Consider the following initial boundary value problem with the region of
interest being inside a right circular cylinder of radius $a$ and length $L$,
$\nabla^2 \psi(\rho, \theta, z, t) - \frac{1}{D_0} \frac{\partial \psi(\rho, \theta, z, t)}{\partial t} = 0$
$\psi(\rho, \theta, z = 0, t) = 0$
$\psi(\rho, \theta, z = L, t) = 0$
$\psi(\rho = a, \theta, z, t) = G(\theta, z)$
$\psi(\rho, \theta, z, t = 0) = H(\rho, \theta, z)$
1. Find a closed form unique solution of the above IBVP in terms of $G$ and
$H$ (20 pts),
2. Find the unique solution for $G$ an $H$ below (5 pts)
$G(\rho, \theta) = \frac{1}{\rho} \delta(\rho)$, $H(\rho, \theta, z) = 0$
3. Find the unique solution for $G$ an $H$ below (5 pts)
$G(\rho, \theta) = 0$, $H(\rho, \theta, z) = \frac{1}{\rho} \delta(\rho)$.