Consider the following simple Fokker-Planck equation: $$\partial_t f(x,t) = a \partial_x^2 f(x,t) $$ which holds on the intervals $x\in(0,c)$ and $(c,b)$. with $0<c<b$. $0$ and $b$ are absorbing boundaries, while $c$ is a continuity boundary. $f(x,t)$ is a pdf for $x$, so the boundary conditions for $t>0$ are: $$f(0,t)=0$$ $$f(b,t)=0$$ $$\int_0^b f(x,t) dx=1$$ and $f(x,t)$ must be continuous at $x=c$. Finally, there is an initial condition $h(x)$, so the last boundary condition is $f(x,0)=h(x)$.
This is modeling a system where probability exits at the $[0,b]$ boundaries and reenters at $c$. I would like to know if there is a general solution to this problem (or failing that, suggestions to find solutions for simple $h(x)$ initial conditions.) This is often impossible with PDEs, but two facts give me hope: the stationary distribution is easily solved by pen and paper, and without the integrating condition this reduces to a heat equation which does have a general solution.
Thanks!