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1、August7,201221:05c10Sheetnumber1Pagenumber589cyanblackCHAPTER10PartialDifferentialEquationsandFourierSeriesInmanyimportantphysicalproblemstherearetwoormoreindependentvariables,sothecorrespondingmathematicalmodelsinvolvepartial,ratherthanordinary,differentialequations.Thischaptertre
2、atsoneimportantmethodforsolvingpar-tialdifferentialequations,amethodknownasseparationofvariables.Itsessentialfeatureisthereplacementofthepartialdifferentialequationbyasetofordinarydifferentialequations,whichmustbesolvedsubjecttogiveninitialorboundarycon-ditions.The?rstsectionofthis
3、chapterdealswithsomebasicpropertiesofboundaryvalueproblemsforordinarydifferentialequations.Thedesiredsolutionofthepar-tialdifferentialequationisexpressedasasum,usuallyanin?niteseries,formedfromsolutionsoftheordinarydifferentialequations.Inmanycasesweultimatelyneedtodealwithaserieso
4、fsinesand/orcosines,sopartofthechapterisdevotedtoadiscussionofsuchseries,whichareknownasFourierseries.Withthenecessarymath-ematicalbackgroundinplace,wethenillustratetheuseofseparationofvariablesinavarietyofproblemsarisingfromheatconduction,wavepropagation,andpotentialtheory.10.1Two
5、-PointBoundaryValueProblemsUptothispointinthebookwehavedealtwithinitialvalueproblems,consistingofadifferentialequationtogetherwithsuitableinitialconditionsatagivenpoint.Atypicalexample,whichwasdiscussedatlengthinChapter3,isthedifferentialequationy+p(t)y+q(t)y=g(t),(1)589August7,
6、201221:05c10Sheetnumber2Pagenumber590cyanblack590Chapter10.PartialDifferentialEquationsandFourierSerieswiththeinitialconditionsy(t)=y,y(t)=y.(2)0000Physicalapplicationsoftenleadtoanothertypeofproblem,oneinwhichthevalueofthedependentvariableyoritsderivativeisspeci?edattwodifferent
7、points.Suchconditionsarecalledboundaryconditionstodistinguishthemfrominitialconditionsthatspecifythevalueofyandyatthesamepoint.Adifferentialequationandsuitableboundaryconditionsformatwo-pointboundaryvalueproblem.Atypicalexampleisthedifferentialequationy+p(x)y+q(x)y=g(x)(3)witht
8、heboundaryconditionsy(α)=y
