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

EDN

SBNQSQ

DOI:

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

Funding information

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).

Authors info

  • 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

  • Olga V. Evdokimova

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

  • Olga M Babeshko

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

References

  1. Бабешко В.А., Евдокимова О.В., Бабешко О.М. Топологический метод решения граничных задач и блочные элементы // ДАН. 2013. Т. 449. № 4. С. 657–660. [Babeshko V.A., Evdokimova O.V., Babeshko O.M. Topologicheskiy metod resheniya granichnykh zadach i blochnye elementy [Topological method for solving boundary problems and block elements]. Doklady Akademii nauk [Rep. of the Academy of Sciences], 2013, vol. 449, no. 4, pp. 657–60. (In Russian)]
  2. Бабешко В.А., Евдокимова О.В., Бабешко О.М. О топологических структурах граничных задач в блочных элементах // ДАН. 2016. Т. 470. № 6. С. 650–654. [Babeshko V.A., Evdokimova O.V., Babeshko O.M. O topologicheskikh strukturakh granichnykh zadach v blochnykh elementakh [On topological structures of boundary value problems in block elements]. Doklady Akademii nauk [Rep. of the Academy of Sciences], 2016, vol. 470, no. 6, pp. 650–54. (In Russian)]
  3. Зорич В.А. Математический анализ. Ч. 2. М.: МЦНМО, 2002. 788 с. [Zorich V.A. Matematicheskiy analiz. Ch. 2 [Mathematical analysis. Pt. 2]. MTsNMO, Moscow, 2002. (In Russian)]
  4. Келли Д. Общая топология. М.: Наука, 1968. 384 с. [Kelli D. Obshchaya topologiya [General topology]. Nauka, Moscow, 1968. (In Russian)]
  5. Мищенко А.С., Фоменко А.Т. Краткий курс дифференциальной геометрии и топологии. М.: Физматлит, 2004. 302 с. [Mishchenko A.S., Fomenko A.T. Kratkiy kurs differentsial'noy geometrii i topologii [A short course in differential geometry and topology]. Fizmatlit, Moscow, 2004. (In Russian)]
  6. Голованов Н.Н., Ильютко Д.П., Носовский Г.В., Фоменко А.Т. Компьютерная геометрия. М.: Академия, 2006. 512 с. [Golovanov N.N., Il'yutko D.P., Nosovskiy G.V., Fomenko A.T. Komp'yuternaya geometriya [Computer geometry]. Akademiya, Moscow, 2006. (In Russian)]
  7. Бабешко В.А., Евдокимова О.В., Бабешко О.М. Трещины нового типа и модели некоторых нано материалов // Известия РАН. Механика твердого тела. 2020. № 5. [Babeshko V.A., Evdokimova O.V., Babeshko O.M. Treshchiny novogo tipa i modeli nekotorykh nano materialov [Cracks of a new type and models of some nano materials]. Izvestiya RAN. Mekhanika tverdogo tela [Izvestiya RAN. Rigid body mechanics], 2020, no. 5. (In Russian)]
  8. Новацкий В. Теория упругости. М.: Мир, 1975. 872 с. [Novatskiy V. Teoriya uprugosti [Elasticity theory]. Mir, Moscow, 1975. (In Russian)]
  9. Новацкий В. Динамические задачи термоупругости. М.: Мир, 1970. 256 с. [Novatskiy V. Dinamicheskie zadachi termouprugosti [Dynamic problems of thermoelasticity]. Mir, Moscow, 1970. (In Russian)]
  10. Новацкий В. Электромагнитные эффекты в твердых телах. М.: Мир, 1986. 162 с. [Novatskiy V. Elektromagnitnye effekty v tverdykh telakh [Electromagnetic effects in solids]. Mir, Moscow, 1986. (In Russian)]
  11. Улитко А.Ф. Метод собственных векторных функций в пространственных задачах теории упругости. Киев: Наукова Думка, 1979. 262 с. [Ulitko A.F. Metod sobstvennykh vektornykh funktsiy v prostranstvennykh zadachakh teorii uprugosti [The method of eigenvector functions in spatial problems of elasticity theory]. Naukova Dumka, Kiev, 1979. (In Russian)]
  12. Гринченко В.Т., Улитко А.Ф. Равновесие упругих тел канонической формы. Киев: Наукова Думка, 1985. 280 с. [Grinchenko V.T., Ulitko A.F. Ravnovesie uprugikh tel kanonicheskoy formy [Equilibrium of elastic bodies of canonical form]. Naukova Dumka, Kiev, 1985. (In Russian)]
  13. Головчан В.Т., Кубенко В.Д., Шульга Н.А., Гуз А.Н., Гринченко В.Т. Динамика упругих тел. Киев: Наукова Думка, 1986. 288 с. [Golovchan V.T., Kubenko V.D., Shul'ga N.A., Guz A.N., Grinchenko V.T. Dinamika uprugikh tel [Dynamics of elastic bodies]. Naukova Dumka, Kiev, 1986. (In Russian)]
  14. Гельфанд И.М., Минлос З.А., Шапиро З.Я. Представления группы вращений и группы Лоренца, их применения. М.: Физматлит, 1958. 368 с. [Gel'fand I.M., Minlos Z.A., Shapiro Z.Ya. Predstavleniya gruppy vrashcheniy i gruppy Lorentsa, ikh primeneniya [Representations of the rotation group and the Lorentz group, their applications]. Fizmatlit, Moscow, 1958. (In Russian)]
  15. Кочин Н.Е. Векторное исчисление и начала тензорного исчисления. М.: Наука, 1965. 426 с. [Kochin N.E. Vektornoe ischislenie i nachala tenzornogo ischisleniya [Vector calculus and beginnings of tensor calculus]. Nauka, Moscow, 1965. (In Russian)]

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Issue

Vol. 18 (2021) No. 1

Pages

8-13

Section

Mathematics

Dates

Submitted

February 24, 2021

Accepted

March 1, 2021

Published

March 30, 2021

How to Cite

[1]
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, т. 18, № 1, pp. 8–13. DOI: 10.31429/vestnik-18-1-8-13

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Copyright (c) 2021 Бабешко В.А., Евдокимова О.В., Бабешко О.М.

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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