Distribution Theory: With Applications in Engineering and Physics PDF
Preview Distribution Theory: With Applications in Engineering and Physics
PetreP.Teodorescu,WilhelmW.Kecs, andAntonelaToma DistributionTheory RelatedTitles Forbes,C.,Evans,M.,Hastings,N., Vaughn,M.T. Peacock,B. Introduction toMathematical StatisticalDistributions Physics 2011 2007 ISBN:978-0-470-39063-4 ISBN:978-3-527-40627-2 Fujita,S.,Godoy,S.V. Kusse,B.,Westwig,E.A. Mathematical Physics Mathematical Physics AppliedMathematicsforScientistsand 2009 Engineers ISBN:978-3-527-40808-5 2006 ISBN:978-3-527-40672-2 Masujima,M. AppliedMathematical Johnson,N.L.,Kemp,A.W.,Kotz,S. Methods inTheoretical UnivariateDiscrete Physics Distributions 2009 ISBN:978-3-527-40936-5 2005 ISBN:978-0-471-27246-5 Velten,K. Trigg,G.L.(ed.) Mathematical Modelingand Mathematical Tools for Simulation Physicists IntroductionforScientistsandEngineers 2009 2005 ISBN:978-3-527-40758-3 ISBN:978-3-527-40548-0 Bayin,S.S. Essentialsof Mathematical Methods inScienceand Engineering 2008 ISBN:978-0-470-34379-1 PetreP.Teodorescu,WilhelmW.Kecs,andAntonelaToma Distribution Theory With Applications in Engineering and Physics WILEY-VCH Verlag GmbH & Co. KGaA TheAuthors AllbookspublishedbyWiley-VCHarecarefully produced.Nevertheless,authors,editors,and Prof.PetreP.Teodorescu publisherdonotwarranttheinformation UniversityofBucharest containedinthesebooks,includingthisbook,to FacultyofMathematics befreeoferrors.Readersareadvisedtokeepin Bucharest,Romania mindthatstatements,data,illustrations, proceduraldetailsorotheritemsmay inadvertentlybeinaccurate. Prof.WilhelmW.Kecs UniversityofPetrosani LibraryofCongressCardNo.: FacultyofScience appliedfor Petrosani,Hunedoara,Romania [email protected] BritishLibraryCataloguing-in-PublicationData: Acataloguerecordforthisbookisavailable fromtheBritishLibrary. Prof.AntonelaToma University’Politehnica’ Bibliographicinformationpublishedbythe Bucharest,Romania DeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhis publicationintheDeutscheNationalbibliografie; CoverPicture detailedbibliographicdataareavailableonthe SpieszDesign,Neu-Ulm Internetathttp://dnb.d-nb.de. ©2013WILEY-VCHVerlagGmbH&Co.KGaA, Boschstr.12,69469Weinheim,Germany Allrightsreserved(includingthoseoftranslation intootherlanguages).Nopartofthisbookmay bereproducedinanyform–byphotoprinting, microfilm,oranyothermeans–nortransmitted ortranslatedintoamachinelanguagewithout writtenpermissionfromthepublishers. Registerednames,trademarks,etc.usedinthis book,evenwhennotspecificallymarkedas such,arenottobeconsideredunprotectedby law. PrintISBN 978-3-527-41083-5 ePDFISBN 978-3-527-65364-5 ePubISBN 978-3-527-65363-8 mobiISBN 978-3-527-65362-1 oBookISBN 978-3-527-65361-4 CoverDesign Grafik-DesignSchulz, Fußgönheim Typesetting le-texpublishingservicesGmbH, Leipzig PrintingandBinding MarkonoPrintMedia PteLtd,Singapore PrintedinSingapore Printedonacid-freepaper V Contents Preface XI 1 IntroductiontotheDistributionTheory 1 1.1 ShortHistory 1 1.2 FundamentalConceptsandFormulae 2 1.2.1 NormedVectorSpaces:MetricSpaces 3 1.2.2 SpacesofTestFunctions 6 1.2.2.1 TheSpaceDm(Ω) 9 1.2.2.2 TheSpaceD(Ω) 10 1.2.2.3 TheSpaceE 11 1.2.2.4 TheSpaceD (theSchwartzSpace) 11 1.2.2.5 TheSpaceS(theSpaceFunctionswhichDecreaseRapidly) 14 1.2.3 SpacesofDistributions 15 1.2.3.1 EqualityofTwoDistributions:SupportofaDistribution 20 1.2.4 CharacterizationTheoremsofDistributions 27 1.3 OperationswithDistributions 31 1.3.1 TheChangeofVariablesinDistributions 31 1.3.2 Translation,SymmetryandHomothetyofDistributions 36 1.3.3 DifferentiationofDistributions 40 1.3.3.1 PropertiesoftheDerivativeOperator 45 1.3.4 TheFundamentalSolutionofaLinearDifferentialOperator 58 1.3.5 TheDerivationoftheHomogeneousDistributions 62 1.3.6 DiracRepresentativeSequences:CriteriafortheRepresentativeDirac Sequences 69 1.3.7 DistributionsDependingonaParameter 81 1.3.7.1 DifferentiationofDistributionsDependingonaParameter 81 1.3.7.2 IntegrationofDistributionsDependingonaParameter 84 1.3.8 Direct Product and Convolution Product of Functions and Distributions 88 1.3.8.1 PropertiesoftheDirectProduct 90 1.3.8.2 TheConvolutionProductofDistributions 92 1.3.8.3 The Convolution of Distributions Depending on a Parameter: Properties 102 VI Contents 1.3.8.4 ThePartialConvolutionProductforFunctionsandDistributions 105 1.3.9 PartialConvolutionProductofFunctions 111 2 IntegralTransformsofDistributions 113 2.1 FourierSeriesandSeriesofDistributions 113 2.1.1 SequencesandSeriesofDistributions 113 2.1.2 ExpansionofDistributionsintoFourierSeries 116 2.1.3 ExpansionofSingularDistributionsintoFourierSeries 128 2.2 FourierTransformsofFunctionsandDistributions 129 2.2.1 FourierTransformsofFunctions 129 2.2.2 FourierTransformandtheConvolutionProduct 131 2.2.3 PartialFourierTransformofFunctions 132 2.2.4 FourierTransformofDistributionsfromtheSpacesS0andD0(Rn) 133 2.2.4.1 PropertiesoftheFourierTransform 136 2.2.4.2 FourierTransformoftheDistributionsfromtheSpaceD0(Rn) 139 2.2.4.3 FourierTransformandthePartialConvolutionProduct 144 2.3 LaplaceTransformsofFunctionsandDistributions 145 2.3.1 LaplaceTransformsofFunctions 146 2.3.2 LaplaceTransformsofDistributions 149 3 VariationalCalculusandDifferentialEquationsinDistributions 151 3.1 VariationalCalculusinDistributions 151 3.1.1 EquationsoftheEuler–PoissonType 158 3.2 OrdinaryDifferentialEquations 160 3.3 ConvolutionEquations 166 3.3.1 ConvolutionAlgebras 166 3.3.2 ConvolutionAlgebraD0 :ConvolutionEquationsinD0 170 C C 3.4 TheCauchyProblemforLinearDifferentialEquationswithConstant Coefficients 174 3.5 PartialDifferentialEquations:FundamentalSolutionsandSolvingthe CauchyProblem 177 3.5.1 FundamentalSolutionfortheLongitudinalVibrationsofViscoelastic BarsofMaxwellType 180 3.6 WaveEquationandtheSolutionoftheCauchyProblem 184 3.7 HeatEquationandCauchyProblemSolution 187 3.8 PoissonEquation:FundamentalSolutions 189 3.9 Green’sFunctions:MethodsofCalculation 190 3.9.1 HeatConductionEquation 190 3.9.1.1 GeneralizedPoissonEquation 194 3.9.2 Green’sFunctionfortheVibratingString 197 4 Representation in Distributions of Mechanical and Physical Quantities 201 4.1 RepresentationofConcentratedForces 201 4.2 RepresentationofConcentratedMoments 206 4.2.1 ConcentrateMomentofLinearDipoleType 208 Contents VII 4.2.2 RotationalConcentratedMoment(CenterofRotation) 210 4.2.3 ConcentratedMomentofPlaneDipoleType(Center ofDilatationor Contraction) 212 4.3 RepresentationinDistributionsoftheShearForcesandtheBending Moments 213 4.3.1 ConcentratedForceofMagnitudePAppliedatthePointc 2[a,b] 217 4.3.2 Concentrated Moment of Magnitude m Applied at the Point c 2 [a,b] 217 4.3.3 DistributedForcesofIntensityq2 L1 ([a,b]) 218 loc 4.4 Representation by Distributions of the Moments of a Material System 220 4.5 RepresentationinDistributionsofElectricalQuantities 225 4.5.1 VolumeandSurfacePotentialoftheElectrostaticField 225 4.5.2 ElectrostaticField 228 4.5.3 ElectricPotentialofSingleandDoubleLayers 232 4.6 Operational Form of the Constitutive Law of One-Dimensional ViscoelasticSolids 235 5 ApplicationsoftheDistributionTheoryinMechanics 241 5.1 NewtonianModelofMechanics 241 5.2 TheMotionofaHeavyMaterialPointinAir 243 5.3 LinearOscillator 246 5.3.1 TheCauchyProblemandthePhenomenonofResonance 246 5.4 Two-PointProblem 249 5.5 BendingoftheStraightBars 250 6 ApplicationsoftheDistributionTheorytotheMechanicsoftheLinear ElasticBodies 253 6.1 TheMathematicalModeloftheLinearElasticBody 253 6.2 EquationsoftheElasticityTheory 256 6.3 Generalized Solution in D0(R2) Regarding the Plane Elastic Problems 258 6.4 Generalized Solutionin D0(R) of the Static Problem of the Elastic Half-Plane 264 6.5 GeneralizedSolution,inDisplacements,fortheStaticProblemofthe ElasticSpace 268 7 ApplicationsoftheDistributionTheorytoLinearViscoelasticBodies 273 7.1 TheMathematicalModelofaLinearViscoelasticSolid 273 7.2 ModelsofOne-DimensionalViscoelasticSolids 275 7.3 ViscoelasticityTheoryEquations:CorrespondencePrinciple 281 7.3.1 ViscoelasticityTheoryEquations 281 7.3.2 CorrespondencePrinciple 283 8 ApplicationsoftheDistributionTheoryinElectricalEngineering 285 8.1 StudyoftheRLCCircuit:CauchyProblem 285 8.2 CoupledOscillatingCircuit:CauchyProblem 290 VIII Contents 8.3 AdmittanceandImpedanceoftheRLCCircuit 294 8.4 Quadrupoles 295 9 ApplicationsoftheDistributionTheoryintheStudyofElasticBars 301 9.1 LongitudinalVibrationsofElasticBars 301 9.1.1 Equationsof Motion Expressed in Displacements and in Stresses: FormulationoftheProblemswithBoundary–InitialConditions 301 9.1.2 Forced LongitudinalVibrationsof a Bar with BoundaryConditions ExpressedinDisplacements 303 9.1.3 Forced Vibrationsof aBar with BoundaryConditionsExpressed in Stresses 306 9.2 TransverseVibrationsofElasticBars 308 9.2.1 DifferentialEquationinDistributionsofTransverseVibrationsofElastic Bars 308 9.2.2 FreeVibrationsofanInfiniteBar 311 9.2.3 ForcedTransverseVibrationsoftheBars 312 9.2.4 BendingofElasticBarsonElasticFoundation 315 9.3 TorsionalVibrationoftheElasticBars 331 9.3.1 DifferentialEquationofTorsionalVibrationsoftheElasticBarswith CircularCross-Section 331 9.3.2 TheAnalogybetweenLongitudinalVibrationsandTorsionalVibrations ofElasticBars 335 9.3.3 TorsionalVibrationofFreeBarsEmbeddedatOneEndandFreeatthe OtherEnd 336 9.3.4 ForcedTorsionalVibrationsofaBarEmbeddedattheEnds 338 10 ApplicationsoftheDistributionTheoryintheStudyofViscoelasticBars 343 10.1 TheEquationsoftheLongitudinalVibrationsoftheViscoelasticBars intheDistributionsSpaceD0 343 C 10.2 LongitudinalVibrationsofMaxwellTypeViscoelasticBars:Solutionin theDistributionsSpaceD0 345 C 10.3 Steady-StateLongitudinalVibrationsfortheMaxwellBar 347 10.4 Quasi-StaticProblemsofViscoelasticBars 349 10.4.1 BendingoftheViscoelasticBars 350 10.4.2 BendingofaViscoelasticBarofKelvin–VoigtType 352 10.4.3 BendingofaViscoelasticBarofMaxwellType 352 10.5 TorsionalVibrationsEquationofViscoelasticBarsintheDistributions SpaceD0 354 C 10.6 Transverse Vibrations of Viscoelastic Bars on Viscoelastic Foundation 357 10.6.1 The Generalized Equation in Distributions Space D0(R2) of the Transverse Vibrations of Viscoelastic Bars on Viscoelastic Foundation 357 10.6.2 GeneralizedCauchyProblemSolutionforTransverseVibrationsofthe ElasticBarsonKelvin–VoigtTypeViscoelasticFoundation 359 Contents IX 10.6.2.1 ParticularCases 363 11 ApplicationsoftheDistributionTheoryinPhysics 365 11.1 ApplicationsoftheDistributionTheoryinAcoustics 365 11.1.1 DopplerEffectforaOne-DimensionalSoundSource 365 11.1.2 DopplerEffectinthePresenceofWind 367 11.2 ApplicationsoftheDistributionTheoryinOptics 371 11.2.1 ThePhenomenonofDiffractionatInfinity 371 11.2.2 DiffractionofFresnelType 374 References 377 Index 379 XI Preface The solution to many theoretical and practical problems is closely connected to themethodsapplied,andtothemathematicaltoolswhichareused.Inthemathe- maticaldescriptionofmechanicalandphysicalphenomena,andinthesolutionof thecorrespondingboundaryvalueandlimitproblems,difficultiesmayappearow- ingtoadditionalconditions.Sometimes,theseconditionsresultfromthelimited rangeofapplicabilityofthemathematicaltoolwhichisinvolved;ingeneral,such conditionsmaybeneithernecessarynorconnectedtothemechanicalorphysical phenomenonconsidered. Themethodsofclassicalmathematicalanalysisareusuallyemployed,buttheir applicabilityisoftenlimited.Thus,thefactthatnotallcontinuousfunctionshave derivativesisasevererestrictionimposedonthemathematicaltool;itaffectsthe unityandthegeneralityoftheresults.Forexample,itmayleadtotheconclusion ofthenonexistenceofthevelocityofaparticleatanymomentduringthemotion, aconclusionwhichobviouslyisnottrue. On the other hand, the development of mechanics, of theoretical physics and particularly of modern quantum mechanics, the study of various phenomena of electromagnetism,optics,wavepropagationandthesolutionofcertainboundary valueproblemshaveallbroughtabouttheintroductionofnewconceptsandcom- putations, which cannot be justified within the frame of classical mathematical analysis. Inthisway,in1926 Diracintroducedthedeltafunction(denotedby δ),which fromaphysicalpointofview,representsthedensityofaloadequaltounitylocated atonepoint.Aformalismhasbeenworkedoutforthefunction,anditsusejusti- fiesandsimplifiesvariousresults.Exceptforasmallnumberofincipientinvesti- gations,itwasonlyduringthe1960sthatthetheoryofdistributionswasincluded asanewchapteroffunctionalanalysis.Thistheoryrepresentsamathematicaltool applicabletoalargeclassofproblems,whichcannotbesolvedwiththeaidofclas- sicalanalysis.Thetheoryofdistributionsthuseliminatestherestrictionswhichare notimposedbythephysicalphenomenonandjustifiesprocedureandresults,e.g., thosecorrespondingtothecontinuousanddiscontinuousphenomena,whichcan thusbestatedinaunitaryandgeneralform. Thismonographpresentselementsofthetheoryofdistributions,aswellasthe- orems with possibilityof application.Whilerespecting the mathematicalrigor, a
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