# Partial DE - heated cylinder

## Homework Statement

A very long homogeneous cylinder is cut in half along its axis. One half s than equally heated, while the other half is equally cooled. How does the temperature change when the two parts are joined back together, if the cylinder is well isolated?

## The Attempt at a Solution

Hmmm,
I know I can work with $$T(r,\varphi ,t)=\sum _{m,n}J_{m,n}(\xi _{m,n}\frac r R)[B_m cos(m\varphi )+C_msin(m\varphi )]e^{-i\omega t}$$where ##\xi _{m,n}## is m-th zero of n-th Bessel function, but what I do not understand at this point is why all ##B_m## are zero?

Because if I am not mistaken, one boundary condition is $$j=-\lambda \frac{\partial }{\partial r}T(r=R,\varphi t)=0$$ and the other should be something at ##\varphi =0## or ##\varphi =\pi ##. I guess? I assume this second bondary condition will also tell me why the ##cos## is gone in the first equation.