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This article is cited in 5 scientific papers (total in 5 papers)
Approximation of Solutions of the Two-Dimensional Wave Equation with Variable Velocity and Localized Right-Hand Side Using Some “Simple” Solutions
S. Yu. Dobrokhotovab, A. Yu. Anikinabc a Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
b Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
c Bauman Moscow State Technical University
Abstract:
Asymptotic solutions based on the characteristics and the modified Maslov canonical operator of the two-dimensional wave equation with variable coefficients and right-hand side corresponding to: (a) an instantaneous source; (b) a rapidly acting, but “time spread,” source, are compared. An algorithm for approximating a (more complicated) solution of problem (b) by linear combinations of the derivatives of the (simpler) solution of problem (a) is proposed. Numerical calculations showing the accuracy of this approximation are presented. The replacement of the solutions of problem (b) by those of problem (a) becomes especially important in the case where the wave equation is considered in the domain with boundary on which the velocity of the wave equation vanishes. Then the characteristics of the problem become singular (nonstandard) and solutions of type (a) generalize to the case referred to above in a much simpler and effective way than solutions of type (b). Such a situation arises in problems where long waves (for example, tsunami waves) are incident on a sloping seashore.
Keywords:
asymptotic solution, wave equation, Maslov canonical operator, nonstandard characteristics.
Received: 02.09.2016
Citation:
S. Yu. Dobrokhotov, A. Yu. Anikin, “Approximation of Solutions of the Two-Dimensional Wave Equation with Variable Velocity and Localized Right-Hand Side Using Some “Simple” Solutions”, Mat. Zametki, 100:6 (2016), 825–837; Math. Notes, 100:6 (2016), 796–806
Linking options:
https://www.mathnet.ru/eng/mzm11366https://doi.org/10.4213/mzm11366 https://www.mathnet.ru/eng/mzm/v100/i6/p825
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Abstract page: | 400 | Full-text PDF : | 60 | References: | 52 | First page: | 25 |
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