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Math Help - Question on Heat problem in a disk

  1. #1
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    Question on Heat problem in a disk

    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:



    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.

    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 as I don't think you can assume \frac{\partial u}{\partial t}=0. Please explain to me.

    Thanks
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  2. #2
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    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.
    Thanks from Alan0354
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    Re: Question on Heat problem in a disk

    Quote Originally Posted by Rebesques View Post
    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 \nabla^2u=h(r,\theta,t) where h(r,\theta,t)=\frac{\partial{u}}{\partial{t}}.
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