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Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain

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Abstract

A method for finding an approximate solution of the homogeneous Dirichlet problem for the Helmholtz operator in a 2-D domain is under consideration. In the polar coordinates, an approximate solution is represented as a finite sum, where each term is proportional to the product of the corresponding Bessel function depending on the radial variable and the trigonometric function of the polar angle. The under consideration method is compared with the FDM and FEM. The methods are compared as on domains that allow the exact solution of the problem, as on domains where such a solution has not yet been found. Based on the calculation results, a conclusion about the effectiveness of the proposed method is made.

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Authors and Affiliations

  1. Peter the Great St. Petersburg Polytechnic University, 195251, St. Petersburg, Russia

    E. G. Apushkinskiy, V. A. Kozhevnikov & A. V. Biryukov

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  1. E. G. Apushkinskiy
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  2. V. A. Kozhevnikov
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  3. A. V. Biryukov
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Correspondence to E. G. Apushkinskiy, V. A. Kozhevnikov or A. V. Biryukov.

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(Submitted by N. B. Pleshchinskii)

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Apushkinskiy, E.G., Kozhevnikov, V.A. & Biryukov, A.V. Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain. Lobachevskii J Math 44, 3989–3997 (2023). https://doi.org/10.1134/S1995080223090044

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