External barycentric coordinates for arbitrary polygons and an approximate method for calculating them
Background. In the article, the concept of external barycentric coordinates is introduced to generalize the applicability of the barycentric method in solving external boundary value and initial boundary value problems of mathematical physics. Aim of the work is to form a simple analytical relation...
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| Format: | Article |
| Language: | English |
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Povolzhskiy State University of Telecommunications & Informatics
2024-12-01
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| Series: | Физика волновых процессов и радиотехнические системы |
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| Online Access: | https://journals.ssau.ru/pwp/article/viewFile/28164/11049 |
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| _version_ | 1832592966582009856 |
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| author | Ivan S. Polyansky |
| author_facet | Ivan S. Polyansky |
| author_sort | Ivan S. Polyansky |
| collection | DOAJ |
| description | Background. In the article, the concept of external barycentric coordinates is introduced to generalize the applicability of the barycentric method in solving external boundary value and initial boundary value problems of mathematical physics. Aim of the work is to form a simple analytical relation that allows calculating barycentric coordinates external to a given arbitrary polygonal area with a given accuracy. Methods. The corresponding ratio is formed when drawing up an approximate analytical calculation rule, which is based on the solution by the Fredholm method of the external Dirichlet problem for the Laplace equation. The basis of this solution is the decomposition of the kernel of the Fredholm integral equation of the second kind by Legendre polynomials of the first and second kind, formed using the Heine formula. Results. The convergence rate of the obtained approximate analytical calculation of the external barycentric coordinates is estimated when establishing exponential convergence in Hilbert space and polynomial convergence in the space of continuous functions. The algorithmic features of the implementation of an approximate analytical solution with a structured representation of pseudocodes of programs for calculating external barycentric coordinates, formed mainly for the MathCad computer algebra system, are clarified. The efficiency is demonstrated by specific examples. Conclusion. The author of the article hopes that the detailed results of the algorithmic implementation of the calculation of external barycentric coordinates will arouse interest and make the publication material more accessible to a wide range of readers, which will lead to the development of the barycentric method in solving boundary and initial boundary value problems of mathematical physics. |
| format | Article |
| id | doaj-art-e1d3661cc1764b9c9416101c8517eea6 |
| institution | Kabale University |
| issn | 1810-3189 2782-294X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Povolzhskiy State University of Telecommunications & Informatics |
| record_format | Article |
| series | Физика волновых процессов и радиотехнические системы |
| spelling | doaj-art-e1d3661cc1764b9c9416101c8517eea62025-01-20T21:08:59ZengPovolzhskiy State University of Telecommunications & InformaticsФизика волновых процессов и радиотехнические системы1810-31892782-294X2024-12-01274293910.18469/1810-3189.2024.27.4.29-398896External barycentric coordinates for arbitrary polygons and an approximate method for calculating themIvan S. Polyansky0https://orcid.org/0000-0002-1282-1522Academy of the Federal Guard Service of the Russian FederationBackground. In the article, the concept of external barycentric coordinates is introduced to generalize the applicability of the barycentric method in solving external boundary value and initial boundary value problems of mathematical physics. Aim of the work is to form a simple analytical relation that allows calculating barycentric coordinates external to a given arbitrary polygonal area with a given accuracy. Methods. The corresponding ratio is formed when drawing up an approximate analytical calculation rule, which is based on the solution by the Fredholm method of the external Dirichlet problem for the Laplace equation. The basis of this solution is the decomposition of the kernel of the Fredholm integral equation of the second kind by Legendre polynomials of the first and second kind, formed using the Heine formula. Results. The convergence rate of the obtained approximate analytical calculation of the external barycentric coordinates is estimated when establishing exponential convergence in Hilbert space and polynomial convergence in the space of continuous functions. The algorithmic features of the implementation of an approximate analytical solution with a structured representation of pseudocodes of programs for calculating external barycentric coordinates, formed mainly for the MathCad computer algebra system, are clarified. The efficiency is demonstrated by specific examples. Conclusion. The author of the article hopes that the detailed results of the algorithmic implementation of the calculation of external barycentric coordinates will arouse interest and make the publication material more accessible to a wide range of readers, which will lead to the development of the barycentric method in solving boundary and initial boundary value problems of mathematical physics.https://journals.ssau.ru/pwp/article/viewFile/28164/11049external barycentric coordinatesexternal dirichlet problemlaplace equationarbitrary polygonlogarithmic potential of the double layerfredholm equationlegendre polynomials |
| spellingShingle | Ivan S. Polyansky External barycentric coordinates for arbitrary polygons and an approximate method for calculating them Физика волновых процессов и радиотехнические системы external barycentric coordinates external dirichlet problem laplace equation arbitrary polygon logarithmic potential of the double layer fredholm equation legendre polynomials |
| title | External barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| title_full | External barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| title_fullStr | External barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| title_full_unstemmed | External barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| title_short | External barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| title_sort | external barycentric coordinates for arbitrary polygons and an approximate method for calculating them |
| topic | external barycentric coordinates external dirichlet problem laplace equation arbitrary polygon logarithmic potential of the double layer fredholm equation legendre polynomials |
| url | https://journals.ssau.ru/pwp/article/viewFile/28164/11049 |
| work_keys_str_mv | AT ivanspolyansky externalbarycentriccoordinatesforarbitrarypolygonsandanapproximatemethodforcalculatingthem |