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 Mat. Zametki, 2011, Volume 90, Issue 2, Pages 254–268 (Mi mz8626)

On the Solvability of Certain Spatially Nonlocal Boundary-Value Problems for Linear Hyperbolic Equations of Second Order

A. I. Kozhanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The present paper studies the solvability of spatially nonlocal boundary-value problems with Samarskii boundary condition with variable coefficients for linear hyperbolic equations of second order in the one-dimensional case. In the case of boundary conditions with constant coefficients for the equation
$$u_{tt}-u_{xx}+c(x)u=f(x,t),$$
similar problems were studied earlier by other authors; a significant aspect of their papers was the use of the Fourier method, which dictated a special form of the equation as well as the constancy of the coefficients of the boundary conditions. The method used here does not involve such constraints and allows us to study more general problems.

Keywords: linear hyperbolic equation of second order, spatially nonlocal boundary-value problem, Samarskii boundary condition, Young's inequality

DOI: https://doi.org/10.4213/mzm8626

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English version:
Mathematical Notes, 2011, 90:2, 238–249

Bibliographic databases:

UDC: 517.946
Revised: 29.12.2010

Citation: A. I. Kozhanov, “On the Solvability of Certain Spatially Nonlocal Boundary-Value Problems for Linear Hyperbolic Equations of Second Order”, Mat. Zametki, 90:2 (2011), 254–268; Math. Notes, 90:2 (2011), 238–249

Citation in format AMSBIB
\Bibitem{Koz11} \by A.~I.~Kozhanov \paper On the Solvability of Certain Spatially Nonlocal Boundary-Value Problems for Linear Hyperbolic Equations of Second Order \jour Mat. Zametki \yr 2011 \vol 90 \issue 2 \pages 254--268 \mathnet{http://mi.mathnet.ru/mz8626} \crossref{https://doi.org/10.4213/mzm8626} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2918441} \transl \jour Math. Notes \yr 2011 \vol 90 \issue 2 \pages 238--249 \crossref{https://doi.org/10.1134/S0001434611070236} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000294363500023} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80052068435} 

• http://mi.mathnet.ru/eng/mz8626
• https://doi.org/10.4213/mzm8626
• http://mi.mathnet.ru/eng/mz/v90/i2/p254

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. E. Tokmagambetov, “Gellerstedt Equation with the Perturbation of the Cauchy Condition”, J. Math. Sci., 213:6 (2016), 910–916
2. Tokmagambetov N. Nalzhupbayeva G., “Operator perturbed Cauchy problem for the Gellerstedt equation”, ADVANCEMENTS IN MATHEMATICAL SCIENCES: Proceedings of the International Conference on Advancements in Mathematical Sciences (Antalya, Turkey, 5–7 November 2015), AIP Conference Proceedings, 1676, ed. Ashyralyev A. Malkowsky E. Lukashov A. Basar F., Amer Inst Physics, 2015, 020098
3. R. R. Safiullova, “On a solvability of the nonlinear inverse problem for the hyperbolic equation”, J. Math. Sci., 228:4 (2018), 431–448
4. Pulkina L.S., “Nonlocal Problems For Hyperbolic Equations With Degenerate Integral Conditions”, Electron. J. Differ. Equ., 2016, 193
5. K. U. Khubiev, “Ob odnoi nelokalnoi zadache dlya uravneniya smeshannogo giperbolo-parabolicheskogo tipa”, Matematicheskie zametki SVFU, 24:3 (2017), 12–18
6. A. I. Kozhanov, G. A. Lukina, “Nelokalnye kraevye zadachi s chastichno integralnymi usloviyami dlya vyrozhdayuschikhsya differentsialnykh uravnenii s kratnymi kharakteristikami”, Sib. zhurn. chist. i prikl. matem., 17:3 (2017), 37–51
7. Sofiane D., Abdelfatah B., Taki-Eddine O., “Study of Solution For a Parabolic Integrodifferential Equation With the Second Kind Integral Condition”, Int. J. Anal. Appl., 16:4 (2018), 569–593
8. Pulkina L.S., Beylin A.B., “Nonlocal Approach to Problems on Longitudinal Vibration in a Short Bar”, Electron. J. Differ. Equ., 2019, 29
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