Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics 2011-07, Vol.51 (7), p.1194-1207
Main Authors: Alexeyeva, L. A., Zakir’yanova, G. K.
Format: Article
Language:English
Subjects:
Boundaries
Boundary conditions
Boundary value problems
Computational mathematics
Computational Mathematics and Numerical Analysis
Hyperbolic systems
Initial value problems
Integral equations
Integrals
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Shock waves
Studies
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Summary:The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.
ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542511070037