On eigenfunctions and eigenvalues of some boundary value problems for a nonlocal biharmonic operator
Anahtar Kelimeler:
Spectral problem, nonlocal operator, biharmonic operator, Dirichlet problem, Neumann problem, Samarsky-Ionkin problem, eigenfunctions, eigenvalues, attached functions, completeness.Özet
In this note, the concept of a nonlocal biharmonic operator is introduced. When
introducing this operator, mappings of the type of involution are used. Namely, in the differential
expression of this operator, in addition to the variables x (x1, x2 ,..., xn ) , transformed arguments
with mappings of the form 1 1 1 ,..., , , ,..., ,1 j j j j j n S x x x p x x x j n and their multiplication
also involved. Spectral problems with Dirichlet and Neumann-type boundary conditions are
considered in an n-dimensional parallelepiped for a given nonlocal biharmonic operator. The
eigenfunctions and eigenvalues of the problems under consideration are explicitly constructed.
When constructing these elements, eigenfunctions and eigenvalues of the classical biharmonic
operator with Dirichlet and Neumann type boundary conditions are essentially used. Theorems on
the orthonomization and completeness of the systems of eigenfunctions of the problems under
consideration are proved. Examples of the corresponding parameters for special cases involved in
the problems under consideration are given. In addition, in the two-dimensional case for the
corresponding nonlocal biharmonic operator, spectral issues of boundary value problems of the
Samarsky-Ionkin type are also investigated. The proper and attached functions of the problem under
consideration are found and theorems on the completeness of these systems are proved.
Keywords: Spectral problem, nonlocal operator, biharmonic operator, Dirichlet
Referanslar
Karachik V.V., Sarsenbi A.M., Turmetov B.Kh. On the solvability of the main boundary
value problems for a nonlocal Poisson equation// Turkish journal of mathematics. – 2019.
– Vol.43, No.3. – P.1604 – 1625. doi:10.3906/mat-1901-71
Турметов Б.Х., Карачик В. В. О разрешимости краевых задач Дирихле и Неймана
для уравнения Пуассона с множественной инволюцией // Вестник Удмуртского
университета. Математика. Механика. Компьютерные науки. – 2021. – Т.31, № 4. –
P. 651 – 667. DOI: 10.35634/vm210409.
Turmetov B., Karachik V. On Eigenfunctions and Eigenvalues of a Nonlocal Laplace
Operator with Multiple Involution // Symmetry. – 2021. – Vol.13, No. 1781. – P. 1 – 20.
https://doi.org/ 10.3390/ sym13101781.
Turmetov B.Kh., Karachik V.V. Solvability of nonlocal Dirichlet problem for generalized
Helmholtz equation in a unit ball// Complex Variables Elliptic Equation. – 2023. –
Vol.68, No.7. – P. 1204–1218. DOI: 10.1080/17476933.2022.2040021.
Yarka U., Fedushko S., Vesely P. The Dirichlet Problem for the Perturbed Elliptic
Equation// Mathematics. – 2020. – Vol.8, No.2108. – P. 1 – 13.
doi:10.3390/math8122108.
Turmetov B.K., Kadirkulov B.J. On the solvability of an initial-boundary value problem
for a fractional heat equation with involution// Lobachevskii Journal of Mathematics. –
– Vol.43, No.1. – P. 249 – 262. doi.org/10.1134/S1995080222040217.
Турметов Б.Х., Кадиркулов Б.Ж. О разрешимости некоторых краевых задач для
дробного аналога нелокального уравнения Лапласа// Итоги науки и техники. Серия
«Современная математика и ее приложения. Тематические обзоры». – 2022. – Т.211. – С.14 – 28. DOI: https://doi.org/10.36535/0233-6723-2022-211-14-28.
Турметов Б., Шалхар А. О спектральных вопросах некоторых краевых задач для нелокального оператора Лапласа в прямоугольнике// Известия Международного казахско-турецкого университета имени Х.А. Ясави. Серия Математика, Физика, Информатика. – 2022. No.1. – P. 79 – 96.
Aziz S., Malik S.A. Identification of an unknown source term for a time fractional fourth-order parabolic equation// Electronic journal of differential equations. – 2016. – 2016, No.293.– P.1–20.
Kerbal S., Kadirkulov B.J., Kirane M. Direct and Inverse Problems for a Samarskii-Ionkin Type Problem for a Two-Dimensional Fractional Parabolic Equation// Progress in Fractional Differentiation and Applications. – 2018. –Vol. 4, No.3. –P.147–160. doi:10.18576/pfda/040301.
Muratbekova, M., Kadirkulov B., Koshanova M., Turmetov B. On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator// Fractal and Fractional. – 2023. – Vol.7, No.404. – P.1–18. https://doi.org/10.3390/fractalfract7050404.
Михлин С. Г. Линейные уравнения в частных производных. Учебное пособие для вузов.М:, «Высшая школа». 1977. 431 с. 13. Владимиров В.С.Уравнения математической физики. М:, Наука, 1988. 512с.
Ионкин Н.И. Решение одной краевой задачи теории теплопроводности с неклассическим краевым условием//Дифференциальные уравнения. – 1977. – Т.13, № 2 –С.294 – 304.
Ионкин Н.И., Морозова В. А. Двумерное уравнение теплопроводности с нелокальными краевыми условиями//Дифференциальные уравнения. – 2000, – Т.36, № 7, – С. 884–888. DOI:https://doi.org/10.1007/BF02754498.
Karachik V.V., Sarsenbi A.M., Turmetov B.Kh. On the solvability of the main boundary value problems for a nonlocal Poisson equation// Turkish journal of mathematics. – 2019. – Vol.43, No.3. – P.1604 – 1625. doi:10.3906/mat-1901-71
Turmetov B. Kh., Karachik V. V. On the solvability of Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution// Vestnik Udmurt University. Mathematics. Mechanics. Computer science. – 2021. – Т. 31, № 4. – P. 651 – 667. DOI: 10.35634/vm210409. [In Russian].
Turmetov B., Karachik V. On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution // Symmetry. – 2021. – Vol.13, No. 1781. – P. 1 – 20. https://doi.org/ 10.3390/ sym13101781.
Turmetov B.Kh., Karachik V.V. Solvability of nonlocal Dirichlet problem for generalized Helmholtz equation in a unit ball// Complex Variables Elliptic Equation. – 2023. – Vol.68, No.7. – P. 1204–1218. DOI: 10.1080/17476933.2022.2040021.
Yarka U., Fedushko S., Vesely P. The Dirichlet Problem for the Perturbed Elliptic Equation// Mathematics. – 2020. – Vol.8, No.2108. – P. 1 – 13. doi:10.3390/math8122108.
Turmetov B.K., Kadirkulov B.J. On the solvability of an initial-boundary value problem for a fractional heat equation with involution// Lobachevskii Journal of Mathematics. – 2022. – Vol.43, No.1. – P. 249 – 262. doi.org/10.1134/S1995080222040217.
Turmetov B.Kh., Kadirkulov B.Zh. On the solvability of some boundary value problems for the fractional analogue of the nonlocal Laplace equation // Results of science and technology. The series «Modern Mathematics and its applications. Thematic reviews». – 2022. – Т.211. – P.14 – 28. DOI: https://doi.org/10.36535/0233-6723-2022-211-14-28. [In Russian].
Turmetov B., Shalkhar A. On spectral questions of some boundary value problems for a non-local Laplace operator in a rectangle // Proceedings of the International Kazakh-Turkish University named after H.A. Yasavi. Series Mathematics, Physics, Computer Science.– 2022. No.1. – P. 79 – 96. [In Russian].
Aziz S., Malik S.A. Identification of an unknown source term for a time fractional fourth-order parabolic equation// Electronic journal of differential equations. – 2016. – 2016, No.293.– P.1–20.
Kerbal S., Kadirkulov B.J., Kirane M. Direct and Inverse Problems for a Samarskii-Ionkin Type Problem for a Two-Dimensional Fractional Parabolic Equation// Progress in Fractional Differentiation and Applications. – 2018. –Vol. 4, No.3. –P.147–160. doi:10.18576/pfda/040301.
Muratbekova, M., Kadirkulov B., Koshanova M., Turmetov B. On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator// Fractal and Fractional. – 2023. – Vol.7, No.404. – P.1–18. https://doi.org/10.3390/fractalfract7050404.
Mikhlin S. G. Linear partial differential equations. Study guide for universities.М:, «High School ». 1977. 431 p. [In Russian]. 13. Vladimirov V.S. Equations of mathematical physics. М:, The science, 1988. 512p. [In Russian].
Ionkin N.I. Solution of one boundary value problem of the theory of thermal conductivity with a non-classical boundary condition // Differential equations. – 1977. – Т.13, № 2 –P.294 – 304. [In Russian].
Ionkin N.I., Morozova V. A. Two-dimensional heat equation with non-local boundary conditions // Differential equations. –2000, –Т.36, №7, –P. 884–888. DOI:https://doi.org/10.1007/BF02754498. [In Russian].
