Is wave and heat equation with zero boundary a Poisson Equation?

I have two questions:
(1)As the tittle, if \[u(a,\theta,t)=0\], is

\[\frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}\]

and

\[\frac{\partial^2{u}}{\partial {t}^2}=\frac{\partial^2{u}}{\partial {r}^2} +\frac{1}{r} \frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}\]

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

\[ h(r,\theta,t)=\frac{\partial{u}}{\partial {t}}\]

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


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

That is

\[ u(r,\theta,t)=u_1+u_2\]

where

\[\nabla^2u_1=h(r,\theta,t)\] with \[\;u(a,\theta,t)=0\]

and

\[\nabla^2u_2=0\] with \[u(a,\theta,t)=f(r,\theta,t)\]

Thanks

Alan,



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