Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions

847 Pages·2009·7.588 MB·English

by  Victor Henner, Tatyana Belozerova, Kyle Forinash

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Mathematical
Methods in Physics
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Mathematical
Methods in Physics
Partial Differential Equations,
Fourier Series, and Special Functions
Victor Henner
Tatyana Belozerova
Kyle Forinash

AKPeters,Ltd.
Wellesley,Massachusetts
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Editorial,Sales,andCustomerServiceOffice
AKPeters,Ltd.
888WorcesterStreet,Suite230
Wellesley,MA02482
www.akpeters.com
Copyright c 2009byAKPeters,Ltd.
(cid:13)
All rights reserved. No part of the material protected by this copyright
notice may be reproduced or utilized in any form, electronic or mechani-
cal,includingphotocopying,recording,orbyanyinformationstorageand
retrievalsystem,withoutwrittenpermissionfromthecopyrightowner.
LibraryofCongressCataloging-in-PublicationData
Henner,Victor.
Mathematicalmethodsinphysics:partialdifferentialequations,Fourierseries,
andspecialfunctions/VictorHenner,TatyanaBelozerova,KyleForinash.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-56881-335-6(alk. paper)
1. Mathematical physics–Textbooks. I. Belozerova, Tatyana. II. Forinash,
Kyle. III.Title.
QC20.H4872009
530.15–dc22
2008022076
PrintedinIndia
1312111009 10987654321
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To Bruce Adams,
a great friend

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Contents
Introduction xi
1 FourierSeries 1
1.1 PeriodicProcessesandPeriodicFunctions . . . . . . . . 1
1.2 FourierFormulas . . . . . . . . . . . . . . . . . . . . . 3
1.3 OrthogonalSystemsofFunctions . . . . . . . . . . . . . 7
1.4 ConvergenceofFourierSeries . . . . . . . . . . . . . . 9
1.5 FourierSeriesforNonperiodicFunctions . . . . . . . . 16
1.6 FourierExpansionsonIntervalsofArbitraryLength . . . 16
1.7 FourierSeriesinCosineorSineFunctions . . . . . . . . 18
1.8 TheComplexFormoftheFourierSeries . . . . . . . . . 30
1.9 ComplexGeneralizedFourierSeries . . . . . . . . . . . 33
1.10 FourierSeriesforFunctionsofSeveralVariables . . . . 35
1.11 UniformConvergenceofFourierSeries . . . . . . . . . 36
1.12 TheGibbsPhenomenon . . . . . . . . . . . . . . . . . . 41

1.13 CompletenessofaSystemofTrigonometricFunctions . 42
1.14 General Systems of Functions: Parseval’s Equality and
Completeness . . . . . . . . . . . . . . . . . . . . . . . 44
1.15 ApproximationofFunctionsintheMean . . . . . . . . . 45
1.16 FourierSeriesofFunctionsGivenatDiscretePoints . . . 49
1.17 SolutionofDifferentialEquationsbyUsingFourierSeries 51
1.18 FourierTransforms . . . . . . . . . . . . . . . . . . . . 55
1.19 TheFourierIntegral . . . . . . . . . . . . . . . . . . . . 60
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
vii
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viii Contents
2 Sturm-LiouvilleTheory 79
2.1 TheSturm-LiouvilleProblem . . . . . . . . . . . . . . . 79
2.2 MixedBoundaryConditions . . . . . . . . . . . . . . . 90
2.3 ExamplesofSturm-LiouvilleProblems . . . . . . . . . 95
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3 One-DimensionalHyperbolicEquations 113
3.1 DerivationoftheBasicEquations . . . . . . . . . . . . 113
3.2 BoundaryandInitialConditions . . . . . . . . . . . . . 116
3.3 Other Boundary Value Problems: Longitudinal Vibra-
tionsofaThinRod . . . . . . . . . . . . . . . . . . . . 121
3.4 TorsionalOscillationsofanElasticCylinder . . . . . . . 124
3.5 AcousticWaves . . . . . . . . . . . . . . . . . . . . . . 127
3.6 WavesinaShallowChannel . . . . . . . . . . . . . . . 132
3.7 ElectricalOscillationsinaCircuit . . . . . . . . . . . . 135
3.8 TravelingWaves: D’AlembertMethod . . . . . . . . . . 138
3.9 Semi-infinite String Oscillations and the Use of Sym-
metryProperties . . . . . . . . . . . . . . . . . . . . . . 157
3.10 FiniteIntervals: TheFourierMethodforOne-Dimensional
WaveEquations . . . . . . . . . . . . . . . . . . . . . . 180
3.11 GeneralizedFourierSolutions . . . . . . . . . . . . . . 211
3.12 EnergyoftheString . . . . . . . . . . . . . . . . . . . . 213
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4 Two-DimensionalHyperbolicEquations 241
4.1 DerivationoftheEquationsofMotion . . . . . . . . . . 242
4.2 OscillationsofaRectangularMembrane . . . . . . . . . 247
4.3 TheFourierMethodAppliedtoSmallTransverseOscil-
lationsofaCircularMembrane . . . . . . . . . . . . . . 277
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5 One-DimensionalParabolicEquations 327
5.1 Physical Problems Described by Parabolic Equations:
BoundaryValueProblems . . . . . . . . . . . . . . . . 327
5.2 The Principle of the Maximum, Correctness, and the
GeneralizedSolution . . . . . . . . . . . . . . . . . . . 340
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Contents ix
5.3 The Fourier Method of Separation of Variables for the
HeatConductionEquation . . . . . . . . . . . . . . . . 344
5.4 HeatConductioninanInfiniteBar . . . . . . . . . . . . 383
5.5 HeatEquationforaSemi-infiniteBar . . . . . . . . . . 394
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
6 ParabolicEquationsforHigher-DimensionalProblems 419
6.1 HeatConductioninMorethanOneDimension . . . . . 419
6.2 HeatConductionwithinaFiniteRectangularDomain . . 430
6.3 HeatConductionwithinaCircularDomain . . . . . . . 460
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
7 EllipticEquations 489
7.1 EllipticPartialDifferentialEquationsandRelatedPhys-
icalProblems . . . . . . . . . . . . . . . . . . . . . . . 489
7.2 The Dirichlet Boundary Value Problem for Laplace’s
EquationinaRectangularDomain . . . . . . . . . . . . 508
7.3 Laplace’sandPoisson’sEquationsforTwo-Dimensional
DomainswithCircularSymmetry . . . . . . . . . . . . 527
7.4 Laplace’sEquationinCylindricalCoordinates . . . . . . 561
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
8 BesselFunctions 613
8.1 BoundaryValueProblemsLeadingtoBesselFunctions . 613
8.2 BesselFunctionsoftheFirstKind . . . . . . . . . . . . 620
8.3 PropertiesofBesselFunctionsoftheFirstKind: J (x) . 622
n
8.4 BesselFunctionsoftheSecondKind . . . . . . . . . . . 627
8.5 BesselFunctionsoftheThirdKind . . . . . . . . . . . . 629
8.6 ModifiedBesselFunctions . . . . . . . . . . . . . . . . 630
8.7 TheEffectofBoundariesonBesselFunctions . . . . . . 632
8.8 OrthogonalityandNormalizationofBesselFunctions . . 634
8.9 TheFourier-BesselSeries . . . . . . . . . . . . . . . . . 638
8.10 FurtherExamplesofFourier-BesselSeriesExpansions . 655
8.11 SphericalBesselFunctions . . . . . . . . . . . . . . . . 663
8.12 TheGammaFunction . . . . . . . . . . . . . . . . . . . 667
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670
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x Contents
9 LegendreFunctions 677
9.1 BoundaryValueProblemsLeadingtoLegendrePolyno-
mials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
9.2 GeneratingFunctionforLegendrePolynomials . . . . . 686
9.3 RecurrenceRelations . . . . . . . . . . . . . . . . . . . 687
9.4 OrthogonalityofLegendrePolynomials . . . . . . . . . 689
9.5 TheMultipoleExpansioninElectrostatics . . . . . . . . 692
9.6 AssociatedLegendreFunctionsPm(x) . . . . . . . . . . 696
n
9.7 OrthogonalityandtheNormofAssociatedLegendreFunc-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
9.8 Fourier-LegendreSeriesinLegendrePolynomials . . . . 702
9.9 Fourier-Legendre Series in Associated Legendre Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
9.10 Laplace’sEquationinSphericalCoordinatesandSpher-
icalFunctions . . . . . . . . . . . . . . . . . . . . . . . 713
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
A EigenvaluesandEigenfunctionsoftheSturm-LiouvilleProb-
lem 743
B Auxiliary Functions for Different Types of Boundary Con-
ditions 747
C TheSturm-LiouvilleProblemandtheLaplaceEquation 751
D VectorCalculus 757
E HowtoUsetheSoftwareAssociatedwiththisBook 769
E.1 ProgramOverview . . . . . . . . . . . . . . . . . . . . 770
E.2 ExamplesUsingtheProgramTrigSeries . . . . . . . . . 773

E.3 ExamplesUsingtheProgramWaves . . . . . . . . . . . 781
E.4 ExamplesUsingtheProgramHeat . . . . . . . . . . . . 797
E.5 ExamplesUsingtheProgramLaplace . . . . . . . . . . 811
E.6 ExamplesUsingtheProgramFourierSeries . . . . . . . 822
Bibliography 831
Index 833
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Introduction
Thetopicsofthisbookarepartialdifferentialequations(PDEs)ofmathe-
matical physics and boundary value problems, Fourier series, and special
functions. This is the core content of many courses in the fields of engi-
neering,physics,mathematics,andappliedmathematics.
Thebook,alongwiththecompanionsoftware,representsaninnovative
teachingandlearningprojectthatdoesnotexistinthecurrentliteraturein
partialdifferentialequationsofmathematicalphysics. Thebookissignifi-
cantlymoredetailedthanthetypicalintroductiontoPDE.Itcontainsmany
examplesthatshowhowtosetupphysicalproblemsasmathematicalones;
how to solve partial differential equations under different types of bound-
ary conditions; how to work with special functions; and how to carry out
aFourieranalysisusingthesefunctions. Thetopicsdiscussedinthisbook
arepresentedinfull—notmerelywithbriefexplanationsandsolutionsof
basicexamples. Thetextalsocontainsmanyphysicalapplications,which
areimportantnotonlybecauseoftheirsignificanceinphysicsbutalsobe-

cause they demonstrate how the same mathematical approaches may be
usedfordifferentphysicalproblems. Thisfeatureprovidesthereaderwith
waysofextendingthesemethodstootherproblemsintherealworld. The
authorshavemanyyearsofexperienceinteachingbothphysicsandmath-
ematicsandbelievethatthetextisareasonablecombinationofastringent
mathematicalapproachandphysicalintuition.
The features of the book allow the presentation of a large amount
of material of very significant depth during a one-semester undergradu-
ate or graduate course for physicists, mathematicians, or engineers. The
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xii Introduction
book can also be used to teach a sequence of undergraduate and graduate
courses. Thetextissubstantiallybroaderthanmostmathematicalphysics
textbooksinPDEs,anaspectthatwillgreatlyaidinstructorsintheirselec-
tionoftopicsforclassesofdifferentcapabilities. Numerousproblemsare
presented, from fairly simple to interestingly complex—an attribute that
allowsalecturertosuggestproblemsofdifferentlevelstomatchtheabil-
ities of the students. Solutions to some of the more difficult problems are
included;othersareleftforthereadertosolve.
Thedetailedexplanationsandabundantexamples,alongwiththecom-
panion software, also make this book useful for self-study, where it is
important to illustrate all the steps of a solution and provide a tool for
theextensionofmathematicalmethodstomorerealisticandsophisticated
problems. In keeping with practices currently recommended by physics
educationresearch, anumberofReadingExercisesarescatteredthrough-
out the text. These exercises are designed to keep the reader engaged in
the flow of arguments presented in the text and are particularly useful for
self-study and self-evaluation. The reader is strongly encouraged to read
the text with pencil and paper at hand, filling in the steps of the exercises
while working through the text. The software provides a laboratory en-
vironment that allows the user to generate and model different physical
situations and learn by experimentation. From this standpoint, the book
along with the software can also be used as a reference for students and
professionalsalikeonPDEs,Fourierseries,andspecialfunctions.
The companion problem-solving software is a very important and in-
trinsic feature of the book and represents an approach to mathematical
physics that integrates text, computational environment, and visualiza-
tion. In typical undergraduate and even graduate courses in mathematical
physics,studentsarelimitedtoasmallnumberofsimpleproblemsdueto
timeconstraints, whichprohibitthestudyofavastnumberofinteresting,

butmorecomplex,applications. Thesoftwareaccompanyingthistextnot
onlyprovidesvisualizationandanimationofvariousclassesofmathemat-
icalproblems,butalsoguidesthereaderandshowsthesequenceofallthe
steps needed to solve the problem. Once the problem is solved, the soft-
wareallowsadeeperinvestigationoftheproblem—aninvestigationofthe
dependenceofthesolutionontheparameters,theaccuracyofthesolution,
thespeedofaseriesconvergence,andrelatedorsimilarquestions. Italso
allows the reader to experiment with a much larger number of problems
thanaretypicallytreatedinastandardcourse.
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