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1、2010–4–30StabilityAnalysisofNumericalMethodsforNonlinearFunctionalDifferentialandFunctionalEquationsCandidateSupervisorCollegeProgramSpecializationDegreeUniversityDateZhongyanLiuProfessorYuexinYuMathematicsandComputationalScienceComputationalMathematicsTheoryandApplica
2、tionofNumericalMethodsforStiffDifferentialEquationsMasterofScienceXiangtanUniversityApril30th,2010,,,,,,,..,.,.Runge−Kutta.:(1)Runge−KuttaD(α,β1,β2,γ1,γ2,δ),.:,(k,l)−.Runge−Kutta(2)−D(α,β1,β2,γ1,γ2,δ),..;;Runge−Kutta;;;;IAbstractFunctionaldiffe
3、rentialequations(FDEs)arisewidelyinthefieldsofbiology,controltheory,physics,chemistry,economicsandsoon.Itismeaningfultoin-vestigatethetheoryandapplicationofnumericalmethodsforFDEs.Inrecent30years,thetheoryofcomputationalmethodsforFDEshasbeenwidelydiscussedbymanyauthorsa
4、ndagreatdealofresultshavebeenfound.Functionaldifferentialandfunctionalequations(FDFEs)areaclassofhybridsystemsthataremorecom-plexthanthatofFDEs.Inparticular,functionaldifferentialequationsofneutraltypecanbetransformedintoFDFEs.Thenumericalstabilityanalysisofneutralfuncti
5、onaldifferentialequationshasbeenstudiedextensivelyinrecentyears.AsforthesystemsofFDFEs,theasymptoticstabilityofnumericalmethodsforlinearFDFEshasbeendiscussedbyseveralauthors.However,littleattentionhasbeenfocusedonthenonlinearsystemsofFDFEs.Forthesereasons,thepresentpape
6、risdevotedtothestabilityanalysisofnumericalmethodsfornonlinearFDFEs.Runge−Kuttamethodsandgenerallinearmethodsarefullydiscussed.Themainworkofthispaperare:(1)ApplyingRunge−Kuttamethodstosolvethefunctionaldifferentialandfunc-tionalequationsoftheclassD(α,β1,β2,γ1,γ2,δ),ther
7、esultsshowthatRunge-Kuttamethodsof(k,l)−algebraicstabilityarestableandasymptoticstableundersuitableconditions.Numericaltestsaregiventoconfirmthetheoreticalresults.(2)ThemoreextensivemethodsofgenerallinearmethodsareadaptedforsolvingaclassofD(α,β1,β2,γ1,γ2,δ),aseriesofsta
8、bleandasymptoticstableconditionsarederived.Numericaltestsaregiventodemonstratethetheoreticalresults.KeyWords:Function
