Defense of the dissertation of Dinara Kamenovna Koikelova for the degree of Doctor of Philosophy (PhD) in the specialty «6D070500 - Mathematical and computer modeling»

L.N. Gumilyov Eurasian National University, a dissertation defense for the degree of Doctor of Philosophy (PhD) by Койкелова Динара Каменовна on the topic «Asymptotic modeling of a three-dimensional incompressible medium by an compressible medium» in the field of «6D070500 - Mathematical and computer modeling».

 

The dissertation was carried out at the «Mathematical and Computer Modeling» of L.N. Gumilyov Eurasian National University.

The language of defense is kazakh

 

Official reviewers:

Ramazanov Murat Ibraevich - Doctor of Physical and Mathematical Sciences, Professor, Honored Professor of the Department of Mathematical Analysis and Differential Equations, E.A. Buketov Karaganda University (Karaganda).

Baishagirov Khairulla Zhambaevich - Doctor of Technical Sciences, Professor of the Department of Mathematics, Physics and Computer Science, Sh. Ualikhanov Kokshetau University (Kokshetau).

 

Temporary members of the Dissertation Committee:

Mukanova Balgaisha Gafurovna - Doctor of Physical and Mathematical Sciences, Professor of Astana IT University (Nur-Sultan);

Mukhambetzhanov Saltanbek - Doctor of Physical and Mathematical Sciences, professor of the Department of Computational Sciences and Statistics, Al-Farabi Kazakh National University (Almaty);

Rysbayuly Bolatbek - Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematical and Computer Modeling, International University of Information Technologies (Almaty);

Ashirbaev Nurgali Kudiyarovich - Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics, M.Auezov South Kazakhstan University (Shymkent).

 

Academic Advisors:

Bukenov Mahat Mukhamedievich - Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University (Astana).

Fatyanov Alexey Gennadievich - Chief Researcher, Doctor of Physical and Mathematical Sciences, Institute of Computational Mathematics and Mathematical Geophysics of the Siberian Branch of the Russian Academy of Sciences (Russian Federation, Novosibirsk).

The defense will take place on April 05, 2024, at 03:00 PM in the Dissertation Council for the training direction «8D061 - Information and communication technologies» in the specialty «6D070500 - Mathematical and computer modeling» of L.N. Gumilyov Eurasian National University. The online broadcast will take place on the Microsoft Teams platform.

Link: http://surl.li/qqztt 

Address: Astana, st. Kazhymukan, 13, room № 205, educational building № 3.

Abstract (English): The aim is to obtain asymptotic estimates for λ⟶∞ proximity of solutions of a compressible and incompressible medium.An asymptotic approximation of a dynamic problem for an incompressible medium by solving a dynamic problem for a compressible medium at λ⟶∞. The object of the study is the study of the behavior of the solution of a compressible medium: statics and dynamics at λ⟶∞. In accordance with the stated purpose of the dissertation, the following research objectives are set: -proximity of solutions to the Stokes problem and static compressible medium, dynamic compressible medium; -obtaining an estimate of the proximity of solutions to these problems at λ⟶∞; -estimation of the proximity of solving a dynamic problem for an incompressible medium and a dynamic problem for a compressible medium at λ⟶∞; -improving the accuracy of the approximate solution using the Richardson extrapolation method for the parameter 1/ λ. The novelty of the work. Fluid flow is a branch of hydrodynamics. This work is devoted to the asymptotics of the solution of an elastic compressible medium at λ⟶∞ (statics, dynamics), obtaining estimates of proximity to the solution of an incompressible medium, and improving the accuracy of the approximate solution with respect to the parameter 1/λ. The study involves the development of new methods for solving the problem for an incompressible medium. The reliability and validity of scientific provisions, conclusions and results of the study are confirmed by publications in indexed international journals, and in publications recommended by the Committee for Control in Education and Science of the Ministry of Education and Science of the Republic of Kazakhstan for the publication of the main results of scientific activity, as well as in conference proceedings. Theoretical and practical significance. The developed algorithm can be used for approximate solution of an incompressible medium, in hydrodynamics and static fluid equilibrium problem. Personal contribution of a doctoral student: Obtaining asymptotic estimates of the proximity of an approximate solution for an incompressible medium, using grid methods. Approbation of work. The main results of the dissertation work were reported and discussed at: - Materials of the International Scientific Conference "Theoretical and Applied Issues of Mathematics, Mechanics and Informatics", dedicated to the 70th anniversary of the Doctor of Physical and Mathematical Sciences, Professor Murat Ibraevich Ramazanov (Karaganda, E.A. Buketov KarSU, 2019); Scientific seminar of the Department of Mathematical and Computer Modeling L.N.Gumilyov ENU (Nur-Sultan, 2020); On the topic of the study, 5 papers were published in collaboration with scientific consultants, including 3 publications in scientific journals recommended by the Committee for Ensuring Control in the Sphere of Education and Science of the Ministry of Education and Science of the Republic of Kazakhstan for the publication of the main results of scientific activity, 1 publication in scientific journals indexed by the database Scopus and 1 publication in the proceedings of domestic conferences. Scientific provisions submitted for defense. -estimates of the proximity, solutions of a compressible medium, to the solution of an incompressible medium at λ⟶∞ are proved. -asymptotic estimates of the proximity of the solution of the dynamic problem for compressible media to the solution of the dynamic problem for incompressible media at λ⟶∞, at the differential and difference levels. - the Richardson extrapolation method with respect to the parameter 1/λ was applied to improve the accuracy of the approximate solution . The appendix provides numerical calculations for a two-dimensional dynamic problem for an incompressible medium for which an analytical solution is known, while the numerical implementation of the solution to this problem is obtained through solving the difference problem of the dynamic problem of the theory of elasticity for a compressible medium at sufficiently large values of the parameter λ. Calculations were carried out on a sequence of grids; through the grid steps τ, h, the value λ was selected. In this case, the optimal value λ at which the proximity of solutions was minimal coincides with the theoretical estimate. Scientific internships. Institute of Computational Mathematics and Geophysics SB RAS, Russia (Novosibirsk), 01.10-29.12, 2019 Publications. Based on Scopus: D.K. Koikelova, M.M. Bukenov, A.M.Kankenova,R. Muratkhan, A.B. Serikbayeva Asymptotic of solving dynamic problem of elasticity theory for an incompressible medium // Journal of Theoretical and Applied Information Technology. 30.04.2022.Vol.100.No 8, 2687-2695p.p., (Q3 30%). In the publications recommended by the CQASE: 2. D.K.Koikelova, A.A.Adamov, M.M.Bukenov. Approaching of the solution of a static compressible medium to the solution of an incompressible medium // Bulletin of the E.A.Buketov KarSU, Mathematics Series, No.3 (95), Karaganda, RK, 2019, pp. 19-26 3. D.K.Koikelova, M.M.Bukenov Modeling of a static incompressible medium // Bulletin of the E.A. Buketov KarSU. Mathematics Series, No.4 (96), Karaganda, RK, 2019, 39-44 pp. 4.D.K. Koikelova, M.M. Bukenov, A.Rakhymova, A.M.Kankenova Improving the accuracy of approximate solutions for incompressible media according to Richardson // Bulletin of KazNPU named after Abai, Series of physical and mathematical naki, No. 1 (81), Almaty, RK, 2023. In conference materials: 5. D.K.Koikelova, M.M. Bukenov Modeling of the dynamic problem of the theory of elasticity for an incompressible medium// Proceedings of the International Scientific Conference "Theoretical and Applied Issues of Mathematics, Mechanics and Informatics", timed to coincide with the 70th anniversary of the Doctor of Physical and Mathematical Sciences, Professor Murat Ibraevich Ramazanov (Karaganda, KarSU named after E.A.Buketov, 2019. - P.127-129). Dissertation structure. The dissertation consists of 95 pages, an introduction, six sections, a conclusion and a list of cited literature, appendices. The introduction includes an analysis and review of existing scientific papers on the topic of research, the relevance of the topic, the purpose of the dissertation, the object and objectives of the study, novelty, theoretical and practical significance, information about published works on the topic of the dissertation. The main difficulties in the solvability of the Navier-Stokes equations are considered. The first section contains the statement of the problem in stresses. In the second section, stationary linearized equations of a weakly compressible fluid are considered, and the convergence of the solution for ε⟶0 is shown. The third section presents numerical methods for calculating the motions of an incompressible fluid. Finite-difference approximations and methods for their implementation are given. The fourth section provides an overview of existing numerical methods for solving the Navier-Stokes equations. The advantages and disadvantages of the considered algorithms are indicated. A sustainability study has been conducted. In the fifth section, an asymptotic estimate of the proximity at λ⟶∞ of the solution of a static, compressible medium to the solution of the Stokes problem is obtained.An asymptotic estimate of the proximity at λ⟶∞ of the dynamic problem for a compressible medium and the solution of the dynamic problem for an incompressible medium is obtained.It is shown that the obtained estimates are unimprovable in order λ. In the sixth section, the Richardson extrapolation method with respect to the parameter 1/λ is applied to improve the accuracy of the approximate solution for both static and dynamic problems. In conclusion -proximity estimates are justified, where u ̅^λ is the solution of an elastic compressible medium satisfying Hooke's law, and u ̅,p are solutions of an incompressible medium satisfying the Stokes problem, for λ⟶∞; -asymptotic estimates of the proximity of the solution of the dynamic problem for compressible media to the solution of the dynamic problem for incompressible media at λ⟶∞, also at the differential and difference levels; - the Richardson extrapolation method with respect to the parameter 1/ λ was applied to improve the accuracy of the approximate solution . The author expresses her deep gratitude to the scientific adviser ph-m.s.c., docent Bukenov Makhat Makhamedievich, ph-m.s.doctor of Institute of Computational Mathematics and Geophysics of the SB RAS, Russia, Fatyanov Alexei Gennadievich.

Dissertation