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

Authors

  • T. Canor,

    Corresponding author
    1. F.R.S.-FNRS, National Fund for Scientific Research, 1000 Bruxelles, Belgium
    • Department of Architecture, Geology, Environment and Constructions, Structural Engineering Division, University of Liege, 4000 Liege, Belgium
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  • V. Denoël

    1. Department of Architecture, Geology, Environment and Constructions, Structural Engineering Division, University of Liege, 4000 Liege, Belgium
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Correspondence to: T. Canor, Department of Architecture, Geology, Environment and Constructions, Structural Engineering Division, University of Liege, Chemin des Chevreuils 1, B52/3, 4000 Liege, Belgium.

E-mail: t.canor@ulg.ac.be

SUMMARY

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.

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