Dimension Polynomials and the Einstein’s Strength of Some Systems of Quasi-linear Algebraic Difference Equations


In this paper we present a method of characteristic sets for inversive difference polynomials and apply it to the analysis of systems of quasi-linear algebraic difference equations. We describe characteristic sets and compute difference dimension polynomials associated with some such systems. Then we apply our results to the comparative analysis of difference schemes for some PDEs from the point of view of their Einstein’s strength. In particular, we determine the Einstein’s strength of standard finite-difference schemes for the Murray, Burgers and some other reaction–diffusion equations.

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  1. 1.

    Cherniha, R.M.: New ansätze and exact solutions for nonlinear reaction–diffusion equationsarising in mathematical biology. Symmetry Nonlinear Math. Phys. 1, 138–146 (1997)

  2. 2.

    Crank, J., Nicholson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Adv. Comput. Math. 6, 207–226 (1996)

  3. 3.

    Einstein, A.: The Meaning of Relativity, 4th edn. Princeton University Press, pp. 133–165 (1953) Appendix II (Generalization of Gravitation Theory)

  4. 4.

    Evgrafov, A.A.: Standardization and Control of the Quality of Transfusion Liquids. Ph. D. Thesis. Sechenov First Moscow State Medical University (1998)

  5. 5.

    Gao, X.S., Luo, Y., Yuan, C.: A characteristic set method for ordinary difference polynomial systems. J. Symb. Comput. 44(3), 242–260 (2009)

  6. 6.

    Gao, X.S., Yuan, C., Zhang, G.: Ritt-Wu’s characteristic set method for ordinary difference polynomial systems with arbitrary ordering. Acta Math. Sci. 29(3–4), 1063–1080 (2009)

  7. 7.

    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, Cambridge (1973)

  8. 8.

    Kondrateva, M.V., Levin, A.B., Mikhalev, A.V., Pankratev, E.V.: Differential and Difference Dimension Polynomials. Kluwer Academic Publishers, Dordrecht (1998)

  9. 9.

    Levin, A.B.: Characteristic polynomials of filtered difference modules and of difference field extensions. Rus. Math. Surv. 33(3), 165–166 (1978)

  10. 10.

    Levin, A.B.: Characteristic polynomials of inversive difference modules and some properties of inversive difference dimension. Rus. Math. Surv. 35(1), 217–218 (1980)

  11. 11.

    Levin, A.B.: Type and dimension of inversive difference vector spaces and difference algebras. VINITI (Moscow, Russia), no. 1606–82, pp. 1–36 (1982)

  12. 12.

    Levin, A.B.: Difference Algebra. Springer, Berlin (2008)

  13. 13.

    Levin, A.B.: Multivariate dimension polynomials of inversive difference field extensions. Lect. Notes Comput. Sci. 8372, 146–163 (2014)

  14. 14.

    Levin, A.B., Mikhalev, A.V.: Type and dimension of finitely generated G-algebras. Contemp. Math. 184, 275–280 (1995)

  15. 15.

    Lim, J.: Stability of solutions to a reaction diffusion system based upon chemical reaction kinetics. J. Math. Chem. 43(3), 1134–1140 (2008)

  16. 16.

    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin (2009)

  17. 17.

    Zhang, G.L., G.L.; Gao, X.S.: Properties of Ascending Chians fr Partial Difference Polynomial Systems. In: ASCM 2007: Computer Mathematics. LNAI 5081, 307–321, Springer, Berlin (2008)

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Correspondence to Alexander Levin.

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This work was completed with the support of the NSF Grant CCF-1714425.

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Evgrafov, A., Levin, A. Dimension Polynomials and the Einstein’s Strength of Some Systems of Quasi-linear Algebraic Difference Equations. Math.Comput.Sci. (2020) doi:10.1007/s11786-019-00430-7

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  • Difference polynomial
  • Autoreduced set
  • Characteristic set
  • Einstein’s strength
  • Difference scheme

Mathematics Subject Classification

  • Primary 12H05
  • Secondary 12H10
  • 39A05
  • 35K57