Numerical evaluation of path-integral solutions to Fokker-Planck equations. III. Time and functionally dependent coefficients

Abstract

A path-integral solution to truly nonlinear Fokker-Planck equations is derived. Such equations exhibit in the drift and diffusion coefficients a functional dependence on the distribution function. This type of implicit time dependence is shown to introduce terms into the propagator function of the exact same order in the time step τ, as does an explicit time dependence if the functional dependence is sufficiently smooth. A standard discrete lattice formulation of the path integral is then used to reproduce the appropriate, truly nonlinear Fokker-Planck equation. This discrete formulation provides a basis for an efficient numerical algorithm and is applied with excellent results to several example problems where exact solutions can be calculated.