I have two questions:

(1)As the tittle, if $\displaystyle u(a,\theta,t)=0$, is

$\displaystyle \frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}$

and

$\displaystyle \frac{\partial^2{u}}{\partial {t}^2}=\frac{\partial^2{u}}{\partial {r}^2} +\frac{1}{r} \frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}$

Just Poisson Equation $\displaystyle \nabla^2u=h(r,\theta,t)$ Where

$\displaystyle h(r,\theta,t)=\frac{\partial{u}}{\partial {t}}$

or $\displaystyle \;h(r,\theta,t)=\frac{\partial^2{u}}{\partial {t}^2}$ respectively.

(2)AND if if $\displaystyle u(a,\theta,t)=f(r,\theta,t)$, then we have to use superposition of Poisson with zero boundary plus Dirichlet with $\displaystyle u(a,\theta,t)=f(r,\theta,t)$?

That is

$\displaystyle u(r,\theta,t)=u_1+u_2$

where

$\displaystyle \nabla^2u_1=h(r,\theta,t)\;\hbox { with }\;u(a,\theta,t)=0$

and

$\displaystyle \nabla^2u_2=0\;\hbox { with }\;u(a,\theta,t)=f(r,\theta,t)$

Thanks