Mathematical Modeling and Simulation - Introduction for Scientists and Engineers

365 Pages·2009·3.16 MB·English

by  Kai Velten

Kai Velten
Mathematical Modeling
and Simulation
Introduction for Scientists and Engineers

KaiVelten
MathematicalModeling
andSimulation
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Kai Velten
Mathematical Modeling
and Simulation
Introduction for Scientists and Engineers
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V
Contents
Preface XIII
1 PrinciplesofMathematicalModeling 1
1.1 AComplexWorldNeedsModels 1
1.2 Systems,Models,Simulations 3
1.2.1 TeleologicalNatureofModelingandSimulation 4
1.2.2 ModelingandSimulationScheme 4
1.2.3 Simulation 7
1.2.4 System 7
1.2.5 ConceptualandPhysicalModels 8
1.3 MathematicsasaNaturalModelingLanguage 9
1.3.1 Input–OutputSystems 9
1.3.2 GeneralFormofExperimentalData 10
1.3.3 DistinguishedRoleofNumericalData 10
1.4 DefinitionofMathematicalModels 11
1.5 ExamplesandSomeMoreDefinitions 13
1.5.1 StateVariablesandSystemParameters 15
1.5.2 UsingComputerAlgebraSoftware 18
1.5.3 TheProblemSolvingScheme 19
1.5.4 StrategiestoSetupSimpleModels 20
1.5.4.1 MixtureProblem 24
1.5.4.2 TankLabelingProblem 27
1.5.5 LinearProgramming 30
1.5.6 ModelingaBlackBoxSystem 31
1.6 EvenMoreDefinitions 34
1.6.1 PhenomenologicalandMechanisticModels 34
1.6.2 StationaryandInstationarymodels 38
1.6.3 DistributedandLumpedmodels 38
1.7 ClassificationofMathematicalModels 39
1.7.1 FromBlacktoWhiteBoxModels 40
1.7.2 SQMSpaceClassification:SAxis 41
1.7.3 SQMSpaceClassification:QAxis 42
VI Contents
1.7.4 SQMSpaceClassification:MAxis 43
1.8 EverythingLooksLikeaNail? 45
2 PhenomenologicalModels 47
2.1 ElementaryStatistics 48
2.1.1 DescriptiveStatistics 48
2.1.1.1 UsingCalc 49
2.1.1.2 UsingtheRCommander 51
2.1.2 RandomProcessesandProbability 52
2.1.2.1 RandomVariables 53
2.1.2.2 Probability 53
2.1.2.3 DensitiesandDistributions 55
2.1.2.4 TheUniformDistribution 57
2.1.2.5 TheNormalDistribution 57
2.1.2.6 ExpectedValueandStandardDeviation 58
2.1.2.7 MoreonDistributions 60
2.1.3 InferentialStatistics 60
2.1.3.1 IsCropA’sYieldReallyHigher? 61
2.1.3.2 StructureofaHypothesisTest 61
2.1.3.3 Thettest 62
2.1.3.4 TestingRegressionParameters 63
2.1.3.5 AnalysisofVariance 63
2.2 LinearRegression 65
2.2.1 TheLinearRegressionProblem 65
2.2.2 SolutionUsingSoftware 66
2.2.3 TheCoefficientofDetermination 68
2.2.4 InterpretationoftheRegressionCoefficients 70
2.2.5 UnderstandingLinRegEx1.r 70
2.2.6 NonlinearLinearRegression 72
2.3 MultipleLinearRegression 74
2.3.1 TheMultipleLinearRegressionProblem 74
2.3.2 SolutionUsingSoftware 76
2.3.3 Cross-Validation 78
2.4 NonlinearRegression 80
2.4.1 TheNonlinearRegressionProblem 80
2.4.2 SolutionUsingSoftware 81
2.4.3 MultipleNonlinearRegression 83
2.4.4 ImplicitandVector-ValuedProblems 86
2.5 NeuralNetworks 87
2.5.1 GeneralIdea 87
2.5.2 Feed-ForwardNeuralNetworks 89
2.5.3 SolutionUsingSoftware 91
2.5.4 InterpretationoftheResults 92
2.5.5 GeneralizationandOverfitting 95
2.5.6 SeveralInputsExample 97
Contents VII
2.6 DesignofExperiments 99
2.6.1 CompletelyRandomizedDesign 100
2.6.2 RandomizedCompleteBlockDesign 103
2.6.3 LatinSquareandMoreAdvancedDesigns 104
2.6.4 FactorialDesigns 106
2.6.5 OptimalSampleSize 108
2.7 OtherPhenomenologicalModelingApproaches 109
2.7.1 SoftComputing 109
2.7.1.1 FuzzyModelofaWashingMachine 110
2.7.2 DiscreteEventSimulation 111
2.7.3 SignalProcessing 113
3 MechanisticModelsI:ODEs 117
3.1 DistinguishedRoleofDifferentialEquations 117
3.2 IntroductoryExamples 118
3.2.1 ArchaeologyAnalogy 118
3.2.2 BodyTemperature 120
3.2.2.1 PhenomenologicalModel 120
3.2.2.2 Application 121
3.2.3 AlarmClock 122
3.2.3.1 NeedforaMechanisticModel 122
3.2.3.2 ApplyingtheModelingandSimulationScheme 123
3.2.3.3 SettingUptheEquations 125
3.2.3.4 ComparingModelandData 126
3.2.3.5 ValidationFails – WhatNow? 127
3.2.3.6 ADifferentWaytoExplaintheTemperatureMemory 128
3.2.3.7 LimitationsoftheModel 129
3.3 GeneralIdeaofODE’s 130
3.3.1 IntrinsicMeaningofπ 130
3.3.2 exSolvesanODE 130
3.3.3 InfinitelyManyDegreesofFreedom 131
3.3.4 IntrinsicMeaningoftheExponentialFunction 132
3.3.5 ODEsasaFunctionGenerator 134
3.4 SettingUpODEModels 135
3.4.1 BodyTemperatureExample 135
3.4.1.1 FormulationofanODEModel 135
3.4.1.2 ODERevealstheMechanism 136
3.4.1.3 ODE’sConnectDataandTheory 137
3.4.1.4 ThreeWaystoSetupODEs 138
3.4.2 AlarmClockExample 139
3.4.2.1 ASystemofTwoODEs 139
3.4.2.2 ParameterValuesBasedonAprioriInformation 140
3.4.2.3 ResultofaHand-fit 141
3.4.2.4 ALookintotheBlackBox 142
3.5 SomeTheoryYouShouldKnow 143