This repository is a Matlab implemented solver for two dimensional Fokker-Planck equation using alternating direction implicit (ADI) method

1. Uncomment one of the parameters setting section in solve_2D_FPE.m:
   • heat diffusion
   • linear oscillator
   • bimodal oscillator
   • van der pol oscillator
2. Run solve_2D_FPE.m
3. Run movie_scpt.m to see a movie of the solution

Assume that we want to solve a 2D FPE like this:

[dx1; dx2] = [f1(x1,x2); f2(x1,x2)]dt + [D1; D2] dW(t)

where W(t) is a 1D wiener process, f1 and f2 defines the drifting force for x1 and x2 respectively. In the code, parameter x1_dt represents f1 and parameter x2_dt represent f2.

For x1_dt and x2_dt, each row defines a term in f1 and f2. Say f1 = 0.5*x1+x2, x1_dt will have two rows since f1 have two terms.

Each row should have three elements, representing coefficent, x1 dependency and x2 dependency. Say a row = [a, b, c], this row represents: a*(x1^b)*(x2^c)

As a result, for f1 = 0.5*x1+x2, x1_dt should be: [0.5, 1, 0; 1, 0, 1] (Same method for x2_dt)

• Parameter D = [D1; D2]
• Parameter dt = unit of time ; T_end = total time
• Parameter M1 = left and right boundary for x1 (x1 = [-M1, M1])
• Parameter M2 = left and right boundary for x2 (x2 = [-M2, M2])
• Parameter N1 = # of grids for x1
• Parameter N2 = # of grids for x2
• Parameter mu = mean of Gaussian (initial condition)
• Parameter sigma = covariance of Gaussian (initial condition)

Linear Oscillator:

Bimodal Oscillator:

Van der Pol Oscillator:

Pichler, Lukas, Arif Masud, and Lawrence A. Bergman. "Numerical solution of the Fokker–Planck equation by finite difference and finite element methods—a comparative study." Computational Methods in Stochastic Dynamics. Springer, Dordrecht, 2013. 69-85.