中mathematics工具箱使用Word格式文档下载.docx
- 文档编号:18787338
- 上传时间:2023-01-01
- 格式:DOCX
- 页数:22
- 大小:116.43KB
中mathematics工具箱使用Word格式文档下载.docx
《中mathematics工具箱使用Word格式文档下载.docx》由会员分享,可在线阅读,更多相关《中mathematics工具箱使用Word格式文档下载.docx(22页珍藏版)》请在冰豆网上搜索。
SolvesTheseKindsofProblems(求解问题)
Method(方法)
ode45
Nonstiffdifferentialequations(非刚性微分方程)
Runge-Kutta
ode23
ode113
Nonstiffdifferentialequations(非刚性微分方程)
Adams
ode15s
StiffdifferentialequationsandDAEs(非刚性微分-代数方程)
NDFs(BDFs)
ode23s
Stiffdifferentialequations(刚性微分方程)
Rosenbrock
ode23t
ModeratelystiffdifferentialequationsandDAEs
(中等刚性微分方程和代数方程)
Trapezoidalrule
ode23tb
Stiffdifferentialequations(刚性微分方程)
TR-BDF2
ode15i
Fullyimplicitdifferentialequations(全隐式微分方程)
BDFs
EvaluationandExtension(赋值和延拓)
YoucanusethefollowingfunctionstoevaluateandextendsolutionstoODEs.
你能应用如下函数对ODE的数值解解进行赋值和延拓。
Function
Description
deval
EvaluatethenumericalsolutionusingtheoutputofODEsolvers
(用ODE输出对数值解进行赋值)
odextend
ExtendthesolutionofaninitialvalueproblemforanODE
对ODE初值问题的解进行延拓
SolverOptions(求解器选项)
AnoptionsstructurecontainsnamedpropertieswhosevaluesarepassedtoODEsolvers,andwhichaffectproblemsolution.Usethesefunctionstocreate,alter,oraccessanoptionsstructure.
选项结构包含署名属性,其值传递给ODE求解器以影响问题求解。
用这些函数可以创建,改变和接受选项结构。
Function(函数)
Description(描述)
odeset
CreateoralteroptionsstructureforinputtoODEsolver.(创建和改变选项)
odeget
Extractpropertiesfromoptionsstructurecreatedwithodeset.(提取属性选项)
OutputFunctions(输出函数)
Ifanoutputfunctionisspecified,thesolvercallsthespecifiedfunctionaftereverysuccessfulintegrationstep.YoucanuseodesettospecifyoneofthesesamplefunctionsastheOutputFcnproperty,oryoucanmodifythemtocreateyourownfunctions.
如果输出函数被指定,则求解器在每步积分后调用该函数进行输出。
你能够用odeset指定这些例子函数之一作为OutputFcn属性,或创建自己的函数对其进行修改。
odeplot
Time-seriesplot(时间序列图形)
odephas2
Two-dimensionalphaseplaneplot(2-维相平面图形)
odephas3
Three-dimensionalphaseplaneplot(3-维相平面图形)
odeprint
Printtocommandwindow(打印到命令窗)
FirstOrderODEs(一阶ODEs)
Anordinarydifferentialequation(ODE)containsoneormorederivatives
ofadependentvariableywithrespecttoasingleindependentvariablet,
usuallyreferredtoastime.Thederivativeofywithrespecttotisdenoted
asy′,thesecondderivativeasy′′,andsoon.Ofteny(t)isavector,having
elementsy1,y2,...,yn.
MATLABsolvershandlethefollowingtypesoffirst-orderODEs:
•ExplicitODEsoftheformy′=f(t,y)
形如y′=f(t,y)的显式ODEs
•LinearlyimplicitODEsoftheformM(t,y)y′=f(t,y),whereM(t,y)is
amatrix
形如M(t,y)y′=f(t,y)的线性隐式ODEs
•FullyimplicitODEsoftheformf(t,y,y′)=0(ode15ionly):
形如f(t,y,y′)=0的全隐式ODEs
HigherOrderODEs(高阶ODEs)
MATLABODEsolversacceptonlyfirst-orderdifferentialequations.Touse
thesolverswithhigher-orderODEs,youmustrewriteeachequationasan
equivalentsystemoffirst-orderdifferentialequationsoftheform
y′=f(t,y)
Youcanwriteanyordinarydifferentialequation
y(n)=f(t,y,y′,...,y(n−1))
asasystemoffirst-orderequationsbymakingthesubstitutions
y1=y,y2=y′,...,yn=y(n−1)
y1=y,y2=y’,...,yn=y(n−1)
Theresultisanequivalentsystemofnfirst-orderODEs.
Rewritethesecond-ordervanderPolequation
asasystemoffirst-orderODEs.
InitialValues(初值问题)
Generallytherearemanyfunctionsy(t)thatsatisfyagivenODE,and
additionalinformationisnecessarytospecifythesolutionofinterest.In
aninitialvalueproblem,thesolutionofinterestsatisfiesaspecificinitial
condition,thatis,yisequaltoy0atagiveninitialtimet0.Aninitialvalue
problemforanODEisthen
Ifthefunctionf(t,y)issufficientlysmooth,thisproblemhasoneandonlyone
solution.Generallythereisnoanalyticexpressionforthesolution,soitis
necessarytoapproximatey(t)bynumericalmeans.
NonstiffProblems(刚性问题)
Therearethreesolversdesignedfornonstiffproblems:
对于非刚性问题求解有3个求解器。
ode45BasedonanexplicitRunge-Kutta(4,5)formula,theDormand-Princepair.Itisaone-stepsolver–incomputingy(tn),itneedsonlythesolutionattheimmediatelyprecedingtimepoint,y(tn–1).Ingeneral,ode45isthebestfunctiontoapplyasa“firsttry”formostproblems.
ode45基于显式Runge--Kutta(4,5)公式,Dormand-Prince对,它是计算y(tn)的单步求解器,只需前一步的解y(tn-1).一般说来,ode45是对大多数问题的“第一试”的最好的函数。
ode23BasedonanexplicitRunge-Kutta(2,3)pairofBogackiandShampine.Itmaybemoreefficientthanode45atcrudetolerancesandinthepresenceofmildstiffness.Likeode45,ode23isaone-stepsolver.
ode113VariableorderAdams-Bashforth-MoultonPECEsolver.Itmaybemoreefficientthanode45atstringenttolerancesandwhentheODEfunctionisparticularlyexpensivetoevaluate.ode113isamultistepsolver—itnormallyneedsthesolutionsatseveralprecedingtimepointstocomputethecurrentsolution.
StiffProblems(刚性问题)
Notalldifficultproblemsarestiff,butallstiffproblemsaredifficultfor
solversnotspecificallydesignedforthem.Solversforstiffproblemscanbe
usedexactlyliketheothersolvers.However,youcanoftensignificantly
improvetheefficiencyofthesesolversbyprovidingthemwithadditional
informationabouttheproblem.(See“IntegratorOptions”onpage10-9.)
Therearefoursolversdesignedforstiffproblems:
并不是所有困难的问题都是刚性的,但是所有的刚性问题对于非专门为此设计的求解器来说都是困难的。
SolverSyntax(求解语法)
AlloftheODEsolverfunctions,exceptforode15i,shareasyntaxthatmakes
iteasytotryanyofthedifferentnumericalmethods,ifitisnotapparent
whichisthemostappropriate.Toapplyadifferentmethodtothesame
problem,simplychangetheODEsolverfunctionname.Thesimplestsyntax,
commontoallthesolverfunctions,is
求解函数调用
[t,y]=solver(odefun,tspan,y0,options)
wheresolverisoneoftheODEsolverfunctionslistedpreviously.
其中,solver是如前列举的ODE求解函数
Thebasicinputargumentsare
举例:
vanderPolEquation(Nonstiff)
ThisexampleillustratesthestepsforsolvinganinitialvalueODEproblem:
该例说明了求解ODE初值问题的步骤:
1Rewritetheproblemasasystemoffirst-orderODEs.Rewritethe
vanderPolequation(second-order)
1.把高阶方程表示为一阶方程组的等价形式。
vanderPol方程(二阶)
whereμ>
0isascalarparameter,bymakingthesubstitutiony’1=y2.The
resultingsystemoffirst-orderODEsis
为标量常数。
做代换
,得到对应的一阶方程组
2Codethesystemoffirst-orderODEs.Onceyourepresenttheequation
asasystemoffirst-orderODEs,youcancodeitasafunctionthatanODE
solvercanuse.Thefunctionmustbeoftheform
2.对一阶ODE方程组编写ODEfun函数,其形式为
dydt=odefun(t,y)
odefun函数程序
functiondydt=vdp1(t,y)
dydt=[y
(2);
(1-y
(1)^2)*y
(2)-y
(1)];
3.Applyasolvertotheproblem.(调用求解函数求解)
Decidewhichsolveryouwanttousetosolvetheproblem.Thencallthe
solverandpassitthefunctionyoucreatedtodescribethefirst-ordersystem
ofODEs,thetimeintervalonwhichyouwanttosolvetheproblem,and
aninitialconditionvector.
决定用哪个求解函数求解问题,然后调用求解器,把创建的描述方程组的函数,求解区间,和初始条件传递给求解函数。
ForthevanderPolsystem,youcanuseode45ontimeinterval[020]with
initialvaluesy
(1)=2andy
(2)=0.
对于VanderPol系统,你可以用ode45在时间区间[0,20]进行积分,初值条件为y
(1)=2和y
(2)=0,其调用求解函数的方法为
[t,y]=ode45(@vdp1,[020],[2;
0]);
4Viewthesolveroutput.Youcansimplyusetheplotcommandtoview
thesolveroutput.
4.视图求解输出。
你能够用plot命令视图解输出。
plot(t,y(:
1),'
-'
t,y(:
2),'
--'
)
title('
SolutionofvanderPolEquation,\mu=1'
);
xlabel('
timet'
ylabel('
solutiony'
legend('
y_1'
y_2'
Asanalternative,youcanuseasolveroutputfunctiontoprocesstheoutput.
ThesolvercallsthefunctionspecifiedintheintegrationpropertyOutputFcn
aftereachsuccessfultimestep.UseodesettosetOutputFcntothedesired
function.SeeSolverOutputProperties,inthereferencepageforodeset,for
moreinformationaboutOutputFcn.
作为选择,你能够用求解器输出处理输出。
vanderPolEquation(Stiff)(刚性vanderPol方程)
Thisexamplepresentsastiffproblem.Forastiffproblem,solutionscan
changeonatimescalethatisveryshortcomparedtotheintervalof
integration,butthesolutionofinterestchangesonamuchlongertimescale.
Methodsnotdesignedforstiffproblemsareineffectiveonintervalswherethe
solutionchangesslowlybecausetheyusetimestepssmallenoughtoresolve
thefastestpossiblechange.
该例表示的是一个刚性问题。
对于刚性问题,解在相对积分区间非常小的时间尺度上变化,感兴趣的解在更大的时间尺度上变化。
不是专为刚性问题设计的方法在解变换缓慢的区间上是无效的,因为它用的时间步长非常小以适应分辨快速变化。
Whenμisincreasedto1000,thesolutiontothevanderPolequation
changesdramaticallyandexhibitsoscillationonamuchlongertimescale.
Approximatingthesolutionoftheinitialvalueproblembecomesamore
difficulttask.Becausethisparticularproblemisstiff,asolverintendedfor
nonstiffproblems,suchasode45,istooinefficienttobepractical.Asolver
suchasode15sisintendedforsuchstiffproblems.
当增加到1000,vdp方程的解急剧变化,在更大时时间尺度上展示振荡。
近似初值问题的解是一个困难的问题。
因为这个特别的问题是刚性的,对于如ode45这样的非刚性求解器实际上是效率非常低的。
象ode15s是适用于刚性问题的。
Thevdp1000functionevaluatesthevanderPolsystemfromtheprevious
example,butwithμ=1000.
vdp1000函数对
的vdp方程组进行赋值。
functiondydt=vdp1000(t,y)
1000*(1-y
(1)^2)*y
(2)-y
(1)];
Nowusetheode15sfunctiontosolvetheproblemwiththeinitialcondition
vectorof[2;
0],butatimeintervalof[03000].Forscalingreasons,plot
justthefirstcomponentofy(t).
现在用ods15s求解具有初值向量[2;
0]和积分区间[0,3000]上的初值问题。
为尺度原因,仅画出
的第一分量。
[t,y]=ode15s(@vdp1000,[03000],[2;
SolutionofvanderPolEquation
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- mathematics 工具箱 使用