數(shù)值計(jì)算常微分方程求解實(shí)驗(yàn)

1、實(shí)驗(yàn)五 常微分方程求解實(shí)驗(yàn)一、實(shí)驗(yàn)?zāi)康? 解初值問題各種方法比較2 常微分方程形態(tài)和龍格庫塔法穩(wěn)定性3 剛性方程計(jì)算二、實(shí)驗(yàn)題目1、解初值各種方法比較實(shí)驗(yàn)題目：給定初值問題取精確解為，按（1）歐拉法，步長h=0.025,h=0.1；（2）改進(jìn)歐拉法，步長h=0.05,h=0.01；（3）四階標(biāo)準(zhǔn)龍格庫塔法，步長h=0.1；求在節(jié)點(diǎn)處的數(shù)值解及誤差，比較各個方法的優(yōu)缺點(diǎn)。3、剛性方程計(jì)算實(shí)驗(yàn)題目：給定剛性微分方程取精確解為。任取一顯示方式，取不同的步長求解，并分析計(jì)算結(jié)果。三、實(shí)驗(yàn)原理：一階常微分方程初值問題的數(shù)值解法，根據(jù)常微分方程解的存在唯一性定，在滿足一定的條件下，解函數(shù)是唯一存在的.取步

2、長，記，按一定的遞推公式依次求得各節(jié)點(diǎn)上解函數(shù)值的近似值，稱為初值問題的數(shù)值解.常微分方程初值問題的數(shù)值解法一般分為兩大類：一步法：這類方法在計(jì)算時只用到，即前一步的值.因此在有了初值之后就可以逐步往下計(jì)算，其代表是龍格庫塔法.多步法：這類方法在計(jì)算時除了用到以外，還要用到，即前面步的值，其代表是亞當(dāng)斯方法歐拉格式由數(shù)值微分的向前差商公式可以解決初值問題（6.1）中的導(dǎo)數(shù)的數(shù)值計(jì)算問題：由此可得（6.1）實(shí)際上給出于是有再由得 （6.2）遞推公式（6.2）稱為歐拉格式。改進(jìn)歐拉格式先對歐拉格式（6.2）對進(jìn)行計(jì)算，并將結(jié)果記為，再代入（6.7）可得“預(yù)報(bào)-校正”形式的差分格式：公式（6.8）稱

3、為改進(jìn)歐拉格式。四階經(jīng)典龍格-庫塔格式：實(shí)驗(yàn)步驟：1、%maeuler1.mfunction x, y=maeuler1(dyfun,xspan,y0,h% ͨ;£º¸Ä½øÅ·À­¸ñʽ½â³£Î¢·Ö·½³Ìy'=f(x,y, y(x0=y0% ¸ñÊ

4、½£ºx,y=maeuler(dyfun,xspan,y0,h dyfunΪº¯Êýf(x,y, xspanΪ% Çó½âÇø¼äx0,xn, y0Ϊ³õÖµy(x0, hΪ²½³¤, x·µ»Ø½Úµã

5、;£¬y·µ»ØÊýÖµ½âformat short;x=xspan(1:h:xspan(2; y(1=y0;for n=1:(length(x-1k1=feval(dyfun,x(n,y(n;y(n+1=y(n+h*k1;endx=x' y=y'function shiyan51h=0.1;dyfun=inline('y./x+x*exp(x'x,y1=maeuler1(dyfun,1,2,0,h;x,y2=maeuler(dyfun,1,2

6、,0,h;x,y3=marunge4(dyfun,1,2,0,h;y=x.*(exp(x-exp(1;err1=abs(y-y1;err2=abs(y-y2;err3=abs(y-y3;x,err1,err2,err33、function shiyan53clch=0.0001format longfun=inline('-600*y+1199.8*exp(-0.1*x-600'tic;x,y1=marunge4(fun,0,5,2,h;yb1=exp(-600*x+2*exp(-0.1*x-1;t1=tocerr1=abs(yb1-y1;s1=sum(err1.2%tic;x

7、,y2=marunge4(fun,0,5,2,h;yb2=exp(-600*x+2*exp(-0.1*x-1;t2=tocerr2=abs(yb2-y2;s2=sum(err2.2實(shí)驗(yàn)結(jié)果：1、h=0.1ans =Columns 1 through 3 1.00000000000000 0 01.50000000000000 0.32284614010794 0.005274889066251.60000000000000 0.42626522205571 0.006398723084441.70000000000000 0.54597705499551 0.007517577613841.8

8、0000000000000 0.68378242658891 0.00861692278559Column 4 00.000000923008970.000001782501610.000002596336140.000003376506500.000004131243470.000004866270110.000005585578480.000006291924020.000006987147120.00000767238581H=0.01ans =Columns 1 through 3 1.00000000000000 0 01.02000000000000 0.0008286688644

9、0 0.000002238131351.06000000000000 0.00260227125193 0.000006792190081.07000000000000 0.00307039495316 0.000007946380051.08000000000000 0.00354858475895 0.000009106658001.09000000000000 0.00403693130605 0.000010272932881.35000000000000 0.02062722245075 0.000042397898481.37000000000000 0.0222439068916

10、1 0.000044986938581.38000000000000 0.02307225284104 0.000046286327141.39000000000000 0.02391411206383 0.000047588820881.40000000000000 0.02476961968956 0.000048894313301.42000000000000 0.02652212893538 0.000051513863511.43000000000000 0.02741940913044 0.000052827703511.44000000000000 0.0283308949112

11、9 0.000054144106571.45000000000000 0.02925672982210 0.000055462961241.54000000000000 0.03825951150297 0.000067424038981.55000000000000 0.03933712270019 0.000068761032771.56000000000000 0.04043078033191 0.000070099176151.58000000000000 0.04266689901224 0.000072778406401.59000000000000 0.0438096970311

12、3 0.000074119237161.60000000000000 0.04496921546606 0.000075460705241.62000000000000 0.04733910958953 0.000078145025221.63000000000000 0.04854983840282 0.000079487609321.64000000000000 0.04977799392466 0.000080830295051.66000000000000 0.05228731434816 0.000083515418701.67000000000000 0.0535688492179

13、4 0.000084857576231.68000000000000 0.05486855077778 0.000086199274561.70000000000000 0.05752321789624 0.000088880714501.72000000000000 0.06025286589314 0.000091558562941.74000000000000 0.06305908122511 0.000094231615981.75000000000000 0.06449140751745 0.000095565960011.76000000000000 0.0659434873945

14、0 0.000096898640401.77000000000000 0.06741552971328 0.000098229498241.78000000000000 0.06890774574697 0.00009955837271Column 4 00.000000000011010.000000000021900.000000000032700.000000000043390.000000000053980.000000000064480.000000000074890.000000000085220.000000000095460.000000000105610.0000000001

15、15690.000000000125700.000000000135630.000000000145490.000000000155280.000000000165010.000000000174670.000000000184270.000000000193810.000000000203290.000000000212720.000000000222100.000000000231420.000000000240690.000000000249910.000000000259080.000000000268210.000000000277290.000000000286330.000000

16、000295330.000000000304290.000000000313210.000000000322090.000000000330930.000000000339730.000000000348500.000000000357240.000000000365940.000000000374610.000000000383250.000000000391860.000000000400440.000000000408990.000000000417510.000000000426000.000000000434470.000000000442910.000000000451330.00

17、0000000459720.000000000468090.000000000476430.000000000484760.000000000493060.000000000501330.000000000509590.000000000517820.000000000526040.000000000534240.000000000542410.000000000550570.000000000558710.000000000566830.000000000574930.000000000583020.000000000591090.000000000599140.00000000060717

18、0.000000000615190.000000000623200.000000000631190.000000000639160.000000000647120.000000000655070.000000000663000.000000000670910.000000000678820.000000000686710.000000000694590.000000000702450.000000000710300.000000000718140.000000000725970.000000000733780.000000000741580.000000000749370.000000000757150.000000000764920.000000000772670.000000000780410.000000000788150.000000000795870.000000000803580.000000000811280.000000000818970.000000000826650.000000000834310.000000000841970.000000000849620.000000000857260.000000000