NuclearPhynance Archive

Semi-Analytic Solution to Fokker Planck Partial Differential Equations.



Here I have derived a semi-analytic solution to Fokker Planck PDEs. It is based on simple calculus and the solution does not borrow from stochastics. The semi-analytic solution is explicit and only requires integrations and does not require any implicit solution of finite-difference equations. Briefly, I found the time evolution of constant CDF curves which make up the solution to Fokker-Planck equation. For detailed equations and derivation please view post 811 here: https://lnkd.in/deA-sD7

Please bookmark the cited thread as I would be posting a worked out code for the solution of Fokker Planck equations with this logic in matlab in a few days.

Solution of Fokker Planck PDE using Hermite Polynomials.



We solve the Fokker-Planck PDE of X(t) by converting the derivatives of the density of X(t) into derivatives with respect to standard normal variable and hermite polynomials. Finally the normal density is cancelled throughout the equation and we are left with an evolution equation solely in terms of derivatives of X(t) with respect to standard normal and hermite polynomials. So we have effectively eliminated the density and we are now directly dealing with how the density is changing/scaling as a function of standard normal Z. There can be an infinite amount of analytics that can be done on this form of equation. This method is different from another method I proposed three days ago. For detailed equations and derivation please view post 812 here: https://lnkd.in/deA-sD7

Hannes Risken, The Fokker-Planck Equation, 2nd Ed, Springer, 1989. See section 4.4.1 for his use of Hermite Polynomials. It might be somewhat related to what you are doing, but I could easily be wrong. It's been a very long time since I've done anything material in this area, but hopefully it will give you some ideas.