微分方程数值解法课程设计--A组

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1、微分方程数值解法课程设计----A组计算022班3号许**1.用复化梯形计算积分解：算法思想：根据复化梯形公式:取步长，有，写出程序，如下所示：N=100;a=0;b=10;h=(b-a)/N;T=zeros(0,N);forj=1:NT(j)=0.5*h*[exp(-(a+h*(j-1)).^2)+exp(-(a+h*j).^2)]t=sum(T)end运行结果：t=0.88622.用Euler、改进的Euler法、梯形法、R-K法解解：根据算法思想：（1）Euler法（2）改进的Euler法（3）梯形法（4）R-K法我们可以写出程序如下所示：formatlon

2、g;N=100;h=1/N;%步长U=zeros(1,N+1);%eulerT=zeros(1,N+1);%真解X=zeros(1,N+1);R=zeros(4,N+1);%误差G=zeros(1,N+1);%改进euleO=zeros(1,N+1);%梯形法K=zeros(1,N+1)%R-K法c=zeros(1,4);a=0;forj=2:N+1;X(j)=h*(j-1);endU(1)=1T(1)=1;G(1)=1;X(1)=0;O(1)=1;K(1)=1;forj=1:NU(j+1)=U(j)+h*(U(j)+X(j));%euler法公式a=G(j)+h

3、*(G(j)+X(j));G(j+1)=G(j)+h/2*(G(j)+X(j)+a+X(j+1));%改进的euler法公式b=O(j)+h*(O(j)+X(j));fori=1:10O(j+1)=O(j)+h/2*(G(j)+X(j)+b+X(j+1));%梯形法公式b=O(j+1);endc(1)=K(j)+X(j);c(2)=K(j)+h/2*c(1)+(X(j)+h/2);c(3)=K(j)+h/2*c(2)+(X(j)+h/2);c(4)=K(j)+h*c(3)+(X(j)+h);K(j+1)=K(j)+h/6*(c(1)+2*c(2)+2*c(3)+c

4、(4));%四阶R-K法公式T(j+1)=-X(j+1)-1+2*exp(X(j+1));%真实值R(1,j+1)=U(j+1)-T(j+1);%EULER法误差R(2,j+1)=G(j+1)-T(j+1);%改进的EULER法误差R(3,j+1)=O(j+1)-T(j+1);%梯形法误差R(4,j+1)=K(j+1)-T(j+1);%四阶R-K法误差end真实值T：11.01011.02041.03091.04161.05251.06371.0751.08661.09831.11031.12261.1351.14771.16051.17371.1871.2006

5、1.21441.22851.24281.25741.27221.28721.30251.31811.33391.34991.36631.38291.39971.41691.43431.45191.46991.48811.50671.52551.54461.5641.58361.60361.62391.64451.66541.68661.70811.731.75211.77461.79741.82061.84411.86791.8921.91651.94131.96651.99212.0182.04422.07092.09792.12522.1532.18112.2

6、0962.23852.26782.29742.32752.3582.38892.42022.45192.4842.51662.54952.58292.61682.65112.68582.7212.75662.79272.82932.86632.90382.94182.98033.01923.05863.09863.1393.183.22143.26343.30593.34893.39253.4366欧拉法U值：11.011.02021.03061.04121.0521.0631.07431.08571.09741.10921.12131.13371.14621.1

7、5891.17191.18521.19861.21231.22621.24041.25481.26941.28431.29951.31491.33051.34641.36261.3791.39571.41271.42991.44741.46521.48321.50151.52021.53911.55821.57771.59751.61761.6381.65861.67961.70091.72251.74451.76671.78931.81221.83541.85891.88281.9071.93161.95651.98182.00742.03342.05972.0

8、8642.