I'm trying to work through Polyanin's solution in Handbook of Linear Partial Differential Equations for Engineers and Scientists for the following PDE:
$$ \frac{\partial C}{\partial t} = \frac{1}{\delta^2(t)}\frac{\partial^2 C}{\partial x^2} + \frac{P(1-x)}{\delta(t)}\frac{\partial C}{\partial x} $$
subject to initial and boundary conditions: $$ \begin{align} C(x,0) &= c_0\\ C_x(0,t) &= 0\\ C_x(1,t) &= \delta(t)\phi_0 \end{align} $$
The PDE matches pattern 1.8.7.4 on pg. 147:
$$ \frac{\partial C}{\partial t} = n(t)\frac{\partial^2 C}{\partial x^2} + [xf(t) + g(t)]\frac{\partial C}{\partial x} $$
with
$$ n(t) = \frac{1}{\delta^2},\quad f(t) = -\frac{P}{\delta},\quad g(t) = \frac{P}{\delta} $$
The solution is given as a change of variables:
$$C(x,t) = \exp[x\alpha(t) + \beta(t)]u(z,\tau),\quad \tau=\varphi(t),\quad z = x\psi(t)+\chi (t) $$
where (bear with me):
$$ \begin{align} \psi &= C_1\exp\left(\int f\ dt\right)\\ \varphi &= \int n\psi^2\ dt + C_2\\ \alpha &= C_3 \psi\\ \beta &= \int(n\alpha^2 + g\alpha)\ dt + C_4\\ \chi &= \int(2n\alpha + g)\psi\ dt + C_5 \end{align} $$
This apparently reduces the original problem to
$$\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial z^2}$$
and we are told to check the solutions from section 1.1.1 for the solution; for example, on pg. 48 has the solution for Neumann BCs in terms of a given Green's function. The only problem is, what happens to the boundary conditions after this variable change? If $z = x\psi(t)+\chi (t)$, then aren't the boundaries moving in time, which renders the methods in this book obsolete? Maybe an actual boundary value problem can't be solved this way; how can I go directly from the original PDE to a solution with Green's functions? If I have to solve it numerically that's fine. I just wanted to check an analytical solution against a finite differences approach if possible.