Partial Differential Equations and Boundary Value Problems with Maple Second Edition George A ...

733 Pages·2009·5.07 MB·English

Partial Differential Equations
and Boundary Value Problems
with Maple
Second Edition
Thispageintentionallyleftblank
Partial Differential Equations
and Boundary Value Problems
with Maple
Second Edition
George A. Articolo
AMSTERDAM•BOSTON•HEIDELBERG•LONDON
NEWYORK•OXFORD•PARIS•SANDIEGO
SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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LibraryofCongressCataloging-in-PublicationData
Articolo,GeorgeA.
PartialdifferentialequationsandboundaryvalueproblemswithMaple/GeorgeA.Articolo.–2nded.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-12-374732-7(pbk.:alk.paper)1.Differentialequations,Partial—Dataprocessing.
2.Boundaryvalueproblems—Dataprocessing.3.Maple(Computerfile)I.Title.
QA377.A822009
515’.3530285–dc22
2009010098
BritishLibraryCataloguing-in-PublicationData
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ISBN:978-0-12-374732-7
ForinformationonallAcademicPresspublications
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09 10 11 9 8 7 6 5 4 3 2 1
Contents
Preface............................................................................... ix
Chapter0:BasicReview...............................................................1
0.1 PreparationforMapleWorksheets...................................................... 1
0.2 PreparationforLinearAlgebra ......................................................... 4
0.3 PreparationforOrdinaryDifferentialEquations....................................... 8
0.4 PreparationforPartialDifferentialEquations.........................................10
Chapter1:OrdinaryLinearDifferentialEquations .................................. 13
1.1 Introduction.............................................................................13
1.2 First-OrderLinearDifferentialEquations.............................................14
1.3 First-OrderInitial-ValueProblem .....................................................19
1.4 Second-OrderLinearDifferentialEquationswithConstantCoefficients............23
1.5 Second-OrderLinearDifferentialEquationswithVariableCoefficients ............28
1.6 FindingaSecondBasisVectorbytheMethodofReductionofOrder...............32
1.7 TheMethodofVariationofParameters—Second-OrderGreen’sFunction.........36
1.8 Initial-ValueProblemforSecond-OrderDifferentialEquations .....................45
1.9 FrobeniusMethodofSeriesSolutionstoOrdinaryDifferentialEquations..........49
1.10 SeriesSineandCosineSolutionstotheEulerDifferentialEquation................51
1.11 FrobeniusSeriesSolutiontotheBesselDifferentialEquation.......................56
ChapterSummary ......................................................................63
Exercises................................................................................65
Chapter2:Sturm-LiouvilleEigenvalueProblemsandGeneralizedFourierSeries ..... 73
2.1 Introduction.............................................................................73
2.2 TheRegularSturm-LiouvilleEigenvalueProblem ...................................73
2.3 Green’sFormulaandtheStatementofOrthonormality ..............................75
2.4 TheGeneralizedFourierSeriesExpansion............................................81
2.5 ExamplesofRegularSturm-LiouvilleEigenvalueProblems ........................86
v
vi Contents
2.6 NonregularorSingularSturm-LiouvilleEigenvalueProblems.................... 129
ChapterSummary .................................................................... 146
Exercises.............................................................................. 147
Chapter3:TheDiffusionorHeatPartialDifferentialEquation.....................161
3.1 Introduction........................................................................... 161
3.2 One-DimensionalDiffusionOperatorinRectangularCoordinates................ 161
3.3 MethodofSeparationofVariablesfortheDiffusionEquation..................... 163
3.4 Sturm-LiouvilleProblemfortheDiffusionEquation............................... 165
3.5 InitialConditionsfortheDiffusionEquationinRectangularCoordinates ........ 168
3.6 ExampleDiffusionProblemsinRectangularCoordinates ......................... 170
3.7 VerificationofSolutions—Three-StepVerificationProcedure..................... 186
3.8 DiffusionEquationintheCylindricalCoordinateSystem ......................... 190
3.9 InitialConditionsfortheDiffusionEquationinCylindricalCoordinates......... 194
3.10 ExampleDiffusionProblemsinCylindricalCoordinates .......................... 196
ChapterSummary .................................................................... 205
Exercises.............................................................................. 206
Chapter4:TheWavePartialDifferentialEquation.................................217
4.1 Introduction........................................................................... 217
4.2 One-DimensionalWaveOperatorinRectangularCoordinates .................... 217
4.3 MethodofSeparationofVariablesfortheWaveEquation......................... 219
4.4 Sturm-LiouvilleProblemfortheWaveEquation ................................... 221
4.5 InitialConditionsfortheWaveEquationinRectangularCoordinates............. 224
4.6 ExampleWaveEquationProblemsinRectangularCoordinates................... 228
4.7 WaveEquationintheCylindricalCoordinateSystem.............................. 244
4.8 InitialConditionsfortheWaveEquationinCylindricalCoordinates ............. 249
4.9 ExampleWaveEquationProblemsinCylindricalCoordinates.................... 251
ChapterSummary .................................................................... 261
Exercises.............................................................................. 262
Chapter5:TheLaplacePartialDifferentialEquation...............................275
5.1 Introduction........................................................................... 275
5.2 LaplaceEquationintheRectangularCoordinateSystem .......................... 276
5.3 Sturm-LiouvilleProblemfortheLaplaceEquationinRectangular
Coordinates............................................................................ 278
5.4 ExampleLaplaceProblemsintheRectangularCoordinateSystem ............... 284
5.5 LaplaceEquationinCylindricalCoordinates....................................... 299
5.6 Sturm-LiouvilleProblemfortheLaplaceEquationinCylindrical
Coordinates............................................................................ 301
Contents vii
5.7 ExampleLaplaceProblemsintheCylindricalCoordinateSystem ................ 307
ChapterSummary .................................................................... 325
Exercises.............................................................................. 327
Chapter6:TheDiffusionEquationinTwoSpatialDimensions......................339
6.1 Introduction........................................................................... 339
6.2 Two-DimensionalDiffusionOperatorinRectangularCoordinates................ 339
6.3 MethodofSeparationofVariablesfortheDiffusionEquationin
TwoDimensions ...................................................................... 341
6.4 Sturm-LiouvilleProblemfortheDiffusionEquationinTwoDimensions ........ 342
6.5 InitialConditionsfortheDiffusionEquationinRectangularCoordinates ........ 347
6.6 ExampleDiffusionProblemsinRectangularCoordinates ......................... 351
6.7 DiffusionEquationintheCylindricalCoordinateSystem ......................... 365
6.8 InitialConditionsfortheDiffusionEquationinCylindricalCoordinates......... 371
6.9 ExampleDiffusionProblemsinCylindricalCoordinates .......................... 374
ChapterSummary .................................................................... 394
Exercises.............................................................................. 395
Chapter7:TheWaveEquationinTwoSpatialDimensions..........................409
7.1 Introduction........................................................................... 409
7.2 Two-DimensionalWaveOperatorinRectangularCoordinates .................... 409
7.3 MethodofSeparationofVariablesfortheWaveEquation..........................411
7.4 Sturm-LiouvilleProblemfortheWaveEquationinTwoDimensions............. 412
7.5 InitialConditionsfortheWaveEquationinRectangularCoordinates............. 417
7.6 ExampleWaveEquationProblemsinRectangularCoordinates................... 420
7.7 WaveEquationintheCylindricalCoordinateSystem.............................. 437
7.8 InitialConditionsfortheWaveEquationinCylindricalCoordinates ............. 443
7.9 ExampleWaveEquationProblemsinCylindricalCoordinates.................... 447
ChapterSummary .................................................................... 466
Exercises.............................................................................. 467
Chapter8:NonhomogeneousPartialDifferentialEquations ........................477
8.1 Introduction........................................................................... 477
8.2 NonhomogeneousDiffusionorHeatEquation...................................... 477
8.3 InitialConditionConsiderationsfortheNonhomogeneousHeatEquation ....... 488
8.4 ExampleNonhomogeneousProblemsfortheDiffusionEquation................. 490
8.5 NonhomogeneousWaveEquation................................................... 510
8.6 InitialConditionConsiderationsfortheNonhomogeneousWaveEquation ...... 520
8.7 ExampleNonhomogeneousProblemsfortheWaveEquation ..................... 523
ChapterSummary .................................................................... 546
Exercises.............................................................................. 547
viii Contents
Chapter9:InfiniteandSemi-infiniteSpatialDomains...............................557
9.1 Introduction........................................................................... 557
9.2 FourierIntegral....................................................................... 557
9.3 FourierSineandCosineIntegrals ................................................... 561
9.4 NonhomogeneousDiffusionEquationoverInfiniteDomains ..................... 564
9.5 ConvolutionIntegralSolutionfortheDiffusionEquation ......................... 568
9.6 NonhomogeneousDiffusionEquationoverSemi-infiniteDomains............... 570
9.7 ExampleDiffusionProblemsoverInfiniteandSemi-infiniteDomains ........... 573
9.8 NonhomogeneousWaveEquationoverInfiniteDomains.......................... 586
9.9 WaveEquationoverSemi-infiniteDomains ........................................ 588
9.10 ExampleWaveEquationProblemsoverInfiniteandSemi-infiniteDomains ..... 594
9.11 LaplaceEquationoverInfiniteandSemi-infiniteDomains ........................ 606
9.12 ExampleLaplaceEquationoverInfiniteandSemi-infiniteDomains.............. 612
ChapterSummary .................................................................... 619
Exercises.............................................................................. 621
Chapter10:LaplaceTransformMethodsforPartialDifferentialEquations.........639
10.1 Introduction........................................................................... 639
10.2 LaplaceTransformOperator......................................................... 639
10.3 InverseTransformandConvolutionIntegral........................................ 641
10.4 LaplaceTransformProceduresontheDiffusionEquation......................... 642
10.5 ExampleLaplaceTransformProblemsfortheDiffusionEquation................ 646
10.6 LaplaceTransformProceduresontheWaveEquation ............................. 666
10.7 ExampleLaplaceTransformProblemsfortheWaveEquation .................... 671
ChapterSummary .................................................................... 693
Exercises.............................................................................. 694
References..........................................................................709
Index ...............................................................................711
Preface
ThisisthesecondeditionofthetextPartialDifferentialEquationsandBoundaryValue
ProblemswithMaple,AcademicPress,1998.ThetexthasbeenupdatedfromMaplerelease4
torelease12.Inaddition,basedonrecommendationsandsuggestionsofthemanyhelpful
reviewersofthefirstedition,thetextincorporatesmoreofthemacrocommandsinMapletobe
usedasameansofcheckingsolutions.Similartowhatwasdoneinthefirstedition,Icontinued
thepresentationofthesolutionstoproblemsusingthetraditional,fundamental,mathematical
approachsothatthestudentgetsafirmunderstandingofthemathematicalbasisofthe
developmentofthesolutions.Themacrocommandsarenotintendedtobeusedasameansof
teachingthemathematics—theyareusedonlyasaquickmeansofchecking.
Ifthereeverweretobeaperfectunionincomputationalmathematics,onebetweenpartial
differentialequationsandpowerfulsoftware,Maplewouldbeclosetoit.Thistextisan
attempttojointhetwotogether.
Manyyearsago,Irecallsittinginapartialdifferentialequationsclasswhentheprofessorwas
discussingaheat-flowboundaryvalueproblem.Usingapieceofchalkattheblackboard,he
wasmakingaseeminglydesperateattempttogethisstudentstovisualizethespatial-time
developmentofthethree-dimensionalsurfacetemperatureofaplatethatwasallowedtocool
downtoasurroundingequilibriumtemperature.Youcanimaginethefrustrationthathe,and
manyprofessorsbeforehim,experiencedatdoingthistask.Now,withthepowerful
computationaltoolsandgraphicscapabilitiesathand,thiseraofdifficultyisover.
Thistextpresentstheformalmathematicalconceptsneededtodevelopsolutionstoboundary
valueproblems,anditdemonstratesthecapabilitiesofMaplesoftwareasbeingapowerful
computationaltool.Thegraphicsandanimationcommandsallowforaccuratevisualizationof
thespatial-timedevelopmentofthesolutionsonthecomputerscreen—whatstudentscould
onlyimaginemanyyearsagocannowbeviewedinrealtime.
Thetextistargetedforusebysenior-graduatelevelstudentsandpractitionersinthedisciplines
ofphysics,mathematics,andengineering.Typically,thesepeoplehavealreadyhadsome
exposuretocoursesinbasicphysics,calculus,linearalgebra,andordinarydifferential
equations.TheneedforpreviousexposuretotheMaplesoftwareisnotnecessary.InChapter0,
weprovideanintroductiontosomesimpleMaplecommands,whichisallthatisnecessaryfor
ix