试验常微分方程数值解

1、实验4常微分方程数值解化工系 化51王培丞2015011830实验目的.练习数值微分的计算；.掌握用MATLAB软件求微分方程初值问题数值解的方法；.通过实例学习用微分方程模型解决简化的实际问题；.了解欧拉方法和龙格库塔方法的基本思想和计算公式，及稳定性等概念.实验内容第3题：小型火箭初始质量为1400kg ,其中包括1080kg燃料，火箭竖直向上发射时燃 料燃烧率为I8kg/s ,由此产生32000N的推刀，火箭引擎在燃料用尽时关闭。 设火箭上升时空气阻力正比于速度的平方，比例系数为0.4kg/m ,求引擎关闭瞬 间火箭的高度和加速度，并画出高度、速度、加速度随时间变化的图形。模型：分别用m

2、,F,f,h,v,a表示火箭某一时刻的质量、合力、阻力、高度、速度、加速 度。在引擎关闭前：F = 32000 0.4v2 nigF = ma=a=v in = 1400 18t 火箭在60s时用尽燃料 程序代码：rocket.mfunction dx=rocket(t,x)*微分方程k=0.4;m0=1400;r=18;F=32000;g=9.8;%其中x(l)对应高度X , x(二)对应速度Vdx=x (2) ; (F-k*x (2).人2) / (mO-r*t)-g;命令行：ts=1:60; h0=0,0;t,h=ode45(rocket,ts,h0);输出数据：006.72362837

3、46311313.489314238027227.046378098540627.179697745522861.116769899172440.9629798125824108.96643968089854.7235735106777170.51659382227768.3485248344128245.57891384663981.7283329559018333.85555642207894.7569502247682434.943782391953107.336539226424548.344445203535119.384358002977673.492057919145130.83

4、0573373987809.770375414086141.621971014618956.512428961777151.7219742425811113.00050717090161.1106481054431278.47334771285169.7871167062691452.24613985764177.7566418635321633.66819905803185.0314122885191822.08083816547191.6364914758692016.81736743009197.6098177035802217.20299174365203.00225534886324

5、22.61356416237207.8701083944492632.62187814993212.2453829387762846.82958203848216.1608632043703064.82306998241219.6587514750633286.17348195828222.7906680959923510.43867920705225.6174200697303737.29390842366228.1858017721203966.58329682873230.5095956193834198.13749935537232.6074359125794431.762364663

6、63234.5071607806694667.23893514049236.245812180511490432928341114237.8684679999245142.91962226208239.3952285123355382.98457439949240.8231215012985624.47082251417242.1573278536825867.30149141681243.411392148185611137614803792244.6072226553346356.57928189365245.7729090431066602.89136687278246.91105022

7、76266850.32468727750248.0128678371217098.86801212328249.0774342121357348.49213600433250.1107369857717599.14987909357251.1256790836947850.78494843906252.1393580531808103.40569836400253.1473701325378357.02926266563254.1398414165358611.65575648531255.1137944409718867.27112965916256.0725700105319123.847

8、16671812257.0258271987849381.34898724964257.9871686927099639.78955057549258.9500543809529899.19107993557259.89995690505610159.5579229200260.83180583959710420.8769438211261.75024149753210683.1175236331262.66961493020410946.2483016321263.606108481043112103003857861264.54054591952311475.2928536841265.4

9、5650570920411741.2184090832266.35592134765412008.0443317145267.258843484582结论：放射性废物的处理。圆桶容积55gal ,装满时527.436lbf ,在海水中受到的浮力 为470.327lbf ,下沉时受到的阻力与下沉速度成正比，比例系数为0.081 lbfs/ft , 大量实验发现圆桶速度超过40ft/s时，就会破裂。(1 )建立解决上述问题的微分方程模型.换算成国际单位：容积没用。桶重 2344.6N(239.2kg),浮力 2090.7N ,比例系数 1.18N s/m边际速度12.19m/s,水深91.44mF

10、 = 2344.6 - 2090.7 - 0.081vF = ma微分方程为：dv=a = 1.061 - 0.0003386vdtdh(2 )数值解：下面开始估计时间的试验范围.既然存在争议，那么最终速度应该偏差12.19m/s不大，故下落时间一定 小于以12.19*0.5为最终速度匀加速运动的时间(14.0s )试验范围为0 : 0.1 : 14代码如下：drop.m:funcxion dd = drop 仁，d)*微分方程al=L. 061;a2=-0.0003386;+其中d(l)对应深度d , d (二)对应速度vdd=d(2);al+a2*d(2);endodescript.m:t

11、s=0:0.1:14;h0=0/0;tr h=ode45(drop,ts,hO);tdV0000.10.005304940.1060982040.20.0212195210.212192815030.0477433830.3182838340.40.0848761680.4243712610.50.1326175160.5304550960.60.1909670670.6365353390.70.2599244640.742611990.80.3394893460.8486850490.90.4296613540.95475451710.5304401291.0608203931.10.641

12、8253131.1668826781.20.7638165451.2729413721.30.8964134671.3789964741.41.039615721.4850479861.51.1934229451.5910959071.61.3578347821.6971402371.71.5328508731.8031809771.81.7184708581.9092181261.91.9146943782.01525168422.1215210752.1212816532.12.3389505892.2273080312.22.5669825622.33333082232.80561663

13、32.4393500182.43.0548524462.5453656272.53.3146896392.6513776462.63.5851278552.7573860762.73.8661667352.8633909162.84.1578059192.9693921672.94.4600450483.07538982934.7728837653.1813839023.15.0963217093.2873743853.25.4303585223.3933612813.35.7749938463.4993445873.46.130227323.6053243053.56.4960585873.

14、7113004353.66.8724872873.8172729763.77.2595130623.9232419293.87.6571355544.0292072943.98.06535440244841692484.241127264.18.91357973443470818624.29.3535855014.453032876439.804186194.5589803034.410.265381444.6649241424.510.73717094.7708643944.611.21955424.8768010594.711.712530994.98273413

15、74.812.216100915.0886636284.912.73026365.194589533513.25501875.3005118515.113.790365855.4064305825.214.336304695.5123457275.314.892834875.6182572865.415.459956035.7241652595.516.037667815.8300696465.616.625969845.9359704475.717.224861786.0418676625.817.834343256.1477612915.918.454413916.253651335619

16、.08507346.3595377946.119.726321356.4654206686.220.378157416.5712999566.321.040581236.6771756596.421.713592436.7830477786.522.397190666.8889163116.623.091375576.994781266.723.796146797.1006426256.824.511503987.2065004056.925.237446767312354601725.973974787.4182052127.126.721087687.524052247.227.47878

17、517.6298956837328.24706677.7357355437.429.025932097.8415718197.529.815380947.9474045127.630.615412888.0532336217.731.426027548.1590591477.832.247224598.264881097.933.079003648.370699449833.921364368.4765142268.134.774306378.582325428.235.637829328.6881330318.336.511932868.793937068.437.396616618.899

18、7375068.538.291880249.0055343698.639.197723379.1113276518.740.114145659.217117358.841.041146729.3229034688.941.978726229.428686003942.92688389.5344649579.143.885619099.6402403299.244.854931759.746012129.345.83482149.8517803299.446.825287699.9575449589.547.8263302710.0633069.648.8379487710.169063479.

19、749.8601428510.274817369.850.8929121310.380567669.951.9362562610.486314381052.9901748810.5920575310.154.0546676410.6977970910.255.1297341810.8035330710.356.2153741410.9092654710.457.3115871611.014994310.558.4183728811.1207195410.659.5357309511.226441210.760.663661113321592810.861.8021626811.43787379

20、10.962.9512356411.543584711164.1108795111.6492920611.165.2810939311.7549958211.266.4618785511.8606960111.367.6532330111.9663926211.468.8551569612.0720856411.570.0676500212.1777750911.671.2907118612.2834609611.772.524342112.3891432611.873.7685403912.4948219711.975.0233063712.600497111276.2886396912.7

21、061686712.177.5645399912.8118366512.278.851006912.9175010512.380.1480400813.0231618712.481.4556391613.1288191212.582.7738037813.2344727912.684.1025335913.3401228812.785.4418282413.445769412.886.7916873513.5514123312.988.1521105913.65705171389.5230975713.7626874813.190.9046479613.8683196913.292.29676

22、13913.9739483213.393.6994375114.0795733713.495.1126759514.1851948513.596.5364763614.2908127513.697.9708383814.3964270713.799.4157616514.5020378213.8100.871245814.60764513.9102.337290514.7132485914103.813895414.81884862可知深度达91.44m时，下沉速度已经大于13m/s ,工程师胜利。解析解：civ dF =1.061 - 0.0003386v该方程为一阶线性常微分方程，下面用常

23、数变易法求解析解解对应的线性齐次方程勺=-0.0003386得： atV =-0.00033861常数变易：v =(:？一03386,代入祟=1061 - 0.0003386得C(t) = 3133.5e3386t + c(3133.5e00003386t+ C)e-0X)0033861代入初值(0,0)得C=3133.5v = (3133.5 - 3133.56-。3386t第6题： 一只小船渡过宽为d的河流，目标是起点A正对着的另一岸B点，已知河水流速v1与船在导水中的速度v2之比为k.(1 )建立描述小船航线的数学模型，求其解析解；设小船的坐标(x,y) , A(0,0),B(0,Y),若v2=v1，小船无论如何都无法到达对岸B点，此时轨迹为一条先曲后直的线，最终的直线段有平行于河岸的渐近线。而题目中的结果是渡过了河流此处必须假设v1v2 ,即k1 vy=v2cos0,vx=v1-v2sin0,0=arctanx/(Y-y)曳= -2 ( - y)dt J(y-y)2 + %2(2 )dxv2x=kl?2人 人出J(y-y)2+/得Y-ydy_ Y-y _