I'd like to find a solution to the heat equation, with a variable diffusion coefficient that depends linearly on position:

$$ \frac{\partial C}{\partial t} = \frac{\partial }{\partial x}\left((ax + b) \frac{\partial C}{\partial x} \right).$$

Here, $a$ and $b$ are positive constants, and I'm interested in solutions for non-negative $x$, with boundary condition $\left. \frac{\partial C}{\partial x} \right|_{x=0} = 0$ (i.e., reflecting boundary), and initial condition $C(x,t=0) = \delta(x - x_0)$.

My motivation is that I want to find an analytical solution to use as a reference when evaluating numerical solutions. Inspired by a similar question, I've tried simplifying by saying that $a = 1$ and $b=0$, and then introducing a new variable either $U = Cx$ or $U = C/x$, but I didn't get anywhere with that.

Any help or hint or reference to literature is greatly appreciated.


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