# Heat Equation on a Cylinder

Please note this is a homework question, and I just want some discussion on the choice of the separation parameter.

Suppose we have a cylinder which satisfies the following steady-state heat equation:

$$u_{rr}+\frac{1}{r}u_r+u_{zz}=0$$,

with the following boundary conditions:

\begin{align} u(r,0)&=0\\ u(r,20)&=20\\ [u_r+u]|_{r=4}&=0 \end{align}

Separation gives two very standard ODEs - the radial part gives a parametric Bessel equation of order $$0$$, and the other gives either a hyperbolic equation or a trigonometric equation. In other examples available to me it is stated that making $$Z(z)$$ be hyperbolic "makes sense in the context," with no explanation of what exactly is meant by that. I'm pretty sure the choice here depends entirely on how we expect $$Z(z)$$ to behave. If we expect the $$Z$$ part to be hyperbolic I have absolutely no idea why we expect it to be hyperbolic.

If one insists that the $$Z(z)$$ function is oscillatory, then $$R(r)$$ obeys the modified Bessel equation of order 0 instead which does have solutions $$I_0(\lambda x), K_0(\lambda x)$$. The second solution diverges at the origin so we can't use it to build a set of solutions. The problem with the first solution is that, unfortunately, it doesn't exhibit oscillatory behavior and in fact it is always positive. As a result the 3rd boundary condition cannot be satisfied or has a finite number of functions that can satisfy it, which in turn stiffens the boundary condition in the z-direction, in the sense that most boundary problems you come up with have no solution.