9 Methods for Ordinary Differential Equations_One step

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1、MethodsforOrdinaryDifferentialEquationsGraduateCourseofFudanUniversityByProf.XuanZengOutlineInitialValueProblemsexamples–Signalpropagation(twotimescales)–LCcircuit(oscillator)–FeedbackSystem(Unstablesystem)Lookatthesimplefinite-differencemethod–ForwardEuler(FE),BackwardEuler(BE)andT

2、rapezoidalRule(TR)–Lookattheapproximationsandalgorithms–ExperimentsgeneratedmanyquestionsDescribedConvergenceTheoremforMethods–SmallhconsiderationsAnalyzedStiffStabilityandOscillationPreservation–LargehconsiderationsMicroelectronicsDepartment,FudanUniversityApplicationProblemsSignal

3、TransmissioninanIntegratedCircuitSignalWireWirehasresistanceWireandgroundplaneformacapacitorLogicLogicGateGateGroundPlane•MetalWirescarrysignalsfromgatetogate.•Howlongisthesignaldelayed?MicroelectronicsDepartment,FudanUniversityApplicationProblemsSignalTransmissioninanIC–CircuitModelr

4、esistorcapacitorConstructingtheModel•Cutthewireintosections.•Modelwireresistancewithresistors.•Modelwire-planecapacitancewithcapacitors.MicroelectronicsDepartment,FudanUniversityApplicationProblemsSignalTransmissioninanIC–2x2examplevivConstitutiveConservation1R22EquationsLawsiC1iR1R2i

5、R3iC2iCdvciii0cCRRdt112C1R1R3C21ivRRiiiCRR2320RNodalEquationsYields2x2Systemdv1111Cv110dtRR12R20Cvdv111222dtRRR232MicroelectronicsDepartment,FudanUniversityAnAsideonEigenanalysisdxt()ConsideranODE:Axt(),x(0)x0dt1100

6、Eigendecomposition:AEEE12nn00EE12E00nE1Changeofvariables:Eyt()(xt)(yt)Ext()dEyt()Substituting:AEyt(),Ey(0)x0dt0011dyt()1y()tMultiplyby:EEAEty()00dt00nMicroelectronicsDepartment,FudanUniversityAnAsideonEigenanalysis00dyt()

7、1DecoupledFromlastslide:00y()tdtEquations!00ndyti()tiDecouplign:yt()ytey()(0)iiidtdxt()StepsforsolvingAxt(),x(0)x0dt11)Determine,E2)Compute(0)yEx01te003)Compute()yt00(0)y4)()xtEyt()nt00eMicroelectronicsDepartment,FudanUniversityApplicationProblemsSi

8、gnalT