On Determination of Minor Coefficient in a Parabolic Equation of the Second Order
E.I. Safonov Ugra State University, Khanty-Mansyisk, Russian Federation E-mail: dc.gerz.hd@gmail.com. Е.И. Сафонов Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: dc.gerz.hd@gmail.com An inverse problem of recovering the minor time-dependent coefficient in a par...
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ftsusuniv:oai:oai:dspace.susu.ru:0001.74/1234:0001.74/27079 2023-05-15T17:02:58+02:00 On Determination of Minor Coefficient in a Parabolic Equation of the Second Order Об определении младшего коэффициента в параболическом уравнении второго порядка Safonov, E.I. Сафонов, Е.И. 2018 application/pdf http://dspace.susu.ru/xmlui/handle/0001.74/27079 https://doi.org/10.14529/mmph180404 other unknown Издательский центр ЮУрГУ Вестник ЮУрГУ. Серия Математика. Механика. Физика Vestnik Ûžno-Ural’skogo gosudarstvennogo universiteta. Seriâ Matematika. Mehanika. Fizika Bulletin of SUSU Математика. Механика. Физика;Том 10 Safonov E.I. On Determination of Minor Coefficient in a Parabolic Equation of the Second Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics. 2018, vol. 10, no. 4, pp. 30-40. DOI:10.14529/mmph180404. Сафонов, Е.И. Об определении младшего коэффициента в параболическом уравнении второго порядка / Е.И. Сафонов // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. 2018. Т. 10, № 4. С. 30-40. DOI:10.14529/mmph180404 2075-809Х 2409-6547 http://dspace.susu.ru/xmlui/handle/0001.74/27079 doi:10.14529/mmph180404 УДК 519.633.2 MSC 35K10 finite element method parabolic equation inverse problem метод конечных элементов параболическое уравнение обратная задача Article 2018 ftsusuniv https://doi.org/10.14529/mmph180404 2022-10-21T14:27:57Z E.I. Safonov Ugra State University, Khanty-Mansyisk, Russian Federation E-mail: dc.gerz.hd@gmail.com. Е.И. Сафонов Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: dc.gerz.hd@gmail.com An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considered. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this timedependent coefficient using data of the initial-boundary value problem and point conditions of overdetermination. Cases of the Dirichlet boundary conditions and oblique derivative conditions are considered. Conditions under which the theorem of existence and solution uniqueness is applicable for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element method and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geometry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the finite difference method. Results of numerical experiments are presented, and numerical solution of the model inverse problem is constructed in the case of Neumann boundary conditions; dependency of an error in calculation of the controlling parameter on the variation of the equation coefficients and the noise level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations show a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, ... Article in Journal/Newspaper khanty South Ural State University: Electronic Archive Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 10 4 30 40 |
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South Ural State University: Electronic Archive |
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УДК 519.633.2 MSC 35K10 finite element method parabolic equation inverse problem метод конечных элементов параболическое уравнение обратная задача |
spellingShingle |
УДК 519.633.2 MSC 35K10 finite element method parabolic equation inverse problem метод конечных элементов параболическое уравнение обратная задача Safonov, E.I. Сафонов, Е.И. On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
topic_facet |
УДК 519.633.2 MSC 35K10 finite element method parabolic equation inverse problem метод конечных элементов параболическое уравнение обратная задача |
description |
E.I. Safonov Ugra State University, Khanty-Mansyisk, Russian Federation E-mail: dc.gerz.hd@gmail.com. Е.И. Сафонов Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: dc.gerz.hd@gmail.com An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considered. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this timedependent coefficient using data of the initial-boundary value problem and point conditions of overdetermination. Cases of the Dirichlet boundary conditions and oblique derivative conditions are considered. Conditions under which the theorem of existence and solution uniqueness is applicable for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element method and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geometry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the finite difference method. Results of numerical experiments are presented, and numerical solution of the model inverse problem is constructed in the case of Neumann boundary conditions; dependency of an error in calculation of the controlling parameter on the variation of the equation coefficients and the noise level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations show a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, ... |
format |
Article in Journal/Newspaper |
author |
Safonov, E.I. Сафонов, Е.И. |
author_facet |
Safonov, E.I. Сафонов, Е.И. |
author_sort |
Safonov, E.I. |
title |
On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
title_short |
On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
title_full |
On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
title_fullStr |
On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
title_full_unstemmed |
On Determination of Minor Coefficient in a Parabolic Equation of the Second Order |
title_sort |
on determination of minor coefficient in a parabolic equation of the second order |
publisher |
Издательский центр ЮУрГУ |
publishDate |
2018 |
url |
http://dspace.susu.ru/xmlui/handle/0001.74/27079 https://doi.org/10.14529/mmph180404 |
genre |
khanty |
genre_facet |
khanty |
op_relation |
Вестник ЮУрГУ. Серия Математика. Механика. Физика Vestnik Ûžno-Ural’skogo gosudarstvennogo universiteta. Seriâ Matematika. Mehanika. Fizika Bulletin of SUSU Математика. Механика. Физика;Том 10 Safonov E.I. On Determination of Minor Coefficient in a Parabolic Equation of the Second Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics. 2018, vol. 10, no. 4, pp. 30-40. DOI:10.14529/mmph180404. Сафонов, Е.И. Об определении младшего коэффициента в параболическом уравнении второго порядка / Е.И. Сафонов // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. 2018. Т. 10, № 4. С. 30-40. DOI:10.14529/mmph180404 2075-809Х 2409-6547 http://dspace.susu.ru/xmlui/handle/0001.74/27079 doi:10.14529/mmph180404 |
op_doi |
https://doi.org/10.14529/mmph180404 |
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