Blasius方程的数值解matlab程序Word下载.doc

Blasius方程的数值解matlab程序Word下载.doc

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0.8000000000000000.1060972955378420.2647059995571820.327381783215304

1.0000000000000000.1655630308847090.3297737923612960.322998326329412

1.2000000000000000.2379442372687800.3937674819581910.316583378968660

1.4000000000000000.3229688368951930.4562584560706150.307858224411080

1.6000000000000000.4202943904016780.5167580586411450.296652046000281

1.8000000000000000.5294908391591200.5747519024337040.282921170876929

2.0000000000000000.6500090694701540.6297515231340940.266750311286562

2.2000000000000000.7811822070295200.6812962493004080.248352130535080

2.4000000000000000.9222728056314760.7289734807030940.228082678389961

2.6000000000000001.0724798222828610.7724470697584080.206450263528860

2.8000000000000001.2309501374214950.8114975840686250.184014296855294

3.0000000000000001.3967879095445580.8460296386569050.161369816626103

3.2000000000000001.5690699253450890.8760707526453200.139126737097636

3.4000000000000001.7469154487035860.9017600525255140.117863021963319

3.6000000000000001.9294876861519370.9233315164928120.098081245701445

3.8000000000000002.1160004034510310.9411092427556170.080139603642537

4.0000000000000002.3057193897850770.9555059580929050.064250507286456

4.2000000000000002.4980051682669550.9669559021448410.050519855519064

4.4000000000000002.6923282496160700.9758675850986780.038977900754526

4.6000000000000002.8882185732850130.9826747519817110.029499001922806

4.8000000000000003.0852829165367120.9877958568329080.021859471891081

5.0000000000000003.2832378195729690.9915456498326390.015900519799442

5.1999999999999993.4818387609016070.9942360903768370.011361976112965

5.4000000000000003.6808868432804850.9961530675400220.007933845886624

5.6000000000000003.8802589909031930.9974749309396950.005439882672210

5.8000000000000004.0798506041197960.9983722945980170.003657568419394

6.0000000000000004.2795889371535250.9989716023318460.002407067561230

6.1999999999999994.4794254608023780.9993613586634180.001555212097374

6.4000000000000004.6793245667901770.9996114936251530.000983390226568

6.6000000000000004.8792638411282270.9997679250728730.000610241632299

6.8000000000000005.0792278579105120.9998642056028700.000370938799775

7.0000000000000005.2792069916857190.9999222231417720.000220942955724

7.2000000000000005.4791951859411130.9999563977383570.000129191951169

7.4000000000000005.6791886128001270.9999762929671100.000073770638928

7.6000000000000005.8791851213661380.9999874576967800.000041576427492

7.8000000000000006.0791832861942990.9999937774180380.000022702183282

8.0000000000000006.2791824324459440.9999971163979160.000012416083429

8.1999999999999996.4791820464355970.9999989986192810.000006410840147

8.4000000000000006.6791819596757440.9999999086952430.000003431483786

8.6000000000000006.8791819969198921.0000004088733780.000001736982564

8.8000000000000017.0791821042849741.0000006683416040.000000832235283

9.0000000000000007.2791822572191481.0000007653700940.000000489499078

查看叠代过程，可在命令窗口执行

Y

Y=

0.3000000000000000.4000000000000000.3331153697266140.332050425572608

x

x=

0.9345673820190331.1321572580226841.0021378204450411.000000765370094

%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

0.332050425572608

0.33206

1

0.165563030884709

0.16577

0.329773792361296

0.32979

0.322998326329412

0.32301

2

0.650009069470154

0.65003

0.629751523134094

0.62977

0.266750311286562

0.26675

3

1.396787909544558

1.39682

0.846029638656905

0.84605

0.161369816626103

0.16136

4

2.305719389785077

2.30576

0.955505958092905

0.95552

0.064250507286456

0.06423

5

3.283237819572969

3.28329

0.991545649832639

0.99115

0.015900519799442

0.01591

6

4.279588937153525

4.27964

0.998971602331846

0.99868

0.002407067561230

0.00240

7

5.279206991685719

5.27926

0.999922223141772

0.99922

0.000220942955724

0.00022

8

6.279182432445944

6.27923

0.999997116397916

1.00000

0.000012416083429

0.00001

8.8

7.079182104284974

7.07923

1.000000668341604

0.000000832235283

0.00000

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：

0.300000000000000

0.100000000000000

0.934567382019033

0.448860281057583

0.400000000000000

1.000000000000000

1.132157258022684

2.085623264677945

0.333115369726614

0.403052886711193

1.002137820445041

1.137910855452804

0.333571519947438

1.000000765370094

1.003052493932121

0.332043411061855

0.999986681577226

0.332050044486290

1.000000000224372

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,clf

Y

（1）=input（'

'

）

[t,y1]=ode45（'

q'

[0:

0.2:

9],[0,0,Y

（1）]）;

x

（1）=y1（length（t）,2）;

Y

（2）=input（'

[t,y2]=ode45（'

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（'

9],[0,0,Y（3）]）;

x（3）=y3（length（t）,2）;

n=0;

while1

n=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（'

9],[0,0,Y（n+3）]）;

x（n+3）=y（length（t）,2）;

ifabs（x（n+3）-1）<

10e-7

break;

end

plot（t,y,'

-'

）;

xlabel（'

η'

）,ylabel（'

f/df/df^2'

title（'

Blasius方程数值解'

gridon;

axis（[0,6,-0.5,2]）;

q.m：

functiondy=q（x,y）

dy=zeros（3,1）;

dy

（1）=y

（2）;

dy

（2）=y（3）;

dy（3）=-0.5*y

（1）*y（3）;