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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Conservation Laws in Thomas's Model of Ion Exchange in a Heterogeneous Solution

Discontinuity, Nonlinearity, and Complexity 2(2) (2013) 147--158 | DOI:10.5890/DNC.2013.04.004

N.H. Ibragimov$^{1}$; Raisa Khamitova$^{2}$

$^{1}$ Laboratory “Group analysis of mathematical models in natural and engineering sciences”, Ufa StateAviation Technical University, 450 000 Ufa, Russia

$^{2}$ Department ofMathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden

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Physically significant question on calculation of conservation laws of the Thomas equation is investigated. It is demonstrated that the Thomas equation is nonlinearly self-adjoint. Using this property and applying the theorem on nonlocal conservation laws the infinite set of conservation laws corresponding to the symmetries of the Thomas equation is computed. It is shown that the Noether theorem provides only one of these conservation laws.


We acknowledge a financial support of the Government of Russian Federation through Resolution No. 220, Agreement No. 11.G34.31.0042. Raisa Khamitova thanks also the Department of Mathematics and Science of Blekinge Institute of Technology for a partial financial support of her research visit to the Laboratory “Group analysis of mathematical models in natural and engineering sciences” at Ufa State Aviation Technical University.


  1. [1]  Thomas, H.C. (1944), Heterogeneous ion exchange in a flowing system, Journal of the American Chemical Society 66, 1664-1666.
  2. [2]  Rosales, R.R. (1978), Exact solutions of some nonlinear evolution equations, Studies in Applied Mathematics 59, 117-151.
  3. [3]  Wei, G.M., Gao, Y.T., and Zhang, H. (2002), On the Thomas equation for the ion-exchange operations, Czechoslovak Journal of Physics, 52 (6), 749-751.
  4. [4]  Yan, Z.Y. (2003), Study of the Thomas equation: a more general transformation (auto-Bäcklund tarnsformation) and exact solutions, Czechoslovak Journal of Physics, 53 (4), 297-300.
  5. [5]  Ibragimov, N.H. (2011), Nonlinear self-adjointness and conservation laws, Journal of Physics A: Mathematical and Theoretical 44, 432002.
  6. [6]  Ibragimov, N.H. (2011), Nonlinear self-adjointness in constructing conservation laws, arXiv:1109.1728v1[mathph] 1-104. See also Archives of ALGA 7/8 (2010-2011), 1-99.
  7. [7]  N.H. Ibragimov and R. Khamitova, Nonlinear self-adjointness and conservation laws of the Thomas equation, Archives of ALGA 7/8 (2010-2011), 147-159.