数值计算常微分方程求解实验

1、实验五 常微分方程求解实验一、实验目的1 解初值问题各种方法比较2 常微分方程形态和龙格库塔法稳定性3 刚性方程计算二、实验题目1、解初值各种方法比较实验题目：给定初值问题取精确解为，按（1）欧拉法，步长h=0.025,h=0.1；（2）改进欧拉法，步长h=0.05,h=0.01；（3）四阶标准龙格库塔法，步长h=0.1；求在节点处的数值解及误差，比较各个方法的优缺点。3、刚性方程计算实验题目：给定刚性微分方程取精确解为。任取一显示方式，取不同的步长求解，并分析计算结果。三、实验原理：一阶常微分方程初值问题的数值解法，根据常微分方程解的存在唯一性定，在满足一定的条件下，解函数是唯一存在的.取步

2、长，记，按一定的递推公式依次求得各节点上解函数值的近似值，称为初值问题的数值解.常微分方程初值问题的数值解法一般分为两大类：一步法：这类方法在计算时只用到，即前一步的值.因此在有了初值之后就可以逐步往下计算，其代表是龙格库塔法.多步法：这类方法在计算时除了用到以外，还要用到，即前面步的值，其代表是亚当斯方法欧拉格式由数值微分的向前差商公式可以解决初值问题（6.1）中的导数的数值计算问题：由此可得（6.1）实际上给出于是有再由得 （6.2）递推公式（6.2）称为欧拉格式。改进欧拉格式先对欧拉格式（6.2）对进行计算，并将结果记为，再代入（6.7）可得“预报-校正”形式的差分格式：公式（6.8）称

3、为改进欧拉格式。四阶经典龙格-库塔格式：实验步骤：1、%maeuler1.mfunction x, y=maeuler1(dyfun,xspan,y0,h% y=f(x,y, y(x0=y0% x,y=maeuler(dyfun,xspan,y0,h dyfunf(x,y, xspan% x0,xn, y0y(x0, h, xyformat 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;dy

4、fun=inline(y./x+x*exp(x;x,y1=maeuler1(dyfun,1,2,0,h;x,y2=maeuler(dyfun,1,2,0,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=ex

5、p(-600*x+2*exp(-0.1*x-1;t1=tocerr1=abs(yb1-y1;s1=sum(err1.2%tic;x,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实验结果：1、h=0.1ans =Columns 1 through 3 1.00000000000000 0 01.10000000000000 0.04264443219022 0.000973565803121.20000000000000 0.09520401466700 0.00

6、1997909375701.30000000000000 0.15865732017504 0.003063154845261.40000000000000 0.23411778154437 0.004159099739751.50000000000000 0.32284614010794 0.005274889066251.60000000000000 0.42626522205571 0.006398723084441.70000000000000 0.54597705499551 0.007517577613841.80000000000000 0.68378242658891 0.00

7、8616922785591.90000000000000 0.84170305352887 0.009680430019012.00000000000000 1.02200658351062 0.01068965912814Column 4 00.000000923008970.000001782501610.000002596336140.000003376506500.000004131243470.000004866270110.000005585578480.000006291924020.000006987147120.00000767238581H=0.01ans =Columns

8、 1 through 3 1.00000000000000 0 01.01000000000000 0.00040956013886 0.000001115747971.02000000000000 0.00082866886440 0.000002238131351.03000000000000 0.00125740933334 0.000003367058581.04000000000000 0.00169586593212 0.000004502438251.05000000000000 0.00214412428325 0.000005644179131.06000000000000

9、0.00260227125193 0.000006792190081.07000000000000 0.00307039495316 0.000007946380051.08000000000000 0.00354858475895 0.000009106658001.09000000000000 0.00403693130605 0.000010272932881.10000000000000 0.00453552650384 0.000011445113641.11000000000000 0.00504446354262 0.000012623109101.12000000000000

10、0.00556383690209 0.000013806828011.13000000000000 0.00609374236023 0.000014996178961.14000000000000 0.00663427700241 0.000016191070371.15000000000000 0.00718553923075 0.000017391410421.16000000000000 0.00774762877379 0.000018597107091.17000000000000 0.00832064669647 0.000019808068071.18000000000000

11、0.00890469541030 0.000021024200731.19000000000000 0.00949987868383 0.000022245412121.20000000000000 0.01010630165340 0.000023471608921.21000000000000 0.01072407083410 0.000024702697421.22000000000000 0.01135329413106 0.000025938583481.23000000000000 0.01199408085089 0.000027179172511.24000000000000

12、0.01264654171350 0.000028424369431.25000000000000 0.01331078886402 0.000029674078661.26000000000000 0.01398693588512 0.000030928204061.27000000000000 0.01467509780946 0.000032186648961.28000000000000 0.01537539113241 0.000033449316071.29000000000000 0.01608793382510 0.000034716107481.30000000000000

13、0.01681284534756 0.000035986924641.31000000000000 0.01755024666226 0.000037261668321.32000000000000 0.01830026024775 0.000038540238581.33000000000000 0.01906301011265 0.000039822534771.34000000000000 0.01983862180981 0.000041108455471.35000000000000 0.02062722245075 0.000042397898481.36000000000000

14、0.02142894072032 0.000043690760791.37000000000000 0.02224390689161 0.000044986938581.38000000000000 0.02307225284104 0.000046286327141.39000000000000 0.02391411206383 0.000047588820881.40000000000000 0.02476961968956 0.000048894313301.41000000000000 0.02563891249802 0.000050202696981.42000000000000

15、0.02652212893538 0.000051513863511.43000000000000 0.02741940913044 0.000052827703511.44000000000000 0.02833089491129 0.000054144106571.45000000000000 0.02925672982210 0.000055462961241.46000000000000 0.03019705914018 0.000056784155021.47000000000000 0.03115202989331 0.000058107574301.48000000000000

16、0.03212179087729 0.000059433104361.49000000000000 0.03310649267375 0.000060760629331.50000000000000 0.03410628766820 0.000062090032161.51000000000000 0.03512133006829 0.000063421194631.52000000000000 0.03615177592242 0.000064753997281.53000000000000 0.03719778313849 0.000066088319391.54000000000000

17、0.03825951150297 0.000067424038981.55000000000000 0.03933712270019 0.000068761032771.56000000000000 0.04043078033191 0.000070099176151.57000000000000 0.04154064993716 0.000071438343141.58000000000000 0.04266689901224 0.000072778406401.59000000000000 0.04380969703113 0.000074119237161.60000000000000

18、0.04496921546606 0.000075460705241.61000000000000 0.04614562780834 0.000076802678981.62000000000000 0.04733910958953 0.000078145025221.63000000000000 0.04854983840282 0.000079487609321.64000000000000 0.04977799392466 0.000080830295051.65000000000000 0.05102375793674 0.000082172944641.66000000000000

19、0.05228731434816 0.000083515418701.67000000000000 0.05356884921794 0.000084857576231.68000000000000 0.05486855077778 0.000086199274561.69000000000000 0.05618660945506 0.000087540369341.70000000000000 0.05752321789624 0.000088880714501.71000000000000 0.05887857099039 0.000090220162231.72000000000000

20、0.06025286589314 0.000091558562941.73000000000000 0.06164630205086 0.000092895765261.74000000000000 0.06305908122511 0.000094231615981.75000000000000 0.06449140751745 0.000095565960011.76000000000000 0.06594348739450 0.000096898640401.77000000000000 0.06741552971328 0.000098229498241.78000000000000

21、0.06890774574697 0.000099558372711.79000000000000 0.07042034921078 0.000100885100991.80000000000000 0.07195355628835 0.000102209518241.81000000000000 0.07350758565827 0.000103531457581.82000000000000 0.07508265852103 0.000104850750051.83000000000000 0.07667899862627 0.000106167224591.84000000000000

22、0.07829683230029 0.000107480707991.85000000000000 0.07993638847397 0.000108791024881.86000000000000 0.08159789871097 0.000110097997671.87000000000000 0.08328159723622 0.000111401446551.88000000000000 0.08498772096485 0.000112701189401.89000000000000 0.08671650953136 0.000113997041841.90000000000000

23、0.08846820531916 0.000115288817121.91000000000000 0.09024305349047 0.000116576326121.92000000000000 0.09204130201654 0.000117859377331.93000000000000 0.09386320170824 0.000119137776781.94000000000000 0.09570900624700 0.000120411328011.95000000000000 0.09757897221611 0.000121679832061.96000000000000

24、0.09947335913232 0.000122943087421.97000000000000 0.10139242947795 0.000124200889981.98000000000000 0.10333644873316 0.000125453033031.99000000000000 0.10530568540883 0.000126699307172.00000000000000 0.10730041107957 0.00012793950031Column 4 00.000000000011010.000000000021900.000000000032700.0000000

25、00043390.000000000053980.000000000064480.000000000074890.000000000085220.000000000095460.000000000105610.000000000115690.000000000125700.000000000135630.000000000145490.000000000155280.000000000165010.000000000174670.000000000184270.000000000193810.000000000203290.000000000212720.000000000222100.000

26、000000231420.000000000240690.000000000249910.000000000259080.000000000268210.000000000277290.000000000286330.000000000295330.000000000304290.000000000313210.000000000322090.000000000330930.000000000339730.000000000348500.000000000357240.000000000365940.000000000374610.000000000383250.000000000391860

27、.000000000400440.000000000408990.000000000417510.000000000426000.000000000434470.000000000442910.000000000451330.000000000459720.000000000468090.000000000476430.000000000484760.000000000493060.000000000501330.000000000509590.000000000517820.000000000526040.000000000534240.000000000542410.000000000550570.000000000558710.000000000566830.000000000574930.000000000583020.000000000591090.00000