# How to solve diffusion equation in Fourier space when mobility is not constant

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.

• What does $\delta c$ stands for? – nicoguaro Dec 21 '17 at 16:37
• 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. – Endulum Dec 21 '17 at 18:03
• $M$ depends on c. – RAHUL KUMAR SAINI Dec 21 '17 at 20:05

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);