(cid:2) AdvancedNumericalandSemi-Analytical MethodsforDifferentialEquations (cid:2) (cid:2) (cid:2) (cid:2) Advanced Numerical and Semi-Analytical Methods for Differential Equations SnehashishChakraverty,NishaRaniMahato, PerumandlaKarunakar,andTharasiDilleswarRao (cid:2) (cid:2) NationalInstituteofTechnology Rourkela,Odisha,India (cid:2) (cid:2) Thiseditionfirstpublished2019 ©2019JohnWiley&Sons,Inc. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfrom thistitleisavailableathttp://www.wiley.com/go/permissions. TherightofSnehashishChakraverty,NishaRaniMahato,PerumandlaKarunakar,andTharasi DilleswarRaotobeidentifiedastheauthorsofthisworkhasbeenassertedinaccordance withlaw. 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LibraryofCongressCataloging-in-PublicationDataisAppliedFor HardbackISBN:9781119423423 Coverdesign:Wiley Coverimage:CourtesyofSnehashishChakraverty Setin10/12ptWarnockProbySPiGlobal,Chennai,India PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 (cid:2) (cid:2) 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 (cid:2) (cid:2) vi Contents 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 (cid:2) (cid:2) Contents vii 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 (cid:2) (cid:2) viii Contents 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 (cid:2) (cid:2) 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 (cid:2) (cid:2) 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 (cid:2) (cid:2) 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 (cid:2)