Article Contents
Article Contents

# Symmetries and solutions of a third order equation

• In this paper we study a new third order evolution equation discovered a couple of years ago using a genetic programming. We show that the Lie symmetries of this equation corresponds to space and time translations, as well as a dilation on the space of independent variables and another one with respect to the depend variable. From its symmetries, explicit solutions of the equation are obtained, some of them expressed in terms of the solutions of the Airy equation and Abel equation of the second kind. Additionally, by using the direct method we establish three conservation laws for the equation, one of them new.
Mathematics Subject Classification: Primary: 76M60, 58J70, 35A30, 70G65.

 Citation:

•  [1] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences 81, Springer, New York, 1989. [2] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences 154, Springer, New York, 2002. [3] S. Anco and G. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873. [4] S. Anco and G. Bluman, Integrating factors and first integrals for ordinary differential equations, European J. Appl. Math., 9 (1998), 245-259. [5] S. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications, European J. Appl. Math., 13 (2002), 545-566. [6] S. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. II. General treatment, European J. Appl. Math., 13 (2002), 567-585. [7] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. [8] G. Bluman, A. Cheviakov and S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer Applied Mathematics Series 168, Spring, New York, 2010. [9] N. H. Ibragimov, Transformation groups applied to mathematical physics, Translated from the Russian Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1985. [10] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, Chirchester, 1999. [11] A. Sen, D. P. Ahalpara, A. Thyagaraja and G. S. Krishnaswami, A KdV-like advection-dispersion equation with some remarkable properties, Commun. Nonlin. Sci. Num. Simul., 17 (2012), 4115-4124. [12] P. J. Olver, Applications of Lie groups to differential equations, Springer, New York, 1986. [13] A. D. Polyanin and V. F. Zaitsev, Handbook of exact solutions for ordinary differential equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton, 2003.
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