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Symmetry Analysis of the Fokker Planck Equation

Received: 30 January 2020     Accepted: 20 February 2020     Published: 28 May 2020
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Abstract

In this work, the infinitesimal criterion of invariance for determining symmetries of partial differential equations is applied to the Fokker Planck equation. The maximum rang condition being satisfied, we determine the Lie point symmetries of this equation. Due to the nature of infinitesimal generators of these symmetries and the stability of Lie brackets, we obtain an infinite number of solutions from which we find examples of solutions for the Fokker Planck equation: other solutions are generated given a particular solution of the equation. Then, the Fokker Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation. We show that this system admits six and an infinite number of infinitesimal generators of point symmetries giving rise to two potential symmetries of the Fokker Planck equation. We then use those potential symmetries to determine solutions of the associated system and therefore provide other solutions of the Fokker Planck equation. Note that these are essentially obtained on the basis of the invariant surface conditions. With respect to these conditions and from the potential symmetries that we have found, we finally show that in particular, some solutions of the considered Fokker Planck equation reduced to the trivial solution (solutions that are zero).

Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 2)
DOI 10.11648/j.ajmcm.20200502.14
Page(s) 51-60
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Fokker-Planck Equation, Symmetry Analysis, Lie Point Aymmetry, Potential Symmetry

References
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[2] Brics M.; kaupˇzs J.; Mahnke R., How to solve Fokker-Planck equation treating mixed eigenvalue spectrum, Condensed Matter Physics, Vol. 16, n°1 (2013) 1-13.
[3] Carrillo J. A.; Cordier S. and Mancini S., A decision-making Fokker-Planck model in computational neuroscience, J. Math. Biol., 63, pp. 801-830, (2011).
[4] Carrillo J. A.; Cordier S. and Mancini S., One dimensional Fokker-Planck re-duced dynamics of decision making models in Computational Neuroscience, Com¬mun. Math. Sci., 11 (2), pp. 523-540, (2013).
[5] Chancelier J.-P.; Cohen De Lara M.; Pacard F., Equation de Fokker-Planck pour la densité d’un processus aléatoire dans un ouvert régulier. Comptes rendus de l’Académie des Sciences. Série 1, Mathématique, 1995, vol. 321, n°9, pp. 1251-1256.
[6] Gardiner C., Stochastic Methods: A Handbook for Natural and Social Sciences. 4th Ed. Springer, 2009.
[7] Hesam S.; Nazemi A. R. and Haghbin A., Analytical solution for the Fokker-Planck equation by differential transform method, Scientia Iranica, Vol. 19, Issue 4, (2012), pp. 1140-1145.
[8] Hottovy S., The Fokker-Planck Equation, http://www.math.wisc.edu/ ˜shottovy/NumPDEreport.pdf, (2011).
[9] Ibragimov N. H., Transformation Groups Applied to Mathematical Physics. Reidel Publishing Company: Dordrecht, Netherlands, 1985.
[10] Jia Zheng, Lie Symmetry Analysis and Invariant Solutions of a Nonlinear Fokker-Planck Equation Describing Cell Population Growth, Advances in Mathematical Physics, https://doi.org/10.1155/2020/4975943, (2020).
[11] Lie Sophus, Theories der Transformations gruppen. Teubner, Leipzig, 3, (1893).
[12] Martin Ph. A., Introduction aux Processus Stochastiques en Physique. Nonequilibrium Statistical Mechanics, 2006.
[13] Olver P. J., Applications of Lie Groups to Differential Equations. 2nd Ed., GTM, Vol. 107, Springer Verlag, New York, 1993.
[14] Olver P. J., Equivalence, Invariants and Symmetry, Cambridge University Press, 1995.
[15] Ouhadan A.; El Kinani E. H.; Rahmoune M. and Awane A., Symétries ponctuelles et potentielles de l’équation de Fokker-Planck. African Journal of Mathematical Physics, Vol 5 (2007) 33-41.
[16] Ouhadan A.; El Kinani E. H.; Rahmoune M.; Awane A.; Ammar A. and Essabab S., Generalized Symmetries and Some new Solution of the Fokker-Planck Equation. African Journal Of Mathematical Physics, Volume 7, No 1 (2009) 9-17.
[17] Ovsiannikov L. V., Group Analysis of Differential Equations. Academic Press, New York, 1982.
[18] Pucci E. and Saccomandi G., Potential symmetries and solutions by reduction of partial differential equations. J. Phys.: Math. Gen. 26 (1993) 681-690.
[19] Tanski I. A., Fundamental solution of Fokker-Planck equation., http://arxiv.org/ pdf/nlin/0407007.pdf, (2004).
[20] Till D. F., Nonlinear Fokker-Planck equations: Fundamentals and Applications. Springer-Verlag, Berlin Heidlberg, 2005.
[21] Winter S., Group classification and symmetry reductions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja Mittal entropy, arXiv: 1904.01307v1 [math. AP] 2 Apr 2019.
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    Faya Doumbo Kamano, Bakary Manga, Joël Tossa. (2020). Symmetry Analysis of the Fokker Planck Equation. American Journal of Mathematical and Computer Modelling, 5(2), 51-60. https://doi.org/10.11648/j.ajmcm.20200502.14

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    ACS Style

    Faya Doumbo Kamano; Bakary Manga; Joël Tossa. Symmetry Analysis of the Fokker Planck Equation. Am. J. Math. Comput. Model. 2020, 5(2), 51-60. doi: 10.11648/j.ajmcm.20200502.14

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    AMA Style

    Faya Doumbo Kamano, Bakary Manga, Joël Tossa. Symmetry Analysis of the Fokker Planck Equation. Am J Math Comput Model. 2020;5(2):51-60. doi: 10.11648/j.ajmcm.20200502.14

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  • @article{10.11648/j.ajmcm.20200502.14,
      author = {Faya Doumbo Kamano and Bakary Manga and Joël Tossa},
      title = {Symmetry Analysis of the Fokker Planck Equation},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {5},
      number = {2},
      pages = {51-60},
      doi = {10.11648/j.ajmcm.20200502.14},
      url = {https://doi.org/10.11648/j.ajmcm.20200502.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20200502.14},
      abstract = {In this work, the infinitesimal criterion of invariance for determining symmetries of partial differential equations is applied to the Fokker Planck equation. The maximum rang condition being satisfied, we determine the Lie point symmetries of this equation. Due to the nature of infinitesimal generators of these symmetries and the stability of Lie brackets, we obtain an infinite number of solutions from which we find examples of solutions for the Fokker Planck equation: other solutions are generated given a particular solution of the equation. Then, the Fokker Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation. We show that this system admits six and an infinite number of infinitesimal generators of point symmetries giving rise to two potential symmetries of the Fokker Planck equation. We then use those potential symmetries to determine solutions of the associated system and therefore provide other solutions of the Fokker Planck equation. Note that these are essentially obtained on the basis of the invariant surface conditions. With respect to these conditions and from the potential symmetries that we have found, we finally show that in particular, some solutions of the considered Fokker Planck equation reduced to the trivial solution (solutions that are zero).},
     year = {2020}
    }
    

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    T1  - Symmetry Analysis of the Fokker Planck Equation
    AU  - Faya Doumbo Kamano
    AU  - Bakary Manga
    AU  - Joël Tossa
    Y1  - 2020/05/28
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    N1  - https://doi.org/10.11648/j.ajmcm.20200502.14
    DO  - 10.11648/j.ajmcm.20200502.14
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 51
    EP  - 60
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20200502.14
    AB  - In this work, the infinitesimal criterion of invariance for determining symmetries of partial differential equations is applied to the Fokker Planck equation. The maximum rang condition being satisfied, we determine the Lie point symmetries of this equation. Due to the nature of infinitesimal generators of these symmetries and the stability of Lie brackets, we obtain an infinite number of solutions from which we find examples of solutions for the Fokker Planck equation: other solutions are generated given a particular solution of the equation. Then, the Fokker Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation. We show that this system admits six and an infinite number of infinitesimal generators of point symmetries giving rise to two potential symmetries of the Fokker Planck equation. We then use those potential symmetries to determine solutions of the associated system and therefore provide other solutions of the Fokker Planck equation. Note that these are essentially obtained on the basis of the invariant surface conditions. With respect to these conditions and from the potential symmetries that we have found, we finally show that in particular, some solutions of the considered Fokker Planck equation reduced to the trivial solution (solutions that are zero).
    VL  - 5
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Author Information
  • Department of Research, Distance Learning High Institut, Conakry, Guinea

  • Department of Mathematics and Computer Sciences, University of Cheikh Anta Diop, Dakar, Senegal

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