% Program 8.5 Finite difference solver for 2D Poisson equation 
%   with Dirichlet boundary conditions on a rectangle 
% Input: rectangle domain [xl,xr]x[yb,yt] with MxN space steps 
% Output: matrix w holding solution values 
% Example usage: w=poisson(0,1,1,2,4,4) 
function w=poisson(xl,xr,yb,yt,M,N) 
f=@(x,y) 0; % define input function data 
g1=@(x) log(x.^2+1); % define boundary values 
g2=@(x) log(x.^2+4); % Example 8.8 is shown 
g3=@(y) 2*log(y);
g4=@(y) log(y.^2+1); 
m=M+1;n=N+1; mn=m*n; 
h=(xr-xl)/M;h2=h^2;k=(yt-yb)/N;k2=k^2; 
x=xl+(0:M)*h; % set mesh values 
y=yb+(0:N)*k; 
A=zeros(mn,mn);b=zeros(mn,1); 
for i=2:m-1 % interior points 
  for j=2:n-1 
    A(i+(j-1)*m,i-1+(j-1)*m)=1/h2;A(i+(j-1)*m,i+1+(j-1)*m)=1/h2; 
    A(i+(j-1)*m,i+(j-1)*m)=-2/h2-2/k2; 
    A(i+(j-1)*m,i+(j-2)*m)=1/k2;A(i+(j-1)*m,i+j*m)=1/k2; 
    b(i+(j-1)*m)=f(x(i),y(j)); 
  end 
end 
for i=1:m % bottom and top boundary points 
  j=1;A(i+(j-1)*m,i+(j-1)*m)=1;b(i+(j-1)*m)=g1(x(i)); 
  j=n;A(i+(j-1)*m,i+(j-1)*m)=1;b(i+(j-1)*m)=g2(x(i)); 
end 
for j=2:n-1 % left and right boundary points 
  i=1;A(i+(j-1)*m,i+(j-1)*m)=1;b(i+(j-1)*m)=g3(y(j)); 
  i=m;A(i+(j-1)*m,i+(j-1)*m)=1;b(i+(j-1)*m)=g4(y(j)); 
end 
v=A\b; % solve for solution in v labeling 
w=reshape(v(1:mn),m,n); %translate from v to w 
surf(x,y,w') 