Look at the example in section
2.6. This heat problem is solved there. The interesting thing is the convection condition at the right side.
The solution was found as

where the eigenvalues are solutions of the equation

For the parameter combination

the first five values are:
2.862772588
5.760557933
8.708313831
11.70267808
14.73347234
Here are animations of the solution for the time
t = 0 to 1 with parameters
a = 1,
k = 1 and
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.