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

so that

(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

This leaves

(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.