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

1. ## 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 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

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

and

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

Thanks

2. ## Re: Is wave and heat equation with zero boundary a Poisson Equation?

Originally Posted by Alan0354
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

$h(r,\theta,t)=\frac{\partial{u}}{\partial {t}}$
No, this is NOT a Poisson equation because h depends upon the derivative of u with respect to t, not just t.

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

(2)AND if 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

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

and

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

Thanks

3. ## Re: Is wave and heat equation with zero boundary a Poisson Equation?

Thanks,

But I just notice the methodology of solving Heat and Wave equation with non zero boundaries is exactly the same as solving Poisson with non zero boundary. They both use superposition that sum:

1) Dirichlet problem with non zero boundary by assuming $\frac{\partial{u}}{\partial {t}}=0$ and use non zero boundary.

2) Using $\frac{\partial{u_{2}}}{\partial {t}}=\frac{\partial^2{u_{2}}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u_{2}}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u_{2}}}{\partia l {\theta}^2}$ with zero boundary.