Separation of variables in the Kramers equation

Zhdanov, Renat; Zhalij, Alexander

doi:10.1088/0305-4470/32/20/315

Cited by 4 publications

(3 citation statements)

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“…hold for all the cases 1-11 in (13). Solving (9) with respect to B j ( x), i = 1, 2, 3 we get (see, also [12])…”

Section: Conical Coordinate Systemmentioning

confidence: 99%

“…It has been successfully applied to solving variable separation problem in the wave [7] and Schrödinger equations [8]- [12] with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…Our analysis is based on the direct approach to variable separation in linear PDEs suggested in [7]- [9]. It has been successfully applied to solving variable separation problem in the wave [7] and Schrödinger equations [8]- [12] with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

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On separable Fokker-Planck equations with a constant diagonal diffusion matrix

Zhalij

1999

J. Phys. A: Math. Gen.

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We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients B 1 ( x), B 2 ( x), B 3 ( x) providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients B( x) must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed.

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“…hold for all the cases 1-11 in (13). Solving (9) with respect to B j ( x), i = 1, 2, 3 we get (see, also [12])…”

Section: Conical Coordinate Systemmentioning

confidence: 99%

“…It has been successfully applied to solving variable separation problem in the wave [7] and Schrödinger equations [8]- [12] with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

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On separable Fokker-Planck equations with a constant diagonal diffusion matrix

Zhalij

1999

J. Phys. A: Math. Gen.

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Symmetries and Separation of Variables

Rosenhaus,

Shankar,

Squellati

2024

J Nonlinear Math Phys

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In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.

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