Blasius方程的数值解matlab程序

1、程序说明该程序用于求解边值问题的非线性blasius方程的数值解，该方程用来描述通过一块无限大 平板的不可压缩的两维稳定流问题。2f + #'=0rj = qj' = 0j'=0 = ooj' = 该程序用matlab编写，由s.m和q.m两个程序组成，s.m为主程序，采用了打靶法和rkf（龙格一库塔一费尔伯格）法，在matlab中，rkf法选用函数ode45。在用打靶法解题 过程中需要先选定两个初始值（即/”的值）用于壳代计算，在叠代过程屮，第一次采用的 是一次多项式插值法，以后各次均采用的二次多项式插值法。具体程序执行如下：执行s.m;命令窗口提不：请输入第

2、一个任意初始值：键入 0.3 i命令窗口提示：请输入第二个任意初始值：键入 0.4 i程序执行完毕。得到数值解图像blasius方程数值解»ityl003320504255726080.2000000000000000.0066406847857570.0664064190007350.3319769302816470.4000000000()00000.0265592724911030327613973480840.331462940642616().6()()0000000()0000 0.8000000(x)0000001.0000()00000()0000 1.2000000

3、000000001.4000000000000001.600000000000000 1.800000000000000 2.0000()00000()0000 2.2()00000000000002.4000000000000002.600000000000000 2.8000000(x)000000 3.0000()00000()0000 3.2000000000000003.4000000000000003.600000000000000 3.800000000000000 4.0000()00000()0000 4.2()0000000000000 4.400000000(x)00()

4、04.600000000000000 4.8000000(x)000000 5.0000000000000005.1999999999999995.4000000000000005.600000000000000 5.8000000(x)000000 6.0000()00000()00006.199999999999999 6.4000000000()00006.600000000000000 6.8000000(x)000000 7.000000000000000 7.2()00000000000007.400000000000000 7.6()()0000000()0000 7.80000

5、00(x)000000 8.00000(xxxx)0(xx)08.199999999999999 8.4()0000000000000 &6000000000()0000 8.800(x)(xxxxx)(xx)l0.05973(x)613687400.1060972955378420 j 655630308847090.2379442372687800.3229688368951930.4202943904016780.5294908391591200.6500090694701540.7811822070295200.922272805631476l0724798222828611.

6、23095013742149513967879095445581.5690699253450891.746915448703586l929487686i5i9372.11600040345103123057193897850772.4980051682669552.6923282496160702.8882185732850133.0852829165367123.2832378195729693.4818387609016073.6808868432804853.8802589909031934.0798506041197964.2795889371535254.47942546080237

7、84.6793245667901774.8792638411282275.0792278579105125.2792069916857195.4791951859411135.6791886128001275.8791851213661386.0791832861942996.2791824324459446.4791820464355976.6791819596757446.8791819969198927.0791821042849740989342481540270.2647059995571820.32977379236129603937674819581910.45625845607

8、06150.5167580586411450.5747519024337040.6297515231340940.6812962493004080.7289734807030940.7724470697584080.8114975840686250.8460296386569050.8760707526453200.9017600525255140.9233315164928120.3300728033338630.3273817832153040322998326329412031658337896866003078582244110800.2966520460002810.28292117

9、08769290.2667503112865620.2483521305350800.2280826783899610.2064502635288600.1840142968552940.1613698166261030.1391267370976360.1178630219633190.0980812457014450.94ii092427556170.0801396036425370.9555059580929050.9669559021448410.9758675850986780.9826747519817110.9877958568329080.9915456498326390.99

10、42360903768370.9961530675400220.9974749309396950.9983722945980170.9989716023318460.9993613586634180.9996114936251530.9997679250728730.9998642056028700.9999222231417720.9999563977383570.9999762929671100.9999874576967800.9999937774180380.9999971163979160.9999989986192810.9999999086952431.000(x)0408873

11、3781.0000006683416040.0642505072864560.0505198555190640.0389779(x)7545260.0294990019228060.0218594718910810.0159005197994420.0113619761129650.0079338458866240.0054398826722100.0036575684193940.0024070675612300.0015552120973740.0009833902265680.0006102416322990.0003709387997750.0002209429557240.00012

12、91919511690.0000737706389280.0(xx)415764274920.0000227021832820.0000124160834290.0000064108401470.0000034314837860.000001736982564().()000008322352839.0000000000000007.2791822572191481.000()007653700940.000000489499078查看魂代过程，可在命令窗口执行» y0.3000000000000000.40000000000000003331153697266140.3320504

13、255726080.9345673820190331.1321572580226841.0021378204450411.000000765370094%y表示二0时的值，x表示二* （实际为9）时对应于“下/的值。 从中可以看到趋近过程。数值解和精确解的比较ff ff数值解精确解数值解精确解数值解精确解000000.3320504255726080.3320610.1655630308847090.165770.3297737923612960.329790.3229983263294120.3230190.6500090694701540.650030.6297515231340940.6

14、29770.2667503112865620.2667531.3967879095445581.396820.8460296386569050.846050.1613698166261030.1613642.3057193897850772.305760.9555059580929050.955520.0642505072864560.0642353.2832378195729693.283290.9915456498326390.991150.0159005197994420.0159164.2795889371535254.279640.9989716023318460.998680.00

15、24070675612300.0024075.2792069916857195.279260.9999222231417720.999220.0002209429557240.0002286.2791824324459446.279230.999997116397916.000000.0000124160834290.000018.87.0791821042849747.079231.000(x)06683416()41 .()00000.000000832235283().(x)00()从上表对比可知，数量解和精确解的结果几乎一致。该程序最大的优点就是可以对输入不同的初始值，来观察叠代次数和

16、对结果的影响。例如：当输入初始值为0.1和1时，与前面的比较见下表：ff0.3000000000000000.1000000000000000.9345673820190330.4488602810575830.4000000000000001.0000000000000001.1321572580226842.0856232646779450.3331153697266140.4030528867111931.0021378204450411.1379108554528040.3320504255726080.3335715199474381.0000007653700941.0030524

17、9393212103320434110618550.9999866815772260.3320500444862901.000000000224372很明显可以看出叠代次数不一样，第二次比第一次多计算两次，但对结果的影响儿乎可以忽 略不计。说明叠代初值对叠代次数的影响非常大，如选初值为3和40,则需要计算172次,但仍然能够将f（0）精确计算到0.332050043715133。s.m：clear,elfy(l)=inputciw输入第一个任意初始值l,y 1 =ode45(q,0:029,0,0, y( 1);x(l)=yl(length(t),2);y(2)=i叩utc请输入第二个任意初始值:jt,y2=ode45('q',0:029,0,0,y(2);x(2)=y2(length(t),2);y(3)=polyval(polyfit(x(l) x,y(1) y(2),l),l);t,y3=ode45(q,0:029,0,0,y(3);x =y3(length(t),2);n=0;while 1n=n+l;y(n+3)=polyval(polyfit(x(n) x(n+l) x(n+2),y(n) y(n+1) y(n+2),2),1