Abstract:
In the paper we study a loaded degenerate hyperbolic equation of the second order with variable coefficients. The principal part of the equation is the Gellerstedt operator. The loaded term is given in the form of the trace of desired solution on the degenerate line. The latter is located in the inner part of the domain. We investigate a boundary value problem. The boundary conditions are given on a characteristics line of the equation under study. For the model equation (when all subordinated coefficients are zero) we construct an explicit representation for solution of the Goursat problem. In the general case, we reduce the problem to an integral Volterra equation of the second kind. We apply the method realized by Sven Gellerstedt solving the second Darboux problem. In both cases, model and general, we use widely properties of the Green–Hadamard function.
Keywords:
Goursat problem, loaded equation, hyperbolic equation, degenerate equation, Gellerstedt operator, the Green–Hadamard's function method.
Original article submitted 13/X/2015 revision submitted – 23/X/2015
Citation:
A. H. Attaev, “Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 7–21
\Bibitem{Att16}
\by A.~H.~Attaev
\paper Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part
\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 7--21
\mathnet{http://mi.mathnet.ru/vsgtu1452}
\crossref{https://doi.org/10.14498/vsgtu1452}
\zmath{https://zbmath.org/?q=an:06964468}
\elib{https://elibrary.ru/item.asp?id=26898032}
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https://www.mathnet.ru/eng/vsgtu/v220/i1/p7
This publication is cited in the following 6 articles:
A. V. Pskhu, M. T. Kosmakova, K. A. Izhanova, “Cauchy Problem for a Loaded Fractional Diffusion Equation”, Lobachevskii J Math, 45:9 (2024), 4574
K. U. Khubiev, “Zadachi so smescheniem dlya nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa s operatorom drobnoi diffuzii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 82–90
A. Kh. Attaev, “Characteristic problems for a loaded equation of hyperbolic type with a wave operator in the principal part”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, eds. A. Ashyralyev, A. Lukashov, M. Sadybekov, Amer. Inst. Phys., 2018, UNSP 020022-1
K. U. Khubiev, “Boundary-Value Problem for a Loaded Equation of Hyperbolic-Parabolic Type with Degeneracy of Order in the Domain of Hyperbolicity”, J. Math. Sci. (N. Y.), 250:5 (2020), 830–834
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