Distributed approximating functional approach to the Fokker–Planck equation: Time propagation

The Fokker–Planck equation is solved by the method of distributed approximating functionals via forward time propagation. Numerical schemes involving higher-order terms in Δt are discussed for the time discretization. Three typical examples (a Wiener process, an Ornstein–Uhlenbeck process, and a bistable diffusion model) are used to test the accuracy and reliability of the present approach, which provides solutions that are accurate up to ten significant figures while using a small number of grid points and a reasonably large time increment. Two sets of solutions for the bistable system, one computed using the eigenfunction expansion of a preceding paper and the other using the present time-dependent treatment, agree to no fewer than five significant figures. It is found that the distributed approximating functional method, while simple in its implementation, yields the most accurate numerical solutions yet available for the Fokker–Planck equation.