Результаты поиска по 'nonlinear Fokker−Planck−Kolmogorov equation':
Найдено статей: 2
  1. Rezaev R.O., Trifonov A.Y., Shapovalov A.V.
    The Einstein−Ehrenfest system of (0, M)-type and asymptotical solutions of the multidimensional nonlinear Fokker−Planck−Kolmogorov equation
    Computer Research and Modeling, 2010, v. 2, no. 2, pp. 151-160

    Semiclassical approximation formalism is developed for the multidimensional Fokker–Planck–Kolmogorov equation with non-local and nonlinear drift vector with respect to a small diffusion coefficient D, D→0, in the class of trajectory concentrated functions. The Einstein−Ehrenfest system of (0, M)-type is obtained. A family of semiclassical solutions localized around a point driven by the Einstein−Ehrenfest system accurate to O(D(M+1)/2) is found.

    Keywords: nonlinear Fokker−Planck−Kolmogorov equation, semiclassical asymptotics, WKB-Maslov method, Einstein−Ehrenfest system.
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  2. Bogomolov S.V.
    Stochastic formalization of the gas dynamic hierarchy
    Computer Research and Modeling, 2022, v. 14, no. 4, pp. 767-779

    Mathematical models of gas dynamics and its computational industry, in our opinion, are far from perfect. We will look at this problem from the point of view of a clear probabilistic micro-model of a gas from hard spheres, relying on both the theory of random processes and the classical kinetic theory in terms of densities of distribution functions in phase space, namely, we will first construct a system of nonlinear stochastic differential equations (SDE), and then a generalized random and nonrandom integro-differential Boltzmann equation taking into account correlations and fluctuations. The key feature of the initial model is the random nature of the intensity of the jump measure and its dependence on the process itself.

    Briefly recall the transition to increasingly coarse meso-macro approximations in accordance with a decrease in the dimensionalization parameter, the Knudsen number. We obtain stochastic and non-random equations, first in phase space (meso-model in terms of the Wiener — measure SDE and the Kolmogorov – Fokker – Planck equations), and then — in coordinate space (macro-equations that differ from the Navier – Stokes system of equations and quasi-gas dynamics systems). The main difference of this derivation is a more accurate averaging by velocity due to the analytical solution of stochastic differential equations with respect to the Wiener measure, in the form of which an intermediate meso-model in phase space is presented. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. The emphasis is placed on the transparency of assumptions during the transition from one level of detail to another, and not on numerical experiments, which contain additional approximation errors.

    The theoretical power of the microscopic representation of macroscopic phenomena is also important as an ideological support for particle methods alternative to difference and finite element methods.

    Keywords: Boltzmann equation, Kolmogorov – Fokker – Planck equation, Navier – Stokes equations, equations of stochastic gas dynamics and quasi-gas dynamics, stochastic differential equations with respect to Bernulli and Wiener measures, particle methods.

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