electrostaticpainting....................... 83
7.2 Fieldequations .......................... 84
7.3 Boundary conditions . ...................... 86
7.4 Nondimensionalisation ...................... 87
II Mathematical techniques
91
8 Partial differential equations 93
8.1 First-orderequations ....................... 93
8.2 Example:Poissonprocesses ................... 97
8.3 Shocks............................... 99
8.3.1 TheRankine–Hugoniotconditions............101
8.4 Nonlinearequations........................102
8.4.1 Example:sprayforming .................102
9 Case study: trac modelling 105
9.1 Case study: trac modelling . . .................105
9.1.1 Localspeed-densitylaws.................107
9.2 Solutions with discontinuities: shocks and the Rankine–Hugoniot
relations..............................108
9.2.1 Tracjams ........................109
9.2.2 Traclights........................109
CONTENTS
5
10 The delta function and other distributions 111
10.1 Introduction ............................111
10.2Apointforceonastretchedstring;impulses..........112
10.3 Informal definition of the delta and Heaviside functions . . . . 114
10.4Examples .............................117
10.4.1 Apointforceonawirerevisited.............117
10.4.2 Continuous and discrete probability. . . ........117
10.4.3 The fundamental solution of the heat equation . . . . . 119
10.5Balancingsingularities ......................120
10.5.1 TheRankine–Hugoniotconditions............120
10.5.2 Casestudy:cable-laying .................121
10.6Green’sfunctions .........................122
10.6.1 Ordinarydifferentialequations..............122
10.6.2 Partialdifferentialequations...............125
11 Theory of distributions 137
11.1 Test functions ...........................137
11.2Theactionofatestfunction...................138
11.3Definitionofadistribution....................139
11.4Furtherpropertiesofdistributions................140
11.5Thederivativeofadistribution .................141
11.6Extensionsofthetheoryofdistributions ............142
11.6.1 Morevariables.......................142
11.6.2 Fouriertransforms ....................142
12 Case study: the pantograph 155
12.1Whatisapantograph?......................155
12.2Themodel.............................156
12.2.1 Whathappensatthecontactpoint? ..........158
12.3Impulsiveattachment.......................159
12.4Solutionnearasupport......................160
12.5Solutionforawholespan.....................162
III Asymptotic techniques
171
13 Asymptotic expansions 173
13.1 Introduction ............................173
13.2Ordernotation ..........................175
13.2.1 Asymptoticsequencesandexpansions..........177
13.3Convergenceanddivergence ...................178

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