Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method

Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function ψ (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker–Planck–Kolmogorov (FPK) equation. The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation by using the well‐known mesh‐free Lagrangian method, smoothed particle hydrodynamics). Numerical implementation shows notable advantages of this method in an unbounded state space: (1) the conservation of total probability in the state space is explicitly written; (2) no artifact is required to manage far‐field boundary conditions; (3) the positivity of the pdf is ensured; and (4) the extension to higher dimensions is straightforward. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution, just a precise representation of the initial distribution is required.Copyright © 2013 John Wiley & Sons, Ltd.