常微分方程数值解实验报告.doc

常微分方程数值解实验报告姓名： 张 彬 班级： 信计092 学号： 0909290229 实验目的：应用前面实验程序求解常微分方程初值问题，通过改变的初值，重点了解Euler方法及其改进方法的稳定性及收敛性实验内容：；求满足初值条件x=2，y=2的解；x2，4，步长h=0.05，并通过数值与图像进行比较。实验原理：1.前述几种数值计算方法对于初值及步长的选取都有依赖性。2.对于无法获得精确解的常微分方程初值问题，更需要分析计算结果的稳定性与收敛性。实验结果：h=0.05 x=2 y=2欧拉法 表1Yend=自变量欧拉值精确解误差估计22202.051.9659341.964661-0.000652.11.9294291.927094-0.001212.151.8909741.887814-0.001672.21.851111.847387-0.002012.251.8104181.806424-0.002212.31.7695281.765583-0.002232.351.7291091.725564-0.002052.41.689871.68711-0.001642.451.6525581.651004-0.000942.51.6179561.6180636.64E-052.551.5868741.5891370.0014242.61.5601521.5651040.0031642.651.5386511.5468610.0053082.71.5232461.5353260.0078682.751.5148261.5314220.0108372.81.514281.5360760.0141892.851.5224931.5502050.0178772.91.5403351.5747110.021832.951.5686551.6104680.02596331.6082651.6583060.0301763.051.6599321.7190050.0343653.11.7243641.7932780.0384293.151.8021991.8817550.0422773.21.8939871.984970.0458363.252.000182.1033430.0490473.32.1211142.2371680.0518753.352.2569962.3865910.0543013.42.4078892.55160.0563223.452.5736972.7320090.0579473.52.7541552.9274410.0591943.552.9488113.1373220.0600873.63.1570223.3608670.0606533.653.3779433.5970740.0609193.73.6105223.8447220.0609153.753.8534974.1023690.0606663.84.1053984.3683580.0601973.854.3645524.6408210.059533.94.629094.9176970.0586873.954.8969655.1967470.05768645.1659645.4755780.056545图1改进欧拉法： 表2Yend=自变量欧拉改进法精确解误差估计22202.051.9647141.964661-2.7E-052.11.9272041.927094-5.8E-052.151.8879861.887814-9.1E-052.21.8476221.847387-0.000132.251.8067231.806424-0.000172.31.7659471.765583-0.000212.351.7259931.725564-0.000252.41.6876041.68711-0.000292.451.651561.651004-0.000342.51.6186781.618063-0.000382.551.5898061.589137-0.000422.61.565821.565104-0.000462.651.5476191.546861-0.000492.71.5361161.535326-0.000512.751.5322341.531422-0.000532.81.5368991.536076-0.000542.851.5510271.550205-0.000532.91.5755171.574711-0.000512.951.6112411.610468-0.0004831.659031.658306-0.000443.051.7196631.719005-0.000383.11.793851.793278-0.000323.151.8822211.881755-0.000253.21.9853091.98497-0.000173.252.1035342.103343-9.1E-053.32.2371892.237168-9.6E-063.352.3864212.3865917.1E-053.42.5512192.55160.0001493.452.7313952.7320090.0002253.52.9265762.9274410.0002953.553.1361883.1373220.0003623.63.3594463.3608670.0004233.653.5953533.5970740.0004783.73.8426893.8447220.0005293.754.1000144.1023690.0005743.84.3656734.3683580.0006143.854.6378034.6408210.000653.94.9143454.9176970.0006823.955.1930625.1967470.00070945.4715655.4755780.000733图2实验体会：由于这是本学期的第一次课程设计，似乎有更高的热情去完成它。此次课程设计的过程中，我们可以通过分析图片中曲线来与精确解进行对比，并可以近似取值，使我们对3、5数值解这一节加深了印象。在完成课程设计的选定后，随之而来的问题却远比我们想象的要困难的多，没想到这项看起来不需要多少技术的作业却是非常需要耐心与精力的，在这几天的课程设计后，我已明白