Consider an infinite cylinder $0ra,-z$ with zero initial and surface temperature. There is a heat production within the cylinder at the rate of $Ct$ per unit time per unit volume for $t0,$ where $C$ is a constant. The surface of the cylinder dissipates heat by convection to a surrounding fluid at zero temperature and finite convection heat transfer coefficient $h$ for $t0.$ Show that the temperature profile is given by:

$T(r,t)={\displaystyle \underset{s=1}{\overset{}{}}}\frac{J_0({}_{n}r)}{[{}_n^2+(\frac{h}{k}{)}^2]}\frac{J_0({}_{n}R)+J_2({}_{n}R)}{[J_0({}_{n}R){]}^2}[\frac{Ct}{k}-\frac{C}{k}\frac{1}{{}_n^2}(1-e^{-{}_n^2})]$

where $k$ is the thermal conductivity, $=\frac{k}{}{c}_p,$ and ${}_n$ are the roots of ${}_n{J}_1({}_{n}a)=\frac{h}{k}{J}_0({}_{n}a),n=1,2,3,$dots,$$