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&lt;\theta&lt;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
<br /> \frac{\partial f}{\partial t} = \frac 1r \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r}\right) - \frac{9f}{r^2}<br />
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}}.