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Homework Help: Neumann Problem

  1. Jan 6, 2010 #1

    Hootenanny

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    1. The problem statement, all variables and given/known data
    The overall question is to construct an asymptotic approximation for a harmonic field on a rectangle with a small disk. However, I'm having difficulty finding the unperturbed field. Perhaps I've been staring at it for too long, but I can't seem to find a solution.

    The unperturbed field satisfies the following Neumann problem:

    [tex]\Delta u =0 \;\;\;\;\;\;\text{in}\;\;\;\;\;\;\Omega = \left\{\left(x_1,x_2\right)\; : \; \left|x_1\right| < 2 \;,\; \left|x_2\right| < 1 \right\}\;,[/tex]

    [tex]\left.\frac{\partial u}{\partial x_2}\right|_{x_1=\pm2} = 1 \;,[/tex]

    [tex]\left.\frac{\partial u}{\partial x_1}\right|_{x_2=\pm1} = 2 \;,[/tex]​

    2. Relevant equations

    N/A

    3. The attempt at a solution
    The usual method to solving such problems is to make an educated guess, however, I'm having some problems guessing the solution. Clearly u(x) = Const. is not a solution. My first thought was that u(x) must either be a linear combination of x1 and x2, or a linear combination of x1.x2. However, as far as I can see, none of these functions can satisfy either the top and bottom or left and right boundary conditions simultaneously.

    A nudge towards the correct 'guess', or any other help would be very much appreciated.

    Edit: I just thought that I'd add that a solution does exist since the existence condition is clearly satisfied,

    [tex]\oint_{\partial\Omega}\frac{\partial u}{\partial n}dS = 0[/tex]
     
    Last edited: Jan 6, 2010
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  3. Jan 6, 2010 #2

    Dick

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    u(x1,x2)=2*x1+x2?? Am I missing something?
     
  4. Jan 6, 2010 #3

    Hootenanny

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    Gahh! I knew that it was going to be simple and I was just being dense!

    #Bangs head against desk#

    Many thanks Dick!
     
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