I've been trying to solve the diffusion equation on a circle. The problem I am running into is that because of the periodic boundary, the wavevector k (when you Fourier transform) gets quantized stopping further analysis as is done in the regular diffusion equation solution on the real line. Is there a way to analytically solve the time dependent diffusion equation on a circle?
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1$\begingroup$ You mention "further analysis as is done in the regular diffusion equation solution on the real line". Can you add how this would be done to your post? And where the quantized values of $k$ causes problems? $\endgroup$– Marius Ladegård MeyerCommented Apr 30, 2020 at 6:29
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$\begingroup$ It's an absolute normal thing that wavevectors $k$ get discrete when working on a compact space, so don't get discouraged by it. What further analysis do you have in mind? $\endgroup$– Johnny LongsomCommented Apr 30, 2020 at 7:46
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$\begingroup$ To solve it on the real line, we Fourier transform the equation so it becomes a first order ODE in time for the Fourier transform, which we solve easily and then inverse Fourier transform to get the result.The inverse Fourier transform involves a gaussian integral in k space. In the discrete case on the ring, this part of the calculation doesn't work - so how do we recover the real space function $\endgroup$– Mikail KhonaCommented Apr 30, 2020 at 15:44
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