常微分方程数值解函数Word格式.docx
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odefun
Afunctionhandlethatevaluatestherightsideofthedifferentialequations.SeeFunctionHandlesintheMATLABProgrammingdocumentationformoreinformation.Allsolverssolvesystemsofequationsintheformorproblemsthatinvolveamassmatrix,.Theode23ssolvercansolveonlyequationswithconstantmassmatrices.ode15sandode23tcansolveproblemswithamassmatrixthatissingular,i.e.,differential-algebraicequations（DAEs）.
tspan
Avectorspecifyingtheintervalofintegration,[t0,tf].Thesolverimposestheinitialconditionsattspan
（1）,andintegratesfromtspan
（1）totspan（end）.Toobtainsolutionsatspecifictimes（allincreasingoralldecreasing）,usetspan=[t0,t1,...,tf].
Fortspanvectorswithtwoelements[t0tf],thesolverreturnsthesolutionevaluatedateveryintegrationstep.Fortspanvectorswithmorethantwoelements,thesolverreturnssolutionsevaluatedatthegiventimepoints.Thetimevaluesmustbeinorder,eitherincreasingordecreasing.
Specifyingtspanwithmorethantwoelementsdoesnotaffecttheinternaltimestepsthatthesolverusestotraversetheintervalfromtspan
（1）totspan（end）.AllsolversintheODEsuiteobtainoutputvaluesbymeansofcontinuousextensionsofthebasicformulas.Althoughasolverdoesnotnecessarilysteppreciselytoatimepointspecifiedintspan,thesolutionsproducedatthespecifiedtimepointsareofthesameorderofaccuracyasthesolutionscomputedattheinternaltimepoints.
Specifyingtspanwithmorethantwoelementshaslittleeffectontheefficiencyofcomputation,butforlargesystems,affectsmemorymanagement.
y0
Avectorofinitialconditions.
options
Structureofoptionalparametersthatchangethedefaultintegrationproperties.Thisisthefourthinputargument.
[t,y]=solver（odefun,tspan,y0,options）
Youcancreateoptionsusingtheodesetfunction.Seeodesetfordetails.
Thefollowingtableliststheoutputargumentsforthesolvers.
T
Columnvectoroftimepoints.
Y
Solutionarray.EachrowinYcorrespondstothesolutionatatimereturnedinthecorrespondingrowofT.
TE
Thetimeatwhichaneventoccurs.
YE
Thesolutionatthetimeoftheevent.
IE
Theindexioftheeventfunctionthatvanishes.
sol
Structuretoevaluatethesolution.
Description
[T,Y]=solver（odefun,tspan,y0）withtspan=[t0tf]integratesthesystemofdifferentialequationsfromtimet0totfwithinitialconditionsy0.odefunisafunctionhandle.SeeFunctionHandlesintheMATLABProgrammingdocumentationformoreinformation.Functionf=odefun（t,y）,forascalartandacolumnvectory,mustreturnacolumnvectorfcorrespondingto.EachrowinthesolutionarrayYcorrespondstoatimereturnedincolumnvectorT.Toobtainsolutionsatthespecifictimest0,t1,...,tf（allincreasingoralldecreasing）,usetspan=[t0,t1,...,tf].
ParameterizingFunctions,intheMATLABMathematicsdocumentation,explainshowtoprovideadditionalparameterstothefunctionfun,ifnecessary.
[T,Y]=solver（odefun,tspan,y0,options）solvesasabovewithdefaultintegrationparametersreplacedbypropertyvaluesspecifiedinoptions,anargumentcreatedwiththeodesetfunction.CommonlyusedpropertiesincludeascalarrelativeerrortoleranceRelTol（1e-3bydefault）andavectorofabsoluteerrortolerancesAbsTol（allcomponentsare1e-6bydefault）.Ifcertaincomponentsofthesolutionmustbenonnegative,usetheodesetfunctiontosettheNonNegativepropertytotheindicesofthesecomponents.Seeodesetfordetails.
[T,Y,TE,YE,IE]=solver（odefun,tspan,y0,options）solvesasabovewhilealsofindingwherefunctionsof,calledeventfunctions,arezero.Foreacheventfunction,youspecifywhethertheintegrationistoterminateatazeroandwhetherthedirectionofthezerocrossingmatters.Dothisbysettingthe'
Events'
propertytoafunction,e.g.,eventsor@events,andcreatingafunction[value,isterminal,direction]=events（t,y）.Fortheitheventfunctioninevents,
value（i）isthevalueofthefunction.
isterminal（i）=1,iftheintegrationistoterminateatazeroofthiseventfunctionand0otherwise.
direction（i）=0ifallzerosaretobecomputed（thedefault）,+1ifonlythezeroswheretheeventfunctionincreases,and-1ifonlythezeroswheretheeventfunctiondecreases.
CorrespondingentriesinTE,YE,andIEreturn,respectively,thetimeatwhichaneventoccurs,thesolutionatthetimeoftheevent,andtheindexioftheeventfunctionthatvanishes.
sol=solver（odefun,[t0tf],y0...）returnsastructurethatyoucanusewithdevaltoevaluatethesolutionatanypointontheinterval[t0,tf].Youmustpassodefunasafunctionhandle.Thestructuresolalwaysincludesthesefields:
sol.x
Stepschosenbythesolver.
sol.y
Eachcolumnsol.y（:
i）containsthesolutionatsol.x（i）.
sol.solver
Solvername.
IfyouspecifytheEventsoptionandeventsaredetected,solalsoincludesthesefields:
sol.xe
Pointsatwhichevents,ifany,occurred.sol.xe（end）containstheexactpointofaterminalevent,ifany.
sol.ye
Solutionsthatcorrespondtoeventsinsol.xe.
sol.ie
IndicesintothevectorreturnedbythefunctionspecifiedintheEventsoption.Thevaluesindicatewhicheventthesolverdetected.
IfyouspecifyanoutputfunctionasthevalueoftheOutputFcnproperty,thesolvercallsitwiththecomputedsolutionaftereachtimestep.Fouroutputfunctionsareprovided:
odeplot,odephas2,odephas3,odeprint.Whenyoucallthesolverwithnooutputarguments,itcallsthedefaultodeplottoplotthesolutionasitiscomputed.odephas2andodephas3producetwo-andthree-dimensionalphaseplaneplots,respectively.odeprintdisplaysthesolutioncomponentsonthescreen.Bydefault,theODEsolverpassesallcomponentsofthesolutiontotheoutputfunction.YoucanpassonlyspecificcomponentsbyprovidingavectorofindicesasthevalueoftheOutputSelproperty.Forexample,ifyoucallthesolverwithnooutputargumentsandsetthevalueofOutputSelto[1,3],thesolverplotssolutioncomponents1and3astheyarecomputed.
Forthestiffsolversode15s,ode23s,ode23t,andode23tb,theJacobianmatrixiscriticaltoreliabilityandefficiency.UseodesettosetJacobianto@FJACifFJAC（T,Y）returnstheJacobianortothematrixiftheJacobianisconstant.IftheJacobianpropertyisnotset（thedefault）,isapproximatedbyfinitedifferences.SettheVectorizedproperty'
on'
iftheODEfunctioniscodedsothatodefun（T,[Y1,Y2...]）returns[odefun（T,Y1）,odefun（T,Y2）...].Ifisasparsematrix,settheJPatternpropertytothesparsitypatternof,i.e.,asparsematrixSwithS（i,j）=1iftheithcomponentofdependsonthejthcomponentof,and0otherwise.
ThesolversoftheODEsuitecansolveproblemsoftheform,withtime-andstate-dependentmassmatrix.（Theode23ssolvercansolveonlyequationswithconstantmassmatrices.）Ifaproblemhasamassmatrix,createafunctionM=MASS（t,y）thatreturnsthevalueofthemassmatrix,anduseodesettosettheMasspropertyto@MASS.Ifthemassmatrixisconstant,thematrixshouldbeusedasthevalueoftheMassproperty.Problemswithstate-dependentmassmatricesaremoredifficult:
IfthemassmatrixdoesnotdependonthestatevariableandthefunctionMASSistobecalledwithoneinputargument,t,settheMStateDependencepropertyto'
none'
.
Ifthemassmatrixdependsweaklyon,setMStateDependenceto'
weak'
（thedefault）;
otherwise,setitto'
strong'
.Ineithercase,thefunctionMASSiscalledwiththetwoarguments（t,y）.
Iftherearemanydifferentialequations,itisimportanttoexploitsparsity:
Returnasparse.
SupplythesparsitypatternofusingtheJPatternpropertyorasparseusingtheJacobianproperty.
Forstronglystate-dependent,setMvPatterntoasparsematrixSwithS（i,j）=1ifforanyk,the（i,k）componentofdependsoncomponentjof,and0otherwise.
Ifthemassmatrixissingular,thenisasystemofdifferentialalgebraicequations.DAEshavesolutionsonlywhenisconsistent,thatis,ifthereisavectorsuchthat.Theode15sandode23tsolverscansolveDAEsofindex1providedthaty0issufficientlyclosetobeingconsistent.Ifthereisamassmatrix,youcanuseodesettosettheMassSingularpropertyto'
yes'
'
no'
or'
maybe'
.Thedefaultvalueof'
causesthesolvertotestwhethertheproblemisaDAE.Youcanprovideyp0asthevalueoftheInitialSlopeproperty.Thedefaultisthezerovector.IfaproblemisaDAE,andy0andyp0arenotconsistent,thesolvertreatsthemasguesses,attemptstocomputeconsistentvaluesthatareclosetotheguesses,andcontinuestosolvetheproblem.WhensolvingDAEs,itisveryadvantageoustoformulatetheproblemsothatisadiagonalmatrix（asemi-explicitDAE）.
Solver
ProblemType
OrderofAccuracy
WhentoUse
ode45
Nonstiff
Medium
Mostofthetime.Thisshouldbethefirstsolveryoutry.
ode23
Low
Forproblemswithcrudeerrortolerancesorforsolvingmoderatelystiffproblems.
ode113
Lowtohigh
Forproblemswithstringenterrortolerancesorforsolvingcomputationallyintensiveproblems.
ode15s
Stiff
Lowtomedium
Ifode45isslowbecausetheproblemisstiff.
ode23s
Ifusingcrudeerrortolerancestoso
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