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Year
2021
Volume
31
Issue
4
Pages
651-667
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Section Mathematics
Title On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution
Author(-s) Turmetov B.Kh.a, Karachik V.V.b
Affiliations Khoja Akhmet Yassawi International Kazakh-Turkish Universitya, South Ural State Universityb
Abstract Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.
Keywords multiple involution, transformation matrix, nonlocal Laplace operator, Poisson equation, Dirichlet problem, Neumann problem
UDC 517.954
MSC 35A09, 35J05, 35J25
DOI 10.35634/vm210409
Received 14 July 2021
Language Russian
Citation Turmetov B.Kh., Karachik V.V. On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 4, pp. 651-667.
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