Blasius方程的数值解matlab程序(共4页)

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。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（龙格库塔费尔伯格）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）用于叠代计算，在叠代过程中，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;命令窗口提示：请输入第一个任意初始值:键入 0.3命令窗口提示：请输入第二个任意初始值:键入 0.4程序执行完毕。得到数值解图像需要完整数

2、值解，可在命令窗口执行 t,yans = 0 0 0 0.332050425572608 0.200000000000000 0.006640684785757 0.066406419000735 0.331976930281647 0.400000000000000 0.026559272491103 0.132761397348084 0.331462940642616 0.600000000000000 0.059730061368740 0.198934248154027 0.330072803333863 0.800000000000000 0.106097295537842 0.2

3、64705999557182 0.327381783215304 1.000000000000000 0.165563030884709 0.329773792361296 0.322998326329412 1.200000000000000 0.237944237268780 0.393767481958191 0.316583378968660 1.400000000000000 0.322968836895193 0.456258456070615 0.307858224411080 1.600000000000000 0.420294390401678 0.5167580586411

4、45 0.296652046000281 1.800000000000000 0.529490839159120 0.574751902433704 0.282921170876929 2.000000000000000 0.650009069470154 0.629751523134094 0.266750311286562 2.200000000000000 0.781182207029520 0.681296249300408 0.248352130535080 2.400000000000000 0.922272805631476 0.728973480703094 0.2280826

5、78389961 2.600000000000000 1.072479822282861 0.772447069758408 0.206450263528860 2.800000000000000 1.230950137421495 0.811497584068625 0.184014296855294 3.000000000000000 1.396787909544558 0.846029638656905 0.161369816626103 3.200000000000000 1.569069925345089 0.876070752645320 0.139126737097636 3.4

6、00000000000000 1.746915448703586 0.901760052525514 0.117863021963319 3.600000000000000 1.929487686151937 0.923331516492812 0.098081245701445 3.800000000000000 2.116000403451031 0.941109242755617 0.080139603642537 4.000000000000000 2.305719389785077 0.955505958092905 0.064250507286456 4.2000000000000

7、00 2.498005168266955 0.966955902144841 0.050519855519064 4.400000000000000 2.692328249616070 0.975867585098678 0.038977900754526 4.600000000000000 2.888218573285013 0.982674751981711 0.029499001922806 4.800000000000000 3.085282916536712 0.987795856832908 0.021859471891081 5.000000000000000 3.2832378

8、19572969 0.991545649832639 0.015900519799442 5.199999999999999 3.481838760901607 0.994236090376837 0.011361976112965 5.400000000000000 3.680886843280485 0.996153067540022 0.007933845886624 5.600000000000000 3.880258990903193 0.997474930939695 0.005439882672210 5.800000000000000 4.079850604119796 0.9

9、98372294598017 0.003657568419394 6.000000000000000 4.279588937153525 0.998971602331846 0.002407067561230 6.199999999999999 4.479425460802378 0.999361358663418 0.001555212097374 6.400000000000000 4.679324566790177 0.999611493625153 0.000983390226568 6.600000000000000 4.879263841128227 0.9997679250728

10、73 0.000610241632299 6.800000000000000 5.079227857910512 0.999864205602870 0.000370938799775 7.000000000000000 5.279206991685719 0.999922223141772 0.000220942955724 7.200000000000000 5.479195185941113 0.999956397738357 0.000129191951169 7.400000000000000 5.679188612800127 0.999976292967110 0.0000737

11、70638928 7.600000000000000 5.879185121366138 0.999987457696780 0.000041576427492 7.800000000000000 6.079183286194299 0.999993777418038 0.000022702183282 8.000000000000000 6.279182432445944 0.999997116397916 0.000012416083429 8.199999999999999 6.479182046435597 0.999998998619281 0.000006410840147 8.4

12、00000000000000 6.679181959675744 0.999999908695243 0.000003431483786 8.600000000000000 6.879181996919892 1.000000408873378 0.000001736982564 8.800000000000001 7.079182104284974 1.000000668341604 0.000000832235283 9.000000000000000 7.279182257219148 1.000000765370094 0.000000489499078查看叠代过程，可在命令窗口执行

13、YY = 0.300000000000000 0.400000000000000 0.333115369726614 0.332050425572608 xx = 0.934567382019033 1.132157258022684 1.002137820445041 1.000000765370094%Y表示时的值，x表示（实际为9）时对应下的值。从中可以看到趋近过程。数值解和精确解的比较数值解精确解数值解精确解数值解精确解000000.3320504255726080.3320610.1655630308847090.165770.3297737923612960.329790.3229

14、983263294120.3230120.6500090694701540.650030.6297515231340940.629770.2667503112865620.2667531.3967879095445581.396820.8460296386569050.846050.1613698166261030.1613642.3057193897850772.305760.9555059580929050.955520.0642505072864560.0642353.2832378195729693.283290.9915456498326390.991150.015900519799

15、4420.0159164.2795889371535254.279640.9989716023318460.998680.0024070675612300.0024075.2792069916857195.279260.9999222231417720.999220.0002209429557240.0002286.2791824324459446.279230.9999971163979161.000000.0000124160834290.000018.87.0791821042849747.079231.0000006683416041.000000.0000008322352830.0

16、0000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：0.3000000000000000.1000000000000000.9345673820190330.4488602810575830.4000000000000001.0000000000000001.1321572580226842.0856232646779450.3331153697266140.4030528867111931.0021378204450411.1379108554528040.332

17、0504255726080.3335715199474381.0000007653700941.0030524939321210.3320434110618550.9999866815772260.3320500444862901.000000000224372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，但对结果的影响几乎可以忽略不计。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，但仍然能够将精确计算到0.332050043715133。s.m：clear,clfY(1)=input(请输入第一个任意初始值:)t,y1=ode45(q,0:0.2:9,0,0,Y(1);x(1)=y1(length(t),2);Y(2)=input(请输入第二个任意初始值:)t,y2=ode45(q,0:0.2:9,0,0,Y(2);x(2)=y2(length(t),2);Y(3)=polyval(polyfit(x(1) x(2),Y(1) Y(2),1),1);t,y3=ode45(q,0:0.2:9,0,0,Y(3);x(3)=y3(length(t),2);n=0;while 1n=n+1;Y(n+3)=polyval(polyfit(x(n) x(n+1)