In Section 2.5, the following heat problem is solved
with solution

where
Here are animations of the solution with parameters
a = 1 and
k = 1 for the time
t = 0 to 1.
On the left is an animation with equally spaced times from
t = 0 to 1.
On the right, the animation uses logarithmically spaced times. This
makes it easier to see how the solution changes at times near 0.
Notice that the
slope at the right end is always 0.