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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 1, Pages 65–73 (Mi vsgtu1469)

Differential Equations and Mathematical Physics

On problems with displacement in boundary conditions for hyperbolic equation

E. A. Utkina

Kazan (Volga Region) Federal University, Kazan, 420008, Russian Federation

Abstract: We consider three problems for hyperbolic equation on a plane in the characteristic domain. In these problems at least one of the conditions of the Goursat problem is replaced by nonlocal condition on the relevant characteristic. Non-local conditions are the linear combinations of the normal derivatives at points on opposite characteristics. In case of replacement of one condition we solve the problem by reduction to the Goursat problem for which it exists and is unique. To find the unknown Goursat condition author receives the integral equation, rewrite it in operational form and finds its unique solvability cases. To prove the unique solvability of the equation, the author shows the continuous linear operator and the fact, that some degree of the resulting operator is a contraction mapping. It is known that in this case the required Goursat condition can be written as Neumann series. We considered in detail only one of the tasks, but for both the unique solvability theorems are formulated. If the two conditions are changed, the uniqueness of the solution on the assumption that it exists, is proved by the method of a priori estimates. For this purpose, the inner product and the norm in $L_2$ are used. As a result, the conditions were obtained for the coefficients of a hyperbolic equation that ensure the uniqueness of the solution. An example is given, confirming that these conditions are essential. Namely, constructed an equation whose coefficients do not satisfy the conditions of the last theorem, given the conditions on the characteristics and nontrivial solution is built.

Keywords: hyperbolic equation, nonlocal conditions, the problem with displacement

DOI: https://doi.org/10.14498/vsgtu1469

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Bibliographic databases:

UDC: 517.956.3
MSC: 35L20
Original article submitted 25/I/2016
revision submitted – 08/II/2016

Citation: E. A. Utkina, “On problems with displacement in boundary conditions for hyperbolic equation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 65–73

Citation in format AMSBIB
\Bibitem{Utk16} \by E.~A.~Utkina \paper On problems with displacement in boundary conditions for hyperbolic equation \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 1 \pages 65--73 \mathnet{http://mi.mathnet.ru/vsgtu1469} \crossref{https://doi.org/10.14498/vsgtu1469} \zmath{https://zbmath.org/?q=an:06964473} \elib{https://elibrary.ru/item.asp?id=26898102}