Example 3.  Consider the heat equation where  .  The length of the rod is  .  Assume that the ends of the rod are held at the temperature  .  Assume that the initial temperature distribution is

        .
      
Apply the Crank-Nicolson method with    and obtain temperature distributions for  .  Compare the solution with the exact solution:

        .
     
We will use   .  This forces  This forces  .  

Solution 3.

Now set up the table of solutions.

Setting up the tri-diagonal matrx with n rows.  Indeed, we could get away with    rows, but the implementation is nice this way.  The following matrix will usually use  .

Next, solve it.

Compare with the analytic solution.

(c) John H. Mathews 2004