数学实验——常微分方程数值解

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1、数学实验报告实验4常微分方程数值解实验4常微分方程数值解分1黄浩2011011743一、实验目的1.掌握用MATLAB软件求微分方程初值问题数值解的方法；2.通过实例学习用微分方程模型解决简化的实际问题；3.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。二、实验内容1.《数学实验》第一版（问题2）问题叙述：小型火箭初始重量为1400kg，其中包括1080kg燃料。火箭竖直向上发射时燃料燃烧率为18kg/s，由此产生32000N的推力，火箭引擎在燃烧用尽时关闭。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、

2、速度、加速度，及火箭到达最高点时的高度和加速度，并画出高度、速度、加速度随时间变化的图形。模型转换及实验过程：（一）从发射到引擎关闭设火箭总质量为m，上升高度为h，瞬时速度为v，瞬时加速度为a，由燃料燃烧时间t=60s，可列如下的方程组：mt=m0-kt=1400-18th't=vt其中t∈[0,60]v't=atat=F推-F阻m(t)-g=32000-0.4v2t1400-18t-9.8因此，上述方程为二元常微分方程组，选择t为自变量，h和v为因变量进行分析。初值条件：h0=0,v0=0对上述模型，使用ode45()函数求数值解（程序见四.1、四.2），结果如下：

3、xh(t)v(t)a(t)29数学实验报告实验4常微分方程数值解0.000.000.0013.05711.006.5713.1913.30452.0026.4426.5813.45333.0059.7640.0613.49724.00106.5753.5413.43315.00166.7966.8913.26136.00240.2780.0212.98537.00326.7292.8312.61228.00425.79105.2212.15209.00536.99117.1111.616910.00659.80128.4311.021311.00793.63139.1

4、410.380012.00937.85149.189.708313.001091.79158.559.020914.001254.71167.238.330915.001425.93175.227.650216.001604.83182.556.990117.001790.78189.226.359318.001983.13195.275.764619.002181.24200.755.209520.002384.47205.704.694621.002592.36210.184.222022.002804.52214.193.794323.003020.56217.

5、793.412024.003240.08221.013.073025.003462.65223.922.772626.003687.88226.562.504427.003915.58228.972.267728.004145.60231.142.063329.004377.76233.111.889830.004611.86234.911.743331.004847.68236.571.617832.005085.02238.141.506233.005323.85239.611.409534.005564.11240.991.329335.005805.77242

6、.281.265036.006048.72243.501.213937.006292.87244.681.170838.006538.11245.831.130339.006784.48246.961.094740.007031.96248.051.066329数学实验报告实验4常微分方程数值解41.007280.54249.101.045642.007530.19250.121.030843.007780.85251.141.017844.008032.49252.151.002445.008285.12253.160.987646.008538.75254.150

7、.976347.008793.39255.120.969648.009049.01256.070.966349.009305.58257.030.962450.009563.08257.990.952751.009821.52258.950.941252.0010080.93259.900.933753.0010341.30260.830.932854.0010602.62261.750.936355.0010864.86262.670.937056.0011127.98263.610.925857.0011392.04264.540.913858.