Fractional Derivative Modeling in Mechanics and Engineering

381 Pages·2022·6.003 MB·English

by  Wen Chen, HongGuang Sun, Xicheng Li

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