Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations
In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solu...
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/22/4586 |
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| author | Gurami Tsitsiashvili Alexey Gudimenko Marina Osipova |
| author_facet | Gurami Tsitsiashvili Alexey Gudimenko Marina Osipova |
| author_sort | Gurami Tsitsiashvili |
| collection | DOAJ |
| description | In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T16:38:03Z |
| publishDate | 2023-11-01 |
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| spelling | doaj.art-4a22ee623bad4af290bde5a4ab5559352023-11-24T14:54:06ZengMDPI AGMathematics2227-73902023-11-011122458610.3390/math11224586Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate ObservationsGurami Tsitsiashvili0Alexey Gudimenko1Marina Osipova2Institute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, RussiaInstitute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, RussiaInstitute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, RussiaIn this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.https://www.mdpi.com/2227-7390/11/22/4586reference pointsexperiment planningone-soliton solution |
| spellingShingle | Gurami Tsitsiashvili Alexey Gudimenko Marina Osipova Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations Mathematics reference points experiment planning one-soliton solution |
| title | Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations |
| title_full | Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations |
| title_fullStr | Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations |
| title_full_unstemmed | Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations |
| title_short | Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations |
| title_sort | fast method for estimating the parameters of partial differential equations from inaccurate observations |
| topic | reference points experiment planning one-soliton solution |
| url | https://www.mdpi.com/2227-7390/11/22/4586 |
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