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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.], 2015, Volume 19, Number 4, Pages 697–709 (Mi vsgtu1436)

Differential Equations and Mathematical Physics

A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space

I. N. Rodionova, V. M. Dolgopolov

Samara State Aerospace University, Samara, 443086, Russian Federation

Abstract: The second-order hyperbolic type equation is considered in the 3D Euclidean space. Boundary value problem is posed in the infinite cylindrical region bounded by the characteristic surfaces of this equation with data on the related characteristic surfaces of the equation and with conditions mates on the internal non-descriptive plane. The solution is also assumed to be zero when $z\to\infty$ with derivative by variable $z$. By the Fourier transform method the problem reduced to the corresponding planar problem $\Delta_1$ for hyperbolic equation, which in characteristic coordinates is the generalized Euler–Darboux equation with a negative parameter. Authors obtained estimates of the plane problem solution and its partial derivatives up to the second order inclusive. This, in turn, provided an opportunity to impose the conditions to given boundary functions ensuring the existence of a classical solution of the problem in the form of the Fourier transform.

Keywords: integral equations, boundary value problems, second-order hyperbolic type equations
Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1436

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Bibliographic databases:

UDC: 517.956.3
MSC: 35L10
Original article submitted 17/V/2015
revision submitted – 27/VIII/2015

Citation: I. N. Rodionova, V. M. Dolgopolov, “A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:4 (2015), 697–709

Citation in format AMSBIB
\Bibitem{RodDol15} \by I.~N.~Rodionova, V.~M.~Dolgopolov \paper A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2015 \vol 19 \issue 4 \pages 697--709 \mathnet{http://mi.mathnet.ru/vsgtu1436} \crossref{https://doi.org/10.14498/vsgtu1436} \zmath{https://zbmath.org/?q=an:06969188} \elib{https://elibrary.ru/item.asp?id=25687497} 

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This publication is cited in the following articles:
1. M. V. Dolgopolov, I. N. Rodionova, V. M. Dolgopolov, “Ob odnoi nelokalnoi zadache dlya uravneniya Eilera–Darbu”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:2 (2016), 259–275
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