分数阶微分方程混合配置及自适应配置方法

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1、HybridcollocationandadaptivecollocationalgorithmforfractionaldifferentialequationsCandidateSupervisorandRankCollegeProgramFengYunProf.CaoXuenianMathematicsandComputationalScienceComputationalMathematicsSpecializationNumericalMethodsforFractionalDifferentialEquationsDegreeUniversity

2、DateMasterofScienceXiangtanUniversityApril16th,2010..................................................................IAbstract...............................................................II.......................................................1.................................

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4、.......33IAbstractThispaperstudieshybridcollocationandadaptivecollocationalgorithmforfrac-tionaldifferentialequations(FDEs).HybridcollocationmethodissuitableforsolvingaclassofFDEs,thederivativeofthesolutionofwhichhassingularityneartheleftendpointoftheintervalconsid-ered.Thecommonly

5、gradedpolynomialcollocationmethod,ontheonehandcouldnotreflectthesingularityofthesolutionduetotheuseofpiecewisepolynomialap-proximation,andontheotherhandmaycauseseriouslyround-offerrorproblemduetotheuseofextremelynon-uniformmesh.Toreflectthesingularityofthesolution,inthefirstsub-interv

6、al,weapproximatetheunknownfunctionsbyusingnon-polynomialfunctions,whileintheremainingsub-intervals,westillusepolynomialfunctionsap-proximatetheunknowns.Inaddition,toavoidusingextremelysmallsub-intervals,wecorrectthecommonlyusedgradedmeshandobtainanewmesh.Underthetheoreticalanalysi

7、sandcalculations,weobtaintheglobalconvergenceorderofthemethod,thenumericalexperimentsdemonstratetheeffectivenessofthemethod.Inordertoobtainauser-orientederrorestimates,wehaveestablishedadaptivecollocationalgorithmforFDEs.Forthispurpose,aposteriorierrorestimatesofcollo-cationmethodi

8、sderivedandthederivederrorindicat