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Question on Heat problem in a disk

  1. Nov 30, 2013 #1
    1. The problem statement, all variables and given/known data
    This is a question in the book to solve Heat Problem
    [tex]\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}[/tex]

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

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

    165824[/ATTACH]"] 6rlsoj.jpg

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

    I disagree.

    2. Relevant equations

    I think it should use the complete solution shown in (2a), then let t=0 where
    [tex]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)][/tex]


    I don't agree with the first part, you cannot assume [itex]\frac{\partial u}{\partial t}=0[/itex]. Please explain to me.

    Thanks
     

    Attached Files:

  2. jcsd
  3. Dec 2, 2013 #2
    Anyone can help please? I just don't understand the solution manual use Dirichlet problem where [itex]\frac{\partial u}{\partial t}=0[/itex]
     
  4. Dec 2, 2013 #3

    pasmith

    User Avatar
    Homework Helper

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

    This immediately gives us a problem when we look for separable solutions: the condition which we would need to be zero (that at [itex]r = 1[/itex]) 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 [itex]r[/itex] (that at [itex]t = 0[/itex]) so we can work out the coefficients of the resulting eigenfunctions is not.

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

    Of course, with a little more thought we would have realized from the boundary conditions that [itex]r^3 \sin 3\theta[/itex] is the final steady-state solution, and we ought therefore to have worked with the variable [itex]v = u - r^3 \sin 3\theta[/itex] instead of [itex]u[/itex].
     
  5. Dec 2, 2013 #4
    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).
     
  6. Dec 2, 2013 #5
    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 [itex]\nabla^2u=h(r,\theta,t)[/itex] where [tex]h(r,\theta,t)=\frac{\partial{u}}{\partial{t}}[/tex].
     
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