Closedness of the Biharmonic System of Basic Potentials

Authors

  • Markovsky A.N. Kuban State University, Krasnodar, Российская Федерация

UDC

517.518.32+517.956.224+519.635.1

DOI:

https://doi.org/10.31429/vestnik-17-1-2-20-26

Abstract

The shift systems of the fundamental solution of biharmonic equation are considered. The shift sequence belongs to the complement of the bounded single-bond domain $Q$, so that all system functions satisfy the biharmonic equation in $Q$. The space of biharmonic functions $G_{2}(Q)$ is introduced as a closure in the $L_{2}(Q)$ norm of the linear shell of all kinds of shifts of the fundamental solution of biharmonic equation. The following problem is considered: what conditions the countable sequence of shifts must have so that the linear shell closure matches the entire $G _ {2} (Q) $ space, or, in other words, when is the considered system complete in $G _ {2} (Q)$? The problem for the harmonic case was considered by V.G. Lezhnev, he introduced a sufficient basis condition and proved the completeness and linear independence of shift systems of the fundamental solution of Laplace's equation.The work generalizes the basis condition to the biharmonic case and proves the closedness and linear independence of biharmonic system. The basis condition is related to the singularity condition of biharmonic functions. The proof relies on the property of continuity of potentials with a biharmonic nucleus and on the Rikier's boundary value problem solution.

Keywords:

harmonic functions, biharmonic functions, complete potential systems, method of basic potentials (MBP), method of fundamental solutions (MFS), projection methods

Author Info

Aleksey N. Markovsky

канд. физ.–мат. наук, доцент кафедры математических и компьютерных методов Кубанского государственного университета

e-mail: mrkvsk@yandex.ru

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Issue

2020. Vol. 17. No. 1. Part 2

Section

Mathematics

Pages

20-26

Submitted

2019-12-08

Published

2020-03-31

How to Cite

Markovsky A.N. Closedness of the Biharmonic System of Basic Potentials. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2020, vol. 17, no. 1, pp. 20-26. DOI: https://doi.org/10.31429/vestnik-17-1-2-20-26 (In Russian)

Copyright (c) 2020 Markovsky A.N.

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