Fractional Derivative Modeling in Mechanics and Engineering PDF
Preview Fractional Derivative Modeling in Mechanics and Engineering
Wen Chen HongGuang Sun Xicheng Li Fractional Derivative Modeling in Mechanics and Engineering Fractional Derivative Modeling in Mechanics and Engineering · · Wen Chen HongGuang Sun Xicheng Li Fractional Derivative Modeling in Mechanics and Engineering WenChen HongGuangSun CollegeofMechanicsandMaterials CollegeofMechanicsandMaterials HohaiUniversity HohaiUniversity Nanjing,China Nanjing,China XichengLi SchoolofMathematicalSciences UniversityofJinan Jinan,China Editedby HongGuangSun YingjieLiang CollegeofMechanicsandMaterials HohaiUniversity HohaiUniversity Nanjing,China Nanjing,China ThisbookwasfundedbytheNaturalScienceFoundationofJiangsuProvince,grantnumber: BK20190024andtheNationalNaturalScienceFoundationofChina,grantnumbers11972148. ISBN978-981-16-8801-0 ISBN978-981-16-8802-7 (eBook) https://doi.org/10.1007/978-981-16-8802-7 JointlypublishedwithSciencePress TheprinteditionisnotforsaleinChina(Mainland).CustomersfromChina(Mainland)pleaseorderthe printbookfrom:SciencePress. ISBNoftheCo-Publisher’sedition:978-7-03-026857-0 TranslationfromtheChineselanguageedition:Fractionalderivativemodelinginmechanicsandengi- neeringbyWenChen,etal.,©ChinaScienceandTechnologyPress2010.PublishedbyChinaScience andTechnologyPress.AllRightsReserved. ©SciencePress2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublishers,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishersnortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublishersremainneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Wededicatethisbooktothehonorable memoryofProfessorWenChen Preface Classic Newtonian mechanics assumes that space and time are continuous every- where. The basic physical quantities (e.g. speed, acceleration and force) can be describedbyaninteger-orderdifferentialoperator;thus,thephysicalandmechanics evolutionscanbeaccuratelydescribedbyusinginteger-orderdifferentialequations, forexample,theFourierheatconductionequationsandHamiltonequationsofclas- sical mechanics. The above scientific research methods and models have achieved huge success in classical mechanics, acoustics, electromagnetics, heat transfer, diffusion theory and even in modern quantum mechanics and relativity. However, physicists, dynamicists and engineers have found that more and more anomalous phenomena cannot be explained from this viewpoint. For example, Richardson pointedoutthatin1926,aturbulentvelocityfieldisnon-differentiable,whichmay beacriticalobstacleinsolvingturbulenceproblemsandcannotbetackledbyusing thetraditionalNewtonianmechanics.Moreover,alargenumberofexperimentshave shownthatthestressrelaxationoftheviscoelasticmaterial(includingviscoelastic andrheologymaterials)isnon-exponentialdecay(non-Debye)andhasmemoryprop- erties. It causes the conventional integer-order viscoelastic constitutive model can notaccuratelydescribetheirmechanicsbehavior.Anomalousdiffusionhasattracted widespreadconcerninrecentyears,andinvolvesthepropertyofhistorydependency, pathdependencyandglobalcorrelationofphysicalandmechanicsprocesses.Ithas beenconfirmedthatclassicalDarcylaw,theFourierheatconductionlaw,Newtonian viscosityandFickiandiffusionlawcannotaccuratelydescribetheaboveanomalous physicalandmechanicsprocesses. Fromthemechanicsmodelingviewpoint,standardinteger-ordertimederivative bylocallimitdefinitionisnotsuitabletodescribethehistory-dependentprocess.On thecontrary,thetime-fractionalderivativeisactuallyadifferential-integralconvolu- tionoperator;itsintegraltermcanfullyreflectthehistorydependencyofthesystem function,andisapowerfulmathematicaltoolformodelingstrongmemory-dependent processes.Ontheotherhand,thefractionalLaplaceoperator(fractionalLaplacian) isatypicalnon-localspacefractionalderivative,whichcanaccuratelydescribethe vii viii Preface anomalousmechanicsbehavior(suchaspathdependenceandlong-rangecharacter- istics)incomplexfractalspatialstructures.Ithasovercomethetheoryofclassical mechanicsonthebasisofEuclideangeometryandabsolutetimeandspace. Fractionalcalculusisanancientandfreshconcept.Intheearlystagesofinteger- order calculus history, there are some mathematicians who began to consider the meaning of fractional calculus, such as L’Hospital, Leibniz and so on. However, it did not attract more attention and has not been further studied, due to the lack of application background and many other reasons. With the development of the naturalandsocialsciences,thedemandforcomplexengineeringapplications,espe- ciallythefractalstudyofavarietyofcomplexsystemssincethe1970sand1980s, thetheoryoffractionalcalculusanditsapplicationsbegantoreceiveextensiveatten- tion. From the beginning of the twenty-first century, fractional calculus modeling methods and theory have achieved successful applications in high-energy physics, anomalous diffusion, complex viscoelastic material mechanical constitutive rela- tions,systemcontrol,rheology,geophysics,biomedicalengineering,economicsand many other fields. Its unique advantages and irreplaceability are highlighted, and relatedtheoreticalandappliedresearchhasbecomeahotspotworldwide. Meanwhile,thenon-localnatureoffractionalcalculus,resultinginthenumerical simulationcomputationandstoragecapacityofthefractionalderivativegoverning equation increases with the size of the problem. Hence, some effective numer- ical methods designed for integer-order equations are no longer applicable for fractional-orderequations.Moreover,alotoffractionalderivativeequationmodels arephenomenologicaldescriptions,theirphysicalandmechanicalmechanismisnot clear,pendingfurtherresearch. Untilnow,severalEnglishlanguagemonographsontheintroductionoffractional calculushavebeenpublishedaroundtheworld.Forexample,OldhamandSpanier “The Fractional Calculus” (Academic Press, Inc., San Diego, 1974), Samko, etc. “Fractional Intergrals and Derivatives: Theory and Applications” (1987, Russian version,1993publishedinEnglish),MillerandRoss“AnIntroductiontotheFrac- tionalCalculusandFractionalDifferentialEquations”(JohnWiley&Sons,Inc.and, New York, 1993), Podlubny “Fractional Differential Equations” (Academic Press, NewYork,1999)andKilbasetal.“TheoryandApplicationsofaFractionalDiffer- entialEquations”(Elsevier,Amsterdam,2006).However,thesemonographsfocus on the mathematical theory of fractional calculus or its application in a particular areaanddidnotfullytakeintoaccountfractionalcalculus,anditsapplicationsare stillanewthingformostresearchers.Webelievemostresearchersneedaprimeron fractionalcalculustheoryanditsapplication. Thisbookaimstoprovidegraduatestudentswithatextbookandanintroductory bookonthetheoryandapplicationsoffractionalcalculus.Thisbookoffersdetailed knowledge of fractional calculus modeling and numerical simulation of complex mechanicalbehavior,withthecombinationofrecentresearchworksofauthors.This book introduces fractional calculus theory and its applications from the aspects of mathematicalfoundations,fractalandfractionalcalculusrelations,unconventional Preface ix statisticsandanomalousdiffusion,typicalapplicationsoffractionalcalculus,numer- icalsolutionsoffractionaldifferentialequations,etc.Here,wealsofurtherexplore theprospectsforthedevelopmentoffractionalcalculusmodeling. Thebookfocusesontheapplicationsoffractionalcalculusinmechanicsandphys- icalmodeling,emphasizingthephysicalandmechanicsbackgroundandconceptof fractionalcalculusmodeling.Meanwhile,thisbook avoidstoomuchintroductionsof mathematicalbackgroundandrigorousmathematicalproof,andstrivestointroduce thebasicknowledgeoffractionalcalculustoourreaders.Interestedreadersmayrefer totheabove-mentionedmonographsandlistedreferencesaboutdetailedmathemat- icalanalysisandproof.Thisbookalsocontainssomelatestresearchachievements onfractionalcalculustheoryanditsapplication,suchaspositivefractionalderiva- tive,fractalderivative,variable-orderderivative,distributed-orderderivativeandtheir applications, and fractional derivative continuum mechanics model of multi-scale turbulence. Thisbookprovidessomenumericalexamplesandsimulationresultstoenhance theunderstandingoffractionalcalculusdefinitionsandconcepts.Eachchapteralso discusses the relevant aspects of the existing problems. The last chapter is the summary and prospects. The key problems which should be solved in the future are also presented. Those key issues are raised by some scholars in the interna- tionalconference“TheThirdIFACWorkshoponFractionalDifferentiationandits Applications”,heldinTurkeyin2008.Wehopethattheseelementscangivereaders inspiration to deepen their understanding and knowledge of the theory and appli- cations of fractional calculus. Here, we should point out that there are too many papersonthetheoryandapplicationsoffractionalcalculus;herein,itisimpossible toenumerateallofthem;interestedreaderscanrefertothereferencelistofthisbook formoreinformation;wewouldbepleasedtoreceivethefeedbackfromreaders. ThisbookispresidedoverbyWenChen,HongGuangSunandXiChengLi.The frameworkofthisbookwasfirstlyproposedbyWenChen,andwasfinallydetermined afterafulldiscussionofalltheauthors.WenChenmakestheoverallarrangements for the book writing; Chap. 1 is mainly written by Wen Chen; Chap. 2 is mainly writtenbyHongGuangSunandLinjuanYe;Chap.3ismainlywrittenbyShuaiHu and Xiaodi Zhang; Chap. 4 is mainly written by Wen Chen and HongGuang Sun; Chap.5ismainlywrittenbyXiaodiZhang,WenChenandHongGuangSun;Chap.6 ismainlywrittenbyWenChen,HongGuangSunandXiaodiZhang;Chap.7ismainly writtenbyWenChenandXi-ChengLi;AppendixismainlywrittenbyHongGuang Sun and Linjuan Ye. Xicheng Li is responsible for the main work of modification andtypesetting.TheauthorsofthisbookthankProf.YangquanChen,Prof.Keqin Zhu,Prof.ChangpinLi,Dr.DeshunYin,Prof.NingChenandDr.YanLifortheir helpinwritingthisbookanditsmodification;theirsuggestionsandcommentshave improvedthequalityofthisbook. This book was supported by National Basic Research Program of China (973 ProjectNo.2010CB832702),R&DSpecialFundforPublicWelfareIndustry(Hydro- dynamics,GrantNo.201101014),NationalScienceFundsforDistinguishedYoung Scholars (Grant No. 11125208), Natural Science Foundation of China (Grant No. x Preface 11202066) and Program of Introducing Talents of Discipline to Universities (111 project,GrantNo.B12032). Time and knowledge being limited, errors and inadequacies are inevitable; any suggestionsandcommentsarewelcome. Nanjing,China WenChen February2013 Contents 1 Introduction ................................................... 1 1.1 HistoryofFractionalCalculus ............................... 1 1.2 Geometric and Physical Interpretation of Fractional DerivativeEquation ........................................ 5 1.2.1 Frequency-DependentEnergyDissipationProcess ...... 6 1.2.2 FractalDescriptionandPower-LawPhenomena ........ 7 1.2.3 AnomalousDiffusion ............................... 8 1.2.4 ConstitutiveRelationofComplex Viscoelastic Material .......................................... 9 1.2.5 FractionalSchrodingerEquation ..................... 10 1.3 ApplicationinScienceandEngineering ...................... 10 1.4 Anomalous Diffusion Modeling in Environmental Mechanics ................................................ 11 1.5 ConstitutiveRelationofViscoelasticity ....................... 11 1.6 BiomedicalScience ........................................ 12 1.7 SystemControl ........................................... 12 References ..................................................... 12 2 MathematicalFoundationofFractionalCalculus .................. 15 2.1 DefinitionofFractionalCalculus ............................ 16 2.1.1 IntroductionofFractionalCalculusDefinition .......... 16 2.1.2 Riemann–LiouvilleDefinition ....................... 19 2.1.3 Caputo’sDefinition ................................ 20 2.1.4 Grünwald–LetnikovDefinition ....................... 22 2.1.5 RieszDefinitionoftheSpatialFractionalLaplace Operator .......................................... 23 2.2 PropertiesofFractionalCalculus ............................ 25 2.2.1 SomePropertiesofRiemann–LiouvilleOperator ....... 26 2.2.2 FractionalDerivativeofSomeFunctions .............. 27 2.2.3 TheRelationshipsofDifferentDefinitions ............. 29 xi
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