numerical_solution_of_ordinary_differential_equations

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1、NumericalAnalysisDamingLiDepartmentofMathematics,ShanghaiJiaoTongUniversity,Shanghai,200240,ChinaEmail:lidaming@sjtu.edu.cnOctober14,2014DamingLiNumericalAnalysisExistenceandUniquenessofSolutionsConsideraninitial-valueproblem(x′=f(t,x)x(t0)=x0Herexisanunknownfunctionoft,wherex′=dx(t)/dt.Forexamp

2、le,(x′=xtan(t+3)x(−3)=1Theanalyticsolutionisx(t)=sec(t+3).Typically,fortheaboveproblem,analyticsolutionsarenotavailableandnumericalmethodsmustbeemployed.DamingLiNumericalAnalysisExistenceandUniquenessofSolutionsIffiscontinuousinarectangleRcenteredat(t0,x0),sayR={(t,x):

3、t−t0

4、≤α,

5、x−x0

6、≤β}thenthein

7、itial-valueproblemhasasolutionx(t)for

8、t−t0

9、≤min(α,β/M),whereMisthemaximumof

10、f(t,x)

11、intherectangleR.DamingLiNumericalAnalysisExistenceandUniquenessofSolutionsProvethattheinitial-valueproblem(x′=(t+sinx)2x(0)=3hasasolutionontheinterval−1≤t≤1.Takingf(t,x)=(t+sinx)2and(t,x)=(0,3).Therectangleis00R={

12、(t,x):

13、t

14、≤α,

15、x−3

16、≤β}Themagnitudeoffisboundedby

17、f(t,x)

18、≤(α+1)2≡MWewantmin(α,β/M)≥1,andsowecanletα=1.ThenM=4,andourobjectiveismetbylettingβ≥4.DamingLiNumericalAnalysisExistenceandUniquenessofSolutionsIffiscontinuousinthestripa≤t≤b,−∞

19、f(t,x1)−f(t,x2)

20、≤L

21、x1−x2

22、the

23、ntheinitial-valueproblemhasauniquesolutionintheinterval[a,b].DamingLiNumericalAnalysisTaylor-SeriesMethod(x′=cost−sinx+t2x(−1)=3h2h3h4x(t+h)=x(t)+hx′(t)+x(2)(t)++x(3)(t)+x(4)(t)+···2!3!4!x(2)=−sint−x′cosx+2tx(3)=−cost−x(2)cosx+(x′)2sint+2x(4)=sint−x(3)cosx+3x′x(2)sinx+(x′)3cosxSubstitutingallthe

24、derivativesattuptofourthorderandtruncatingtotheorderh4,wecancalculatex(t+h)withthetruncationerrorO(h5).DamingLiNumericalAnalysisTaylor-SeriesMethodAlthoughthetruncationerrorcanbeveryhighbyTaylor-seriesmethod,therearemanydisadvantagesinthismethod.First,themethoddependsonrepeateddifferentiationoft

25、hegivendifferentialequation.Hence,thefunctionf(t,x)mustpossesspartialderivativesintheregionwherethesolutioncurvepassesinthetx-plane.Suchanassumptionis,ofcourse,notnecessaryfortheexistenceofasolution.Secondly,variousderivativ