Table Of ContentYichao Zhu
Equations and
Analytical Tools
in Mathematical
Physics
A Concise Introduction
Equations and Analytical Tools in Mathematical
Physics
Yichao Zhu
Equations and Analytical
Tools in Mathematical
Physics
A Concise Introduction
YichaoZhu
DepartmentofEngineeringMechanics
DalianUniversityofTechnology
Dalian,Liaoning,China
ISBN978-981-16-5440-4 ISBN978-981-16-5441-1 (eBook)
https://doi.org/10.1007/978-981-16-5441-1
JointlypublishedwithSciencePress
TheprinteditionisnotforsaleinChina(Mainland).CustomersfromChina(Mainland)pleaseorderthe
printbookfrom:SciencePress.
TranslationfromtheChineseSimplifiedlanguageedition:ShuXueWuLiFangChengYuJinJieGong
JubyYichaoZhu,©SciencePress2020.PublishedbySciencePress.AllRightsReserved.
©SciencePress2021
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
Preface
The core contents of the present book are partial differential equation (PDE). The
essence of PDE as a modelling tool can be perceived from summarising several
keyfundamentalfeaturesunderlyingvariousphysicalprocesses.Wearelivingina
worldwherethingsandstatesarenormallyspatiallyvaryingandtemporallyevolving.
Fromamathematicalviewpoint,differentiationcanbeemployedtodescribevariation
orevolution.Meanwhile,certainphysicalquantities,suchasenergy,areconserved
duringevolution,andequationoridentitycanbeusedtocaptureconservation.There-
fore,differentialequationsnaturallybecomeaneffectivetoolfordescribingevolving
physicalprocesseswithcertainquantitiesconserved.Whenmultiplefreevariables
areneeded,onenormallyendsupwithpartialdifferentialequations.
TheprocedureofanalysingaphysicalprocesswithPDEcanbegenerallydivided
into three stages. The first stage can be termed as “modelling” (from physics to
mathematics), which seeks to represent the physical processes of interest through
the mathematical formulation of PDE. The second stage can be termed as “analy-
sis”,whichseekstomathematicallysolvethederivedPDEproblems.Thethirdstage
canbetermedas“interpretation”(frommathematicsbacktophysics),whichseeksto
gainphysicalinsightsoutofthemathematicalresults.Themaingoalofthepresent
book is to help the reader develop a habit of such a chained way of thinking, i.e.
“starting from physics, digging in mathematics, and return to physics”. Moreover,
severalmathematicalconceptsandmethods,whichhavebeenproventobeeffective
insolvingPDEproblems,arealsodiscussed.Ithastobementionedthatthemathe-
maticaltoolsthatareoriginallydevisedforsolvingPDEsarequitevastinnumber,
andsomeofthemhavealreadyevolvedtoindividualtopics,suchasfunctionalanal-
ysis.Therefore,perhapsduetotheauthor’slimitedunderstandingoverthesubject,
it is quite difficult to use more specific terms than “analytical tools” to summarise
thosevariousmathematicaltools.Thisishowthetitleofthebookemerges,andonly
selectedtopicsarediscussedhere.
Part I of this book is aimed to introduce three typical PDEs: the wave equa-
tion,theheatequationandPoisson’sequation.Theyarediscussedinparallelinthe
firstthreechapters.Ineachchapter,thecontentscoverthephysicalphenomenathat
can be described by the corresponding PDE, the solution strategies designated for
solving the resulting PDE problems, as well as what can be interpreted from the
v
vi Preface
derivedsolutions.Insuchanillustrativemanner,theessentiallinksbetweenaphys-
icalphenomenonandthemathematicalstructuresunderlyingitscorrespondingPDE
can be addressed. Awareness of such insightful linkages has proven to be crucial
for implementing PDE tools, both theoretically and numerically, for solving real-
worldproblems.Forinstance,implementationofthecentraldifferenceschemefor
approximatingfunctionalderivativesislikelytogiveriseto(unphysical)oscillation
whensimulatingfluidmechanicsproblems.Butsuchbehaviourdoesnotappearin
thecomputationoflinearelastostaticproblems.Thisisbecausethegoverningequa-
tioninfluidmechanicsareusually(first-order)hyperbolic,wheretheinformationis
transmitteddownstream.Mathematically,thisisimpliedbytheso-calledcharacter-
istics.Incontrast,thegoverningdifferentialequationsforelastostaticproblemsare
generalised elliptic systems, which do not exhibit the behaviour of characteristics.
Hence,usingthecentraldifferencefordiscretisingthembecomeseffective.There-
fore,understandingthemathematicalstructuresbehindPDEsshouldenableoneto
choose appropriate numerical schemes, even facing with brand new PDE models.
To this end, the classification of general second-order linear PDEs is discussed in
Chap.4.
PartIIofthisbookcoverstopicsonspecialfunctions,apowerfultoolespecially
foranalysingPDEproblemsdefinedincylindrical/sphericaldomains.Attentionsare
firstpaidtotworepresentativeclassesofspecialfunctions,namelyBesselfunctions
and Legendre functions. Part II of this book concludes with a brief discussion on
hypergeometric functions, which is aimed to address the commonalities of many
specialfunctionsintensivelyusedinthefieldofmathematicalphysics.
The book is suitable for undergraduates of science and engineering, who have
accomplished their courses on calculus and linear algebra. A common impression
aboutcoursesonPDEsandrelatedtopicsis,giventheirimportanceinthesenseof
theirapplicability,theyarequitechallengingforstudentstomanage.Intheauthor’s
opinion,thedifficultiescanbesummarisedasfollows.Firstly,solidunderstanding
of the course materials entails an intensive use of the prerequisite knowledge on
calculus,linearalgebra,etc.Asaresult,studentsoftengetfrustratedwhentheneces-
sary prerequisite knowledge slips their minds. Secondly, some related concepts in
thecoursearequiteabstruse,whilethederivationsinvolvedareoftencumbersome.
Theseintrinsicfeaturesofthecoursetendtodiscouragestudents,whichcouldfurther
preventtheirabsorptionofadvancedknowledgeonusingPDEsintheirownfields.
To cope with such challenging issues, actions are taken in writing this book.
Firstly,whencertainprerequisiteknowledgeisexpectedtobeintensivelyusedfor
elaboratingthecontentsofasection,asubsectionwhichconciselysummarisesthe
key prerequisite knowledge is included, aiming to awake one’s memory over the
prerequisite topic. Secondly, when new concepts are to be introduced, addressing
their linkages with the known (if applicable) is always prioritised. For example,
when the properties of Bessel functions are introduced, a comparative discussion
with that of sinusoidal functions is made. Thirdly, a road map is charted at the
beginningofeachchapter.Theaimistodevelopanoverviewofthecontentsabout
the chapter and their interrelationships. Fourthly, a set of online videos are also
devisedinparallelwiththebook,soastomakethebestuseofmultimediatoolsto
Preface vii
presentthecoursecontents.Insummary,acquiringknowledgeonPDEsandrelated
topicsistechnicallychallengingforstudents,butoneshouldalsoseeitsbrightside.
Itoffersanopportunityto(re-)consolidatetheunderstandingofone’smathematical
knowledgesysteminanapplication-orientedmanner,andthusgetsthereaderready
fortakingadvancedcoursesonquantitativeanalysis.
Ithastobeemphasisedthatasasubject,“PDEs”itselfhasaverywidecoveragein
contents.HereonlyrepresentativetopicsonPDEsarementionedatafairlyelemen-
tarylevel.Hence,thepresentbookisrecommendedasanintroductorytooltofamil-
iarisereaderswithPDEsandspecialfunctions.Besides,technicaltermsarewritten
intheitalicfontwhentheyappearforthefirsttime.
Onwritingthisbook,anumberofclassicaltextbooksonPDEsandspecialfunc-
tions are referred to, and they are listed in the references with my great gratitude.
IwouldliketothankmycolleaguesinDalianUniversityofTechnology(DUT),in
particular,Prof.XuGuoandProf.YonggangZheng,fortheirvaluablesuggestions.
The hard works fromseveral students atDUT, Dingchuan Xue, Jing Luo, Chuang
Ma,QianXiangandZhengchengZhou,onhelpingmearrangingthebookmaterials
aregratefullyacknowledged.AllthestudentsofYunchuang1901classatDUTalso
have my gratitude, as they were forced to trial on the book manuscript, and nice
commentsregardingitsrevisionarereportedfromthem.Amongthem,HongyuMa
and Qikai Zhu who have pointed several typos appearing in the book manuscript
havemyespecialgratitude.Moreover,thefinancialsupportfromtheDepartmentof
EngineeringMechanicsofDUTisacknowledged.Finally,Imustthankmyfamily
for their constant support on my work, in particular my daughter Peien, without
whomthebookshouldhavealreadybeenpublished,butofcourse,withfarlessfun.
Dalian,China YichaoZhu
Contents
PartI Second-OrderLinearPartialDifferentialEquations
1 TheWaveEquation ............................................ 3
1.1 EquationforStringVibration ................................. 4
1.1.1 DerivationoftheEquationforStringVibration ........... 4
1.1.2 InitialandBoundaryConditions ........................ 9
1.1.3 Terminology ........................................ 11
1.2 D’Alembert’sFormula ...................................... 12
1.2.1 D’Alembert’sFormula ................................ 13
1.2.2 Characteristics ....................................... 15
1.2.3 TheCaseofaSemi-infinitelyLongString ............... 17
1.2.4 Duhamel’sPrinciple .................................. 22
1.3 MethodofSeparationofVariables ............................ 24
1.3.1 MakingBoundaryConditionsHomogeneous ............. 24
1.3.2 MethodofSeparationofVariables—ItsProcedure ........ 25
1.3.3 PhysicalImplications ................................. 31
1.3.4 InhomogeneousGoverningEquations ................... 32
1.4 WaveEquationinHigherDimensions ......................... 35
1.4.1 SmallandTransverseVibrationofaMembrane .......... 35
1.4.2 DefiniteProblems .................................... 40
1.4.3 SolutionsforCauchyProblems ........................ 41
1.5 SolutionProperties ......................................... 46
1.5.1 TheEnergyofaVibratingMembrane ................... 47
1.5.2 Solution Uniqueness of Problems for the Wave
Equation ............................................ 48
2 TheHeatEquation ............................................. 57
2.1 ModellingHeatConduction .................................. 57
2.1.1 DerivationoftheHeatEquation ........................ 57
2.1.2 InitialandBoundaryConditions ........................ 61
2.1.3 PhysicalAnalogies ................................... 62
ix
x Contents
2.2 MethodofIntegralTransform ................................ 63
2.2.1 ConvolutionandFourierTransform ..................... 64
2.2.2 SolutionforCauchyProblems ......................... 67
2.2.3 SolutionProperties ................................... 70
2.2.4 InhomogeneousGoverningEquations ................... 72
2.3 ARevisittotheMethodofSeparationofVariables .............. 75
2.3.1 AnExamplewiththeHeatEquation .................... 75
2.3.2 Sturm–LiouvilleSystem .............................. 81
2.3.3 InhomogeneousGoverningEquations ................... 88
2.4 SolutionProperties ......................................... 89
2.4.1 MaximumPrinciple .................................. 89
2.4.2 SolutionUniqueness .................................. 91
2.4.3 Stability ............................................ 92
3 Poisson’sEquation ............................................. 99
3.1 Poisson’sEquationandHarmonicEquation .................... 99
3.1.1 Definitions .......................................... 99
3.1.2 MotivationfromPhysics .............................. 101
3.1.3 BoundaryConditions ................................. 106
3.2 VariationalPrinciple ........................................ 108
3.3 HarmonicFunctionsinPolarSystem .......................... 112
3.3.1 Laplace’sEquationinPolarSystem ..................... 113
3.3.2 RadialSolutionstoLaplace’sEquation .................. 115
3.4 TheMethodofGreen’sFunction ............................. 117
3.4.1 Green’sFormulaeRelatedtoLaplacianOperator ......... 117
3.4.2 FundamentalSolution ................................ 118
3.4.3 DerivationofGreen’sFunction ......................... 120
3.4.4 PropertiesofGreen’sFunction ......................... 122
3.4.5 ProblemsforPoisson’sEquation ....................... 123
3.4.6 FinalRemarks ....................................... 125
3.5 ImageMethodforElectricPotentials .......................... 125
3.5.1 TheCaseinThree-DimensionalHalfSpace .............. 126
3.5.2 TheCaseinaSphericalDomain ....................... 127
3.6 SolutionUniqueness ........................................ 130
3.6.1 Mean-ValueFormula ................................. 130
3.6.2 MaximumPrinciple .................................. 132
3.6.3 StrongMaximumPrinciple ............................ 133
3.6.4 EnergyMethod ...................................... 134
4 Summary over Second-Order Linear Partial Differential
Equation ...................................................... 141
4.1 ClassificationofSecond-OrderLinearPartialDifferential
Equations ................................................. 141
4.1.1 CaseswithTwoVariables ............................. 141
4.1.2 SummaryandExamples .............................. 148
Contents xi
4.1.3 MultivariableSituations ............................... 150
4.2 TopicalDiscussion ......................................... 155
PartII SpecialFunctions
5 BesselFunctions ............................................... 165
5.1 BesselEquationandBesselFunctions ......................... 165
5.1.1 PhysicalMotivation .................................. 165
5.1.2 BesselFunctionoftheFirstKind ....................... 169
5.1.3 BesselFunctionoftheSecondKind .................... 173
5.2 PropertiesofBesselFunctions ................................ 177
5.2.1 RecurrenceFormulae ................................. 177
5.2.2 Zeros ............................................... 179
5.2.3 ApproximatingFormula .............................. 181
5.2.4 Orthogonality ....................................... 181
5.2.5 AnalogieswithSinusoidalFunctions .................... 184
5.3 SolvingPDEProblemswithBesselFunctions .................. 186
5.4 Generalisation ............................................. 195
5.4.1 HankelFunctions .................................... 195
5.4.2 ModifiedBesselFunctions ............................ 195
6 LegendrePolynomial ........................................... 199
6.1 LegendreEquation ......................................... 199
6.2 LegendrePolynomial ....................................... 204
6.2.1 SeriesSolutiontotheLegendreEquation ................ 204
6.2.2 LegendrePolynomial ................................. 207
6.3 PropertiesofLegendrePolynomial ............................ 210
6.3.1 RodriguesFormula ................................... 210
6.3.2 KeyPropertiesataGlance ............................ 210
6.3.3 OrthogonalSystems .................................. 212
6.3.4 DiscussiononGeneralOrthogonalPolynomials .......... 218
6.4 ApplicationswithLegendrePolynomials ...................... 220
6.4.1 SolvingPDEsDefinedinaSphere ...................... 220
6.4.2 Legendre–GaussQuadrature ........................... 222
6.5 TheAssociatedLegendreFunctions ........................... 225
7 IntroductionofHypergeometricFunction ........................ 231
7.1 CommonalitiesofBesselandLegendreFunctions ............... 231
7.2 GaussHypergeometricFunction .............................. 234
7.2.1 Definition ........................................... 234
7.2.2 HypergeometricDifferentialEquation ................... 235
7.2.3 Legendre Function and Legendre
Polynomial—ARevisit ............................... 239
