Grant funding of scientific and technological research for 2020-2022 with a period of implementation of 12 months

AP08955795 «Boundary value problems for the heat equation with load of fractional order»

Актуальность.

In recent years, we have seen an increase in the number of studies of various boundary value problems for loaded equations; the distinguishing feature of this problem is the presence of fractional integro-differentiation operators in the boundary conditions. Of interest are boundary value problems for the loaded heat equation, when the loaded term is represented in the form of a fractional derivative. Currently, such problems are not fully investigated.

Цель. Statement and study of the solvability of boundary value problems for a fractionally loaded heat equation in certain functional classes; reducing the boundary value problems to Volterra integral equations with kernels containing special functions; study of the limiting cases of the derivative order in the loaded term of the equation.

Ожидаемые и достигнутые результаты.

Boundary value problems for the fractionally loaded heat equation are set (the loaded term of the equation is represented as a Caputo’s fractional derivative). The posed boundary value problem are reduced to a Volterra integral equation of the second kind with singularities in a kernel or with a kernel containing special functions;

investigation of the limiting cases of the fractional derivative order of the term with the load in the heat equation of the boundary value problem.

Имена и фамилии членов исследовательской группы с их идентификаторами (Scopus Author ID, Researcher ID, ORCID,  если имеются и ссылками на соответствующие профили.

Список публикаций (со ссылками на них) и патентов

Abstracts abroad

1 Kosmakova M.T., Kasymova L.Zh. To solving the heat equation with a fractional load // Marchuk Scientific Readings – 2020: Abstracts of Intern. Conf., dedic. to the 95th anniversary of the birthday of Academician Guri I. Marchuk / Institute comp. mathematics and math. geophysics SB RAS. ‒ Novosibirsk: PPC NSU, 2020. – P. 16. [in Russian]

http://conf.nsc.ru/files/conferences/msr2020/616453/%D0%A2%D0%B5%D0%B7%D0%B8%D1%81%D1%8B%202020%20DOI_final.pdf

2 Kosmakova M.T., Tuleutaeva Zh.M. To solving the heat equation in a degenerating two-dimensional domain // Marchuk Scientific Readings – 2020: Abstracts of Intern. Conf., dedic. to the 95th anniversary of the birthday of Academician Guri I. Marchuk / Institute comp. mathematics and math. geophysics SB RAS. ‒ Novosibirsk: PPC NSU, 2020. – P. 16-17. [in Russian]

http://conf.nsc.ru/files/conferences/msr2020/616453/%D0%A2%D0%B5%D0%B7%D0%B8%D1%81%D1%8B%202020%20DOI_final.pdf

3 Akhmanova D.M., Kosmakova M.T., Shamatayeva N.K. About boundary problem for essential loaded heat equations // Marchuk Scientific Readings – 2020: Abstracts of Intern. Conf., dedic. to the 95th anniversary of the birthday of Academician Guri I. Marchuk / Institute comp. mathematics and math. geophysics SB RAS. ‒ Novosibirsk: PPC NSU, 2020. – P. 4. [in Russian]

http://conf.nsc.ru/files/conferences/msr2020/616453/%D0%A2%D0%B5%D0%B7%D0%B8%D1%81%D1%8B%202020%20DOI_final.pdf

2021

ИРН, Название проекта.

AP08955795 «Boundary value problems for the heat equation with load of fractional order»

Актуальность.

A wide range of non-classical models of mathematical physics is represented by equations that include the values of the required function and its high-order fractional derivatives on some manifolds from its domain. However, for these problems a rigorous mathematical theory has not yet been created. Within the framework of the project, there are reducing the solvability of the studied boundary value problems to the study of the solvability of Volterra integral equations of the second kind with singularities in the kernel or with kernels containing special functions; and, analyzing the solution of boundary value problems for the heat equation with a fractional load. The boundary value problem includes the loaded term in the form of a fractional derivative, and the kernel of the resulting integral equation contains special functions.

Цель. Statement and study of the solvability of boundary value problems for a fractionally loaded heat equation in certain functional classes; reducing the boundary value problems to Volterra integral equations with kernels containing special functions; solvability investigation of the integral equations depending both on the order of the fractional derivative in the loaded term of the initial boundary value problem and on the behavior character of the load.

Ожидаемые и достигнутые результаты.

Limiting cases of the fractional derivative order of the term with a load in the heat equation of the boundary value problem are studied. The theorems on the existence and uniqueness of solutions to boundary value problems or related integral equations in certain functional classes are established depending on both the intervals of changing the fractional derivative order in the loaded term of the initial boundary value problems and on the behavior character of the load.

Relationship between singularities of the integral equation kernel and the differential part nature of the equation for the formulated boundary-value heat conduction problem are shown when studying the solvability of integral equations.

Имена и фамилии членов исследовательской группы с их идентификаторами (Scopus Author ID, Researcher ID, ORCID,  если имеются и ссылками на соответствующие профили.

Список публикаций (со ссылками на них) и патентов

Abstracts abroad

1 Kosmakova M., Kasymova L. On the solvability of a fractionally loaded heat conduction problem // Modern methods of function theory and related problems: Abstracts of Intern. Conf. – Voronezh, 2021. – P. 160-165. https://vzmsh.math-vsu.ru/files/vzmsh2021.pdf  [in Russian]

2 Kazhkenova N., Orumbayeva N. On a nonlinear boundary value problem for differential equations in partial derivatives of the third order // Modern methods of function theory and related problems: Abstracts of Intern. Conf. – Voronezh, 2021. – P. 133-134. https://vzmsh.math-vsu.ru/files/vzmsh2021.pdf [in Russian]

Abstracts in the Republic of Kazakhstan

1 Kosmakova M.T., Kasymova L.Zh. Dirichlet problem for the heat equation with fractional load // Annual Intern. April Mathematical Conf. in honor of the Day of Science Workers of the Republic of Kazakhstan, dedic. to the 75th anniversary of Acad. NAS RK Kalmenov Т.Sh.: abstracts. – Almaty, 2021. – P. 37. [in Russian]

http://www.math.kz/public/filemanager/userfiles/Abstracts_of_conference_2021.pdf

Articles abroad

Kosmakova M.T., Ramazanov М.I., Kasymova L.Zh. To Solving the Heat Equation with Fractional Load / М.I. Ramazanov, М.Т. Kosmakova, L.Zh. Kasymova // Lobachevskii Journal of Mathematics — 2021. — Vol. 42, No. 12. — P. 2854 - 2866 (in Press). (Scopus 50%, 2019; 48%, 2020), DOI: 10.1134/S1995080221120210

Articles in the Republic of Kazakhstan

1 Kosmakova M.T., Iskakov S.A., Kasymova L.Zh. To solving the fractionally loaded heat equation // Bulletin of the Karaganda University. Mathematics Series. – Karaganda, 2021. – No. 1 (101). - P. 65-77. DOI 10.31489/2021M1/65-77 (Web of science)

https://mathematics-vestnik.ksu.kz/apart/2021-101-1/7.pdf

2 Orumbayeva N.T., Keldibekova A.B. On a solution of a nonlinear semi-periodic boundary value problem for a third-order pseudoparabolic equation // Kazakh Mathematical Journal. Almaty, Kazakhstan. - 2020. – No. 20 (4) - P. 119-132.

http://www.math.kz/media/journal/journal2021-01-2344842.pdf

АР08956033 Boundary value problems of heat conduction in degenerating domains with special boundary conditions.

Relevance: The need to solve boundary value problems for equations of non-stationary transfer in domains with boundaries that change with time is explained by the fact that these problems have a wide practical application. Such problems are of great practical value for studying the thermal effects during crack propagation, when a constant temperature is set on the banks of a propagating crack (a domain with a moving boundary), leading to the destruction of materials, mechanisms, or aircraft; in studies of freezing solutions, soils; in the study of kinetic crystal growth.

Project objective: The goal of the project is to formulate and study the solvability of boundary value problems for the heat equation in non-cylindrical domains, namely, in domains bounded by a conical surface. A feature of the problems under consideration is: changing the boundary of the domain with changing time; degeneration of the domain to a point at the initial moment of time; in the boundary conditions specified on the surface of the cone contains the total derivative with respect to the time variable.

Achieved results:

Statement and transformation of boundary value problems for the two-dimensional spatial variable heat equation with a boundary condition containing the total time derivative on the moving boundary of the domains. Construction of the Green's function. Integral representations of the solution of boundary value problems. Reduction of problems to complete singular integral equations of Volterra type of the second kind.

Construction and solution of the corresponding characteristic singular integral equations of Volterra type of the second kind. Finding resolvents. Estimation of the resolvent of the characteristic equation. Proof of the solvability theorems for characteristic integral equations.

Solving complete Volterra-type singular integral equations of the second kind using the regularization method by solving the characteristic integral equation, that is, the Carleman-Vekua regularization method. Definition of uniqueness classes for a solution. Proof of solvability theorems for complete singular integral equations.

Solving the initial boundary value problem. Determination of uniqueness classes for solutions to the initial problem. Proof of theorems on the solvability of a boundary value problem.

Names and surnames of the members of the research group with their identifiers (Scopus Author ID, Researcher ID, ORCID)

List of publications:

1 Ramazanov M. I., Gulmanov N. K. On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain // Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki. - 2021. - Vol. 31, N 2. - P. 241-252. https://doi.org/10.35634/vm210206.

2 Amangaliyeva M., Jenaliyev M., Iskakov S., Ramazanov M. On a boundary value problem for the heat equation and a singular integral equation associated with it // Applied Mathematics and Computation. - 2021. - Vol. 399. - P. 1-14. https://doi.org/10.1016/j.amc.2021.126009.

3 Ramazanov M. I., Jenaliyev M. T., Tanin A. O. Two-dimensional boundary value problem of heat conduction in a cone with special boundary conditions // Lobachevskii Journal of Mathematics. – 2021. - Vol. 42, N 12. - P. 2913-2925. https://doi.org/10.1134/S1995080221120271. (В печати).

4 Jenaliyev M.T., Ramazanov M.I., Attaev A.Kh.,. Gulmanov N.K. Stabilization of a solution for two-dimensional loaded parabolic equation // Bulletin of the Karaganda University. Mathematics series. - 2020. - No. 4 (100). - P. 55-70. DOI 10.31489/2020M4/55-70.

5 Jenaliyev M.T., Ramazanov M.I., Tanin A.O. To the solution of the Solonnikov-Fasano problem with boundary moving on arbitrary law x = γ(t) // Bulletin of the Karaganda University. Mathematics series. - 2021. - No. 1 (101). - P. 37-49. DOI 10.31489/2021M1/37-49.

6 Дженалиев М.Т., Гульманов Н.К., Рамазанов М.И. К решению одного особого интегрального уравнения типа Вольтерра второго рода // Современные методы теории функций и смежные проблемы: материалы Международной конференции. - Воронеж, 2021. - С. 109.

7 Дженалиев М.Т., Искаков С.А., Гульманов Н.К. Решение краевой задачи в угловой области, симметричной относительно временной оси // Современные методы теории функций и смежные проблемы: материалы Международной конференции. - Воронеж, 2021. - С. 110.

8 Дженалиев М.Т., Рамазанов М.И., Танин А.О. К решению задачи Солонникова-Фазано при движении границы по произвольному закону x = γ(t) // Современные методы теории функций и смежные проблемы: материалы Международной конференции. - Воронеж, 2021. - С. 111.

9 Рамазанов М.И., Дженалиев М.Т., Танин А.О. Двумерная граничная задача теплопроводности в конусе со специальными граничными условиями // Проблемы современной фундаментальной и прикладной математики: Тезисы докладов Международной научно-практической конференции. - Нур-Султан, 2021. - С. 137-138.

10 Рамазанов М.И., Гульманов Н.К. Сингулярное интегральное уравнение Вольтерра краевой задачи теплопроводности в вырождающихся областях // Традиционная Международная апрельская математическая конференция в честь Дня работников науки Республики Казахстан, посвященная 75-летию академика НАН РК Тынысбека Шариповича Кальменова: Тезисы докладов. - Алматы, 2021. - С. 50.