Symmetry Analysis of the Fokker Planck Equation

Faya Doumbo Kamano; Bakary Manga; Joël Tossa

doi:doi:10.11648/j.ajmcm.20200502.14

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Symmetry Analysis of the Fokker Planck Equation

Faya Doumbo Kamano, Bakary Manga, Joël Tossa

Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 2)

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

[1]	Bluman G. W.; Kumei S., Symmetries and Differential Equations. Springer Verlag, New York, 1989.
[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. 4^th 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. 2^nd 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.
[22]	Zorzano M. P.; Mais H.; Vazquez L., Numerical Solution for Fokker-Planck Equa-tion in Accelerators. Physica D 113, (1998) 379-381.

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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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TY  - JOUR
T1  - Symmetry Analysis of the Fokker Planck Equation
AU  - Faya Doumbo Kamano
AU  - Bakary Manga
AU  - Joël Tossa
Y1  - 2020/05/28
PY  - 2020
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
IS  - 2
ER  -

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Author Information

Faya Doumbo Kamano

Department of Research, Distance Learning High Institut, Conakry, Guinea
Bakary Manga

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

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APA Style

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Symmetry Analysis of the Fokker Planck Equation
AU  - Faya Doumbo Kamano
AU  - Bakary Manga
AU  - Joël Tossa
Y1  - 2020/05/28
PY  - 2020
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
IS  - 2
ER  -

Copy | Download