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I am using the FFT method to solve the two dimensional PDE on a $2\pi\times 2\pi$ square. I have boundary conditions that $u(x,0,t)=0$ and $u(x,2\pi,t)=0$. How do I go about setting these boundaries? I can't just expand in different basis functions because of the way my fft function works.

Thanks in advance.

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Sounds like a job for the discrete sine transform. Any reference on FFT should include something on how to get a canned FFT routine to do DST. Alternatively, you might want to look at FFTW. – J. M. Oct 29 '10 at 15:04

If you don't mind wasting some resources, extend the $y$ coordinate to the interval $[0,4\pi]$. Then only look at solutions that are odd (anti-symmetric) under the flip $y\mapsto 4\pi-y$. In particular, if your PDE has source terms, they have to be anti-symmetrically mirrored from the $[0,2\pi]$ to the $[2\pi,4\pi]$ half of the extended interval. When restricted to $y\in[0,2\pi]$ a periodic solution on the extended interval will satisfy the desired boundary conditions.

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This is essentially what a discrete sine transform does, only with less waste of space due to the exploitation of the antisymmetry. – J. M. Oct 31 '10 at 10:32

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