On the properties of topological discretization of solutions to boundary value problems

Authors

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

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-18-1-8-13

Abstract

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems. The constructions made for the homogeneous Helmholtz equation remain valid for the inhomogeneous one.

Keywords:

boundary value problems, block element method, packed block elements, discrete topological spaces, Helmholtz equation

Acknowledgement

Some fragments of the work were carried out as part of the implementation of the State Task for 2021 of the Ministry of Education and Science (project FZEN-2020-0022), UNC RAS (project 00-20-13) No. gosreg. 01201354241, and with the support of RFBR grants (projects 19-41-230003, 19-41-230004, 19-48-230014, 18-05-80008).

Author Infos

Vladimir A. Babeshko

академик РАН, д-р физ.-мат. наук, заведующий кафедрой математического моделирования Кубанского государственного университета, руководитель научных направлений математики и механики Южного научного центра РАН

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

References

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Issue

Section

Mathematics

Pages

8-13

Submitted

2021-02-24

Published

2021-03-30

How to Cite

Babeshko V.A., Evdokimova O.V., Babeshko O.M. On the properties of topological discretization of solutions to boundary value problems. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2021, vol. 18, no. 1, pp. 8-13. DOI: https://doi.org/10.31429/vestnik-18-1-8-13