Example 6.  Solve Laplace's equation over the square where the boundary conditions are  

     ,  ,  for  ,  
and  
     ,  for  .

Solution 6.

First set up the boundary values.
,  ,  for  ,  and  ,  for  .

Next, solve it.

Plot the solution.

We can make a contour plot of the solution.  However, to get the orientation like the table and the 3D figure requires reversing the rows in the matrix.

The boundary function is continuous for this problem and Mathematica has no trouble computing the contours.  

(c) John H. Mathews 2004