Some exact solutions for a unidimensional fokker-planck equation by using Lie Symmetries
Algunas soluciones exactas para la ecuación unidimensional de fokker-planck usando Simetrías de Lie
Hugo Hernán Ortiz-Álvarez*+, Francy Nelly Jiménez-García†*, Abel Enrique Posso-Agudelo‡*
Abstract ]]>
The Fokker Planck equation appears in the study of diffusion phenomena, stochastics processes and quantum and classical mechanics. A particular case from this equation, ut — uxx — xux — u = 0, is examined by the Lie group method approach. From the invari-ant condition it was possible to obtain the infinitesimal generators or vectors associated to this equation, identifying the corresponding symmetry groups. Exact solution were found for each one of this generators and new solution were constructed by using symmetry properties.
La ecuación de Fokker Planck aparece en el estudio de fenómenos de difusión, procesos estocásticos y mecánica clasica y cuántica. Un caso particular de esta ecuacion, ut — uxx — xux — u = 0, es analizada empleando el método de los grupos de Lie. De la condición de invariación fue posible obtener los generadores infinitesimales o vectores de la ecuación identificando los correspondientes grupos de simetría. Se obtuvieron soluciones exactas para cada uno de estos generadores y se construyeron nuevas soluciones aplicando propiedades de simetría.
Palabras clave: grupos de Lie; ecuaciones diferenciales parciales; soluciones invariantes; ecuación de Fokker Planck. ]]>
Mathematics Subject Classification: 35A30. Ver contenido en pdf.
References
[1] Abramowitz M.; Stegun I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. [ Links ] ]]>
[2] Bluman G. B.; Kumei S. (1989) Symmetries and Differential Equations. Springer Verlag, New York. [ Links ]
[3] Bump D. (2004) Lie Groups. Springer Verlag, New York. [ Links ]
[4] Cherniha R.; Davydovych V. (2011) "Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system", Mathematical and Computer Modellling54(5-6): 1238-1251. [ Links ] ]]>
[5] Cohen A.; (2007) An Introduction to the Lie Theory of one Parameter Groups. Kessinger Publishing, New York. [ Links ]
[6] Emanuel G. (2000) Solution of Ordinary Differential Equations by Continuous Groups. Chapman and Hall/CRC, Boca Raton, FL. [ Links ]
[7] Hanze L.; Jibin L. (2009) "Lie symmetry analysis and exact solutions for the short pulse equation", Nonlinear Analysis: Theory, Methods and Application71(5-6): 2126-2133. [ Links ]
[8] Hereman W. (1997) "Review of symbolic software for Lie symmetrie analysis", Mathematical and Computer Modelling25(8-9): 115-132. [ Links ]
[9] Kolmogorov A.N. ; Fomin S.V. (1931) Analytical Methods in the Theory of Probability. Springer-Verlag, Berlin. [ Links ]
[10] Olver P.J. (1993) Application of Lie Groups to Differential Equations. Springer Verlag, New York. [ Links ]
[11] Ovsiannikov L.V. (1978) Group Analysis of Differential Equations. Academic Press, New York [ Links ] ]]>
[12] Robert C.M. (2002) Partial Differential Equations: Methods and Applications. Prentice Hall, New York. [ Links ]
[13] Ruo-Xia Y.; Zhi-Bin L. (2004) "On new invariant solutions of gener-alized Fokker-Planck equation", Commun, Theor. Phys. 41(5): 665¬668. [ Links ]
[15] Whittaker E. T.; Watson G. N. (1990) Parabolic Cylinder Function in a Course in Modern Analysis. Cambridge University Press, Cambridge. [ Links ] ]]>
*Universidad de Caldas & Universidad Nacional de Colombia, Manizales, Colombia. E-Mail: hugo.ortiz@ucaldas.edu.co †Universidad Autónoma & Universidad Nacional de Colombia, Manizales, Colombia. E-Mail: fnjimenezg@unal.edu.co ‡Universidad Tecnológica, Pereira, Colombia. E-Mail: possoa@utp.edu.co