Practical Applied Mathematics: Modelling, Analysis, Approximation

342 Pages·2005·1.79 MB·English

by  Sam Howison

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Cambridge Texts in Applied Mathematics
Allthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfrom
CambridgeUniversityPress.Foracompleteserieslistingvisit
http://publishing.cambridge.org/stm/mathematics/ctam
ViscousFlow
H.OckendonandJ.R.Ockendon
AFirstCourseintheNumericalAnalysisofDifferentialEquations
AriehIserles
MathematicalModelsintheAppliedSciences
A.C.Fowler
ThinkingAboutOrdinaryDifferentialEquations
RobertE.O'Malley
AModernIntroductiontotheMathematicalTheoryofWaterWaves
R.S.Johnson
RarefiedGasDynamics
CarloCercignani
SymmetryMethodsforDifferentialEquations
PeterE.Hydon
HighSpeedFlow
C.J.Chapman
WaveMotion
J.BillinghamandA.C.King
AnIntroductiontoMagnetohydrodynamics
P.A.Davidson
LinearElasticWaves
JohnG.Harris
VorticityandIncompressibleFlow
A.J.MajdaandA.L.Bertozzi
Infinite-dimensionalDynamicalSystems
J.C.Robinson
IntroductiontoSymmetryAnalysis
BrianJ.Cantwell
Ba¨cklundandDarbouxTransformations
C.RogersandW.K.Schief
FiniteVolumeMethodsforHyperbolicProblems
R.J.Leveque
IntroductiontoHydrodynamicStability
P.G.Drazin
TheoryofVortexSound
M.S.Howe
Scaling
G.I.Barenblatt
ComplexVariables:IntroductionandApplications(Secondedition)
MarkJ.AblowitzandAthanassiosS.Fokas
AFirstCourseinCombinatorialOptimization
JonLee

Practical Applied Mathematics
Modelling, Analysis, Approximation
SAMHOWISON
MathematicalInstitute,OxfordUniversity
DirectorofTheOxfordCentreforIndustrialand
AppliedMathematics
  
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Cambridge University Press
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First published in print format 2005
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Contents
Preface Pagexi
PartI Modellingtechniques
1 Thebasicsofmodelling 3
1.1 Introduction 3
1.2 Whatdowemeanbyamodel? 4
1.3 Principlesofmodelling:physicallawsandconstitutiverelations 6
1.4 Conservationlaws 11
1.5 Generalremarks 12
1.6 Exercises 12
2 Units,dimensionsanddimensionalanalysis 15
2.1 Introduction 15
2.2 Unitsanddimensions 16
2.3 Electricfieldsandelectrostatics 18
2.4 Sourcesandfurtherreading 20
2.5 Exercises 20
3 Nondimensionalisation 28
3.1 Nondimensionalisationanddimensionalparameters 28
3.2 TheNavier–StokesequationsandReynoldsnumbers 36
3.3 Buckingham’sPi-theorem 40
3.4 Sourcesandfurtherreading 42
3.5 Exercises 42
4 Casestudies:hairmodellingandcablelaying 50
4.1 TheEuler–Bernoullimodelforabeam 50
4.2 Hairmodelling 52
4.3 Underseacablelaying 53
4.4 Modellingandanalysis 54
4.5 Sourcesandfurtherreading 58
4.6 Exercises 58
v
vi Contents
5 Casestudy:thethermistor(1) 63
5.1 Heatandcurrentflowinthermistors 63
5.2 Nondimensionalisation 66
5.3 Athermistorinacircuit 67
5.4 Sourcesandfurtherreading 69
5.5 Exercises 69
6 Casestudy:electrostaticpainting 72
6.1 Electrostaticpainting 72
6.2 Fieldequations 73
6.3 Boundaryconditions 75
6.4 Nondimensionalisation 76
6.5 Sourcesandfurtherreading 77
6.6 Exercises 77
PartII Analyticaltechniques
7 Partialdifferentialequations 81
7.1 First-orderquasilinearpartialdifferentialequations:theory 81
7.2 Example:Poissonprocesses 85
7.3 Shocks 87
7.4 Fullynonlinearequations:Charpitt’smethod 90
7.5 Second-orderlinearequationsintwovariables 94
7.6 Furtherreading 97
7.7 Exercises 97
8 Casestudy:trafficmodelling 104
8.1 Simplemodelsfortrafficflow 104
8.2 Trafficjamsandotherdiscontinuoussolutions 107
8.3 Moresophisticatedmodels 110
8.4 Sourcesandfurtherreading 111
8.5 Exercises 111
9 Thedeltafunctionandotherdistributions 114
9.1 Introduction 114
9.2 Apointforceonastretchedstring;impulses 115
9.3 InformaldefinitionofthedeltaandHeavisidefunctions 117
9.4 Examples 120
9.5 Balancingsingularities 122
9.6 Green’sfunctions 125
9.7 Sourcesandfurtherreading 134
9.8 Exercises 134
Contents vii
10 Theoryofdistributions 140
10.1 Testfunctions 140
10.2 Theactionofatestfunction 141
10.3 Definitionofadistribution 142
10.4 Furtherpropertiesofdistributions 143
10.5 Thederivativeofadistribution 143
10.6 Extensionsofthetheoryofdistributions 145
10.7 Sourcesandfurtherreading 148
10.8 Exercises 148
11 Casestudy:thepantograph 157
11.1 Whatisapantograph? 157
11.2 Themodel 158
11.3 Impulsiveattachmentforanundampedpantograph 160
11.4 Solutionnearasupport 162
11.5 Solutionforawholespan 164
11.6 Sourcesandfurtherreading 167
11.7 Exercises 167
PartIII Asymptotictechniques
12 Asymptoticexpansions 173
12.1 Introduction 173
12.2 Ordernotation 175
12.3 Convergenceanddivergence 178
12.4 Sourcesandfurtherreading 180
12.5 Exercises 180
13 Regularperturbationexpansions 183
13.1 Introduction 183
13.2 Example:stabilityofaspacecraftinorbit 184
13.3 Linearstability 185
13.4 Example:thependulum 188
13.5 Smallperturbationsofaboundary 190
13.6 Caveatexpandator 194
13.7 Exercises 195
14 Casestudy:electrostaticpainting(2) 200
14.1 Smallparametersintheelectropaintmodel 200
14.2 Exercises 202
15 Casestudy:pianotuning 205
15.1 Thenotesofapiano:thetonalsystemofWesternmusic 205
viii Contents
15.2 Tuninganidealpiano 208
15.3 Arealpiano 209
15.4 Sourcesandfurtherreading 211
15.5 Exercises 211
16 Boundarylayers 216
16.1 Introduction 216
16.2 Functionswithboundarylayers;matching 216
16.3 Examplesfromordinarydifferentialequations 221
16.4 Casestudy:cablelaying 224
16.5 Examplesforpartialdifferentialequations 225
16.6 Exercises 230
17 Casestudy:thethermistor(2) 235
17.1 Stronglytemperature-dependentconductivity 235
17.2 Exercises 238
18 ‘Lubricationtheory’analysisinlongthindomains 240
18.1 ‘Lubricationtheory’approximations:slendergeometries 240
18.2 Heatflowinabarofvariablecross-section 241
18.3 Heatflowinalongthindomainwithcooling 244
18.4 Advection–diffusioninalongthindomain 246
18.5 Exercises 249
19 Casestudy:continuouscastingofsteel 255
19.1 Continuouscastingofsteel 255
19.2 Exercises 260
20 Lubricationtheoryforfluids 263
20.1 Thinfluidlayers:classicallubricationtheory 263
20.2 Thinviscousfluidsheetsonsolidsubstrates 265
20.3 Thinfluidsheetsandfibres 271
20.4 Furtherreading 275
20.5 Exercises 275
21 Casestudy:turningofeggsduringincubation 285
21.1 Incubatingeggs 285
21.2 Modelling 286
21.3 Exercises 290
22 Multiplescalesandothermethodsfornonlinearoscillators 292
22.1 ThePoincare´–Linstedtmethod 292
22.2 Themethodofmultiplescales 294