can anyone help me solve it? Or help me uncouple the equations? :(
$u_{xx}+u_{yy}=au+bw$
$w_{xx}+w_{yy}=cu+dw$
Also I have to solve this problem using the finite difference method, but once i get the output everything's zero. This is my Matlab code, if anyone could help me with that. this is my code so far
clear; close all;
n=101; h=1/(n-1);
x=0:h:1.01; y=0:h:1.01;
U=zeros(n+1); W=zeros(n+1);
omega=1.7777777;err=1000;
a1=0.5; a2=5; b1=-1; b2=1;
tol=1e-6;
for i=1:n+1 %boundary conditions
U(1,i)=cos(5*pi*y(i)); U(i,1)=cos(5*pi*y(i)); U(n+1,i)=0; U(i,n+1)=0;
W(1,i)=cos(5*pi*y(i)); W(i,1)=cos(5*pi*y(i)); W(n+1,i)=0; W(i,n+1)=0;
end
figure(1);
subplot(1,2,1);mesh(x,y,U);view(130,25);grid on; title('boundary conditions U(x,y)');
subplot(1,2,2);mesh(x,y,W);view(130,15);grid on; title('boundary conditions W(x,y)');
u=U; w=W;
while err>tol
for i=2:n
for j=2:n
u(i,j)=(1-omega)*U(i,j)+omega*(((u(i+1,j)+u(i-1,j)+u(i,j+1)+u(i,j-1))/h^2)-b1*W(i,j))/(a1+(4/h^2));
w(i,j)=(1-omega)*W(i,j)+omega*(((w(i+1,j)+w(i-1,j)+w(i,j+1)+w(i,j-1))/h^2)-a2*U(i,j))/(b2+(4/h^2));
end
end
W=w; U=u; err=max(max(abs(u-U)));
end
figure(2);
subplot(1,2,1);mesh(x,y,U);view(130,25);grid on; title('U(x,y)');
subplot(1,2,2);mesh(x,y,W);view(130,15);grid on; title('W(x,y)');
The problem is this line:
W=w; U=u; err=max(max(abs(u-U)));
You are setting U=u`` first and then looking at the different of uandU```. Reverse the computation and it works.
err=max(max(abs(u-U))); W=w; U=u;