一维抛物线型方程数值解法1附图及matlab程序

1、 一维抛物线偏微分方程数值解法（1）解一维抛物线型方程(理论书籍可以参看孙志忠：偏微分方程数值解法）Ut-Uxx=0, 0<x<1,0<t<=1（Ut-aUxx=f(x,t),a>0) U(x,0)=ex, 0<=x<=1,U(0,t)=et,U(1,t)=e(1+t), 0<t<=1精确解为：U(x,t)=e(x+t);下面给出两个matlab程序，实质一样（用的是向前欧拉格式）第二个程序由之前解线性方程组的G-S迭代法得到，迭代次数k=2（固定）function p u e x t=pwxywxq(h1,h2,m,n)% 解抛物线型一维

2、方程 向前欧拉格式 （Ut-aUxx=f(x,t),a>0)%不用解线性方程组，由下一层(时间层）的值就直接得到上一层的值%m,n为x,t方向的网格数，例如（2-0）/0.01=200;%e为误差，p为精确解u=zeros(n+1,m+1);x=0+(0:m)*h1;t=0+(0:n)*h2;for(i=1:n+1) u(i,1)=exp(t(i); u(i,m+1)=exp(1+t(i);endfor(i=1:m+1) u(1,i)=exp(x(i);endfor(i=1:n+1) for(j=1:m+1) f(i,j)=0; endendr=h2/(h1*h1); %此处r=a*h2

3、/(h1*h1）；a=1 要求r<=1/2差分格式才稳定for(i=1:n) for(j=2:m) u(i+1,j)=(1-2*r)*u(i,j)+r*(u(i,j-1)+u(i,j+1)+h2*f(i,j); endendfor(i=1:n+1) for(j=1:m+1) p(i,j)=exp(x(j)+t(i); e(i,j)=abs(u(i,j)-p(i,j); endend或者：function u e p x t k=paowuxianyiweixq(h1,h2,m,n,kmax,ep)% 解抛物线型一维方程 向前欧拉格式 （Ut-aUxx=f(x,t),a>0)%kma

4、x为最大迭代次数%m,n为x,t方向的网格数，例如（2-0）/0.01=200;%e为误差，p为精确解syms temp;u=zeros(n+1,m+1);x=0+(0:m)*h1;t=0+(0:n)*h2;for(i=1:n+1) u(i,1)=exp(t(i); u(i,m+1)=exp(1+t(i);end for(i=1:m+1) u(1,i)=exp(x(i);endfor(i=1:n+1) for(j=1:m+1) f(i,j)=0; endenda=zeros(n,m-1);r=h2/(h1*h1);%此处r=a*h2/(h1*h1）；a=1 要求r<=1/2差分格式才稳定

5、for(k=1:kmax) for(i=1:n) for(j=2:m) temp=(1-2*r)*u(i,j)+r*(u(i,j-1)+u(i,j+1)+h2*f(i,j); a(i+1,j)=(temp-u(i+1,j)*(temp-u(i+1,j); u(i+1,j)=temp; end end a(i,j)=sqrt(a(i,j); if(k>kmax) break; end if(max(max(a)<ep) break; endendfor(i=1:n+1) for(j=1:m+1) p(i,j)=exp(x(j)+t(i); e(i,j)=abs(u(i,j)-p(i,

6、j); endend 在命令窗口中输入： p u e x t=pwxywxq(0.1,0.005,10,200);>> surf(x,t,u)>> shading interp;>> xlabel('x');ylabel('t');zlabel('u');>> title('一维抛物线方程 向前欧拉法 数值解'); surf(x,t,p)>> shading interp;xlabel('x');ylabel('t');zlabel(

7、9;p');>> title('一维抛物线方程 向前欧拉法 精确解')同理： plot(x,u)>> xlabel('x');ylabel('u');>> title('固定时间 改变x u与x 的关系 数值解')p u e x t=pwxywxq(0.1,0.01,10,100); surf(x,t,u)Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering

8、Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering >> surf(x,t,e)Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Notrendering >> 所以空间步长与