Abstract:
We study some questions of the qualitative theory of solutions to differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations. The right-hand sides of these equations contain discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. It is in particular established that its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts a common approach of investigation which uses the reduction of a system of two first-order equations to a single second-order equation.
Citation:
D. S. Anikonov, S. G. Kazantsev, D. S. Konovalova, “Differential properties of a generalized solution to a hyperbolic system of first-order differential equations”, Sib. Zh. Ind. Mat., 16:2 (2013), 26–39; J. Appl. Industr. Math., 7:3 (2013), 313–325
\Bibitem{AniKazKon13}
\by D.~S.~Anikonov, S.~G.~Kazantsev, D.~S.~Konovalova
\paper Differential properties of a~generalized solution to a~hyperbolic system of first-order differential equations
\jour Sib. Zh. Ind. Mat.
\yr 2013
\vol 16
\issue 2
\pages 26--39
\mathnet{http://mi.mathnet.ru/sjim777}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3203339}
\transl
\jour J. Appl. Industr. Math.
\yr 2013
\vol 7
\issue 3
\pages 313--325
\crossref{https://doi.org/10.1134/S1990478913030046}
Linking options:
https://www.mathnet.ru/eng/sjim777
https://www.mathnet.ru/eng/sjim/v16/i2/p26
This publication is cited in the following 2 articles:
Bedida N., Hermas N., “Some Hyperbolic Conservation Laws on R-N”, Mediterr. J. Math., 17:6 (2020), 197
D. S. Konovalova, “Localization for the discontinuity line of the right-hand side of a differential equation”, J. Appl. Industr. Math., 10:1 (2016), 97–105
Citation in format AMSBIB
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