Scientific supervisor: Aloev Rakhmatillo Juraevich
Implementation period: 04.01.2021-31.12.2023
Project code: UZB-Ind-2021-87
Project type: Practical

Expected results and their importance: The invariance condition using the Lie group method, Lie symmetry analysis, Lie algebra of infinitesimal symmetries, study of symmetric solutions of symmetric variables and quasi-linear hyperbolic systems. Also, effectively present research findings and explain them through graphical representations. Development of numerical algorithms based on analysis of Lyapunov stability of Lie group and hyperbolic systems. In addition, analyzing the results obtained to study the specific results using these numerical methods. Within the framework of the project, the analysis of Lie symmetry and the development of an adequate computational model to find the exact solutions of the mixed problem and the exponential steady-state solution for the system of quasi-linear hyperbolic equations. Application of Lee symmetry analysis and developed adequate calculation model in controlling water movement in irrigation systems of unlined concrete irrigation canals. Analysis of Lie symmetry and construction of adequate computational models for exact and approximate solution of mixed problems of quasi-linear hyperbolic systems. Hence, to prove the stability of Lyapunov differential scheme for computational models.

Key results achieved during the reporting period (at the end of the project): Processes represented by mixed problems applied to one-dimensional linear symmetric hyperbolic systems were analyzed. On the basis of hydrological and meteorological data obtained in the conditions of Uzbekistan, information on solving these issues was collected, and the collected data were summarized and reprocessed. Discrete schemes for finding stable solutions of hyperbolic systems with variable coefficients were constructed, and differential schemes for finding stable solutions of Saint Venant equations were studied by proving their stability. Computational experiments were conducted on the circuits whose stability was proven, and the obtained results were visualized. The obtained results are theoretical and practical in nature and will serve to develop the theory of discrete schemes for mixed problems in hyperbolic systems in the future. It can also be used in numerical solutions of problems such as electric energy transport, fluid flow in open channels, and light propagation in optical fibers.

Patent

International scientific works published in WoS and Scopus database within the framework of the project:

1. Aloev R., Nematova D., Lyapunov numerical stability of a hyperbolic system of linear balance laws with inhomogeneous coefficients. Cite as: AIP Conference Proceedings 2365, 020001 (2021); https://doi.org/10.1063/5.0056862. Published Online: 16 July 2021.
2. Aloev R., Nematova D., Three-dimensional linear hyperbolic system. Cite as: AIP Conference Proceedings 2365, 020002 (2021); https://doi.org/10.1063/5.0056863. Published Online: 16 July 2021.
3. Nematova D., Difference upwind scheme for the numerical calculation of stable solutions for a linear hyperbolic system. Cite as: AIP Conference Proceedings 2365, 020003 (2021); https://doi.org/10.1063/5.0057123. Published Online: 16 July 2021.
4. Eshkuvatov Z., Mamatova H., Ismail Sh., Abdullah I., and Aloev R., Numerical approach for nonlinear system of Fredholm-Volterra integral equations. Cite as: AIP Conference Proceedings 2365, 020010 (2021); https://doi.org/10.1063/5.0057121. Published Online: 16 July 2021.
5. Khudoyberganov M., An adequate computational model for a mixed problem for the wave equation in a domain with an angle. Cite as: AIP Conference Proceedings 2365, 020027 (2021); https://doi.org/10.1063/5.0057039. Published Online: 16 July 2021.
6. Khudoyberganov M., Rikhsiboev D., Rashidov J. About one difference scheme for quasi-linear hyperbolic system. Cite as: AIP Conference Proceedings 2365, 020028 (2021); https://doi.org/10.1063/5.0057131. Published Online:16 July 2021.
7. Akbarova A. Numerical solution of Saint-Venant equations. Cite as: AIP Conference Proceedings 2365, 020026 (2021); https://doi.org/10.1063/5.0056878 Published Online: 16 July 2021
8. Mamatov A., Bakhramov S., Narmamatov A., An approximate solution by the Galerkin method of a quasilinear equation with a boundary condition containing the time derivative of the unknown function. Cite as: AIP Conference Proceedings 2365, 070003 (2021); https://doi.org/10.1063/5.0057126. Published Online: 16 July 2021
9. Aloev, R., Berdyshev, A., Akbarova, A., Baishemirov, Z. Development of an algorithm for calculating stable solutions of the saint-venant equation using an upwind implicit difference scheme. Eastern-European Journal of Enterprise Technologies. Volume 4, Issue 4, Pages 47 – 56August 2021
10. Aloev R.D., Eshkuvatov Z.K., Khudoyberganov M.U., Nematova D.E., The Difference splitting scheme for n-dimensional hyperbolic systems» Malaysian Journal of Mathematical Sciences, Malaysian Journal of Mathematical Sciences Scopus, https://mjms.upm.edu.my/current.php 2022. January 2022
11. Eshkuvatov Z.K., Ismail Sh, Mamatova H.X., Viscarra D.S., Aloev R.D.  Modified HAM for solving linear system of Fredholm-Volterra Integral Equations. Malaysian Journal of Mathematical Sciences Scopus, https://mjms.upm.edu.my/current.php. January 2022
12. R.D.Aloev, S.U. Dadabaev, Stability of the upwind difference splitting scheme for symmetric t-hyperbolic systems with constant coefficients. Results in Applied Mathematics,2022
13. R.D.Aloev, M.U.Hudayberganov. A Discrete Analogue of the Lyapunov Function for Hyperbolic Systems, Journal Journal of Mathematical Sciences DOI 10.1007/s10958-022-06028-y. 2022