Section

Mathematics

Title

Evaluation of the stability of some inverse problems solutions for integrodifferential equations

Author(s)

Safarov Zh.Sh.^{a}

Affiliations

Tashkent University of Information Technology^{a}

Abstract

The paper investigates the stability of inverse problems solutions for two integrodifferential 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 integrodifferential 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 integrodifferential 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 $(u_xhu)\big_{x=0}=0$, $(u_x+Hu)\big_{x=l}=0$, $t>0$. 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

integrodifferential 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 integrodifferential equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 7582.

References

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 Durdiev D.K, Totieva Zh.D. The problem of determining the onedimensional kernel of the viscoelasticity equation, Sib. Zh. Ind. Mat., 2013, vol. 16, no. 2, pp. 7282 (in Russian).
 Safarov J.Sh. Problems of the local solvability of the inverse problem for integrodifferential equations vibrations of an infinite string, Uzbek. Math. J., 2013, no. 2, pp. 100106.
 Safarov J.Sh. Inverse problem for integrodifferential equations of hyperbolic type in the limited area, Uzbek. Math. J., 2012, no. 2, pp. 117124.

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