In Section 2.11, this problem was solved as an example:
The solution was found in the form of a Fourier integral,
Here are animations of the solution with parameters
a = 1 and
k = 1 for the time
t = 0 to 2.
On the left is an animation with equally spaced times from
t = 0 to 2.
On the right, the animation uses logarithmically spaced times. This
makes it easier to see how the solution changes at times near 0.