Table Of Content(cid:2)
FractionalOrderAnalysis
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Fractional Order Analysis
Theory, Methods and Applications
Edited by
Hemen Dutta, Ahmet Ocak Akdemir, and
Abdon Atangana
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Thiseditionfirstpublished2020
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LibraryofCongressCataloging-in-PublicationData
Names:Dutta,Hemen,1981-editor.|Akdemir,AhmetOcak,1985-editor.|
Atangana,Abdon,editor.
Title:Fractionalorderanalysis:theory,methodsandapplications/
editedbyHemenDutta,AhmetOcakAkdemir,AbdonAtangana.
Description:Hoboken,NJ:Wiley,[2020]|Includesbibliographical
referencesandindex.
Identifiers:LCCN2020015384(print)|LCCN2020015385(ebook)|ISBN
9781119654162(cloth)|ISBN9781119654209(adobepdf)|ISBN
9781119654230(epub)
Subjects:LCSH:Fractionalcalculus.
Classification:LCCQA314.F7352020(print)|LCCQA314(ebook)|DDC
515/.83–dc23
LCrecordavailableathttps://lccn.loc.gov/2020015384
LCebookrecordavailableathttps://lccn.loc.gov/2020015385
CoverDesign:Wiley
CoverImage:©oxygen/GettyImages
Setin9.5/12.5ptSTIXTwoTextbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
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v
Contents
Preface xi
ListofContributors xv
AbouttheEditors xix
1 OntheFractionalDerivativeandIntegralOperators 1
MustafaA.Dokuyucu
1.1 Introduction 1
(cid:2) 1.2 FractionalDerivativeandIntegralOperators 2 (cid:2)
1.2.1 PropertiesoftheGrünwald–LetnikovFractionalDerivativeand
Integral 2
1.2.1.1 IntegralofArbitraryOrder 6
1.2.1.2 DerivativesofArbitraryOrder 7
1.2.2 PropertiesofRiemann–LiouvilleFractionalDerivativeandIntegral 9
1.2.2.1 UnificationofInteger-OrderDerivativesandIntegrals 10
1.2.2.2 IntegralsofArbitraryOrder 12
1.2.2.3 DerivativesofArbitraryOrder 14
1.3 PropertiesofCaputoFractionalDerivativeandIntegral 17
1.4 PropertiesoftheCaputo–FabrizioFractionalDerivativeand
Integral 20
1.5 PropertiesoftheAtangana–BaleanuFractionalDerivativeand
Integral 24
1.6 Applications 28
1.6.1 Keller–SegelModelwithCaputoDerivative 28
1.6.1.1 ExistenceandUniquenessSolutions 28
1.6.1.2 UniquenessofSolution 31
1.6.1.3 Keller–SegelModelwithAtangana–BaleanuDerivativeinCaputo
Sense 32
1.6.1.4 UniquenessofSolution 33
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1.6.2 CancerTreatmentModelwithCaputo-FabrizioFractional
Derivative 34
1.6.2.1 ExistenceSolutions 35
1.6.2.2 UniquenessSolutions 38
1.6.2.3 Conclusion 39
Bibliography 40
2 GeneralizedConformableFractionalOperatorsandTheir
Applications 43
MuhammadAdilKhanandTahirUllahKhan
2.1 IntroductionandPreliminaries 43
2.2 GeneralizedConformableFractionalIntegralOperators 46
2.2.1 ConstructionofNewIntegralOperators 47
2.3 GeneralizedConformableFractionalDerivative 52
2.4 ApplicationstoIntegralEquationsandFractionalDifferential
Equations 60
2.4.1 EquivalenceBetweentheGeneralizedNonlinearProblemandthe
VolterraIntegralEquation 61
2.4.2 ExistenceandUniquenessofSolutionfortheNonlinearProblem 61
2.5 ApplicationstotheFieldofInequalities 63
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2.5.1 InequalitiesRelatedtotheLeftSideofHermite–Hadamard
Inequality 65
2.5.1.1 ApplicationstoSpecialMeansofRealNumbers 74
2.5.1.2 ApplicationstotheMidpointFormula 75
2.5.2 InequalitiesRelatedtotheRightSideofHermite–Hadamard
Inequality 76
2.5.2.1 ApplicationstoSpecialMeansofRealNumbers 84
2.5.2.2 ApplicationstotheTrapezoidalFormula 84
Bibliography 86
3 AnalysisofNewTrendsofFractionalDifferential
Equations 91
AbdonAtanganaandAliAkgül
3.1 Introduction 91
3.2 Theory 92
3.3 Discretization 101
3.4 Experiments 103
3.5 StabilityAnalysis 104
3.6 Conclusion 110
Bibliography 111
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Contents vii
4 NewEstimationsforExponentiallyConvexityvia
ConformableFractionalOperators 113
AlperEkinciandSeverS.Dragomir
4.1 Introduction 113
4.2 MainResults 117
Bibliography 130
5 Lyapunov-typeInequalitiesforLocalFractionalProportional
Derivatives 133
ThabetAbdeljawad
5.1 Introduction 133
5.2 TheLocalFractionalProportionalDerivativesandTheirGenerated
NonlocalFractionalProportionalIntegralsandDerivatives 135
5.3 Lyapunov-TypeInequalitiesforSomeNonlocalandLocalFractional
Operators 137
5.4 TheLyapunovInequalityfortheSequentialLocalFractional
ProportionalBoundaryValueProblem 141
5.5 AHigher-OrderExtensionoftheLocalFractionalProportional
OperatorsandanAssociateLyapunovOpenProblem 144
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Acknowledgement 146
Bibliography 147
6 Minkowski-TypeInequalitiesforMixedConformable
FractionalIntegrals 151
ErhanSetandMuhametE.Özdemir
6.1 IntroductionandPreliminaries 151
6.2 ReverseMinkowskiInequalityInvolvingMixedConformable
FractionalIntegrals 158
6.3 RelatedInequalities 160
Bibliography 167
7 NewEstimationsforDifferentKindsofConvexFunctionsvia
ConformableIntegralsandRiemann–LiouvilleFractional
IntegralOperators 169
AhmetOcakAkdemirandHemenDutta
7.1 Introduction 169
7.2 SomeGeneralizationsforGeometricallyConvexFunctions 172
7.3 NewInequalitiesforCo-ordinatedConvexFunctions 179
Bibliography 191
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8 Legendre-SpectralAlgorithmsforSolvingSomeFractional
DifferentialEquations 195
YoussriH.YoussriandWaleedM.Abd-Elhameed
8.1 Introduction 195
8.2 SomePropertiesandRelationsConcernedwithShiftedLegendre
Polynomials 197
8.3 GalerkinApproachforTreatingFractionalTelegraphType
Equation 200
8.4 DiscussionoftheConvergenceandErrorAnalysisoftheSuggested
DoubleExpansion 204
8.5 SomeTestProblemsforFractionalTelegraphEquation 207
8.6 SpectralAlgorithmsforTreatingtheSpaceFractionalDiffusion
Problem 209
8.6.1 TransformationoftheProblem 210
8.6.2 BasisFunctionsSelection 211
8.6.3 ACollocationSchemeforSolvingEq.8.44 213
8.6.4 AnAlternativeSpectralPetrov–GalerkinSchemeforSolving
Eq.(8.44) 214
8.7 InvestigationofConvergenceandErrorAnalysis 214
8.8 NumericalResultsandComparisons 216
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8.9 Conclusion 220
Bibliography 220
9 MathematicalModelingofanAutonomousNonlinear
DynamicalSystemforMalariaTransmissionUsingCaputo
Derivative 225
AbdonAtanganaandSaniaQureshi
9.1 Introduction 225
9.2 MathematicalPreliminaries 227
9.3 ModelFormulation 228
9.4 BasicPropertiesoftheFractionalModel 230
9.4.1 ReproductiveNumber 230
9.4.2 ExistenceandStabilityofDisease-freeEquilibriumPoints 231
9.4.3 ExistenceandStabilityofEndemicEquilibriumPoint 232
9.5 ExistenceandUniquenessoftheSolutions 233
9.5.1 PositivityoftheSolutions 236
9.6 NumericalSimulations 237
9.7 Conclusion 247
Bibliography 250
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Contents ix
10 MHD-freeConvectionFlowOveraVerticalPlatewith
RampedWallTemperatureandChemicalReactioninViewof
NonsingularKernel 253
MuhammadB.Riaz,AbdonAtangana,andSyedT.Saeed
10.1 Introduction 253
10.2 MathematicalModel 254
10.2.1 Preliminaries 256
10.3 Solution 256
10.3.1 ConcentrationFields 257
10.3.1.1 ConcentrationFieldwithCaputoTime-FractionalDerivative 257
10.3.1.2 ConcentrationFieldwithCaputo–FabrizioTime-Fractional
Derivative 257
10.3.1.3 ConcentrationFieldwithAtangana–BaleanuTime-Fractional
Derivative 257
10.3.2 TemperatureFields 258
10.3.2.1 TemperatureFieldwithCaputoTime-FractionalDerivative 258
10.3.2.2 TemperatureFieldwithCaputo–FabrizioTime-Fractional
Derivative 258
10.3.2.3 TemperatureFieldwithAtangana–BaleanuTime-Fractional
Derivative 258
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10.3.3 VelocityFields 259
10.3.3.1 VelocityFieldwithCaputoTime-FractionalDerivative 259
10.3.3.2 VelocityFieldwithCaputo–FabrizioTime-FractionalDerivative 259
10.3.3.3 VelocityFieldwithAtangana–BaleanuTime-Fractional
Derivative 262
10.4 ResultsandDiscussion 263
10.5 Conclusion 263
Bibliography 279
11 ComparisonoftheDifferentFractionalDerivativesforthe
DynamicsofZikaVirus 283
MuhammadAltafKhan
11.1 Introduction 283
11.2 BackgroundofFractionalOperators 284
11.3 ModelFramework 286
11.4 AFractionalZikaModelwithDifferentFractionalDerivatives 287
11.5 NumericalSchemeforCaputo–FabrizioModel 288
11.5.1 SolutionsExistencefortheAtangana–BaleanuModel 289
11.5.2 NumericalSchemeforAtangana–BaleanuModel 291
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11.6 NumericalResults 293
11.7 Conclusion 303
Bibliography 303
Index 307
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