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Section
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Mathematics
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Title
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Evaluation of the stability of some inverse problems solutions for integro-differential equations
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Author(-s)
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Safarov Zh.Sh.a
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Affiliations
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Tashkent University of Information Technologya
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Abstract
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The paper investigates the stability of inverse problems solutions for two integro-differential hyperbolic equations. Theorems of existence and uniqueness of these solutions (in the small) have been obtained and published earlier by author. Thus only stability problems of these solutions are considered in this paper. In Theorem 1 we prove conditional stability of the solution of the following inverse problem: determine the kernel of the integral for integro-differential equation
$$u_{tt}=u_{xx}-\int_0^tk(\tau)u(x,t-\tau)\, d\tau, \qquad (x,t)\in \mathbb{R}\times \mathbb{R}_+,$$
with initial data $u\big|_{t=0}=0$, $u_t\big|_{t=0}=\delta(x),$ and additional information about the direct problem solution $u(0,t)=f_1(t)$, $u_x(0,t)=f_2(t).$ The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. To prove the theorem the method of successive approximations is used. Next, the method of estimating the integral equations and Gronwall's inequality are used.
In a similar manner we prove Theorem 2. It is devoted to estimating the conditional stability of the solution of kernel determination problem for the same integro-differential equation in a bounded domain with respect to $x,$ $x\in(0,l),$ with initial data $u\big|_{t=0}=0$, $u_t\big|_{t=0}=\delta'(x),$ and boundary conditions $(u_x-hu)\big|_{x=0}=0$, $(u_x+Hu)\big|_{x=l}=0$, $t>0$. In this case the additional information about the direct problem solution is given as $u(0,t)=f(t)$, $t\geqslant0$. Here $h$ and $H$ are finite real numbers.
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Keywords
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integro-differential equation, inverse problem, stability, delta function, kernel
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UDC
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517.958
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MSC
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35L70, 58J45
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DOI
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10.20537/vm140307
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Received
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20 May 2014
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Language
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Russian
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Citation
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Safarov Zh.Sh. Evaluation of the stability of some inverse problems solutions for integro-differential equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 75-82.
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References
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- Romanov V.G. Ustoichivost' v obratnykh zadachakh (Stability in inverse problems), Moscow: Nauchnyi mir, 2005, 295 p.
- Bukhgeym A.L. Inverse problems of memory reconstrution, Journal of Inverse and Ill-posed Problems, 1993, vol. 1, issue 3, pp. 193-206.
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- Durdiev D.K. Global solvability of an inverse problem for an integro-differential equation of electrodynamics, Differential Equations, 2008, vol. 44, no. 7, pp. 893-899.
- Durdiev D.K, Safarov Zh.Sh. 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, 2012, issue 4 (29), pp. 37-47 (in Russian).
- Durdiev D.K, Totieva Zh.D. The problem of determining the one-dimensional kernel of the viscoelasticity equation, Sib. Zh. Ind. Mat., 2013, vol. 16, no. 2, pp. 72-82 (in Russian).
- Safarov J.Sh. Problems of the local solvability of the inverse problem for integro-differential equations vibrations of an infinite string, Uzbek. Math. J., 2013, no. 2, pp. 100-106.
- Safarov J.Sh. Inverse problem for integro-differential equations of hyperbolic type in the limited area, Uzbek. Math. J., 2012, no. 2, pp. 117-124.
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Full text
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