info@viam.science.tsu.ge (+995 32) 2 30 30 40 (+995 32) 2 18 66 45

Investigation and Approximate Solution of Some Nonlinear Partial Differential and Integro-Differential Models (FR-21-2101)


Funded by

SRNSFGShota Rustaveli National Science Foundation of Georgia

Start Date: 2022-03-15       End Date: 2025-03-15

In mathematical modeling of many natural processes, the partial differential and integro-differential equations and systems of those equations arise. Most of those models, as a rule, are multi-dimensional and nonlinear. These circumstances significantly complicate the study of initial-boundary value problems posed for those models. Therefore, the investigation of nonlinear mathematical models is very actual.
One important nonlinear nonstationary model is obtained in the mathematical modeling of processes of electromagnetic field penetration into a substance. One main aim of the project is to investigate initial-boundary value problems for a corresponding nonlinear partial differential system of the Maxwell equations for different cases. In the quasi-stationary case, the corresponding system of Maxwell equations can be reduced to the nonlinear partial integro-differential models.
It is important to study the questions of unique solvability, asymptotic behavior of solutions, and numerical solution of initial-boundary value problems for such types of models. In this direction, some investigations have already been performed and the goal of the project is to continue and deepen that research. In particular, the main attention will be paid to the study of multi-dimensional cases. The aforementioned problems mainly are conditioned by physical problems, but part of them is the result of natural mathematical generalization as well.
The purpose of the project is the study of quantitative and qualitative characteristics of solutions of different kinds of initial-boundary value problems as well as construct and investigate finite difference, finite element, Galerkin methods, and decomposition algorithms of approximate solution for the above-mentioned differential and integro-differential models. Together, with the consideration of decomposition schemes regarding spatial variables, significant attention will be paid to the splitting schemes with respect to the physical processes. The project will also focus on the investigation of non-linear models using machine learning algorithms. Besides the studied algorithms, the creation of software packages for carrying out numerical calculations and analyzing obtained results is foreseen. Within the framework of the project is planned to carry out the investigation for more general cases of nonlinearity, by using corresponding methods of numerical analysis and machine learning algorithms. Numerical calculations for different models will be fulfilled and analysis of the obtained results will be carried out. The project belongs to the field of fundamental research. In order to achieve the project goals, fulfilled investigations will have an interdisciplinary character. The novelty of the research consists in the coalescence of classical and modern methods of the theory of partial differential and integro-differential equations, theories of equations of mathematical physics, nonlinear and numerical analysis as well as corresponding fields of computer sciences, such as machine learning, for example.

Project members:

Talks

  • Investigation and Approximate Solution of Nonlinear One-Dimensional Maxwell System, by Mikheil Gagoshidze (Speaker), Temur Jangveladze, Zurab Kiguradze at XIII Annual International Meeting of the Georgian Mechanical Union, 2022, Batumi, Georgia.
  • On Investigation and Numerical Solution of One Nonlinear Fourth-Order Integro-Differential Model, by Tamar Faiqidze (Speaker), Teimuraz Chkhikvadze, Temur Jangveladze at XII International Conference of the Georgian Mathematical Union, 2022, Batumi, Georgia.
  • Asymptotic Properties and Approximate Solution of Initial-Boundary Value Problem for One Nonlinear Partial Differential Model, by Nino Mzhavanadze (Speaker), Mikheil Gagoshidze , Temur Jangveladze at XII International Conference of the Georgian Mathematical Union, 2022, Batumi, Georgia.
  • On investigation and approximate solution of two systems of nonlinear partial differential equations, by Temur Jangveladze (Speaker) at XXXVI International Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics (VIAM) of Ivane Javakhisvili Tbilisi State University (TSU), 2022, Tbilisi, Georgia.
  • Numerical Solution Of One Nonlinear Partial Differential Multidimensional System, by Mikheil Gagoshidze (Speaker) at XXXVII International Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics (VIAM) of Ivane Javakhisvili Tbilisi State University (TSU), Dedicated to the 105th Anniversary of, 2023, Tbilisi, Georgia.
  • On two systems of nonlinear partial differential equations, by Mikheil Gagoshidze (Speaker), Temur Jangveladze, Zurab Kiguradze at Third International Conference MATHEMATICS IN ARMENIA Advances and Perspectives, Dedicated to the 80th anniversary of foundation of Armenian National Academy of Sciences, 2023, Yerevan, Armenia.
  • Some Properties and Numerical Solution of Initial-Boundary Value Problem for One System of Nonlinear Partial Differential Equations, by Mikheil Gagoshidze (Speaker), Temur Jangveladze, Zurab Kiguradze at XIII International Conference of the Georgian Mathematical Union,, 2023, Batumi, Georgia.
  • Numerical Solution of One Two-Dimensional System of Nonlinear Partial Differential Equations, by Besiki Tabatadze (Speaker), Mikheil Gagoshidze , Temur Jangveladze, Zurab Kiguradze at XIII International Conference of the Georgian Mathematical Union, 2023, Batumi, Georgia.
  • On One Nonlinear Parabolic Integro-Differential Model, by Teimuraz Chkhikvadze (Speaker), Mikheil Gagoshidze , Temur Jangveladze, Zurab Kiguradze at XIII International Conference of the Georgian Mathematical Union, 2023, Batumi, Georgia.
  • Two Methods of the Numerical Solution of One System of Nonlinear Partial Differential Equations, by Besiki Tabatadze (Speaker), Mikheil Gagoshidze , Temur Jangveladze, Zurab Kiguradze at Fourth International Conference "MODERN PROBLEMS IN APPLIED MATHEMATICS" Dedicated to the 105th Anniversary of I.Javakhishvili Tbilisi State University (TSU) & 55th Anniversary of I.Vekua Institute of, 2023, Tbilisi, Georgia.
  • On the system of Maxwell's nonlinear partial differential equations, by Temur Jangveladze (Speaker) at XXXVII International Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics (VIAM) of Ivane Javakhisvili Tbilisi State University (TSU), 2023, Tbilisi, Georgia.
  • On Decomposition Method for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Nonlinear Two-Dimensional Second Order Elliptic Equations, by Temur Jangveladze (Speaker) at International Workshop on the Qualitative Theory of Differential Equations, QUALITDE–2023, 2023, Tbilisi, Georgia.
  • On one nonlinear fourth-order integro-differential parabolic equation, by Besiki Tabatadze (Speaker), Teimuraz Chkhikvadze, Temur Jangveladze, Zurab Kiguradze at Third International Conference MATHEMATICS IN ARMENIA: Advances and Perspectives. Dedicated to the 80th anniversary of foundation of Armenian National Academy of Sciences, 2023, Yerevan, Armenia.
  • Numerical Solution of Two Systems of Nonlinear Partial Differential Equations Using Machine Learning, by Mikheil Gagoshidze (Speaker), Temur Jangveladze, Zurab Kiguradze at 7th International Conference on Advances in Natural and Applied Sciences (ICANAS 2024), 2024, Antalya, Turkey.
  • Numerical Solution of One ultidimensional System of Partial Differential Equations Using Machine Learning, by Mikheil Gagoshidze (Speaker) at XXXVII International Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics (VIAM) of Ivane Javakhisvili Tbilisi State University (TSU), 2024, Tbilisi, Georgia.
  • Numerical Solution of One Nonlinear Fourth-Order Integro-Differential Parabolic Equation Using Machine Learning , by Besiki Tabatadze (Speaker), Teimuraz Chkhikvadze, Temur Jangveladze, Zurab Kiguradze at 7th International Conference on Advances in Natural and Applied Sciences (ICANAS 2024), 2024, Antalya, Turkey.
  • On Two Multidimensional Systems of Nonlinear Partial Differential Equations, by Temur Jangveladze (Speaker) at XXXVII International Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics (VIAM) of Ivane Javakhisvili Tbilisi State University (TSU), 2024, Tbilisi, Georgia.

Publications

  • Temur Jangveladze, On the System of Maxwell's Nonlinear Partial Differential Equations, Rep. Enlarged Sess. Semin. I.Vekua Inst. Appl. Math.,V.37, 15-18., Tbilisi University Press, 2023.
  • Temur Jangveladze, On Decomposition Method for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Nonlinear Two-Dimensional Second Order Elliptic Equations, International Workshop on the Qualitative Theory of Differential Equations, QUALITDE–2023, REPORTS OF QUALITDE, V.2, p.66-70, Tbilisi State University Press, 2023.
  • Temur Jangveladze, Zurab Kiguradze, Teimuraz Chkhikvadze, Application of Deep Neural Network for Numerical Approximation for Averaged Nonlinear Integro-Differential Equation, Bulletin of TICMI, V.28 (accepted), Tbilisi University Press, 2024.
  • Temur Jangveladze, Mikheil Gagoshidze , Zurab Kiguradze, Besiki Tabatadze, Tabatadze B.Two Methods of the Numerical Solution of One System of Nonlinear Partial Differential Equations, Lecture Notes of TICMI, V.25, (accepted), Tbilisi University Press, 2024.