eigenvalue problems and fourier series

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1、EigenvalueProblemsandFourierSeries9Inpreviouschapters,wehaveseenthatmanyphysicalsituationscanbemodeledbyeitherordinarydifferentialequationsorsystemsofordinarydifferentialequa-tions.However,tounderstandthemotionofastringataparticularlocationandataparticulartime,thetemperatureinathinwireatapar

2、ticularlocationandaparticulartime,ortheelectrostaticpotentialatapointonaplate,wemustsolvepartialdifferentialequationsaseachofthesequantitiesdependson(atleast)twoindependentvariables.Waveequationc2uuxxttHeatequationuc2utxxLaplace’sequationuxxuyy0InChapter10,weintroduceaparticularmethodfor

3、solvingthesepartialdiffer-entialequations(aswellasothers).Inordertocarryoutthismethod,however,weintroducethenecessarytoolsinthischapter.Webeginwithadiscussionofboundary-valueproblemsandtheirsolutions.9.1Boundary-ValueProblems,EigenvalueProblems,Sturm–LiouvilleProblems9.1.1Boundary-ValueProbl

4、emsInprevioussections,wehavesolvedinitial-valueproblems.However,atthistimewewillconsiderboundary-valueproblemswhicharesolvedinmuchthesame727728Chapter9EigenvalueProblemsandFourierSerieswayasinitial-valueproblemsexceptthatthevalueofthefunctionanditsderiva-tivesaregivenattwovaluesoftheindepend

5、entvariableinsteadofone.Thegen-eralformofasecond-order(two-point)boundary-valueproblemisd2ydya2xa1xa0xyfx,a

6、shomogeneousboundaryconditions.Wealsoconsiderboundary-valueproblemsthatincludeaparameterinthedifferentialequation.Wesolvetheseproblems,calledeigenvalueproblems,inordertoinvestigateseveralusefulpropertiesassociatedwiththeirsolutions.yy0,0

7、ausethecharacteristicequationisk210withrootski,ageneralsolutionofyy0isyccosxcsinxandit1,212followsthatycsinxccosx.Applyingtheboundaryconditions,12wehavey0c0.Then,yccosx.Withthissolution,wehave21yΠcsinΠ0foranyvalueofc.Theref