The relevance of the project.
Non-classical equations of mathematical physics are one of the intensively developing sections of the theory of partial differential equations. Non-classical equations include equations that are not classical, such as mixed-type equations, degenerate equations of elliptic, parabolic, hyperbolic type, integral and differential equations of fractional order, mixed-composite type, composite type equations, higher-order differential equations, etc.
Within the framework of the project, it is planned to study the correctness of a number of new problems for a multidimensional degenerate elliptic differential equation. Recently, degenerate equations have aroused interest in connection with the active use in modeling of such physical processes, where the boundary of the domain significantly affects the phenomena occurring next to its domain. In such cases, when moving from one part of the domain to another, not only the type of equation may change, but also its order. Such equations are used in modeling the phenomenon of filtration of an ideal barotropic gas from an inhomogeneous anisotropic porous medium, filtration processes of two-phase liquids, in particular, during the extrusion of oil from porous media by water, etc.
In many physical phenomena, the actual development of the system is influenced by past values of variables. Correct modeling of this phenomenon naturally leads to differential equations, the so-called memory equations, where the memory term arises as a convolution of the time of an unknown function relative to a suitable memory core. The non-local nature of such models is a complex task of analyzing equations with memory, which has not been sufficiently studied for many decades. Our goal will be an attempt to translate the problem for an integro-differential equation with memory into the solution of an ordinary differential equation.
Memory equations are closely related to hyperbolic equations of the third order. Thus, the next stage of the project will be the study of wave processes in media characterized by the presence of absorption dispersion. This range of issues includes, in particular, the study of acoustic waves in media where wave propagation violates the state of thermodynamic or mechanical equilibrium. The medium, brought out of the equilibrium state, tends to return to it again, but already with new values of parameters changed by the wave.
The purpose of the project.
The main goal of the project is to formulate and study the solvability of new local and non-local boundary value problems for non-classical equations of mathematical physics of various types, as well as to develop methods for constructing solutions to these problems.
| № | Expected results (for 2022) | Achieved results (for 2022) |
| 2. | During this period, the solvability of boundary value problems with local and non-local conditions for non-classical equations of mathematical physics of elliptic, hyperbolic and mixed types will be studied. | Work is underway to study the solvability of boundary value problems with local and non-local conditions for non-classical equations of mathematical physics of elliptic, hyperbolic and mixed types. Formulation of the correct formulation of nonlocal problems for a second-order hyperbolic equation. Finding conditions that ensure regular and strong solvability of the formulated problems. The existence and uniqueness of the solution of local problems for a degenerate elliptic equation is proved. |
| 2.1 | The existence and uniqueness of the solution of the first mixed problem for a hyperbolic equation with memory will be proved. | The existence and uniqueness of the solution of the first mixed problem for a hyperbolic equation with memory is proved. The formulation of the first mixed boundary value problem for a hyperbolic equation with memory is formulated and the theorems of the existence and uniqueness of the solution of the problems are proved. Difference analogues of the above mixed problem are constructed. The issues of convergence and stability of a counterflow difference scheme for a hyperbolic equation with memory are investigated. Computational experiments have been carried out. |
| 2.2 | Solutions of boundary value problems for a third-order hyperbolic equation in a characteristic triangle will be investigated. | Solutions of boundary value problems for a third-order hyperbolic equation in a characteristic triangle are investigated. The issues of solvability of a number of local problems for a hyperbolic equation of the third order, when the entire boundary of the domain is a characteristic of this equation, are studied. Sufficient conditions for unambiguous solvability of the Dirichlet type problem for the above equation are found. Examples are given showing the significance of the conditions found for the unambiguous solvability of the problem. |
| 2.3 | Study of the solvability of a local problem for a mixed parabolic-hyperbolic equation of fractional order in a characteristic triangle. | Work is underway to study the solvability of a local problem for a mixed parabolic-hyperbolic fractional order equation in a characteristic triangle. The properties of an analogue of the first boundary value problem of the parabolic part of the domain for a mixed parabolic-hyperbolic equation of fractional order are investigated. Sufficient conditions have been found for these problems, which ensure the existence and uniqueness of the solution of the formulated problems. |
Composition of the research group:
| № | Full name | Position | Academic degree |
| 1 | Байшемиров Жарасбек Дуйсембекович. | руководитель проекта, ГНС | PhD, ассоц.проф., . Scopus AU-ID: 55817472800; ORCID: 0000-0002-4812-4104 Web of Science Researcher ID: AAD-8778-2021 |
| 2 | Ахтаева НазгульСмайлхановна. | Главный научный сотрудник (ГНС) | PhD, и.о. ассоц.проф. |
| 3 | Рысқан Айнұр Рысқанқызы. | Научный сотрудник (НС) | PhD, ст.преп. Scopus AU-ID: 57209534432 ORCID: 0000-0002-8764-4751 |
| 4 | Абдираманов Жанарс Алда-Онгарович. | Научный сотрудник (НС) | PhD докторант. Scopus AU-ID: 57219802387 |
| 5 | Баймурзаев Джомарт Джумаханович. | Научный сотрудник (НС) | магистр. |
| 6 | Адил Наурызбай. | Научный сотрудник (НС) | PhD докторант. |
List of publications for 2022.
1. Baishemirov Zh., Berdyshev A., Ryskan A. Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation. // Mathematics. 2022. 10(7), 1094; Base: Scopus, SJR – 0.495, CiteScore–2.2; Percentile – 80. Database: Web of Science IF – 2,258; Q1. https://doi.org/10.3390/math10071094
2. Adil N., Berdyshev A.,Eshmatov B., Baishemirov Zh. Solvability and Volterra property of nonlocal problems for mixed fractional order diffusion-wave equation. // Boundary Value Problem. Database: Web of Science IF – 1,793; Q1. (Printed).
3. Berdyshev A.S., Abdiramanov J.A. A difference analogue of a mixed problem for a hyperbolic equation with memory // Computational Models and Technologies: Abstracts of the Uzbekistan-Malaysia international conference, Editor-in-chief: Khudoyberganov M.U., September 16-17, 2022, Tashkent, Uzbekistan.-P. 39.
4. Adil N., Berdyshev A.S., Eshmatov B.E. Solvability of a nonlocal problem for a fractional order wave equation // Computational Models and Technologies: Abstracts of the Uzbekistan-Malaysia international conference, Editor-in-chief: Khudoyberganov M.U., September 16-17, 2022, Tashkent, Uzbekistan.-P. 62.
5. Berdyshev A.S., Abdiramanov Zh.A., Shavkaeva E.E. Problems with the Bitsadze-Samarsky condition for a linear hyperbolic equation with memory // Computational Models and Technologies: Abstracts of the Uzbekistan-Malaysia international conference, Editor-in-chief: Khudoyberganov M.U., September 16-17, 2022, Tashkent, Uzbekistan.- P. 93.
