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 AcademicPressisanimprintofElsevier AcademicPressisanimprintofElsevier 30CorporateDrive,Suite400,Burlington,MA01803,USA 525BStreet,Suite1900,SanDiego,California92101-4495,USA 84Theobald’sRoad,LondonWC1X8RR,UK Copyright©2009,ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopy,recording,oranyinformationstorageandretrievalsystem,without permissioninwritingfromthepublisher. 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PrintedintheUnitedStates 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