Partial differential equation containing the Inverse Laplacian Operator

[Celeritas]

TL;DR Summary

I want to know how can i solve a partial differential equation containing an inverse laplacian operator numerically in fourier space.

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$
where $\phi,g,f$ are fields in 2D, $\nabla_x=\frac{\partial}{\partial{x}}$ and $\nabla^2$ is the laplacian operator. The author mentions that this is solved numerically in Fourier space, by using for example the laplacian in Fourier space
$${\nabla_k}^2=\frac{cos(k_x\Delta{x})cos(k_x\Delta{y})+cos(k_x\Delta{x})+cos(k_y\Delta{y})-3}{\Delta{x}^2}$$
but nothing else is mentioned. My first question is how is the initial equation transformed into Fourier space in the first place? I don't see how the typical "multiply by exponential ikr and integrate over r" would work here.
My second question is about the Laplacian in Fourier space. Where did he get that from? Is it something common that can be found in some book?

Thank you.

 

[wrobel]

This thesis, along with a couple of papers (I can cite them if you wish), have used this technique to successfully explain experiments

would you please to cite a journal in that these results were published

1) To speak about inverse Laplace is senseless without specification of boundary conditions and functional space
2) in general $\Delta^{-1}\Delta\ne \Delta\Delta^{-1}$ and $\Delta^{-1}\Delta\ne \mathrm{id}$ it is so for the Dirichlet boundary conditions for example; so that the notation $\frac{\nabla^2\nabla^2}{\nabla^2}$ does not make sense

 

[Celeritas]

It was originally used by Onuki in 1989:
Onuki, Akira. "Long-range interactions through elastic fields in phase-separating solids." Journal of the Physical Society of Japan 58.9 (1989): 3069-3072.
Later on adopted by many authors:
Ohta, T. "Interface dynamics under the elastic field." Journal of Physics: Condensed Matter 2.48 (1990): 9685.
Müller, Judith, and Martin Grant. "Model of surface instabilities induced by stress." Physical Review Letters 82.8 (1999): 1736.

The boundary conditions depend on the physics. For example the last paper has used periodic boundary conditions in x and y. Anyway that's not the point of my question.
Given that you have that equation and all the pesky mathematical details work out, how would you transform it into Fourier space?
I agree with your second point that the order of it is important, but again that is not the point. From what I understood, if they are just replacing $\nabla^2$ with its Fourier space equivalent, and similar terms for $\nabla_x^2$,$\nabla_y^2$, then you will end up with just a normal ratio of functions of k, but I am not sure.

 

[wrobel]

regarding the initial post I could perhaps say something if the problem would be stated in the way accepted in math