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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.], 2012, Issue 4(29), Pages 37–47 (Mi vsgtu1097)

Differential Equations

The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type

D. K. Durdieva, Zh. Sh. Safarovb

a Bukhara State University
b Tashkent University of Information Technology

Abstract: The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman's boundary condition as Dirac's delta-function concentrated on the boundary of the domain $(x,t)\in \mathbb R^{n+1}$, $z>0$. As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The significant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables $x\in \mathbb R^{n}$. The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable $z$ and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.

Keywords: integro-differential equation, inverse problem, uniqueness, estimate of stability, pulse source, characteristic
Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1097

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Bibliographic databases:

UDC: 517.956.3
MSC: Primary 35R30; Secondary 35L10, 35R10, 35L20
Original article submitted 22/VI/2012
revision submitted – 04/IX/2012

Citation: D. K. Durdiev, Zh. Sh. Safarov, “The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 4(29) (2012), 37–47

Citation in format AMSBIB
\Bibitem{DurSaf12} \by D.~K.~Durdiev, Zh.~Sh.~Safarov \paper The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2012 \vol 4(29) \pages 37--47 \mathnet{http://mi.mathnet.ru/vsgtu1097} \crossref{https://doi.org/10.14498/vsgtu1097} 

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This publication is cited in the following articles:
1. Zh. Sh. Safarov, “Otsenki ustoichivosti reshenii nekotorykh obratnykh zadach dlya integro-differentsialnykh uravnenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 3, 75–82
2. D. K. Durdiev, Zh. D. Totieva, “Zadacha ob opredelenii mnogomernogo yadra uravneniya vyazkouprugosti”, Vladikavk. matem. zhurn., 17:4 (2015), 18–43
3. Durdiev U., Totieva Zh., “a Problem of Determining a Special Spatial Part of 3D Memory Kernel in An Integro-Differential Hyperbolic Equation”, Math. Meth. Appl. Sci., 42:18 (2019), 7440–7451
4. Durdiev U.D., “a Problem of Identification of a Special 2D Memory Kernel in An Integro-Differential Hyperbolic Equation”, Eurasian J. Math. Comput. Appl., 7:2 (2019), 4–19
5. Z. R. Bozorov, “Zadacha opredeleniya dvumernogo yadra uravneniya vyazkouprugosti”, Sib. zhurn. industr. matem., 23:1 (2020), 28–45
6. D. K. Durdiev, A. A. Rahmonov, “The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium”, J. Appl. Industr. Math., 14:2 (2020), 281–295
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