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Sibirskii Zhurnal Industrial'noi Matematiki, 2012, Volume 15, Number 4, Pages 90–101
(Mi sjim755)
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This article is cited in 7 scientific papers (total in 7 papers)
A numerical method for solving the Dirichlet problem for the wave equation
S. I. Kabanikhina, O. I. Krivorot'kob a Institute of Computational Mathematics and Mathematical Geophysics (Computing Center), Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia
Abstract:
A numerical method for solving the Dirichlet problem for the wave equation in the two-dimensional space is constructed. An analysis of the ill-posedness of the problem is carried out and a reguralization algorthm is constructed. The first step in the regularization of the problem consists in expansion in a Forier series with respect to one of the variables and passage to a finite sequence of Dirichlet problems for the wave equation in the one-dimensional space. Each of the Dirichlet problems obtained for the wave equation in the one-dimensional space is reduced to the inverse problem $Aq=f$ to some direct (correct) problem. We accomplish an analysis of the ill-posedness degree of the inverse problem on the basis of the study of the nature of the decay of the singular values of $A$ and its discrete analog $A_{mn}$. For relatively small values $m$ and $n$, we develop a numerical algorithm for constructing $r$-solutions to the inverse problem. For the general case, we apply an optimization method for solving the inverse problem. The results of numerical calculations are given.
Keywords:
Dirichlet problem, wave equation, ill-posedness degree, singular value decomposition.
Received: 18.06.2012
Citation:
S. I. Kabanikhin, O. I. Krivorot'ko, “A numerical method for solving the Dirichlet problem for the wave equation”, Sib. Zh. Ind. Mat., 15:4 (2012), 90–101; J. Appl. Industr. Math., 7:2 (2013), 187–198
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https://www.mathnet.ru/eng/sjim755 https://www.mathnet.ru/eng/sjim/v15/i4/p90
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