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Uzbekistan Tashkent
Section Mathematics
Title Evaluation of the stability of some inverse problems solutions for integro-differential equations
Author(-s) Safarov Zh.Sh.a
Affiliations Tashkent University of Information Technologya
Abstract 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.
Keywords integro-differential equation, inverse problem, stability, delta function, kernel
UDC 517.958
MSC 35L70, 58J45
DOI 10.20537/vm140307
Received 20 May 2014
Language Russian
Citation 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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