Since the boundary conditions are independent of or and , I might suggest switching the PDEs into polar coordinates and looking for solution in terms of the radius only.
If I had to solve this PDE with conditions (another question I have):
I would do it as follows:
Then applying all conditions and skipping a few steps the solution is
But I'm getting confused with the initial conditions here, and how to go from Eqn (1) onwards with the intitial conditions onwards i.e.
(a) for and for
(b) for and for
For the 3d case, I have
Which results in 3 ODE's
which have solutions
So, due to superposition,
which is all I have...I don't understand the form of the initial conditions.