Lie Symmetries and Fundamental Solutions of the Linear Kramers Equation
Keywords:Linear Kramers equation, Fundamental solution, Lie symmetries
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.
N.Kh. Ibragimov, “Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie)”, Uspehi Matematicheskih Nauk, vol. 47, no. 4 (286), pp. 83–144, 1992 (in Russian).
A.V. Aksenov, “Symmetries of linear partial differential equations and fundamental solutions”, Doklady Akademii Nauk, vol. 342, no. 2, pp. 151–153, 1995 (in Russian).
C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences.Berlin,Germany: Springer-Verlag, 2009.
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
E.A. Saied, “On the silimarity solutions for the free Kramers equation. I”, Appl. Math. Comp., vol. 74, no. 1, pp. 59–63, 1996. doi:10.1016/0096-3003(95)00088-7
S.V. Spichak and V.I. Stogniy, “Symmetry classification and exact solutions of the Kramers equation”, J. Math. Phys., vol. 39, no. 6, pp. 3505−3510, 1998. doi:10.1063/1.532447
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).
LicenseCopyright (c) 2017 NTUU KPI Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work