Is this integral equation solvable? If so, how?

Hello, I'm new to the forums (actually, just found you via google) and I have 2 questions (with the other I'll post in a separate thread a little later)!

My questions is this, can the following equation be solved for http://quicklatex.com/cache3/ql_ff59...52ee1b1_l3.png:

http://quicklatex.com/cache3/ql_0a99...31dd8de_l3.png

http://quicklatex.com/cache3/ql_70dc...2c8ed79_l3.png and t sufficiently large that transient/initial conditions don't matter.

We also have charge conservation:

http://quicklatex.com/cache3/ql_b531...6512464_l3.png

I tried approaching this by using a Fourier transform type solution

http://quicklatex.com/cache3/ql_d4b7...b562dc2_l3.png

but when I pursue this I seem to get multiple exponential integrals and things don't seem to simplify.

Any ideas? Are there additional boundary conditions I need to impose? I have almost no experience with integral equations so all suggestions welcome!