Shota Rustaveli National Science Foundation of GeorgiaStart 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.