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chemwayne

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[求助] 求助，matlab求解二元二阶的常微分方程组

现在需要解一个二元二阶的常微分方程组，想用四阶龙哥库塔来解，刚接触Matlab, 希望能得到大家帮助，万分感谢！我想得到的是数值解，z-x-y， z= 0-1,
（原来曾用excel 的solver来解，当时是猜测y0的初值）
方程：

[ Last edited by chemwayne on 2013-5-23 at 08:43 ]

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1楼 2013-05-23 08:40:47

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dingd

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边值问题Matlab可用BVP4C命令，但感觉比较麻烦，下面用1stOpt求解，很简单快捷：

CODE:

Constant Pey=9.73, Nox=8.05, uxuy=3, bd=1, cx1e=2.3,

b1=309.7,b2=2832.5, a1=27.8,a2=2.15,a3=-0.84,a4=0.935;

Variable t=[0:0.025:1],x=1,y'=[0,-Pey*bd*y];

Plot x,y,y';

ODEFunction x'=-(Nox/b1)*(b1*x+cx1e-(a1*y+a2)/(a3*y+a4));

            y''=-Pey*bd*y'-(Nox*Pey*bd*uxuy/b2)*(b1*x+cx1e-(a1*y+a2)/(a3*y+a4));

边值估算:
y(t=0): 0.327786532200411

算法: 龙格-库塔-费尔博格法(Runge-Kutta-Fehlberg Method)
步长值: 0.025
步长数: 40
种群数: 5

结果:
t      x(t)                  y'(t)          y(t)                  x'(t)             y''(t)       y'(t)
0      1                     0                     0.327786532200411 -7.66600291732416 -24.4666416985093    0
1      0.000693840858838046 -0.00586485658562458 0.000602760183379987 -0.00510084725252375 0.0407853057400083 -0.00586485658562458

结果过程:
No. t      x(t)                  y'(t)          y(t)                  x'(t)             y''(t)       y'(t)
0 0      1                     0                     0.327786532200411 -7.66600291732416 -24.4666416985093    0
1 0.025 0.826313919267495 -0.490075360338193    0.321182502343127 -6.27870464940025 -15.2705386363518    -0.490075360338193
2 0.05    0.683848068843495 -0.786264397862707    0.304892497030711 -5.1579190209382      -8.81154521345653    -0.786264397862707
3 0.075 0.566664089408245 -0.947175982341636    0.282992172618731 -4.24801699438317 -4.34185274765138    -0.947175982341636
4 0.1    0.470051559334184 -1.01530304646892 0.258303652175927 -3.50591713326772 -1.3105072646148    -1.01530304646892
5 0.125 0.390249601896457 -1.02128711560973 0.232742495433715 -2.89829612055369 0.68698722317898    -1.02128711560973
6 0.15    0.324233874445947 -0.987077366676523    0.207572500259331 -2.39920298546376 1.94702073906203    -0.987077366676523
7 0.175 0.269556313218175 -0.928258904656385    0.183592425130687 -1.98822461738467 2.68638637716003    -0.928258904656385
8 0.2    0.224224427946598 -0.855766958939663    0.161272572460497 -1.64913802323729 3.06326085759594    -0.855766958939663
9 0.225 0.186609425886391 -0.777149796437229    0.14085444777691      -1.36893219952477 3.19261444219091    -0.777149796437229
10 0.25    0.155375287754656 -0.697500084306226    0.122423216362664 -1.13709376937326 3.15755303846612    -0.697500084306226
11 0.275 0.129423296920018 -0.620141103883988    0.105960033630092 -0.94507770210612 3.01768430066738    -0.620141103883988
12 0.3    0.10784802305656      -0.547133377656043    0.0913796756856091    -0.785907023627595    2.81532466545902    -0.547133377656043
13 0.325 0.0899021385914759    -0.479646400522898    0.0785572032799784    -0.653865484735837    2.58009717001762    -0.479646400522898
14 0.35    0.074968022580219 -0.418231138978627    0.0673466535604279    -0.544257391395793    2.33234938931414    -0.418231138978627
15 0.375 0.0625347128329202    -0.363018610699521    0.057593889044556 -0.453217546785379    2.08569220018259    -0.363018610699521
16 0.4    0.0521791123811746    -0.313863587563825    0.0491452021428162    -0.377559285865171    1.84888302113582    -0.313863587563825
17 0.425 0.0435506113178284    -0.270447635858435    0.0418528608540996    -0.314651994050368    1.62721930221573    -0.270447635858435
18 0.45    0.0363584698185308    -0.232352118108376    0.035578472976028 -0.262321788886875    1.42356581491271    -0.232352118108376
19 0.475 0.0303614453595037    -0.199109100417178    0.0301948156838825    -0.218770614815521    1.23910819590991    -0.199109100417178
20 0.5    0.0253592512680494    -0.170236100523047    0.0255866050433442    -0.182510109417003    1.07390211666811    -0.170236100523047
21 0.525 0.0211855142304534    -0.145259103337654    0.0216505513938588    -0.152307402404205    0.927270211329725    -0.145259103337654
22 0.55    0.0177019615432633    -0.123727136798432    0.0182949507170654    -0.127140604839182    0.798085959370831    -0.123727136798432
23 0.575 0.0147936190504607    -0.10522084841223 0.0154389909459885    -0.106162198564426    0.684973980835008    -0.10522084841223
24 0.6    0.0123648409053087    -0.0893568825522887    0.013011899547216 -0.0886688856491884    0.586448855258726    -0.0893568825522887
25 0.625 0.0103360247349054    -0.0757893783829889    0.0109520199975861    -0.0740767321274094    0.501009027163578    -0.0757893783829889
26 0.65    0.00864089210235943    -0.0642095490478649    0.00920587648845576    -0.0619006582113814    0.427198170126278    -0.0642095490478649
27 0.675 0.00722423559559995    -0.0543440348572125    0.00772726568799934    -0.0517375017368198    0.363643217620131    -0.0543440348572125
28 0.7    0.00604005138059095    -0.0459525242309598    0.00647639968487763    -0.0432520224132175    0.309075883307506    -0.0459525242309598
29 0.725 0.00504999039082492    -0.0388249890925048    0.00541911382433476    -0.0361653286316663    0.262342698160537    -0.0388249890925048
30 0.75    0.00422207307997283    -0.0327787733794859    0.00452614590083615    -0.0302453015167734    0.222407244050448    -0.0327787733794859
31 0.775 0.00352962231729211    -0.0276556945654436    0.00377248824180515    -0.0252986667688346    0.188347254882715    -0.0276556945654436
32 0.8    0.00295037694023563    -0.0233192611702517    0.00313681097020956    -0.0211644268962893    0.159348504512056    -0.0233192611702517
33 0.825 0.00246575500715863    -0.0196520685790281    0.00260095269653461    -0.0177084172807521    0.134696842800273    -0.0196520685790281
34 0.85    0.00206024116846864    -0.0165534068636294    0.00214947371190719    -0.0148187912247088    0.113769329539156    -0.0165534068636294
35 0.875 0.00172087700369442    -0.0139370945264954    0.00176926617464773    -0.0124022733798921    0.0960251142514258 -0.0139370945264954
36 0.9    0.00143683682371022    -0.0117295387756349    0.00144921561232049    -0.0103810490993766    0.0809964905399249 -0.0117295387756349
37 0.925 0.00119907445004718    -0.00986801430609513 0.00117990816225777    -0.00869018040220058 0.068280395780233    -0.00986801430609513
38 0.95    0.00100002897020513    -0.00829914727748304 0.00095337824605355    -0.00727545828585919 0.0575305147328749 -0.00829914727748304
39 0.975 0.000833379522126888 -0.00697758825381183 0.000762891746903158 -0.00609161680467778 0.0484500671551354 -0.00697758825381183
40 1      0.000693840858838046 -0.00586485658562458 0.000602760183379987 -0.00510084725252375 0.0407853057400083 -0.00586485658562458