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I am very confused. So I have Laplace's equation $\nabla^2\phi(x,y)=0$

and B.C.'s $\phi(x,0)=f(x); \,\,\,\,\,\, \phi(x,1)\equiv0$ where

I have to solve it by Fourier transform.

So I take the Fourier transform of the equation $-k^2\Phi(k,y)=0$ But I don't expect $\Phi$ to be identically $0$...?

Also, I have no idea how to proceed and it's driving me nuts. PLease help! Thanks.

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2 Answers

up vote 2 down vote accepted

If $\Phi(k,y)$ is the Fourier transform of $\phi(x,y)$ with respect to $x$, the equation $\nabla^2 \phi(x,y) = 0$ becomes $\displaystyle -k^2 \Phi(k,y) + \frac{\partial^2 \Phi}{\partial y^2}(k,y) = 0$. Treating $k$ as constant, you want to solve this as an ordinary differential equation with boundary conditions $\Phi(k,0) = F(k)$ and $\Phi(k,1) = 0$.

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Either you forgot to "transform the $y$-variable" or "treat the $y$-derivative".

Moreover, it is quite important how the domain for $(x,y)$ looks like. Probably $0\leq y\leq 1$ and $x\in\mathbb{R}$?

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Thanks, Dirk. Yes the domain is exactly that you suggested. I don't know how to do this at all... Could you provide a few steps to illustrate what I am supposed to do? Thanks. –  confused Nov 16 '11 at 16:35
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