Advanced numerical and semi-analytical methods for differential equations

238 Pages·2019·1.598 MB·English

by  Chakraverty S

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AdvancedNumericalandSemi-Analytical
MethodsforDifferentialEquations
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Advanced Numerical and Semi-Analytical
Methods for Differential Equations
SnehashishChakraverty,NishaRaniMahato,
PerumandlaKarunakar,andTharasiDilleswarRao
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NationalInstituteofTechnology
Rourkela,Odisha,India
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Thiseditionfirstpublished2019
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HardbackISBN:9781119423423
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v
Contents
Acknowledgments xi
Preface xiii
1 BasicNumericalMethods 1
1.1 Introduction 1
1.2 OrdinaryDifferentialEquation 2
1.3 EulerMethod 2
1.4 ImprovedEulerMethod 5
(cid:2) 1.5 Runge–KuttaMethods 7 (cid:2)
1.5.1 MidpointMethod 7
1.5.2 Runge–KuttaFourthOrder 8
1.6 MultistepMethods 10
1.6.1 Adams–BashforthMethod 10
1.6.2 Adams–MoultonMethod 10
1.7 Higher-OrderODE 13
References 16
2 IntegralTransforms 19
2.1 Introduction 19
2.2 LaplaceTransform 19
2.2.1 SolutionofDifferentialEquationsUsingLaplaceTransforms 20
2.3 FourierTransform 25
2.3.1 SolutionofPartialDifferentialEquationsUsingFourier
Transforms 26
References 28
3 WeightedResidualMethods 31
3.1 Introduction 31
3.2 CollocationMethod 33
3.3 SubdomainMethod 35
3.4 Least-squareMethod 37
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3.5 GalerkinMethod 39
3.6 ComparisonofWRMs 40
References 42
4 BoundaryCharacteristicsOrthogonalPolynomials 45
4.1 Introduction 45
4.2 Gram–SchmidtOrthogonalizationProcess 45
4.3 GenerationofBCOPs 46
4.4 Galerkin’sMethodwithBCOPs 46
4.5 Rayleigh–RitzMethodwithBCOPs 48
References 51
5 FiniteDifferenceMethod 53
5.1 Introduction 53
5.2 FiniteDifferenceSchemes 53
5.2.1 FiniteDifferenceSchemesforOrdinaryDifferentialEquations 54
5.2.1.1 ForwardDifferenceScheme 54
5.2.1.2 BackwardDifferenceScheme 55
5.2.1.3 CentralDifferenceScheme 55
5.2.2 FiniteDifferenceSchemesforPartialDifferentialEquations 55
(cid:2) 5.3 ExplicitandImplicitFiniteDifferenceSchemes 55 (cid:2)
5.3.1 ExplicitFiniteDifferenceMethod 56
5.3.2 ImplicitFiniteDifferenceMethod 57
References 61
6 FiniteElementMethod 63
6.1 Introduction 63
6.2 FiniteElementProcedure 63
6.3 GalerkinFiniteElementMethod 65
6.3.1 OrdinaryDifferentialEquation 65
6.3.2 PartialDifferentialEquation 71
6.4 StructuralAnalysisUsingFEM 76
6.4.1 StaticAnalysis 76
6.4.2 DynamicAnalysis 78
References 79
7 FiniteVolumeMethod 81
7.1 Introduction 81
7.2 DiscretizationTechniquesofFVM 82
7.3 GeneralFormofFiniteVolumeMethod 82
7.3.1 SolutionProcessAlgorithm 83
7.4 One-DimensionalConvection–DiffusionProblem 84
7.4.1 GridGeneration 84
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7.4.2 SolutionProcedureofConvection–DiffusionProblem 84
References 89
8 BoundaryElementMethod 91
8.1 Introduction 91
8.2 BoundaryRepresentationandBackgroundTheoryofBEM 91
8.2.1 LinearDifferentialOperator 92
8.2.2 TheFundamentalSolution 93
8.2.2.1 HeavisideFunction 93
8.2.2.2 DiracDeltaFunction 93
8.2.2.3 FindingtheFundamentalSolution 94
8.2.3 Green’sFunction 95
8.2.3.1 Green’sIntegralFormula 95
8.3 DerivationoftheBoundaryElementMethod 96
8.3.1 BEMAlgorithm 96
References 100
9 Akbari–Ganji’sMethod 103
9.1 Introduction 103
9.2 NonlinearOrdinaryDifferentialEquations 104
(cid:2) 9.2.1 Preliminaries 104 (cid:2)
9.2.2 AGMApproach 104
9.3 NumericalExamples 105
9.3.1 UnforcedNonlinearDifferentialEquations 105
9.3.2 ForcedNonlinearDifferentialEquation 107
References 109
10 Exp-FunctionMethod 111
10.1 Introduction 111
10.2 BasicsofExp-FunctionMethod 111
10.3 NumericalExamples 112
References 117
11 AdomianDecompositionMethod 119
11.1 Introduction 119
11.2 ADMforODEs 119
11.3 SolvingSystemofODEsbyADM 123
11.4 ADMforSolvingPartialDifferentialEquations 125
11.5 ADMforSystemofPDEs 127
References 130
12 HomotopyPerturbationMethod 131
12.1 Introduction 131
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12.2 BasicIdeaofHPM 131
12.3 NumericalExamples 133
References 138
13 VariationalIterationMethod 141
13.1 Introduction 141
13.2 VIMProcedure 141
13.3 NumericalExamples 142
References 146
14 HomotopyAnalysisMethod 149
14.1 Introduction 149
14.2 HAMProcedure 149
14.3 NumericalExamples 151
References 156
15 DifferentialQuadratureMethod 157
15.1 Introduction 157
15.2 DQMProcedure 157
15.3 NumericalExamples 159
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References 165
16 WaveletMethod 167
16.1 Introduction 167
16.2 HaarWavelet 168
16.3 Wavelet–CollocationMethod 170
References 175
17 HybridMethods 177
17.1 Introduction 177
17.2 HomotopyPerturbationTransformMethod 177
17.3 LaplaceAdomianDecompositionMethod 182
References 186
18 PreliminariesofFractalDifferentialEquations 189
18.1 IntroductiontoFractal 189
18.1.1 TriadicKochCurve 190
18.1.2 SierpinskiGasket 190
18.2 FractalDifferentialEquations 191
18.2.1 HeatEquation 192
18.2.2 WaveEquation 194
References 194
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Contents ix
19 DifferentialEquationswithIntervalUncertainty 197
19.1 Introduction 197
19.2 IntervalDifferentialEquations 197
19.2.1 IntervalArithmetic 198
19.3 GeneralizedHukuharaDifferentiabilityofIDEs 198
19.3.1 ModelingIDEsbyHukuharaDifferentiability 199
19.3.1.1 SolvingbyIntegralForm 199
19.3.1.2 SolvingbyDifferentialForm 199
19.4 AnalyticalMethodsforIDEs 201
19.4.1 Generalformofnth-orderIDEs 202
19.4.2 MethodBasedonAdditionandSubtractionofIntervals 202
References 206
20 DifferentialEquationswithFuzzyUncertainty 209
20.1 Introduction 209
20.2 SolvingFuzzyLinearSystemofDifferentialEquations 209
20.2.1 𝛼-CutofTFN 209
20.2.2 FuzzyLinearSystemofDifferentialEquations(FLSDEs) 210
20.2.3 SolutionProcedureforFLSDE 211
References 215
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21 IntervalFiniteElementMethod 217
21.1 Introduction 217
21.1.1 Preliminaries 218
21.1.1.1 ProperandImproperInterval 218
21.1.1.2 IntervalSystemofLinearEquations 218
21.1.1.3 GeneralizedIntervalEigenvalueProblem 219
21.2 IntervalGalerkinFEM 219
21.3 StructuralAnalysisUsingIFEM 223
21.3.1 StaticAnalysis 223
21.3.2 DynamicAnalysis 225
References 227
Index 231
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xi
Acknowledgments
Thefirstauthorgreatlyappreciatesthepatience,support,andencouragement
providedbyhisfamilymembers,inparticular,hiswifeShewli,anddaughters
Shreyati and Susprihaa. The book may not have been possible without the
blessings of his parents late Sh. Birendra K. Chakraborty and Smt. Parul
Chakraborty.Thesecondauthor’swarmestgratitudegoestoherfamilymem-
bers for their continuous motivation and support, especially Sh. Devendra
Mahato, Smt. Premshila, Tanuja, Devasish, and Satish. Further, the third
author would like to thank for the support and encouragement provided by
(cid:2) all his family members, in particular, his parents Sh. Veeraiah Perumandla (cid:2)
and Smt. Alivela Perumandla, and his wife Madhavi as well as sons Charan
SaiandHarshavardhan.Finally,thefourthauthorwouldliketoacknowledge
the blessings and motivation provided by his family members, especially his
parents Sh. Tharasi Rama Rao and Smt. Tharasi Mahalaxmi. Also second,
third,andfourthauthorsappreciatetheinspirationofthefirstauthorandhis
family.
Oursincereacknowledgmentgoestothereviewersfortheirfruitfulsugges-
tionsandappreciationsinthebookproposal.Further,alltheauthorsdoappre-
ciatethesupportandhelpofthewholeteamofWiley.Finally,wearegreatly
indebted to the authors/researchers mentioned in the bibliography sections
givenattheendofeachchapter.
S.Chakraverty
N.R.Mahato
P.Karunakar
T.D.Rao
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