Numerical Solution of Two-Point Boundary Value Pro

NumericalSolutionofTwo-PointBoundaryValueProblemsB.S.c.ThesisbyGabriellaSebestyenMathematicsB.S.c.,MathematicalAnalystSupervisor:IstvanFaragoProfessorattheDepartmentofAppliedAnalysisandComputationalMathematicsEotvosLorandUniversityBudapest20111Contents1Introduction32SolvabilityofBoundaryValueProblems53ShootingMethod73.1Introduction..............................73.2Intervallum-BisectionMethod....................73.3ChordMethod............................83.4NewtonMethod............................94FiniteDierenceSchemes114.1FiniteDierences...........................114.2SolutionwithFiniteDierencesMethod..............124.3TheSolutionofLinearBoundaryValueProblemswithFiniteDierences...............................145NumericalsolutionwithMATLAB255.1WhyMATLAB?...........................255.2SolvingtheexamplewithMATLAB................255.2.1FiniteDierencesMethod..................255.2.2ShootingMethod.......................286Summary3221IntroductionTheobjectofmydissertationistopresentthenumericalsolutionoftwo-pointboundaryvalueproblems.Insomecases,wedonotknowtheinitialconditionsforderivativesofacertainorder.Instead,weknowinitialandnalvaluesfortheunknownderivativesofsomeorder.Thesetypeofproblemsarecalledboundary-valueproblems.Mostphysicalphenomenasaremodeledbysystemsofordinaryorpartialdif-ferentialequations.Usually,theexactsolutionoftheboundaryvalueproblemsaretoodicult,sowehavetoapplynumericalmethods.WeuseddierentnumericalmethodsfordeterminingthenumericalsolutionsofCauchy-problem.OneofthemistheExplicitEulermethod,whichisthesimplestscheme.TheImprovedEulermethodisthesimplestofafamilyofsimilarpredictor-correctormethodsfollowingtheformofasinglepredictorstepandoneormorecorrectorsteps.OnesubgroupofthisfamilyaretheRunge-Kuttamethodswhichuseaxednumberofcorrectorsteps.TheImprovedEulermethodisthesimplestofthissubgroup.Allthemethodsgivencanbeappliedtohigherofordinarydierentialequa-tions,provideditispossibletowriteanexplicitexpressionforthehighestorderderivativeandthesystemhasacompletesetofinitialconditions.MotivationThefollowingexampledescribesaphysicaltaskwhatwecanuseinpracticeanditoriginatesfromatwo-pointboundaryvalueproblem.ShootingProblemWelaunchacannonballfromaxedplace.Lety(t)betheheightofthecan-nonballandx(t)thedistancefromthexedplaceatt0moment.Moreoverwesupposethatthehorizontalspeedisconstant,theverticaldistancedependsonlyonthegravitation,sowedispensewiththedragco-ecient.Thegoalistodeterminetheangleofthelaunch,ifthecannonballislocatedinplacex=L!Thecontinousmathematicalmodeloftheproblemisthefollowing:x0(t)=v;y00(t)=g;x(0)=0;y(0)=0:(1)Thesolutionofthesecondequationwiththeinitialconditionis:3x(t)=vt,whichmeansthatt=xv:LetusintroduceanewfunctionY(x)asfollows:y(t)=y(xv)=Y(x):Then,usingthechainrule,wegety0(t)=dYdxdxdt=dYdxv;y00(t)=vd2Ydx2v=v2d2Ydx2:(2)Hence,weobtaintheproblem:Y00(x)=gv2;x2(0;L);Y(0)=0;Y(L)=0:(3)Thisproblemcanbeeasilysolvedandthesolutionis:Y(x)=gx2v2(Lx):(4)Thenwecandeterminetheangleofthelaunchfromthefollowingrelation:tan()=Y0(0)=gL2v2:(5)Summarizetheshootingproblem,accordingto(5)thesolutionalwaysexists.Itmeanswecanlauchthecannonballtoeverydistance.Butitisaparadoxinthereality.42SolvabilityofBoundaryValueProblemsInthissection,weappoitthetwo-pointboundaryvalueproblemgenerally.Denition1Letf:R3!Rgivenfunctionand;aregivennumbers.Theproblemu00=f(t;u;u0);t2(a;b);(6)u(a)=;u(b)=(7)iscalledtwo-pointboundaryvalueproblem.WhenweanalyzetheCauchyproblemfortheordinarydierentialequationofrstorderintheformu0(t)=g(t;u);(8)u(t0)=u0:(9)wehaveseenthatthesolvabilitydependsonlyonthefunctionf.Thefollowingtheoremgivessucientconditionoftheexistance.Thetheoremconnecttothisproblem:Theorem1Supposeg:[t0;t0+]B(a;)!RiscontinousandboundedbyM.Suppose,furthermore,thatg(t;
)isLipschitzcountinouswithLipschitzconstantLforeveryt2[t0;t0+]:Thentheproblem(8),(9)hasauniquesolutionu(t)denedon[t0b;t0+b],whereb=minn;Mo:Ifweanalyzeboundaryvalueproblems(6),(7)thesituationisdierent.Aswewillsee,boththefunctionfandtheboundaryvaluedeterminetheresulttogether.ExampleLetf(t;u;u0)=1u,hencetheequation(6)hastheformu00+u=1:Thearbitrarysolutionofthisdierentalequationisu(t)=c1cost+c2sint+1,wherec1;c2areconstants.Weanalyzedierentboundaryconditionsin(7),whichshoulddenetheseconstans.5Firstly,letusputa=4;=2;b=;=2:Thesolutionisunique:c1=1;c2=p2+1:Hence,theuniquesolutionisu(t)=1cost+(p2+1)sint+1:Ifweputa=4;=2;b=54;=2;thentheproblemhasnosolution,becausedonotexistsuchconstantsc1;c2forwhichtheboundaryvalueistrue.Thefollowingtheoremgivessucientcondition.Theorem2Supposethat,T=(t1;s1;s2):t2[a;b];s1;s2;2Randf:R3!Risagivenfunctionwiththepropertiesf2C(T);@1f;@2f2C(T);@2f0onT,thereexistsanonnegativeconstantMsuchthatj@3fjMonT.Undertheseconditionsthetwo-pointboundaryvalueproblem(6),(7)hasuniquesolution.CorollaryLetusconsiderthespecialcasewhenfislinear,u00=f(t;u;u0)=p(t)u0+q(t)u+r(t);t2(a;b);u(a)=;u(b)=wherep;q;r2C[a;b]arecontinousfunctions.Ifq(t)0forallt2[a;b];thenthelinearboundaryvalueproblemhasauniquesolution,becauset