The Dirichlet–Neumann problem for linear nonelliptic partial differential equations with constant coefficients
DOI:
https://doi.org/10.15407/dopovidi2015.02.024Keywords:
constant coefficients, Dirichlet–Neumann problem, linear nonelliptic partial differential equationAbstract
In the domain, which is the Cartesian product of a segment and a multidimensional torus, we study the boundary value-problem with Dirichlet-Neumann conditions with respect to the selected variable and conditions of periodicity with respect to other coordinates for general (regardless of type) linear partial differential equations of a high order with constant coefficients, isotropic in the order of differentiation with respect to independent variables. We establish conditions for the unique solvability of the problem and structurally built the solution in the form of a series in a system of orthogonal functions. To estimate the small denominators arising in the construction of a solution to the problem from below, we use the metric approach.
Downloads
References
Bilusiak N. I., Komarnytska L. I., Ptashnyk B. Y. Ukr. mat. zhurn., 2002, 54, No 12: 1592–1602 (in Ukrainian).
Bobyk I. O., Symotiuk M. M. Visn. Nac. Univ. “Lvivska politekhnika”. Fiz.-mat. nauky, 2010, Iss. 687: 11–19 (in Ukrainian).
Symotiuk M. M. Mat. visnyk NTS, 2005, 2: 199–212 (in Ukrainian).
Ptashnyk B.I. Incorrect boundary value problems for differential equations with partial derivatives, Kiev: Nauk. Dumka, 1984 (in Russian).
Ptashnyk B. Y, Repetylo S. M. Mat. metody ta fiz.-mekh. polia, 2013, 56, No 3: 15–28 (in Ukrainian).
Pavlenko V. N., Petrash T. A. Tr. Int. mathematiki i mekhaniki UrO RAN, 2012, 18, No 2: 199–204 (in Russian).
Gentile G., Mastropietro V., Procesi M. Commun. Math. Phys., 2005, 256, No 2: 437–490. DOI: https://doi.org/10.1007/s00220-004-1255-8
Rudakov I. A. Russian Math., 2007: 51, No 2: 44–52. DOI: https://doi.org/10.3103/S1066369X07020065
Korzyuk V. I., Konopel'ko O. A. Different. Equat., 2010, 46, No 5: 690–701. DOI: https://doi.org/10.1134/S0012266110050083
Zikirov O. S. Math. Modelling and Analysis., 2009, 47, No 3: 407–421. DOI: https://doi.org/10.3846/1392-6292.2009.14.407-421
Gorbachuk V. I., Gorbachuk M. L. Boundary value problems for operator-differential equations, Kiev: Nauk. Dumka, 1984 (in Russian).
Faddeev D. K., Sominskii I. S. Collection of problems on higher algebra. Moscow: Nauka, 1972 (in Russian).
Naimark M. A. Linear differential operators, Moscow: Nauka, 1969 (in Russian).
Sprindzhuk V. G. Metrical theory of the Diophantine approximations, Moscow: Nauka, 1977 (in Russian).
Groshev A. V. Dokl. AN SSSR, 1938, 19, No 3: 151–152 (in Russian).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Reports of the National Academy of Sciences of Ukraine
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
