Question on Heat problem in a disk

[yungman]

Homework Statement

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

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

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

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

I disagree.

Homework Equations

I think it should use the complete solution shown in (2a), then let t=0 where
$$u_{1}(r,\theta,0)=\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}J_{m}(\lambda_{mn}r)[a_{mn}\cos (m\theta)+b_{mn}\sin (m\theta)]$$I don't agree with the first part, you cannot assume $\frac{\partial u}{\partial t}=0$. Please explain to me.

Thanks

 

[yungman]

Anyone can help please? I just don't understand the solution manual use Dirichlet problem where $\frac{\partial u}{\partial t}=0$

 

[pasmith]

It seems obvious from the boundary condition that we will have $u = f(r,t)\sin 3\theta$. This gives
$$\frac{\partial f}{\partial t} = \frac 1r \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r}\right) - \frac{9f}{r^2}$$
subject to $f(1,t) = 1$ and $f(r,0) = 0$ (and the implied condition of finiteness at $r = 0$).

This immediately gives us a problem when we look for separable solutions: the condition which we would need to be zero (that at $r = 1$) so we can apply Sturm-Liouville theory to get an infinite strictly increasing sequence of real eigenvalues is not, and the condition which we would need to be a non-zero function of $r$ (that at $t = 0$) so we can work out the coefficients of the resulting eigenfunctions is not.

This prompts us to write $f(r,t) = f_1(r) + f_2(r,t)$ where $f_1(1) = 1$ and $f_2(1,t) = 0$ and $f_2(r,0) = -f_1(r)$, which is exactly the book's method!

Of course, with a little more thought we would have realized from the boundary conditions that $r^3 \sin 3\theta$ is the final steady-state solution, and we ought therefore to have worked with the variable $v = u - r^3 \sin 3\theta$ instead of $u$.

 

[yungman]

thanks for the reply. I don't understand, I got the separation of variable in the given condition and got the general solution show in (2a). All I have to do is to apply the boundary condition at t=0. Why change to another form or even use Sturm-Liouville theory?

I thought we do separation of variables and just apply boundary condition as shown in (2b).

 

[yungman]

I was thinking, do I 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 $\nabla^2u=h(r,\theta,t)$ where $$h(r,\theta,t)=\frac{\partial{u}}{\partial{t}}$$.