I'd like to solve the heat equation for a cylinder in 3D, in cylindrical coordinates with no azimuthal dependence. The equation is homogeneous but the bc at the cylinder wall has an arbitrary dependence on the axial coordinate. This bc does not depend on time.
Separation of variables is problematic because of the inhomogeneous bc. I have not been able to find a good reference for this particular problem, using google. That is all I have tried so far.
Advice on a numerical solution would be welcome.
What you can do is to find a basis of the solution to the heat equation without azimuthal dependence. Each solution will have its own (also azimuthal) boundary value. Now expand the given boundary data in terms of the boundary value of that basis. The desired solution will be the corresponding linear combination of the basis solution.
To obtain such a basis, you can do separation of variables. Indeed, let $u=f(r)g(z)$. Then
$$0=\frac{\Delta u}{u}=\frac{1}{f(r)}\left( \frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r} \right)f(r)+\frac{g''(z)}{g(z)}$$.
The two parts depend on $r$ and $z$ respectively, so they must both be constants. This implies $g$ is a sinusoidal function, and $f$ is the eigenfunction of the Laplacian on the disk, which can be written in terms of $J_0$, the Bessel functions of order 0.