One can also try this problem with defining new variable q(x,t) such that
and choosing q(0)=q(a)=0.
Then the equation above can be written as
Then perhaps we can assume solution of the form
and so the equation becomes
Of course, I'm not sure how to evaluate
Mathematica gives that
But I don't see how I can get from that... as the expression becomes the incredibly ugly sum:
Somehow I would suspect that the terms with x have to all go away (or my guess of solution is incorrect, but exponential should for complete set and the boundary conditions are well defined so it should be a sum). Another thing is to try expansion in terms of bessel functions instead of exponential...
Also, perhaps summation has to be not from n=1 but from n=- infinity... but q_0 = 0, for certain (no constant component).