Lie Symmetries and Fundamental Solutions of the Linear Kramers Equation

Валерій Іванович Стогній, Інна Миколаївна Kопась, Cергій Сергійович Коваленко

Abstract


Background. The group-theoretical analysis of fundamental solutions of the one-dimensional linear Kramers equation was carried out in the article.

Objective. The aim of the paper is to find the algebra of invariance of fundamental solutions of the equation under study using the Aksenov–Berest approach, and construct a fundamental solution of the one in the explicit form taking into account the algebra of Lie symmetries to be found.

Methods. The group-theoretical methods of analysis of partial differential equations are used. In particular, the Aksenov–Berest method of constructing in explicit form of fundamental solutions of linear partial differential equations is applied.

Results. The Lie algebra of non-trivial symmetries of the one-dimensional linear Kramers equation under consider was found. The fundamental solution in the explicit form of the equation was constructed. The effectiveness of using of symmetry methods in investigating of fundamental solutions of linear Kolmogorov–Fokker–Planck equations was shown.

Conclusions. Using the Aksenov–Berest approach, the algebra of invariance of fundamental solutions of one one-dimensional linear Kramers equation was found. The operators of the algebra were used in the process of constructing of invariant fundamental solutions of the equation. It was shown that the fundamental solution found early by S. Chandrasekhar without using the methods of symmetry analysis of differential equations is the weak invariant fundamental solution.


Keywords


Linear Kramers equation; Fundamental solution; Lie symmetries

References


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W.M. Shtelen and V.I. Stogny, “Symmetry properties of one- and two-dimensional Fokker–Planck equations”, J. Rhys. A: Math. Gen., vol. 22, no. 13, pp. L539–L543, 1989. doi:10.1088/0305-4470/22/13/002

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Yu.Yu. Berest, “Group analysis of linear differential equations in distributions and the construction of fundamental solutions”, Differencial'nye Uravnenija, vol. 29, no. 11, pp. 1598–1970, 1993 (in Russian).

S. Chandrasekhar, “Stochastic problems in physics and astronomy”, Rev. Mod. Phys., vol. 15, no. 1, pp. 1−89, 1943. doi:10.1103/RevModPhys.15.1

Yu.Yu. Berest, “Weak invariants of local transformation groups”, Differencial'nye Uravnenija, vol. 29, no. 10, pp. 1796–1803, 1993 (in Russian).


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DOI: https://doi.org/10.20535/1810-0546.2016.4.77034

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