boundary value problems and sturm liouville theory

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1、August7,201221:05c11Sheetnumber1Pagenumber677cyanblackCHAPTER11BoundaryValueProblemsandSturm–LiouvilleTheoryAsaresultofseparatingvariablesinapartialdifferentialequationinChapter10,werepeatedlyencounteredthedifferentialequationX+λX=0,0

2、blemistheprototypeofalargeclassofproblemsthatareimportantinappliedmathematics.TheseproblemsareknownasSturm–Liouvilleboundaryvalueproblems.InthischapterwediscussthemajorpropertiesofSturm–Liouvilleproblemsandtheirsolutions;intheprocessweareabletogeneralizesomewhatthemethodofseparationofvariablesforpa

3、rtialdifferentialequations.11.1TheOccurrenceofTwo-PointBoundaryValueProblemsInChapter10wedescribedthemethodofseparationofvariablesasameansofsolvingcertainproblemsinvolvingpartialdifferentialequations.Theheatconductionproblemconsistingofthepartialdifferentialequationα2u=u,00(1)xxtsubjecttothe

4、boundaryconditionsu(0,t)=0,u(L,t)=0,t>0(2)677August7,201221:05c11Sheetnumber2Pagenumber678cyanblack678Chapter11.BoundaryValueProblemsandtheinitialconditionu(x,0)=f(x),0≤x≤L(3)istypicaloftheproblemsconsideredthere.Acrucialpartoftheprocessofsolv-ingsuchproblemsistofindtheeigenvaluesandeigenfunctionsof

5、thedifferentialequationX+λX=0,0

6、zetheresultsofChapter10.OurmaingoalistoshowhowthemethodofseparationofvariablescanbeusedtosolveproblemssomewhatmoregeneralthanthatofEqs.(1),(2),and(3).Weareinterestedinthreetypesofgeneralizations.First,wewishtoconsidermoregeneralpartialdifferentialequations—forexample,theequationr(x)ut=[p(x)ux]x−q(x

7、)u+F(x,t).(7)Thisequationcanarise,asindicatedinAppendixAofChapter10,inthestudyofheatconductioninabarofvariablematerialpropertiesinthepresenceofheatsources.Ifp(x)andr(x)areconstants,andifthesourcetermsq(x)ua