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A numerical method for solving the Dirichlet problem for the wave equation

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Abstract

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem Aq = f with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator A in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem Aq = f on a uniform grid is reduced to a system of linear algebraic equations A ll q = F. Using the singular value decomposition, we find singular values of the matrix A ll and develop a numerical algorithm for constructing the r-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated r-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.

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Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, pr. Lavrent’eva 6, Novosibirsk, 630090, Russia

    S. I. Kabanikhin

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    S. I. Kabanikhin & O. I. Krivorot’ko

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  1. S. I. Kabanikhin
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  2. O. I. Krivorot’ko
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Correspondence to S. I. Kabanikhin.

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Original Russian Text © S.I. Kabanikhin, O.I. Krivorot’ko, 2012, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2012, Vol. XV, No. 4, pp. 90–101.

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Kabanikhin, S.I., Krivorot’ko, O.I. A numerical method for solving the Dirichlet problem for the wave equation. J. Appl. Ind. Math. 7, 187–198 (2013). https://doi.org/10.1134/S1990478913020075

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