# Thread: Question on Heat problem in a disk

1. ## Question on Heat problem in a disk

This is a question in the book to solve Heat Problem
$\displaystyle \frac{\partial \;u}{\partial\; t}=\frac{\partial^2 u}{\partial\; r^2}+\frac{1}{r}\frac{\partial\; u}{\partial\;r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}$

With 0<r<1, $\displaystyle 0<\theta<2\pi$, t>0. And $\displaystyle u(1,\theta,t)=\sin(3\theta),\;u(r,\theta,0)=0$

The solution manual gave this which I don't agree:

What the solution manual did is for $\displaystyle u_1$, it has to assume $\displaystyle \frac{\partial \;u}{\partial\; t}=0$ in order using Dirichlet problem to get (1a) shown in the scanned note.

I disagree.

I think it should use the complete solution shown in (2a), then let t=0 where
$\displaystyle u_{1}(r,\theta,0)=\sum_{m=0}^{\infty}\sum_{n=1}^{\ infty}J_{m}(\lambda_{mn}r)[a_{mn}\cos (m\theta)+b_{mn}\sin (m\theta)]$

I don't agree with the first part as I don't think you can assume $\displaystyle \frac{\partial u}{\partial t}=0$. Please explain to me.

Thanks

2. ## Re: Question on Heat problem in a disk

I don't think you can assume

Well, you don't actually 'assume' it. You decompose the solution into two parts: the plainly homogeneous part (u_1) and what's left of the original function (u_2).
Thankfully (great job, superposition) it works.

3. ## Re: Question on Heat problem in a disk

Originally Posted by Rebesques
Well, you don't actually 'assume' it. You decompose the solution into two parts: the plainly homogeneous part (u_1) and what's left of the original function (u_2).
Thankfully (great job, superposition) it works.
Do you mean you treat this as just a Poisson problem with non zero boundary? That you decomposes into a Poisson problem with zero boundary PLUS a Dirichlet problem with non zero boundary?

That you just treat this as Poisson problem $\displaystyle \nabla^2u=h(r,\theta,t)$ where $\displaystyle h(r,\theta,t)=\frac{\partial{u}}{\partial{t}}$.