Example 1.  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 forward difference method with    and  obtain temperature distributions for  .
We will use   .  This forces  .

Solution 1.

Now set up the table of solutions.

Next, solve it.

To see the numerical values enter the command:

(c) John H. Mathews 2004