On double reductions from symmetries and conservation laws. (English) Zbl 1179.35038

Summary: We present the theory of double reductions of PDEs with two independent variables that admit a Lie point symmetry and a conserved vector invariant under the symmetry. The theory is applied to a third order nonlinear partial differential equation which describes the filtration of a visco-elastic liquid with relaxation through a porous medium.


35B06 Symmetries, invariants, etc. in context of PDEs
76S05 Flows in porous media; filtration; seepage
Full Text: DOI


[1] Kara, A.; Mahomed, F., The relationship between symmetries and conservation laws, Int. J. theor. phys., 39, 1, 23-40, (2000) · Zbl 0962.35009
[2] Kara, A.; Mahomed, F., A basis of conservation laws for partial differential equations, J. nonlinear math. phys, 9, Suppl. 2, 60-72, (2002) · Zbl 1362.35024
[3] Sjöberg, A., Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. math. comput., 184, 608-616, (2007) · Zbl 1116.35004
[4] Euler, N.; Steeb, W.-H., Continuous symmetries, Lie algebras and differential equations, (1992), B.I. Wissenschaftsverlag Mannheim, Leipzig, Wien, Zürich · Zbl 0764.35098
[5] Steeb, W.-H., Continuous symmetries, Lie algebras, differential equations and computer algebra, (1996), World Scientific Publishing Co. Pty. Ltd. Singapore · Zbl 0916.34001
[6] Olver, P., ()
[7] Stephani, H., Differential equations: their solutions using symmetries, (1989), Cambridge University Press Cambridge
[8] Bluman, G.; Kumei, S., ()
[9] V. Baikov, Filtration of a non-Newtonian liquid in porous media: Models, symmetries and solutions, in: Interdisciplinary Workshop on Symmetry Analysis and Mathematical Modelling, University of the North-West, Mmabatho, 1988
[10] Ö. Kartal, Visco-elastic liquid with relaxation: Symmetries, conservation laws and solutions, Dissertation, University of Johannesburg, Auckland Park Campus, Gauteng, 2004
[11] Sjöberg, A.; Kartal, Ö., Filtration of a visco-elastic liquid with relaxation: A note on Lie point symmetries and reductions, J. nonlinear math. phys., 15, Suppl. 1, 203-210, (2008) · Zbl 1362.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.