Solving Steady Fokker-Planck Equation in 2D using standard software

[sokrates]

Hello all,

I am trying to solve the FPE in 2D,

The equation is (with implied summation over repeated indices):

$0 = \left(- \displaystyle \frac{d}{dx_i} D_i^{(1)} + \frac{d}{dx_i}\frac{d}{dx_j} D_{ij}^{(2)}\right) ( n )$where my diffusion coefficients (the D are functions of position; i.e they may not be constants. I don't have them analytically, but I have another microscopic model that calculates these diffusion coefficients and gives them to me in a discrete array. These are smooth functions in general, therefore I can interpolate them if I need to give "continuous" descriptions of them.

I am trying to crack pdetool Box from MATLAB; but I am having trouble whether the FPE fits into an "elliptic" function category; or would I ever be able to put Drift and Diffusion coefficients that are changing in position?

I attend a public university, and I have access to standard software such as Mathematica, Matlab, Maple, Comsol or any other that comes to mind. Does anyone have experience with a problem like this?

Any help would be greatly appreciated.

Sokrates.

 

[jvicens]

Hello Sokrates,

Thank you for sharing your problem with us. The equation you are trying to solve is known as the Fokker-Planck equation (FPE) and it is commonly used in statistical mechanics and diffusion processes. It is a type of partial differential equation (PDE) and can be solved using numerical methods.

In order to solve the FPE in 2D, you will need to discretize the equation and then solve it using software such as Mathematica, Matlab, or Comsol. The FPE can be classified as an elliptic function, and you can use the pdetool Box from Matlab to solve it. However, you will need to specify the diffusion and drift coefficients as functions of position, as they are not constant in this case.

I would suggest using the interpolation method to obtain a continuous description of the diffusion coefficients from your discrete array. You can then input these functions into the pdetool Box and solve the equation numerically.

If you are having trouble with the pdetool Box, you can also try using other software such as Comsol or Maple. These software packages also have tools for solving PDEs and may be more suitable for your problem.

I hope this helps and please let me know if you have any further questions. Good luck with your research!