# Stability of time dependent PDE

Recently in another post I asked a question regarding typing the following PDE to solve in Mathematica.

I have the following PDE:

$$\qquad \frac{\partial P(x_1,x_2,t)}{\partial t} = -\frac{\partial}{\partial x_1}[F_1(x_1,x_2)P] - \frac{\partial}{\partial x_2}[F_2(x_1,x_2)P] + D(\frac{\partial^2 P}{\partial x_1^2} + \frac{\partial^2 P}{\partial x_2^2} )$$

where

$$\qquad F_1 = \frac{\epsilon^2 + x_1^2}{(1+x_1^2)(1+x_2)}-ax_1$$ and $$F_2 = \frac{1}{\tau_0}(b-\frac{x_2}{1+cx_1^2})$$.

The values of parameters $$\epsilon, a, b, c, \tau_0, D$$ are 0.1, 0.1, 0.1, 100, 5.0, 0.001 respectively.

Thanks to helpful comments, especially by Andrew, I can let Mathematica solve the equation. However, I observe instability issue because probability becomes negative at some regions when time evolves.

There is a book chapter summarizing methods to solve the type of aforementioned PDE (which is Fokker-Planck equation).

Hence, I would like to ask how we can:

1. Code in Mathematica to overcome the issue or
2. Translate methods in the chapter into Mathematica code or MATLAB code to solve equation
• There are no initial and boundary conditions. What are we going to discuss - a chapter from a book? – Alex Trounev Jan 8 at 0:32
• Please look at another link (hyperlink here) for how the code is implemented in Mathematica. When I ran the code, I noticed stability issue and now would like to know how to fix the issue. Incase we can't fix the issue, the book chapter suggests other approaches which maybe helpful. – canhochoi Jan 8 at 6:45
• This is a common problem for all numerical methods. We can solve this problem in a particular case, but we cannot solve it in the general case. – Alex Trounev Jan 8 at 12:02