Exact solutions of the Navier boundary value problem for a biharmonic equation with a special right-hand side in an infinite layer
https://doi.org/10.18384/2949-5067-2023-3-6-14
Abstract
Aim. Purpose is to find exact solutions of the boundary value problem for the biharmonic equation in an infinite 𝑛𝑛-dimensional layer with Navier boundary conditions.
Methodology. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer 𝑥 ∈ Rn, 0 < y < a with Navier boundary conditions. This problem reduces to the sequential solution of two Dirichlet problems for the Poisson equation, the explicit solutions of which were obtained earlier by the authors using the Fourier transform of generalized functions of slow growth.
Results. Exact solutions of the Navier boundary value problem are obtained for a biharmonic equation whose right-hand side is a polyharmonic function in 𝑥𝑥, in particular a polynomial. In this case, the solution is also a polyharmonic function in 𝑥, in particular a polynomial.
Research implications. They consist in obtaining exact solutions of the Navier boundary value problem for a biharmonic equation in an infinite 𝑛𝑛-dimensional layer.
About the Authors
O. Algazin
Bauman Moscow State Technical University
Russian Federation
Russian Federation
Oleg D. Algazin, Cand. Sci. (Phys.-Math.), Assoc. Prof.
Department of Computational Mathematics and Mathematical Physics
105005
ulitsa Vtoraya Baumanskay 5 build. 1
Moscow
A. Kopaev
Bauman Moscow State Technical University
Russian Federation
Russian Federation
Anatoliy V. Kopaev, Cand. Sci. (Phys.-Math.), Assoc. Prof.
Department of Higher Mathematics
105005
ulitsa Vtoraya Baumanskay 5 build. 1
Moscow
References
1. Sweers G. A survey on boundary conditions for the biharmonic. In: Complex Variables and Elliptic Equations, 2009, vol. 54, iss. 2, pp. 79–93. DOI: 10.1080/17476930802657640.
2. Gazzola F., Grunau H., Sweers G. Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Berlin, Springer, 2010. 423 p. (Series: Lecture Notes in Mathematics. Vol. 1991).
3. Meleshko V. V. Selected topics in the history of the two-dimensional biharmonic problem. In: Applied Mechanics Reviews, 2003, vol. 56, iss. 1, pp. 33–85. DOI: 10.1115/1.1521166.
4. Matevossian O. A. [On solutions of the Neumann problem for the biharmonic equation in unbounded domains]. In: Matematicheskiye zametki [Mathematical Notes], 2015, vol. 98, no. 6, pp. 944–947. DOI: 10.4213/mzm10980.
5. Karachik V. V., Torebek B. T. [On the Dirichlet – Riquier problem for biharmonic equations]. In: Matematicheskiye zametki [Mathematical Notes], 2017, vol. 102, no. 1, pp. 39–51. DOI: 10.4213/mzm11035.
6. Zveryayev Ye. M., Kovalenko M. D., Abrukov D. A., Menshova I. V., Kerzhayev A. P. [Examples of exact solutions for problems of bending plate with free face planes]. In: Preprinty IPM im. M. V. Keldysha [KIAM Preprint], 2019, no. 46, 17 p. Available at: https://keldysh.ru/papers/2019/prep2019_46.pdf (accessed: 12. 05. 2023). DOI: 10.20948/prepr-2019-46.
7. Algazin O. D., Kopayev A. V. [Solution of the Dirichlet Problem for the Poisson’s Equation in a Multidimensional Infinite Layer]. In: Matematika i Matematicheskoye modelirovaniye (setevoye izdaniye MGTU im. N. E. Baumana) [Mathemathics and Mathematical Modelling (electronic journal of the Bauman MSTU)], 2015, no. 4, pp. 41–53. Available at: https://elpub.ru/elpub-article/mathm/24 (accessed: 12. 05. 2023).
8. Algazin O. D. [Polynomial solutions of the boundary value problems for the Poisson equation in a layer]. In: Matematika i Matematicheskoye modelirovaniye (setevoye izdaniye MGTU im. N. E. Baumana) [Mathemathics and Mathematical Modelling (electronic journal of the Bauman MSTU)], 2017, no. 6, pp. 1–18. Available at: https://elpub.ru/elpub-article/mathm/82 (accesssed: 12.05.2023). DOI: 10.24108/mathm.0517.0000082.
9. Algazin O. D., Kopayev A. V. [The solution of the mixed boundary value problem of Dirichlet - Neumann for the Poisson equation in a multidimensional infinite layer]. In: Vestnik Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N. E. Baumana. Seriya: Yestestvennyye nauki [Herald of the Bauman Moscow State Technical University. Series: Natural Sciences], 2016, no. 3, pp. 42–56. DOI: 10.18698/1812-3368-2016-3-42-56.
10. Algazin O. D., Kopayev A. V. [Solution of a mixed boundary value problem for the Moisil-Teodoresku system in an infinite layer]. In: Vestnik Moskovskogo gosudarstvennogo oblastnogo universiteta. Seriya: Fizika-Matematika [Bulletin of the Moscow Region State University. Series: Physics-Mathematics], 2022, no. 2, pp. 6–16. DOI: 10.18384/2310-7251-2022-2-6-16.
Review
Views:
125
Email this article 