Maybe I can help. What you have is the following problem to solve:
You have two difficulties here. One is the boundary condition and two, the source term .
The idea here is: Is is possible to introduce a change of variable
(1) the new BC becomes ? and
(2) the new PDE becomes
Normally we can just achieve (1) but sometimes (as in this case) we can get (2) as well. Let's go after (2) first. When we sub. into the PDE we get
Now we choose
(one of the desired results).
Integrating (*) gives
So at this point we have
Now we go after (1)
(a) u(0,t) = 0 and v(0,t) = 0. From (**), we obtain .
(b) u(L,t) = 1 and v(L,t) = 0. From (**) we obtain
At this point we have the following.
The only thing left is to find out what the new IC is for the problem. Here you'll need to use (***). I'll leave this to you.