Abstract:
One of widespread approaches to solving the Cauchy problem for the Laplace equation is to reduce it to the inverse problem. As a rule, an iterative procedure to solve the latter is used. In this study, an efficient direct method for the numerical solution of the inverse problem in the rectangular form is described. The main idea is based on the expansion of the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of the Laplace operator.
Key words:
Cauchy problem for Laplace equation, inverse problem, numerical solution, efficient direct method.
This work was supported by Basic Research Program of the Department of Mathematical Sciences of
the Russian Academy of Sciences on Modern Computational and Informational Technologies for Large
Problems and RAS Presidium under Program on Intelligent Information Technologies, Mathematical
Modeling, System Analysis, and Automatization.
Citation:
S. B. Sorokin, “An efficient direct method for the numerical solution to the Cauchy problem for the Laplace equation”, Sib. Zh. Vychisl. Mat., 22:1 (2019), 99–117; Num. Anal. Appl., 12:1 (2019), 87–103
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\paper An efficient direct method for the numerical solution to the Cauchy problem for the Laplace equation
\jour Sib. Zh. Vychisl. Mat.
\yr 2019
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\issue 1
\pages 99--117
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\jour Num. Anal. Appl.
\yr 2019
\vol 12
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\pages 87--103
\crossref{https://doi.org/10.1134/S1995423919010075}
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Linking options:
https://www.mathnet.ru/eng/sjvm703
https://www.mathnet.ru/eng/sjvm/v22/i1/p99
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N. E. Sibiryakov, D. Yu. Kochkin, O. A. Kabov, A. L. Karchevsky, “Determination of the heat flux density in the region of the contact line during evaporation of a liquid into a bubble”, J. Appl. Industr. Math., 17:3 (2023), 628–639
S. B. Sorokin, “Difference method for calculating the heat flux at an inaccessible boundary in the problem of heat conduction”, J. Appl. Industr. Math., 17:3 (2023), 651–663
Andrey L. Ushakov, “Investigation of the problem of representation of a linear functional in the form of a scoal product”, Yugra State University Bulletin, 18:3 (2022), 152
A. L. Ushakov, “Analiz zadachi dlya bigarmonicheskogo uravneniya”, J. Comp. Eng. Math., 9:1 (2022), 43–58
A. L. Ushakov, E. A. Meltsaykin, “Analiz bigarmonicheskikh i garmonicheskikh modelei metodami iteratsionnykh rasshirenii”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:3 (2022), 51–66
S. B. Sorokin, “Direct method for solving the inverse coefficient problem
for elliptic equation with piecewise constant coefficients”, J. Appl. Industr. Math., 15:2 (2021), 331–342
A. L. Ushakov, “Chislennyi analiz smeshannoi kraevoi zadachi dlya uravneniya Sofi Zhermen”, J. Comp. Eng. Math., 8:1 (2021), 46–59
A. L. Ushakov, “Analysis of the mixed boundary value problem for the Poisson's equation”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 13:1 (2021), 29–40
A. L. Ushakov, 2020 International Russian Automation Conference (RusAutoCon), 2020, 273
S. B. Sorokin, “An implicit iterative method for numerical solution
of the Cauchy problem for elliptic equations”, J. Appl. Industr. Math., 13:4 (2019), 759–770
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