The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science

435 Pages·2017·11.526 MB·English

by  Carl F. Lorenzo, Tom T. Hartley

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TheFractionalTrigonometry
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The Fractional Trigonometry
WithApplicationstoFractional
DifferentialEquationsandScience
CarlF.Lorenzo
NationalAeronauticsandSpaceAdministration
GlennResearchCenter
Cleveland,Ohio
TomT.Hartley
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TheUniversityofAkron
Akron,Ohio
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Copyright©2017byJohnWiley&Sons,Inc.Allrightsreserved
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LibraryofCongressCataloging-in-PublicationData:
Names:Lorenzo,CarlF.|Hartley,T.T.(TomT.),1964-
Title:Thefractionaltrigonometry:withapplicationstofractional
differentialequationsandscience/CarlF.Lorenzo,NationalAeronautics
andSpaceAdministration,GlennResearchCenter,Cleveland,Ohio,TomT.
Hartley,TheUniversityofAkron,Akron,Ohio.
Description:Hoboken,NewJersey:JohnWiley&Sons,Inc.,[2017]|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2016027837(print)|LCCN2016028337(ebook)|ISBN
9781119139409(cloth)|ISBN9781119139423(pdf)|ISBN9781119139430
(epub)
Subjects:LCSH:Fractionalcalculus.|Trigonometry.
Classification:LCCQA314.L672017(print)|LCCQA314(ebook)|DDC
516.24–dc23
LCrecordavailableathttps://lccn.loc.gov/2016027837
Setin10/12ptWarnockbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
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v
Contents
Preface xv
Acknowledgments xix
AbouttheCompanionWebsite xxi
1 Introduction 1
1.1 Background 2
1.2 TheFractionalIntegralandDerivative 3
1.2.1 GrŭnwaldDefinition 3
1.2.2 Riemann–LiouvilleDefinition 4
1.2.3 TheNatureoftheFractional-OrderOperator 5
1.3 TheTraditionalTrigonometry 6
(cid:2) 1.4 PreviousEfforts 8 (cid:2)
1.5 ExpectationsofaGeneralizedTrigonometryandHyperboletry 8
2 TheFractionalExponentialFunctionviatheFundamentalFractional
DifferentialEquation 9
2.1 TheFundamentalFractionalDifferentialEquation 9
2.2 TheGeneralizedImpulseResponseFunction 10
2.3 RelationshipoftheF-functiontotheMittag-LefflerFunction 11
2.4 PropertiesoftheF-Function 12
2.5 BehavioroftheF-FunctionastheParameteraVaries 13
2.6 Example 16
3 TheGeneralizedFractionalExponentialFunction:TheR-FunctionandOther
FunctionsfortheFractionalCalculus 19
3.1 Introduction 19
3.2 FunctionsfortheFractionalCalculus 19
3.2.1 Mittag-Leffler’sFunction 20
3.2.2 Agarwal’sFunction 20
3.2.3 Erdelyi’sFunction 20
3.2.4 OldhamandSpanier’s,Hartley’s,andMatignon’sFunction 20
3.2.5 Robotnov’sFunction 21
3.2.6 MillerandRoss’sFunction 21
3.2.7 GorenfloandMainardi’s,andPodlubny’sFunction 21
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vi Contents
3.3 TheR-Function:AGeneralizedFunction 22
3.4 PropertiesoftheR (a,t)-Function 23
q,v
3.4.1 DifferintegrationoftheR-Function 23
3.4.2 RelationshipBetweenR andR 25
q,mq q,0
3.4.3 Fractional-OrderImpulseFunction 27
3.5 RelationshipoftheR-FunctiontotheElementaryFunctions 27
3.5.1 ExponentialFunction 27
3.5.2 SineFunction 27
3.5.3 CosineFunction 28
3.5.4 HyperbolicSineandCosine 28
3.6 R-FunctionIdentities 29
3.6.1 Trigonometric-BasedIdentities 29
3.6.2 FurtherIdentities 30
3.7 RelationshipoftheR-FunctiontotheFractionalCalculusFunctions 31
3.7.1 Mittag-Leffler’sFunction 31
3.7.2 Agarwal’sFunction 31
3.7.3 Erdelyi’sFunction 31
3.7.4 OldhamandSpanier’s,andHartley’sFunction 31
3.7.5 MillerandRoss’sFunction 32
3.7.6 Robotnov’sFunction 32
3.7.7 GorenfloandMainardi’s,andPodlubny’sFunction 32
3.8 Example:CoolingManifold 32
3.9 FurtherGeneralizedFunctions:TheG-FunctionandtheH-Function 34
(cid:2) 3.9.1 TheG-Function 34 (cid:2)
3.9.2 TheH-Function 36
3.10 PreliminariestotheFractionalTrigonometryDevelopment 38
3.11 EigenCharacteroftheR-Function 38
3.12 FractionalDifferintegraloftheTimeScaledR-Function 39
3.13 R-FunctionRelationships 39
3.14 RootsofComplexNumbers 40
3.15 IndexedFormsoftheR-Function 41
3.15.1 R-FunctionwithComplexArgument 41
3.15.2 IndexedFormsoftheR-Function 42
3.15.2.1 ComplexityForm 42
3.15.2.2 ParityForm 43
3.16 Term-by-TermOperations 44
3.17 Discussion 46
4 R-FunctionRelationships 47
4.1 R-FunctionBasics 47
4.2 RelationshipsforR inTermsofR 48
m,0 1,0
4.3 RelationshipsforR inTermsofR 50
1∕m,0 1,0
4.4 RelationshipsfortheRationalFormR inTermsofR 51
m∕p,0 1∕p,0
4.5 RelationshipsforR inTermsofR 53
1∕p,0 m∕p,0
4.6 RelatingR totheExponentialFunctionR (b, t)=ebt 54
m∕p,0 1,0
4.7 InverseRelationships–RelationshipsforR inTermsofR 56
1,0 m,k
4.8 InverseRelationships–RelationshipsforR inTermsofR 57
1,0 1∕m,0
4.9 InverseRelationships–Relationshipsforeat =R (a,t)inTermsofR 59
1,0 m∕p,0
4.10 Discussion 61
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Contents vii
5 TheFractionalHyperboletry 63
5.1 TheFractionalR -HyperbolicFunctions 63
1
5.2 R -HyperbolicFunctionRelationship 72
1
5.3 FractionalCalculusOperationsontheR -HyperbolicFunctions 72
1
5.4 LaplaceTransformsoftheR -HyperbolicFunctions 73
1
5.5 Complexity-BasedHyperbolicFunctions 73
5.6 FractionalHyperbolicDifferentialEquations 74
5.7 Example 76
5.8 Discussions 77
6 TheR -FractionalTrigonometry 79
1
6.1 R -TrigonometricFunctions 79
1
6.1.1 R -TrigonometricProperties 81
1
6.2 R -TrigonometricFunctionInterrelationship 88
1
6.3 RelationshipstoR -HyperbolicFunctions 89
1
6.4 FractionalCalculusOperationsontheR -TrigonometricFunctions 89
1
6.5 LaplaceTransformsoftheR -TrigonometricFunctions 90
1
6.5.1 LaplaceTransformofR Cos (a, k, t) 90
1 q.v
6.5.2 LaplaceTransformofR Sin (a, k, t) 91
1 q.v
6.6 Complexity-BasedR -TrigonometricFunctions 92
1
6.7 FractionalDifferentialEquations 94
7 TheR -FractionalTrigonometry 97
2
7.1 R -TrigonometricFunctions:BasedonRealandImaginaryParts 97
2
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7.2 R -TrigonometricFunctions:BasedonParity 102
2
7.3 LaplaceTransformsoftheR -TrigonometricFunctions 111
2
7.3.1 R Cos (a, k, t) 111
2 q,v
7.3.2 R Sin (a, k, t) 112
2 q,v
7.3.3 L{R Cofl (a, k, t)} 113
2 q,v
7.3.4 L{R Flut (a, k, t)} 113
2 q,v
7.3.5 L{R Covib (a, k, t)} 113
2 q,v
7.3.6 L{R Vib (a, k, t)} 113
2 q,v
7.4 R -TrigonometricFunctionRelationships 113
2
7.4.1 R Cos (a, k, t)andR Sin (a, k, t)RelationshipsandFractionalEuler
2 q,v 2 q,v
Equation 114
7.4.2 R Rot (a, t)andR Cor (a, t)Relationships 116
2 q,v 2 q,v
7.4.3 R Cofl (a, t)andR Flut (a, t)Relationships 116
2 q,v 2 q,v
7.4.4 R Covib (a, t)andR Vib (a, t)Relationships 118
2 q,v 2 q,v
7.5 FractionalCalculusOperationsontheR -TrigonometricFunctions 119
2
7.5.1 R Cos (a, k, t) 119
2 q,v
7.5.2 R Sin (a, k, t) 121
2 q,v
7.5.3 R Cor (a, t) 122
2 q,v
7.5.4 R Rot (a, t) 122
2 q,v
7.5.5 R Coflut (a, t) 122
2 q,v
7.5.6 R Flut (a, k, t) 123
2 q,v
7.5.7 R Covib (a, k, t) 123
2 q,v
7.5.8 R Vib (a, k, t) 124
2 q,v
7.5.9 SummaryofFractionalCalculusOperationsontheR -Trigonometric
2
Functions 124
7.6 InferredFractionalDifferentialEquations 127
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viii Contents
8 TheR -TrigonometricFunctions 129
3
8.1 TheR -TrigonometricFunctions:BasedonComplexity 129
3
8.2 TheR -TrigonometricFunctions:BasedonParity 134
3
8.3 LaplaceTransformsoftheR -TrigonometricFunctions 140
3
8.4 R -TrigonometricFunctionRelationships 141
3
8.4.1 R Cos (a,t)andR Sin (a,t)RelationshipsandFractionalEulerEquation 142
3 q,v 3 q,v
8.4.2 R Rot (a,t)andR Cor (a,t)Relationships 143
3 q,v 3 q,v
8.4.3 R Cofl (a,t)andR Flut (a,t)Relationships 144
3 q,v 3 q,v
8.4.4 R Covib (a,t)andR Vib (a,t)Relationships 145
3 q,v 3 q,v
8.5 FractionalCalculusOperationsontheR -TrigonometricFunctions 146
3
8.5.1 R Cos (a,k,t) 146
3 q,v
8.5.2 R Sin (a,k,t) 148
3 q,v
8.5.3 R Cor (a,t) 149
3 q,v
8.5.4 R Rot (a,t) 150
3 q,v
8.5.5 R Coflut (a,k,t) 150
3 q,v
8.5.6 R Flut (a,k,t) 152
3 q,v
8.5.7 R Covib (a,k,t) 153
3 q,v
8.5.8 R Vib (a,k,t) 154
3 q,v
8.5.9 SummaryofFractionalCalculusOperationsontheR -Trigonometric
3
Functions 157
9 TheFractionalMeta-Trigonometry 159
9.1 TheFractionalMeta-TrigonometricFunctions:BasedonComplexity 160
(cid:2) 9.1.1 AlternateForms 161 (cid:2)
9.1.2 GraphicalPresentation–ComplexityFunctions 161
9.2 TheMeta-FractionalTrigonometricFunctions:BasedonParity 166
9.3 CommutativePropertiesoftheComplexityandParityOperations 179
9.3.1 GraphicalPresentation–ParityFunctions 181
9.4 LaplaceTransformsoftheFractionalMeta-TrigonometricFunctions 188
9.5 R-FunctionRepresentationoftheFractionalMeta-TrigonometricFunctions 192
9.6 FractionalCalculusOperationsontheFractionalMeta-Trigonometric
Functions 195
9.6.1 Cos (a,𝛼,𝛽,k,t) 195
q,v
9.6.2 Sin (a,𝛼,𝛽,k,t) 197
q,v
9.6.3 Cor (a,𝛼,𝛽,t) 198
q,v
9.6.4 Rot (a,𝛼,𝛽,t) 198
q,v
9.6.5 Coflut (a,𝛼,𝛽,k,t) 199
q,v
9.6.6 Flut (a,𝛼,𝛽,k,t) 200
q,v
9.6.7 Covib (a,𝛼,𝛽,k,t) 202
q,v
9.6.8 Vib (a,𝛼,𝛽,k,t) 203
q,v
9.6.9 SummaryofFractionalCalculusOperationsontheMeta-Trigonometric
Functions 204
9.7 SpecialTopicsinFractionalDifferintegration 206
9.8 Meta-TrigonometricFunctionRelationships 206
9.8.1 Cos (a,𝛼,𝛽,t)andSin (a,𝛼,𝛽,t)Relationships 206
q,v q,v
9.8.2 Cor (a,𝛼,𝛽,t)andRot (a,𝛼,𝛽,t)Relationships 207
q,v q,v
9.8.3 Covib (a,𝛼,𝛽,t)andVib (a,𝛼,𝛽,t)Relationships 208
q,v q,v
9.8.4 Cofl (a,𝛼,𝛽,t)andFlut (a,𝛼,𝛽,t)Relationships 208
q,v q,v
9.8.5 Cofl (a,𝛼,𝛽,t)andVib (a,𝛼,𝛽,t)Relationships 209
q,v q,v
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Contents ix
9.8.6 Cos (a,𝛼,𝛽,t)andSin (a,𝛼,𝛽,t)RelationshipstoOtherFunctions 211
q,v q,v
9.8.7 Meta-IdentitiesBasedontheInteger-orderTrigonometricIdentities 211
9.8.7.1 Thecos(−x)=cos(x)-BasedIdentityforCos (a,𝛼,𝛽,t) 211
q,v
9.8.7.2 Thesin(−x)=−sin(x)-BasedIdentityfor Sin (a,𝛼,𝛽,t) 212
q,v
9.8.7.3 TheCos (a,𝛼,𝛽,t)⇔Sin (a,𝛼,𝛽,t)Identity 212
q,v q,v
9.8.7.4 Thesin(x)=sin(x±m𝜋/2)-BasedIdentityforSin (a,𝛼,𝛽,t) 213
q,v
9.9 FractionalPoles:StructureoftheLaplaceTransforms 214
9.10 CommentsandIssuesRelativetotheMeta-TrigonometricFunctions 214
9.11 BackwardCompatibilitytoEarlierFractionalTrigonometries 215
9.12 Discussion 215
10 TheRatioandReciprocalFunctions 217
10.1 FractionalComplexityFunctions 217
10.2 TheParityReciprocalFunctions 219
10.3 TheParityRatioFunctions 221
10.4 R-FunctionRepresentationoftheFractionalRatioandReciprocalFunctions 225
10.5 Relationships 226
10.6 Discussion 227
11 FurtherGeneralizedFractionalTrigonometries 229
11.1 TheG-Function-BasedTrigonometry 229
11.2 LaplaceTransformsfortheG-TrigonometricFunctions 230
11.3 TheH-Function-BasedTrigonometry 234
11.4 LaplaceTransformsfortheH-TrigonometricFunctions 235
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IntroductiontoApplications 241
12 TheSolutionofLinearFractionalDifferentialEquationsBasedon
theFractionalTrigonometry 243
12.1 FractionalDifferentialEquations 243
12.2 FundamentalFractionalDifferentialEquationsoftheFirstKind 245
12.3 FundamentalFractionalDifferentialEquationsoftheSecondKind 246
12.4 Preliminaries–LaplaceTransforms 246
12.4.1 FractionalCosineFunction 246
12.4.2 FractionalSineFunction 248
12.4.3 Higher-OrderNumeratorDynamics 248
12.4.3.1 FractionalCosineFunction 248
12.4.3.2 FractionalSineFunction 248
12.4.4 ParityFunctions–TheFlutterFunction 249
12.4.5 AdditionalTransformPairs 250
12.5 FractionalDifferentialEquationsofHigherOrder:UnrepeatedRoots 250
12.6 FractionalDifferentialEquationsofHigherOrder:ContainingRepeatedRoots 252
12.6.1 RepeatedRealFractionalRoots 252
12.6.2 RepeatedComplexFractionalRoots 253
12.7 FractionalDifferentialEquationsContainingRepeatedRoots 253
12.8 FractionalDifferentialEquationsofNon-CommensurateOrder 254
12.9 IndexedFractionalDifferentialEquations:MultipleSolutions 255
12.10 Discussion 256
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x Contents
13 FractionalTrigonometricSystems 259
13.1 TheR-FunctionasaLinearSystem 259
13.2 R-SystemTimeResponses 260
13.3 R-Function-BasedFrequencyResponses 260
13.4 Meta-TrigonometricFunction-BasedFrequencyResponses 261
13.5 FractionalMeta-Trigonometry 264
13.6 ElementaryFractionalTransferFunctions 266
13.7 StabilityTheorem 266
13.8 StabilityofElementaryFractionalTransferFunctions 267
13.9 InsightsintotheBehavioroftheFractionalMeta-TrigonometricFunctions 268
13.9.1 ComplexityFunctionStability 268
13.9.2 ParityFunctionStability 269
13.10 Discussion 270
14 NumericalIssuesandApproximationsintheFractionalTrigonometry 271
14.1 R-FunctionConvergence 271
14.2 TheMeta-TrigonometricFunctionConvergence 272
14.3 UniformConvergence 273
14.4 NumericalIssuesintheFractionalTrigonometry 274
14.5 TheR Cos-andR Sin-FunctionAsymptoticBehavior 275
2 2
14.6 R-FunctionApproximations 276
14.7 TheNear-OrderEffect 279
14.8 High-PrecisionSoftware 281
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15 TheFractionalSpiralFunctions:FurtherCharacterizationoftheFractional
Trigonometry 283
15.1 TheFractionalSpiralFunctions 283
15.2 AnalysisofSpirals 288
15.2.1 DescriptionsofSpirals 288
15.2.1.1 PolarDescription 289
15.2.1.2 ParametricDescription 290
15.2.1.3 Definitions 293
15.2.1.4 AlternateDefinitions 294
15.2.1.5 Examples 294
15.2.2 SpiralLengthandGrowth/DecayRates 294
15.2.2.1 SpiralLength 294
15.2.2.2 SpiralGrowthRatesforccwSpirals 295
15.2.2.3 ComponentGrowthRates 296
15.2.3 ScalingofSpirals 296
15.2.3.1 UniformRectangularScaling 297
15.2.3.2 NonuniformRectangularScaling 297
15.2.3.3 PolarScaling 298
15.2.3.4 RadialScaling 298
15.2.3.5 AngularScaling 298
15.2.4 SpiralVelocities 299
15.2.5 ReferencedSpirals:Retardation 301
15.3 RelationtotheClassicalSpirals 303
15.3.1 ClassicalSpirals 303
15.4 Discussion 307
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Contents xi
16 FractionalOscillators 309
16.1 TheSpaceofLinearFractionalOscillators 309
16.1.1 ComplexityFunction-BasedOscillators 310
16.1.2 ParityFunction-BasedOscillators 311
16.1.3 IntrinsicOscillatorDamping 312
16.2 CoupledFractionalOscillators 314
17 ShellMorphologyandGrowth 317
17.1 Nautiluspompilius 317
17.1.1 Introduction 317
17.1.2 NautilusMorphology 318
17.1.2.1 FractionalDifferentialEquations 323
17.1.3 SpiralLength 325
17.1.4 MorphologyoftheSiphuncleSpiral 325
17.1.5 FractionalGrowthRate 325
17.1.6 NautilusStudySummary 328
17.2 Shell5 329
17.3 Shell6 330
17.4 Shell7 332
17.5 Shell8 332
17.6 Shell9 336
17.7 Shell10 336
17.8 Ammonite 339
17.9 Discussion 340
(cid:2) (cid:2)
18 MathematicalClassificationoftheSpiralandRingGalaxyMorphologies 341
18.1 Introduction 341
18.2 Background–FractionalSpiralsforGalacticClassification 342
18.3 ClassificationProcess 347
18.3.1 SymmetryAssumption 347
18.3.2 GalaxyImagetoSpiralorSpiraltoGalaxyImage 347
18.3.3 Inclination 347
18.3.4 DataPresentation 348
18.4 MathematicalClassificationofSelectedGalaxies 350
18.4.1 NGC4314SB(rs)a 350
18.4.2 NGC1365SBb/SBc/SB(s)b/SBb(s) 350
18.4.3 M95SB(r)b/SBb(r)/SBa/SBb 350
18.4.4 NGC2997Sc/SAB(rs)c/Sc(s) 353
18.4.5 NGC4622(R′)SA(r)apec/Sb 353
18.4.6 M66orNGC3627SAB(s)b/Sb(s)/Sb 353
18.4.7 NGC4535SAB(s)c/SB(s)c/Sc/SBc 355
18.4.8 NGC1300SBc/SBb(s)/SB(rs)bc 355
18.4.9 Hoag’sObject 358
18.4.10 M51Sa+Sc 358
18.4.11 AM0644-741Sc/Stronglypeculiar/ 359
18.4.12 ESO269-G57(R′)SAB(r)ab/Sa(r) 360
18.4.13 NGC1313SBc/SB(s)d/SB(s)d 362
18.4.14 CarbonStar3068 362
18.5 Analysis 362
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