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I want to solve non-classic diffusion equation in Fourier space. The equation is $$∂c/∂t=-∇.J$$ Where $J$ is $$J= -M.∇μ$$ Where M is mobility. It depends on c and $\mu$ is $$ μ= g(c) - \nabla^2c $$ Now i am trying to solve above equation $$∂c/∂t= -∇.(-M.∇μ)$$ $$ =∇.[M.(∇(g(c)-∇^2 c))]$$ Now how to do Fourier transform and what is real and imaginary part of this.

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  • $\begingroup$ What does $\delta c$ stands for? $\endgroup$ – nicoguaro Dec 21 '17 at 16:37
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    $\begingroup$ Does $M$ depend on space and/or time? What does the function $g$ look like? Where did $k'$ come from? You need to clean up your presentation of the problem to get a useful answer. $\endgroup$ – Endulum Dec 21 '17 at 18:03
  • $\begingroup$ $M$ depends on c. $\endgroup$ – RAHUL KUMAR SAINI Dec 21 '17 at 20:05
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It's quite normal to have nonlinearities in equations and still want to use a transformation based method. The way that this is commonly done is to backtransform to calculate the nonlinear parts, i.e. if the equation is solved in phase space, evaluate $g(c)$ as $\mathcal{F}[g(\mathcal{F}^{-1}[c])]$. This paper displays this in section 4 and shows code like

g.*fft(real(ifft(a)).^2);

You can also use something like ApproxFun.jl to calculate approximations of your nonlinear operators in your function space.

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This is a standard problem; in general, you can't solve it purely in Fourier Space (i.e. the representation of the problem is not "diagonal" in the Fourier space basis). However, if you want to get numerical solutions, you can use a pseudospectral method. One very relevant paper, which should give you enough ideas to implement something like this, is:

"Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method" Zhu et al. Phys. Rev. E 1999 https://doi.org/10.1103/PhysRevE.60.3564 (there are also PDF copies floating around the net).

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