计算方法第八章电子教案(32学时-欧阳洁)

12Taylor2TaylorRungeRunge––KuttaKutta**4433112⎪⎩⎪⎨⎧=≤=000)(),(yxyxxyxfdxdyyy((xx))[x0,b]f(x,y(x))3y(x)[x0,b]nnnhxx+=−1L,2,1,=nyn)(nxynx[x0,b]LLnxxxx210L,2,1,1=−=−nxxhnnn1−nxnx4(())1+nynynnxx,1+kpnx−)1;1,2,1(−=−kkpypnL1+nynynnxx,1+),,(1hyxhyynnnnφ+=+),,,(11hyyxhyynnnnn+++=φTaylor522EulerEulerEulerEulerEulerEuler--6),2,1,0(),(1L=+=+nyxhfyynnnnL+′′+′+=+)(2)()()(2nnnnxyhxyhxyhxyEulerEuler)(nxynyh),2,1(0L=+=nnhxxnL+′′++=)(2))(,()(2nnnnxyhxyxhfxynxTaylor)(hxyn+EulerEulernnnhxx+=−1)(nxy),2,1(L=nyn7EulerEuler1()()nnyxyxh+−()))(,()(nnnxyxfxy=′)(nxyny)()(2))()((11nnnxyyhxyxyh′=′′−−+ξ),()(2))()((1)(173.10010xxfhxfxfhxfp∈′′−−=′ξξ),(1nnnnyxfhyy=−+),1,0(),(1L=+=+nyxhfyynnnn))(,(nnxyxf=8))(,()(xyxfxy=′∫∫++=′=−+11))(,()()()(1nnnnxxxxnndxxyxfdxxyxyxy),1,0(),(1L=+=+nyxhfyynnnnEulerEuler21()()(,())(,())2nnnnhyxyxhfxyxfyηη+′=++)(nxynyEulerEuler)(2)()()()(175.2ηfabafabdxxfPba′−+−=∫9),,2,1,0(),(111L=+=+++nyxhfyynnnnL+′′+′−=+++)(2)()()(1211nnnnxyhxyhxyxyEulerEuler)(nxynyhL+′′+−=++++)(2))(,()(12111nnnnxyhxyxhfxy1+nxTaylor)(nxyEulerEulerEulerEuler1+ny1+ny1+ny10111()(,())nnnyxfxyx+++′=1()()nnyxyxh+−())(nxynyEulerEuler))(,()(2))()((1111+++=′′+−nnnnxyxfyhxyxyhξ),(111+++=−nnnnyxfhyy),()(2))()((1)(173.10011xxfhxfxfhxfp∈′′+−=′ξξ),1,0(),(111L=+=+++nyxhfyynnnn11∫∫++=′=−+11))(,()()()(1nnnnxxxxnndxxyxfdxxyxyxy))(,()(xyxfxy=′2111()()(,())(,())2nnnnhyxyxhfxyxfyηη+++′=+−)(nxynyEulerEuler)(2)()()()(175.2ηfabbfabdxxfPba′−−−=∫),1,0(),(111L=+=+++nyxhfyynnnn1+ny12∫∫++=′=−+11))(,()()()(1nnnnxxxxnndxxyxfdxxyxyxy3111()()[(,())(,()](,())212nnnnnnhhyxyxfxyxfxyxfyηη+++′′=++−)(nxyny)],(),([2111+++++=nnnnnnyxfyxfhyy)(21)()]()([2)(3ηfabbfafabdxxfba′′−−+−=∫Euler13⎪⎩⎪⎨⎧=++=+=+++++L,2,1,0)],(),([2),()(11)1(1)0(1kyxfyxfhyyyxhfyyknnnnnknnnnnε−+++)(1)1(1knknyyεn=0,1,…)1(1++kny1+ny)(1+nxyLLipschitz12≤∂∂Lyfh14EulerEuler--⎪⎩⎪⎨⎧++=+=++++)],(),([2),()0(111)0(1nnnnnnnnnnyxfyxfhyyyxhfyyhEulerEuler−−Euler-Euler-()⎪⎪⎩⎪⎪⎨⎧++==++=+121211,),()(2hKyhxfKyxfKKKhyynnnnnn151+nx011,,,,xxxxnnL−ny()nyxnx11111)(nnnyxy−=ε1nx16)(nnxyy=111)(+++−=nnnyxyR1+nxppppp)(1+phOp)())(,(211++++=ppnnnhOhxyxRψ1))(,(+pnnhxyxψ17EulerEulerEulerEuler111)(+++−=nnnyxyR)(22nyhξ′′=)(221nnyhRξ′′=+()(,())nnnyxfxyx′=)(nnxyy=),(1nnnnyxhfyy+=+)(2)()()(21nnnnyhxyhxyxyξ′′+′+=+18∫∫++=′=−+11))(,()()()(1nnnnxxxxnndxxyxfdxxyxyxy)],(),([2111+++++=nnnnnnyxfyxfhyy))(,(12))](,())(,([2)()(3111ηηyfhxyxfxyxfhxyxynnnnnn′′−++=+++)(nnxyy=111)(+++−=nnnyxyR))(,(12)],())(,([231111ηηyfhyxfxyxfhnnnn′′−−=++++19))(,(12)],())(,([2311111ηηyfhyxfxyxfhRnnnnn′′−−=+++++1+nR()11),(1111)(),())(,(1++++++−∂∂=−+nnxnnnnyxyyfyxfxyxfnξ⎥⎥⎦⎤⎢⎢⎣⎡∂∂−′′−=++),(31121))(,(12ξηηnxnyfhyfhR()))(,(12)(21311),(1ηηξyfhyxyyfhnnxn′′−=−⎥⎥⎦⎤⎢⎢⎣⎡∂∂−+++h12),(1∂∂+ξnxyfhyf∂∂20⎥⎥⎦⎤⎢⎢⎣⎡∂∂−′′−=++),(31121))(,(12ξηηnxnyfhyfhR)(212112),(),(11hOyfhyfhnnxx+∂∂+=∂∂−++ξξ⎥⎦⎤⎢⎣⎡′′−⎥⎥⎦⎤⎢⎢⎣⎡+∂∂+=++))(,(12)(2132),(11ηηξyfhhOyfhRnxn)(3hO=EulerEuler21EulerEuler--⎪⎩⎪⎨⎧++=+=++++)],(),([2),()0(111)0(1nnnnnnnnnnyxfyxfhyyyxhfyy()121211,),()(2hKyhxfKyxfKKKhyynnnnnn++==++=+)(nnxyy=111)(+++−=nnnyxyR221(,)nnKfxy=21(,)nnKfxhyhK=++2(,)(,)(,)(,)()nnxnnnnynnfxyhfxyhfxyfxyOh=+++1(,)(,)(,)[]nnnnnnxyxyfffxyhhKxy∂∂=++∂∂2()()()nnyxhyxOh′′′=++2(,)(,)[]()nnnnxyxyfxyhfffOh=+++(,())()nnnfxyxyx′==22222221122(,)(,)(,)1[2]2!nnnnnnxyxyxyfffhhKhKxxyy∂∂∂++++∂∂∂∂L(,)(,)(,)(,)xyyfxyyfxyfxyfxy′′′=⇒=+),(),()(2121211hKyhxfKyxfKKKhyynnnnnn++==++=+23EulerEuler--212()()()()nnnKyxKyxhyxOh′′′′==++112()2nnhyyKK+=++231[()()()()]2!3!nnnhhyxhyxyxyξ′′′′′′=+++31[()()()]()2nnnnnhyyyxyxhyxOh+′′′′=++++Taylor1()nyx+2311()()()()()2!3!nnnnhhyxyxhyxyxyξ+′′′′′′=+++3[(2()())()]2nnnhyyxhyxOh′′′−+++3()Oh=111)(+++−=nnnyxyR24),,(1hyxhyynnnnφ+=+yipschitz)1)((11≥=++phORpn),,(hyxφ)()(111pnnnhOyxy=−=ε2121),(),(yyLyxyx−≤−φφEulerEulerεnf(x,y)yLipschitz2121),(),(yyLyxfyxf−≤−),2,1(21||22L=≤nMhRn)1(2)(20)(−+≤−−LabLabneLhMeεεLLipschitzb−a)(max2xyMbxa′′=≤≤()25EulerEulerRemarkRemark26RemarksRemarks0nxxnh=+nhxxn+=0],[0Nxx)(lim)(0nnnhxyy=∞→→0))((limlim)(0)(0=−=∞→→∞→→nnnhnnhyxyε22EulerEuler-2733f(x,y)h0),2,1(L=+kyknnynδ28Re()0λhλGλh∈GGGhλf(x,y)yLipschitzλyyλ=′hyyλ=′nδ)(nmymnδmδny)(nmnm≤δδ29EulerEuler),2,1,0(),(1L=+=+nyxhfyynnnnnnnnyhyhyy)1(1λλ+=+=+1+ny1+nδ))(1(11nnnnyhyδλδ++=+++nnhδλδ)1(1+=+nnδδ≤+1111≤+=+λδδhnnEuleryyλ=′nnyδ+nδny11≤+λhλ(λ0)λ20−≤≤h30RemarksRemarkshhλEulerhh11≤+λhEuleriyxh+=λ1)1(22≤++yx31EulerEuler),2,1,0(),(111L=+=+++nyxhfyynnnn11≥−hλEuleryyλ=′11+++=nnnyhyyλλhyynn−=+111+ny1+nδ)(1111nnnnyhyδλδ+−=+++1111≤−=+λδδhnnnnyδ+nδnynnhδλδ−=+11132RemarksRemarksEulerRe(λh)0EulerEulerEulerEuleriyxh+=λ1)1(22≥+−yxλEulerh11≥−hλRe(Re(λλ)0)0hh0033yyλ=′),2,1,0()],(),([2111L=++=+++nyxfyxfhyynnnnnn[]112++++=nnnnyyhyyλλnnyhhyλλ2112111−+=+1221≤−+=+λλδδhhnn1+nδnδnnhhδλλδ2112111−+=+122≤−+λλhh34Re(λh)0x=Re(λh)01iyxh+=λiyxiyxhhnn−−++=−+=+22221λλδδ212222))2()2((yxyx+−++=Re(λ)0hh0035TaylorTaylorRungeRunge——KuttaKutta2Taylor2TaylorRungeRunge––KuttaKutta36TaylorTaylory(x)f(x,y(x))TaylorTaylorTaylorp)()!1()(!)()()()(1)(1nppnppnnnyphxyphxyhxyxyξ++++′+=++Lh)(1+phO)()()(nppnxyy≈)(21!!2pnpnnnnyphyhyhyy++′′+′+=+L)()!1()1(11nppnyphRξ++++=37………………),()()(nnyxyy