(adsbygoogle = window.adsbygoogle || []).push({}); 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 aneducated guess, however, I'm having some problems guessing the solution. Clearlyis not a solution. My first thought was thatu(x) = Const.must either be a linear combination ofu(x)andx_{1}, or a linear combination ofx_{2}. 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.x_{1}.x_{2}

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]

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# Homework Help: Neumann Problem

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