How does one solve $$G_{t}(t,x) = \left(\frac{1-p}{2}(x^{k-1}-x)-\frac{1+p}{2}\right)G_{x}(t,0)+\frac{1-p}{2}(x+x^{k-1})G_{x}(t,1)+\left(\frac{1+p}{2}-x\right)G_{x}(t,x).$$ $G$ here is a generating function such that $G(t,x)=\sum_{i}[i](t)x^{i}=\sum_{i=1}^{k-1}[i](t)x^{i},$ where $[i](t)$ is the time dependant coefficient of $x^{i}$.
I have already found the steady state solution, which is $G(x,t)=0$, and was found using the physical definition of the $[i]$, which must all be $0$ in the steady state. Otherwise, I don't think the physical definition is useful, but is a bit complicated.
The issue I find with this PDE is that it explicitly involves what would be the boundary conditions, inside the PDE. I am not sure then what to do with these terms, or how to interpret them, despite this being an otherwise perfectly simple PDE. Can anyone help out, or at least help tell me what to call this so I can suitably google and find an explanation of how to do this?