SHOW ALL STEPS AND CALCULATIONS$)$: Consider an infinite cylinder $0rR,-z$ with zero initial and surface temperature. There is heat production at the rate $Q$ per unit volume for $t0.$ The surface is maintained at zero temperature for $t0.$ Show that the temperature profile is given by:

$T(r,t)=\frac{Q}{4k}(R^2-r^2)-\frac{2Q}{kR}{\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{{}_n^3}{e}^{-{}_n^{2}t}\frac{J_0({}_{n}r)}{J_1({}_{n}R)}$

where $k$ is the thermal conductivity, $=\frac{k}{}{c}_p,$ and ${}_n$ are the roots of $J_0({}_{n}R)=0.$