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Igor A. Tanski

Publications and source records attributed to Igor A. Tanski.

16 recordsLinked to original sources
arXiv2009

Solution of linearized Fokker - Planck equation for incompressible fluid

In this work we construct algebraic equation for elements of spectrum of linearized Fokker - Planck differential operator for incompressible fluid. We calculate roots of this equation using simple numeric method. For all these roots real part is positive, that is corresponding solutions are damping. Eigenfunctions of linearized Fokker - Planck differential operator for incompressible fluid are expressed as linear combinations of eigenfunctions of usual Fokker - Planck differential operator. Poisson's equation for pressure is derived from incompressibility condition. It is stated, that the pressure could be totally eliminated from dynamics equations. The Cauchy problem setup and solution method is presented. The role of zero pressure solutions as eigenfunctions for confluent eigenvalues is emphasized.

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arXiv2009

Fokker - Planck equation for incompressible fluid

In this article we derive Fokker - Planck equation for incompressible fluid and investigate its properties. In version 2 symmetries of linearized equations and some examples of invariant solutions are added.

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arXiv2008

Fundamental solution of Fokker - Planck equation

Fundamental solution of Fokker - Planck equation is built by means of the Fourier transform method. The result is checked by direct calculation. Changes: missed factor in (29), (30), corrected (31), (35), removed former (36), added explanation on prolonged operators (45), (46), (47).

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arXiv2007

Spectral decomposition approach to macroscopic parameters of Fokker-Planck flows: Part 1

In this paper we proceed with investigation of connections between Fokker - Planck equation and continuum mechanics. In spectral decomposition of Fokker - Planck equation solution we preserve only terms with the smallest degree of damping. We find, that macroscopic parameters of Fokker-Planck flows, obtained in this way, have following properties: velocities field possess potential, its potential is proportional to density logarithm and satisfy diffusion equation. We proved, that such a pair of density and velocities field satisfy the set of classic hydrodynamics equations for isothermal compressible fluid with friction mass force, proportional to velocity. We proved also, that the potential velocities field alone, with potential, which satisfy diffusion equation, satisfy Burgers equation without mass forces.

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arXiv2007

Spectral decomposition approach to macroscopic parameters of Fokker-Planck flows: Part 2

In this paper we proceed with investigation of connections between Fokker - Planck equation and continuum mechanics. We base upon expressions from our work [2], based upon the spectral decomposition of Fokker - Planck equation solution. In this decomposition we preserve only terms with the smallest degrees of damping. We find, that macroscopic parameters of Fokker-Planck flows, obtained in this way, satisfy the set of conservation laws of classic hydrodynamics. The expression for stresses (30) contains additional term - this term is negligible in big times limit. We proved also, that the velocities field alone satisfy Burgers equation without mass forces - but with some additional term. This term is also negligible in big times limit. For the zero degree theory, considered in [1], there are no additional terms. But this theory is valid only for the potential velocities field, fully deductible from density - the potential is proportional to density logarithm. In this theory we can not specify initial conditions for velocities independently from density. Taking in account of the next degree terms could partly solve this problem, but result in some loss of exactness.

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arXiv2007

Spectral decomposition of 3D Fokker - Planck differential operator

We construct spectral decomposition of 3D Fokker - Planck differential operator in this paper. We use the decomposition to obtain solution of Cauchy problem - and especially the fundamental solution. Then we use the decomposition to calculate macroscopic parameters of Fokker - Planck flow.

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arXiv2006

Macroscopic parameters of Fokker-Planck flows

The aim of this work is to investigate properties of solutions of Fokker - Planck equation in the context of continuum mechanics. We show that average quantities, calculated for these solutions approximately satisfy equations of isothermal motion of viscous ideal gas.

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arXiv2005

Fokker - Planck equation in curvilinear coordinates. Part 2

The aim of this paper is to derive Fokker - Planck equation in curvilinear coordinates using physical argumentation. We get the same result, as in our previous article [1], but for broader class of arbitrary holonomic mechanical systems.

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arXiv2005

Fokker - Planck equation in curvilinear coordinates

The aim of this paper is to rewrite the Fokker - Planck equation according to transformation of space coordinates. This is nontrivial problem, because transformation of space coordinates induces transformation of velocities. We can use covariant, contravariant or physical velocity components as independent variables in curvilinear coordinate system. These 3 possibilities are considered in this paper and 3 kinds of Fokker - Planck equation in curvilinear coordinates are formulated.

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arXiv2004

The symmetries of the Fokker - Planck equation in one dimension

We calculate all point symmetries of the Fokker - Planck equation in one-dimensional Euclidean space. General expression of symmetry group action on arbitrary solution of Fokker - Planck equation is presented. We propose new notation for the group-theoretic analysis of PDE. The Lie prolongation formula is derived as an example of the new notation.

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