partial differential equations and fourier series

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1、August7,201221:05c10Sheetnumber1Pagenumber589cyanblackCHAPTER10PartialDifferentialEquationsandFourierSeriesInmanyimportantphysicalproblemstherearetwoormoreindependentvariables,sothecorrespondingmathematicalmodelsinvolvepartial,ratherthanordinary,differentialequations.Thischaptertre

2、atsoneimportantmethodforsolvingpar-tialdifferentialequations,amethodknownasseparationofvariables.Itsessentialfeatureisthereplacementofthepartialdifferentialequationbyasetofordinarydifferentialequations,whichmustbesolvedsubjecttogiveninitialorboundarycon-ditions.Thefirstsectionofthis

3、chapterdealswithsomebasicpropertiesofboundaryvalueproblemsforordinarydifferentialequations.Thedesiredsolutionofthepar-tialdifferentialequationisexpressedasasum,usuallyaninfiniteseries,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,oneinwhichthevalueofthedependentvariableyoritsderivativeisspecifiedattwodifferent

7、points.Suchconditionsarecalledboundaryconditionstodistinguishthemfrominitialconditionsthatspecifythevalueofyandyatthesamepoint.Adifferentialequationandsuitableboundaryconditionsformatwo-pointboundaryvalueproblem.Atypicalexampleisthedifferentialequationy+p(x)y+q(x)y=g(x)(3)witht

8、heboundaryconditionsy(α)=y