常微分方程数值解实验报告(2)

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1、《计算方法》实验报告2常微分方程实验报告实验目的：1．熟练编程技巧，体会Runge-kutta方法。2．与前面的实验方法进行对比，了解数值计算方法区别和联系。实验内容：运用Runge-kutta方法求解常微分方程初值问题实验题目：，。实验原理：取四阶Runge-kutta方法进行计算实验程序：%fun为目标函数字符串%x0为自变量初始值。%y0为fun(x0);%bou=[a,b]自变量区间%h为步长fun='x*y+x';bou=[3,6];a=bou(1);b=bou(2);x0=3;y0=8;h=0.1;5常微分方程实验报告n=ceil((b-a)/h);xx=lin

2、space(a,b,n+1)';yy=zeros(1,n+1)';lengthx=length(xx);xx(1)=x0;yy(1)=y0;fori=2:n+1x=xx(i-1);y=yy(i-1);k1=eval(fun);x=xx(i-1)+h/2;y=yy(i-1)+h*k1/2;k2=eval(fun);x=xx(i-1)+h/2;y=yy(i-1)+h*k2/2;k3=eval(fun);x=xx(i-1)+h;y=yy(i-1)+h*k3;k4=eval(fun);yy(i)=yy(i-1)+h/6*(k1+2*k2+2*k3+k4);endYs=dsolve(

3、'Dy=x*y+x','y(3)=8','x');fori=1:lengthxx=xx(i);exacty(i)=eval(Ys);endYY=exacty';yend=[xx,yy,YY]p=plot(xx,yend(:,2),'k-o','LineWidth',1,...'MarkerEdgeColor','k',...'MarkerFaceColor','g',...'MarkerSize',4);holdon;pp=plot(xx,yend(:,3),'r-.+','LineWidth',0.8,...'MarkerEdgeColor','r',...'Marker

4、FaceColor','m',...'MarkerSize',6);legend([p,pp],'lgkt4','jingquejie');5常微分方程实验报告实验结果：表1Runge-kutta方法与精确解的比较节点Runge-kutta方法精确解3883.111.2094211.209633.215.7297315.730353.322.1539222.155323.431.3669431.369763.544.7003944.705773.664.1747364.184693.792.8813692.899453.8135.5904135.62293.9199.724

5、9199.7834296.9356297.03914.1445.6665445.85074.2675.3694675.69774.31033.4841034.074.41597.0951598.1454.52492.562494.4514.63928.8443932.2634.76254.5966260.814.810056.7210068.084.916332.0516352.95526788.9326827.625.144381.5244453.645常微分方程实验报告图1Runge-kutta方法与精确解的比较5常微分方程实验报告表2Euler法、Euler改进法、R

6、unge-kutta法的比较节点Euler法Euler改进法Runge-kutta法精确解388883.110.711.163511.2094211.209633.214.32715.5983115.7297315.730353.319.2316421.8691522.1539222.155323.425.9080830.8132831.3669431.369763.535.0568343.6817544.7003944.705773.647.6767262.3587364.1747364.184693.765.2003489.7043592.8813692.899453.

7、889.69446130.095135.5904135.62293.9124.1584190.2807199.7249199.7834172.9701280.7565296.9356297.03914.1242.5582417.9719445.6665445.85074.2342.417627.9187675.3694675.69774.3486.6522952.00051033.4841034.074.4696.34261456.711597.0951598.1454.51003.1732249.7042492.