Invariance of the Equations of the Theory of Penetration with Respect to Galileo's Algebra and its Extensions

  • Authors

    • Nataliia Ichanska
    • Maria Serova
    • Taras Skliarenko
    2018-10-13
    https://doi.org/10.14419/ijet.v7i4.8.27298
  • Lie algebra, group classification, quasilinear equations, symmetry operator, equations of evolution type, exact solutions.

  • Abstract

    In this work, the symmetry properties of the system of equations of the theory of penetration, which describes the adiabatic motion of an inviscid compressible fluid, are investigated. Maximal invariance algebras of a class of systems that describe the adiabatic motion of an inviscid compressible fluid in the absence of mass forces and in their presence are found. The paper shows that for a given system one can observe a similar effect of the absence and presence of axial symmetry the same as for the known equations of mathematical physics, for example, the Schrödinger equation. It was established that in the absence of axial symmetry, this system is invariant with respect to the generalized Galilean algebra AG2 (1, n) with a fixed power nonlinearity, and in the presence of axial symmetry the system is invariant with respect to the generalized Galilean algebra AG2 (1, n-1) with arbitrary power nonlinearity.

     

  • References

    1. [1] Bluman G.W., Cole J.D. (1969), The general similarity solution of the heat equation. J. Math. Mech., Vol. 18, â„– 11. - P.1025-1042.

      [2] Fushchych W., Shtelen W., Serov N., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Dordrecht: Kluwer Academic Publishers, (1993), 436 p.

      [3] Nikitin A.G., Popovych R.O., (2001), â€Group Classification of Nonlinear Schrodinger Equationsâ€, Ukrainian Mathematical Journal, Vol. 53, â„–8., (2001), P.1255-1265.

      [4] Ovsjannykov, L. V. (1959). Ghruppovыe svojstva uravnenyj nelynejnoj teploprovodnosty. In Dokl. AN SSSR (Vol. 125, No. 3, pp. 492-495).

      [5] Ovsiannikov L.V., Group analysis of differential equations, New York: Academic Press.,(1982), 400 p.

      [6] Serova M., Andreeva N., â€Evolution equations invariant under the conformal algebraâ€, Proceedings of the Second International Conference "Symmetry in Nonlinear Mathematical Phisics".~--- Kyiv: Institute of Mathematics, Vol. 1, (1998), P. 217-221.

      [7] Ichansjka, N. V. (2011). Ghrupova klasyfikacija evoljucijnykh rivnjanj vysokogho porjadku. Visnyk Zaporizjkogho nacionaljnogho universytetu: Zbirnyk naukovykh pracj. Fizyko-matematychni nauky, (1), 43-48.

      [8] Serov, M., Ichansjka, N. (2016). Pro konformnu invariantnistj nelinijnykh evoljucijnykh baghatovymirnykh rivnjanj. Visnyk Kyjivsjkogho nacionaljnogho universytetu imeni Tarasa Shevchenka. Matematyka. Mekhanika, (36), 35-39.

      [9] Saghomonjan A.Ja., Pronykanye, M.: Yzdateljstvo Moskovskogho Unyversyteta, (1974), 300 s.

      [10] Saghomonjan A.Ja., Poruchnykov V.B., Prostranstvennыe zadachy neustanovyvsheghosja dvyzhenyja szhymaemoj zhydkosty, M.: Yzdateljstvo Moskovskogho unyversyteta, (1970), 120 s.

      [11] Fushhych V.I., Serov M.I., Podoshvelev Ju.Gh. (1998) “Konformna symetrija nelinijnogho cylindrychno-symetrychnogho khvyljovogho rivnjannjaâ€, Dopovidi NAN Ukrajiny, â„– 4., S. 64-68.

      [12] Serov M.I., Serova M.M., Ghleba A.V., (1998), “Symetrijni vlas-tyvosti ta dejaki tochni rozv'jazky nelinijnogho cylindrychno-symetrychnogho rivnjannja Shredingheraâ€, Dopovidi NAN Ukrajiny, â„– 5,C. 41-45.

      [13] Cherniha, R., & Serov, M. (2006). Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II. European Journal of Applied Mathematics, 17(5), 597-605. https://doi.org/10.1017/S0956792506006681

      [14] Loburets, A. T., Naumovets, A. G., Senenko, N. B., & Vedula, Y. S. (1997). Surface diffusion and phase transitions in strontium overlayers on W(112). Zeitschrift Fur Physikalische Chemie, 202(1-2), 75-85.

      [15] Gutak, A. D. (2015). Experimental investigation and industrial application of ranque-hilsch vortex tube. International Journal of Refrigeration, 49, 93-98. https://doi.org/10.1016/j.ijrefrig.2014.09.021

      [16] Popova, A. V., Kremenetsky, V. G., Solov'ev, V. V., Chernenko, L. A., Kremenetskaya, O. V., Fofanov, A. D., & Kuznetsov, S. A. (2010). Standard rate constants of charge transfer for nb(V)/Nb(IV) redox couple in chloride-fluoride melts: Experimental and calculation methods. Russian Journal of Electrochemistry, 46(6), 671-679. https://doi.org/10.1134/S1023193510060121

  • Downloads

  • How to Cite

    Ichanska, N., Serova, M., & Skliarenko, T. (2018). Invariance of the Equations of the Theory of Penetration with Respect to Galileo’s Algebra and its Extensions. International Journal of Engineering & Technology, 7(4.8), 517-523. https://doi.org/10.14419/ijet.v7i4.8.27298

    Received date: 2019-02-11

    Accepted date: 2019-02-11

    Published date: 2018-10-13