
SIGMA 2 (2006), 004, 11 pages nlin.SI/0601036
http://dx.doi.org/10.3842/SIGMA.2006.004
On Linearizing Systems of Diffusion Equations
Christodoulos Sophocleous ^{a} and Ron J. Wiltshire ^{b}
^{a)} Department of Mathematics and Statistics,
University of Cyprus, CY 1678 Nicosia, Cyprus
^{b)} The Division of Mathematics and Statistics,
The University of Glamorgan, Pontypridd CF37 1DL, Great Britain
Received November 23, 2005, in final form January 10, 2006; Published online January 16, 2006
Abstract
We consider systems of diffusion equations
that have considerable
interest in Soil Science and Mathematical Biology and focus upon the problem of finding those
forms of this class that can be linearized. In particular we use the equivalence transformations of the
second generation potential system to derive forms of this system that can be linearized.
In turn, these transformations lead to nonlocal mappings that linearize the original system.
Key words:
diffusion equations; equivalence transformations; linearization.
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References
 Akhatov I.Sh., Gazizov R.K., Ibragimov N.Kh., Nonlocal symmetries. Heuristic approach,
J. Soviet. Math., 1991, V.55, 14011450.
 Baikov V.A, Gladkov A.V., Wiltshire R.J., Lie symmetry
classification analysis for nonlinear coupled diffusion, J.
Phys. A: Math. Gen., 1998, V.31, 74837499.
 Bluman G.W., Kumei S., Symmetries and differential equations, New York, Springer, 1989.
 Bluman G.W., Kumei S., On the remarkable nonlinear diffusion equation
(¶/¶x)[a(u+b)^{2}(¶u /¶x)](¶u/¶t)=0, J. Math. Phys., 1980, V.21, 10191023.
 Ibragimov N.H., Torrisi M., Valenti A., Preliminary group classification of equations
v_{tt}=f(x,v_{x})v_{xx}+g(x,v_{x}), J. Math. Phys., 1991, V.32, 29882995.
 Jury W.A., Letey J., Stolzy L.H., Flow of water and energy
under desert conditions, in Water in Desert Ecosystems, Editors D. Evans and J.L. Thames,
Stroudsburg, PA: Dowden, Hutchinson and
Ross, 1981, 92113.
 Ovsiannikov L.V., Group analysis of differential equations, New York, Academic, 1982.
 Philip J.R., de Vries D.A., Moisture movement in porous media
under temperature gradients, Trans. Am. Geophys. Un., 1957,
V.38, 222232.
 Sophocleous C., Wiltshire R.J., Systems of diffusion equations, in Proceedings of 11th
Conference "Symmetry
in Physics", Prague, 2004, 17 pages,
 Wiltshire R.J., The use of Lie transformation groups in the
solution of the coupled diffusion equation, J. Phys. A: Math.
Gen., 1994, V.27, 78217829.

