方程的數(shù)值解matlab程序

1、程序說明該程序用于求解邊值問題的非線性Blasius方程的數(shù)值解，該方程用來描述通過一塊無限大平板的不可壓縮的兩維穩(wěn)定流問題。該程序用MATLAB編寫，由s.m和q.m兩個(gè)程序組成，s.m為主程序，采用了打靶法和RKF（龍格庫塔費(fèi)爾伯格）法，在MATLAB中，RKF法選用函數(shù)ode45。在用打靶法解題過程中需要先選定兩個(gè)初始值（即的值）用于疊代計(jì)算，在疊代過程中，第一次采用的是一次多項(xiàng)式插值法，以后各次均采用的二次多項(xiàng)式插值法。具體程序執(zhí)行如下：執(zhí)行s.m;命令窗口提示：請(qǐng)輸入第一個(gè)任意初始值:鍵入 0.3命令窗口提示：請(qǐng)輸入第二個(gè)任意初始值:鍵入 0.4程序執(zhí)行完畢。得到數(shù)值解圖像需要完整數(shù)

2、值解，可在命令窗口執(zhí)行>> t,yans = 0 0 0 0.332050425572608 1.400000000000000 0.322968836895193 0.456258456070615 0.307858224411080 1.800000000000000 0.529490839159120 0.574751902433704 0.282921170876929 2.000000000000000 0.650009069470154 0.629751523134094 0.266750311286562 2.400000000000000 0.92227280563

3、1476 0.728973480703094 0.228082678389961 2.600000000000000 1.072479822282861 0.772447069758408 0.206450263528860 3.600000000000000 1.929487686151937 0.923331516492812 0.098081245701445 4.400000000000000 2.692328249616070 0.975867585098678 0.038977900754526 5.600000000000000 3.880258990903193 0.99747

4、4930939695 0.005439882672210 5.800000000000000 4.079850604119796 0.998372294598017 0.003657568419394 6.000000000000000 4.279588937153525 0.998971602331846 0.002407067561230 6.800000000000000 5.079227857910512 0.999864205602870 0.000370938799775 7.000000000000000 5.279206991685719 0.999922223141772 0

5、.000220942955724查看疊代過程，可在命令窗口執(zhí)行>> YY =>> xx =%Y表示時(shí)的值，x表示（實(shí)際為9）時(shí)對(duì)應(yīng)下的值。從中可以看到趨近過程。數(shù)值解和精確解的比較數(shù)值解精確解數(shù)值解精確解數(shù)值解精確解000000.3320504255726080.3320610.165770.3297737923612960.329790.3229983263294120.3230120.6500090694701540.650030.6297515231340940.629770.2667503112865620.2667531.3967879095445581.39

6、6820.8460296386569050.846050.1613642.305760.9555059580929050.955520.0642505072864560.0642353.2832378195729693.283290.991150.0159164.2795889371535254.279640.9989716023318460.998680.0024070675612300.0024075.2792069916857195.279260.9999222231417720.999220.0002209429557240.0002286.279230.999997116397916

7、1.000000.000018.87.079231.0000006683416041.000000.0000008322352830.00000從上表對(duì)比可知，數(shù)量解和精確解的結(jié)果幾乎一致。該程序最大的優(yōu)點(diǎn)就是可以對(duì)輸入不同的初始值，來觀察疊代次數(shù)和對(duì)結(jié)果的影響。例如：當(dāng)輸入初始值為0.1和1時(shí)，與前面的比較見下表：0.3000000000000000.9345673820190330.4488602810575830.4000000000000001.0000000000000002.0856232646779450.4030528867111930.3320504255726080.333

8、5715199474381.0000007653700941.0030524939321210.3320434110618550.9999866815772260.3320500444862901.000000000224372很明顯可以看出疊代次數(shù)不一樣，第二次比第一次多計(jì)算兩次，但對(duì)結(jié)果的影響幾乎可以忽略不計(jì)。說明疊代初值對(duì)疊代次數(shù)的影響非常大，如選初值為3和40，則需要計(jì)算172次，但仍然能夠?qū)⒕_計(jì)算到0.332050043715133。s.m：clear,clfY(1)=input('請(qǐng)輸入第一個(gè)任意初始值:')t,y1=ode45('q',0:0.2

9、:9,0,0,Y(1);x(1)=y1(length(t),2);Y(2)=input('請(qǐng)輸入第二個(gè)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) x(n+2),Y(n) Y(n+1) Y(n+2),2),1);t,y=ode45('q',0:0.2:9,0,0,Y(n+3);x(n+3)=y(length(t),2);if abs(x(n+3)-1)<10e-7 break;endendplot(t,y,'-');xlabel(''),ylabel(&#