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Journal of Mathematical Sciences

, Volume 231, Issue 5, pp 598–607 | Cite as

Group Classification of a Class of Kolmogorov Equations with Time-Dependent Coefficients

  • V. V. Davydovych
Article

We propose a group classification for one class of Kolmogorov equations.

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References

  1. 1.
    N. H. Ibragimov, Groups of Transformations in Mathematical Physics [in Russian], Nauka, Moscow (1983).Google Scholar
  2. 2.
    S. Kovalenko, I. Kopas’, and V. Stohnii, “Lie symmetries and fundamental solutions of the Kolmogorov linear equations,” Mat. Visnyk NTSh, 11, 62–72 (2014).Google Scholar
  3. 3.
    V. I. Lahno, S. V. Spichak, and V. I. Stohnii, Symmetry Analysis of the Evolutionary Equations [in Ukrainian], Institute of Mathematics Kyiv (2002).Google Scholar
  4. 4.
    L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).zbMATHGoogle Scholar
  5. 5.
    L. V. Ovsiannikov, “Group properties of nonlinear heat-conduction equation,” Dokl. Akad. Nauk SSSR, 125, No. 3, 492–495 (1959).MathSciNetGoogle Scholar
  6. 6.
    S. V. Spichak, V. I. Stohnii, and I. M. Kopas’, “Symmetry analysis and exact solutions of the Kolmogorov linear equation,” Naukovi Visti Nats. Tekh. Univ. Ukrainy “Kyiv Politekh. Inst.,” No. 4, 93–97 (2011).Google Scholar
  7. 7.
    G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York (1989).CrossRefzbMATHGoogle Scholar
  8. 8.
    R. Cherniha, M. Serov, and I. Rassokha, “Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations,” J. Math. Anal. Appl., 342, No. 2, 1363–1379 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin (1985).Google Scholar
  10. 10.
    A. N. Kolmogoroff, “Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung),” Ann. Math. (Second Ser.), 35, No. 1, 116–117 (1934).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1986).CrossRefzbMATHGoogle Scholar
  12. 12.
    A. Pascucci, “Kolmogorov equations in physics and in finance,” in: H. Brezis (editor), Progress in Nonlinear Differential Equations and Their Applications, Vol. 63: Elliptic and Parabolic Problems, Birkhäuser, Basel (2005), pp. 313–324.Google Scholar
  13. 13.
    O. A. Pocheketa, R. O. Popovych, and O. O. Vaneeva, “Group classification and exact solutions of variable-coefficient generalized Burgers equations with linear damping,” Appl. Math. Comput., 243, 232–244 (2014).MathSciNetzbMATHGoogle Scholar
  14. 14.
    S. Spichak and V. Stogny, “Symmetry analysis of the Kramers equation,” Rep. Math. Phys., 40, No. 1, 125–130 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    O. Vaneeva and A. Zhalij, “Group classification of variable coefficient quasilinear reaction-diffusion equations,” Publications de l’Institut Mathematique, 94, No. 108, 81–90 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    P. Wilmott, S. Howison, and J. Dewynne, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford (1993).zbMATHGoogle Scholar
  17. 17.
    R. Z. Zhdanov and V. I. Lahno, “Group classification of heat conductivity equations with a nonlinear source,” J. Phys., A : Math. Gen., 32, No. 42, 7405–7418 (1999); http://dx.doi.org/ https://doi.org/10.1088/0305-4470/32/42/312.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • V. V. Davydovych
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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