Whether to use Green's Function or Laplace transform to solve the following partial differential equation in 3 variable?

Question

We denote $$ f_x = \frac{\partial f}{\partial x} \quad \text{and similarly} \quad f_{xx} = \frac{\partial^2 f}{\partial x^2} $$ Let $T_a,a,b,c$ be constants.

How do we solve $$ T_{rr} + T_r/r + T_{zz} = T_t/\alpha $$

with boundary conditions:

1. $$k T_z(z=\delta) = \begin{cases} h(T_a–T(r,δ,t)), & rd \le r \le rp, T \ge t \ge 0\\ -q(r,δ,t), & rp \le r \le R, T \ge t \ge 0, q(r,δ,t)= ar(b+ct) \end{cases}$$
2. $kT_r(r= R) = h(T_a – T(R,z,t)), 0 ≤ z ≤ δ, T ≥ t ≥ 0$
3. $T_r(r=rd) = 0, 0 ≤ z ≤ δd , T ≥ t ≥ 0$
4. $T(r,z,0) = T_o , rd≤ r ≤ R , 0 ≤ z ≤ δ$

It is a heat conduction equation representing heat conduction during the time of braking when brake and friction pad are in contact for a time interval $T$.