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Abstract

The article presents "-approximation of hydrodynamics equations’ stationary model along with the proof of a theorem about existence of a hydrodynamics equations’ strongly generalized solution. It was proved by a theorem on the existence of uniqueness of the hydrodynamics equations’ temperature model’s solution, taking into account energy dissipation. There was implemented the Galerkin method to study the Navier–Stokes equations, which provides the study of the boundary value problems correctness for an incompressible viscous flow both numerically and analytically. Approximations of stationary and non-stationary models of the hydrodynamics equations were constructed by a system of Cauchy–Kovalevsky equations with a small parameter ". There was developed an algorithm for numerical modelling of the Navier– Stokes equations by the finite difference method.
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Bibliography

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[5] S.Sh. Kazhikenova, S.N. Shaltakov, D. Belomestny, and G.S. Shai- hova: Finite difference method implementation for numerical integration hydrodynamic equations melts. Eurasian Physical Technical Journal, 17(1), (2020), 50–56.
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[12] T.A. Barannyk, A.F. Barannyk, and I.I. Yuryk: Exact Solutions of the nonliear equation. Ukrains’kyi Matematychnyi Zhurnal, 69(9), (2017), 1180–1186, http://umj.imath.kiev.ua/index.php/umj/article/view/1768.
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[16] S.Sh. Kazhikenova, M.I. Ramazanov, and A.A. Khairkulova: epsilon- Approximation of the temperatures model of inhomogeneous melts with allowance for energy dissipation. Bulletin of the Karaganda University- Mathematics, 90(2), (2018), 93–100, DOI: 10.31489/2018M2/93-100.
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Authors and Affiliations

Saule Sh. Kazhikenova
1
ORCID: ORCID

  1. Head of the Department of Higher Mathematics, Karaganda Technical University, Kazakhstan
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Abstract

The basic objective of the research is to construct a difference model of the melt motion. The existence of a solution to the problem is proven in the paper. It is also proven the convergence of the difference problem solution to the original problem solution of the melt motion. The Rothe method is implemented to study the Navier–Stokes equations, which provides the study of the boundary value problems correctness for a viscous incompressible flow both numerically and analytically.
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Bibliography

[1] R. Lakshminarayana, K. Dadzie, R. Ocone, M. Borg, and J. Reese: Recasting Navier–Stokes equations. Journal of Physics Communications, 3(10), (2019), 13–18, DOI: 10.1088/2399-6528/ab4b86.
[2] S.Sh. Kazhikenova, S.N. Shaltaqov, D. Belomestny, and G.S. Shai- hova: Finite difference method implementation for Numerical integration hydrodynamic equations melts. Eurasian Physical Technical Journal, 17(33), (2020), 50–56.
[3] C. Bardos: A basic example of non linear equations: The Navier– Stokes equations. Mathematics: Concepts and Foundations, III (2002), http://www.eolss.net/sample-chapters/c02/e6-01-06-02.pdf.
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[6] P.M. Gresho: Incompressible fluid dynamics: some fundamental formulation issues. Annual Review of Fluid Mechanics, 23, (1991), 413–453, DOI: 10.1146/annurev.fl.23.010191.002213.
[7] S.E. Rogers, K. Dochan, and K. Cetin: Steady and unsteady solutions of the incompressible Navier–Stokes equations. AIAA Journal, 29(4), (1991), 603–610, DOI: 10.2514/3.10627.
[8] S. Masayoshi, T. Hiroshi, S. Nobuyuki, and N. Hidetoshi: Numerical simulation of three-dimensional viscous flows using the vector potential method. JSME International Journal, 34(2), (1991), 109–114, DOI: 10.1299/jsmeb1988.34.2_109.
[9] E. Sciubba: A variational derivation of the Navier–Stokes equations based on the exergy destruction of the flow. Journal of Mathematical and Physical Sciences, 25(1), (1991), 61–68.
[10] A. Bouziani and R. Mechri: The Rothe’s method to a parabolic integrodifferential equation with a nonclassical boundary conditions. International Journal of Stochastic Analysis, Article ID 519684, (2010), DOI: 10.1155/2010/519684.
[11] N. Merazga and A. Bouziani: Rothe time-discretization method for a nonlocal problem arising in thermoelasticity. Journal of Applied Mathematics and Stochastic Analysis, 2005(1), (2005), 13–28, DOI: 10.1080/00036818908839869.
[12] T.A. Barannyk, A.F. Barannyk, and I.I. Yuryk: Exact solutions of the nonliear equation. Ukrains’kyi Matematychnyi Zhurnal, 69(9), (2017), 1180–1186, http://umj.imath.[K]iev.ua/index.php/umj/article/view/1768.
[13] N.B. Iskakova, A.T. Assanova, and E.A. Bakirova: Numerical method for the solution of linear boundary-value problem for integrodifferential equations based on spline approximations. Ukrains’kyi Matematychnyi Zhurnal, 71(9), (2019), 1176–91, http://umj.imath.[K]iev.ua/index.php/ umj/article/view/1508.
[14] S.L. Skorokhodov and N.P. Kuzmina: Analytical-numerical method for solving an Orr-Sommerfeld type problem for analysis of instability of ocean currents. Zh. Vychisl. Mat. Mat. Fiz., 58(6), (2018), 1022–1039, DOI: 10.7868/S0044466918060133.
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Authors and Affiliations

Saule Sh. Kazhikenova
1
ORCID: ORCID
Sagyndyk N. Shaltakov
1
ORCID: ORCID
Bekbolat R. Nussupbekov
2
ORCID: ORCID

  1. Karaganda Technical University, Kazakhstan
  2. Karaganda University E.A. Buketov, Kazakhstan

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