No takers?

One can also try this problem with defining new variableq(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

Edit:

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).