This question is related to Sturm-Liouville Equation with Uncommon Weight Function, which I will explain later.
I am looking to solve the PDE
$$\frac{\partial c}{\partial t}+Ex\frac{\partial c}{\partial x}=\beta\frac{\partial^2c}{\partial x^2}$$
on the domain $(0,1)$, where $E$ and $\beta$ are positive constants. $c$ is subject to the conditions $c(t=0)=1$, $\frac{\partial c}{\partial x}=0$ at $x=0$, and $\frac{\partial c}{\partial x}=E\beta^{-1}c$ at $x=1$. I have attempted many different techniques and all have hit a metaphorical wall at some point. I will describe the techniques below and why they met with failure. I would greatly appreciate any tips on avenues I should pursue or corrections to the below methods. Thanks!
Separation of Variables / Fourier Series
My first attempt was to directly apply a separation of variables technique. However, this results in needing to solve the equation
$$\frac{d^2\phi}{dx^2}-E\beta^{-1}x\frac{d\phi}{dx}+\lambda\phi=0.$$
This is Hermite's equation, but the Hermite polynomials are not applicable here due to the finite domain and boundary conditions. As such, I have no way to obtain an analytic solution. Attempting a Fourier-series solution yields the question I have posted above, again with no apparent analytic solution.
Asymptotics at small $\beta$
For the cases I am interested in, $\beta\ll1$. Through a method of multiple time scales and introducing a scaled coordinate, I can derive the leading-order equation
$$\frac{\partial c}{\partial \tau}+E\frac{\partial c}{\partial \eta}=\frac{\partial^2 c}{\partial \eta^2}$$
on the domain $(0,\infty)$ with conditions $c(\tau=0)=1$, $\frac{\partial c}{\partial \eta}=-Ec$ at $\eta=0$, and $c\to1$ as $\eta\to\infty$.
From here, I have attempted a similarity solution (stymied by the Robin condition at $\eta=0$ not being self-similar), and a Fourier transform solution. The Fourier transform solution does not work due to the first-order derivative. The standard Fourier kernel introduces $c(\eta=0)$ into the equations, which depends on $\tau$, and it is not possible to find a kernel that doesn't (the solution to the auxiliary equation does not yield a valid kernel).
Again, I appreciate any suggestions anyone can give me. I feel that there should be an analytic solution to this (relatively) simple equation, but I can't seem to find a method that works. Perhaps there is a method I am not aware of.