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Uniqueness of the kernel determination problem in an integro-differential parabolic equation with variable coefficient
D. K. Durdievab, J. Z. Nuriddinovb a V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, 46 University str., Tashkent, 100170 Republic of Uzbekistan
b Bukhara State University, 11 M. Ikbol str., Bukhara, 200117 Republic of Uzbekistan
Abstract:
We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the $n$-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions $k_1(x,t)$ and $k_2(x,t)$ of the stated problem, it is formed an equation for the difference of this solution. Further research is being conducted for the difference $k_1(x,t)-k_2(x,t)$ of solutions of the problem and using the techniques of integral equations estimates. It is shown that if the unknown kernel $k(x,t)$ can be represented as $k(x,t)=\displaystyle\sum\limits_{i=0}^Na_i(x)b_i(t)$, then $k_1(x,t)\equiv k_2(x,t)$. Thus, the theorem on the uniqueness of the solution of the problem is proved.
Keywords:
inverse problem, parabolic equation, Cauchy problem, integral equation, uniqueness.
Received: 29.03.2023 Revised: 29.03.2023 Accepted: 29.05.2023
Citation:
D. K. Durdiev, J. Z. Nuriddinov, “Uniqueness of the kernel determination problem in an integro-differential parabolic equation with variable coefficient”, Izv. Vyssh. Uchebn. Zaved. Mat., 2023, no. 11, 3–14
Linking options:
https://www.mathnet.ru/eng/ivm9913 https://www.mathnet.ru/eng/ivm/y2023/i11/p3
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