Block element method in solving vector boundary problems using scalar

Authors

  • Babeshko V.A. Kuban State University, Krasnodar, Российская Федерация
  • Evdokimova O.V. Southern Scientific Center, Russian Academy of Science, Rostov-on-Don, Российская Федерация
  • Babeshko O.M. Kuban State University, Krasnodar, Российская Федерация

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-17-4-6-13

Abstract

This paper presents for the first time a solution of a vector boundary value problem decomposed over packed block elements that are solutions of scalar boundary value problems in a non-classical domain. Solutions of a number of vector partial differential equations in continuum mechanics, electromagnetic phenomena, and field theory allow representations in the form of decompositions based on solutions of scalar equations. This approach is convenient for solving problems in the entire space. When solving boundary value problems, the difficulty of applying this approach is the difficulty of satisfying boundary conditions. In a number of classical fields, this can be done and exact solutions to boundary value problems can be obtained. These classic areas include the half-space, the ball, the cylinder, and some areas obtained from views of transformation groups spaces. However, for a number of important areas other than classical ones, such as wedge-shaped ones, this approach has not yet been able to build accurate solutions. In this paper, probably for the first time, this approach is used to construct an exact solution in the first quadrant of a plane boundary value problem of the second kind for dynamic Lamé equations. The solution is compared with the obtained direct application of the block element method to the vector boundary value problem. It is known that the unbounded domain makes it not effective to use numerical methods in this boundary value problem. The solution is constructed using the block element method under arbitrary boundary conditions. This makes it possible to study different properties of solutions by changing the effects on the boundary.

Keywords:

boundary value problems, block element method, packed block elements, Lame and Helmholtz equations

Acknowledgement

This work was supported by the Russian Foundation for Basic Research (projects 19-41-230003, 19-41-230004, 19-48-230014, 18-08-00465, 18-01-00384, 18-05-80008), the GZ UNC RAS reg. 01201354241 (project 00-20-13), Ministry of the science Russian Federation (project FZEN-2020-0022) charged on 2020 year.

Author Infos

Vladimir A. Babeshko

академик РАН, д-р физ.-мат. наук, заведующий кафедрой математического моделирования Кубанского государственного университета, директор Научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета, заведующий лабораторией Южного федерального университета

e-mail: babeshko41@mail.ru

Olga V. Evdokimova

д-р физ.-мат. наук, главный научный сотрудник Южного научного центра РАН

e-mail: evdokimova.olga@mail.ru

Olga M. Babeshko

д-р физ.-мат. наук, главный научный сотрудник научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета

e-mail: babeshko49@mail.ru

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Issue

2020. Vol. 17. No. 4

Section

Mechanics

Pages

6-13

Submitted

2020-11-24

Published

2020-12-27

How to Cite

Babeshko V.A., Evdokimova O.V., Babeshko O.M. Block element method in solving vector boundary problems using scalar. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2020, vol. 17, no. 4, pp. 6-13. DOI: https://doi.org/10.31429/vestnik-17-4-6-13

Copyright (c) 2020 Babeshko V.A., Evdokimova O.V., Babeshko O.M.

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