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 Amn. 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.
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
\Bibitem{KabKri12}
\by S.~I.~Kabanikhin, O.~I.~Krivorot'ko
\paper A numerical method for solving the Dirichlet problem for the wave equation
\jour Sib. Zh. Ind. Mat.
\yr 2012
\vol 15
\issue 4
\pages 90--101
\mathnet{http://mi.mathnet.ru/sjim755}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3112602}
\transl
\jour J. Appl. Industr. Math.
\yr 2013
\vol 7
\issue 2
\pages 187--198
\crossref{https://doi.org/10.1134/S1990478913020075}
Linking options:
https://www.mathnet.ru/eng/sjim755
https://www.mathnet.ru/eng/sjim/v15/i4/p90
This publication is cited in the following 7 articles:
A. V Glushak, “ON SOLVABILITY OF INITIAL AND BOUNDARY VALUE PROBLEMS FOR ABSTRACT FUNCTIONAL-DIFFERENTIAL EULER–POISSON–DARBOUX EQUATIONS”, Differencialʹnye uravneniâ, 60:3 (2024), 346
A. V. Glushak, “Uniqueness Criterion for the Solution of Boundary-Value Problems for the Abstract Euler–Poisson–Darboux Equation on a Finite Interval”, Math. Notes, 109:6 (2021), 867–875
Glushak A.V., “On the Solvability of Boundary Value Problems For An Abstract Bessel-Struve Equation”, Differ. Equ., 55:8 (2019), 1069–1076
V. I. Vasilev, A. M. Kardashevskii, V. V. Popov, “Iteratsionnyi metod resheniya zadachi Dirikhle i ee modifikatsii”, Matematicheskie zametki SVFU, 24:3 (2017), 38–51
Holec M., Cotelo M., Velarde P., Liska R., “Application of Discontinuous Galerkin Adaptive Mesh and Order Refinement Method to Energy Transport and Conservation Equation in Radiation-Hydrodynamics”, Proceedings of the 1St Pan-American Congress on Computational Mechanics and Xi Argentine Congress on Computational Mechanics, eds. Idelsohn S., Sonzogni V., Coutinho A., Cruchaga M., Lew A., Cerrolaza M., Int Center Numerical Methods Engineering, 2015, 919–930
Kabanikhin S.I., Krivorotko O.I., “Coupled Inverse Problems and Visualization of Atmosphere-Ocean System”, Coupled Problems in Science and Engineering Vi, eds. Schrefler B., Onate E., Papadrakakis M., Int Center Numerical Methods Engineering, 2015, 921–929
Kabanikhin S., Hasanov A., Marinin I., Krivorotko O., Khidasheli D., “A Variational Approach to Reconstruction of an Initial Tsunami Source Perturbation”, Appl. Numer. Math., 83 (2014), 22–37
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