
A coefficient determination in nonlocal problem for Boussinesq type integrodifferential equation with degenerate kernel
T. K. Yuldashev^{} ^{} Siberian State Aerospace University, 31 Krasnoyarskiy Rabochiy Ave., Krasnoyarsk 660014, Russia
Abstract:
In the threedimensional domain a Boussinesq type linear integrodifferential equation of the fourth order with a restore coefficient and a degenerate kernel is considered. The solution of this integrodifferential equation is considered in the class of continuously differentiable functions. First, we study the classical solvability of a nonlocal direct boundary value problem for the considered Boussinesq integrodifferential equation with a parameter in the integral term. The method of separation of variables and the method of a degenerate kernels are used. A countable system of algebraic equations is obtained. The solution of this algebraic system of equations for regular values of the spectral parameter in the integral term of a given equation allows us to construct a solution of a nonlocal direct boundary value problem for an integrodifferential equation in the form of a Fourier series. A criterion for the unique solvability of a direct boundary value problem is established for fixed values of the restore function. Using the Cauchy–Bunyakovsky inequality and the Bessel inequality, we prove the absolute and uniform convergence of the obtained Fourier series. The continuity of all the derivatives of the solution of the direct boundary value problem for a given equation is also proved. Further, with the help of an additional integral condition, the restore function is uniquely determined in the form of a Fourier series. The criterion of continuity of second order derivatives of the restore function with respect to space variables is established. Based on the found values of the restore function, the main unknown function is uniquely determined as a solution to the inverse problem for the considering integrodifferential equation. In addition, the stability with respect to restore function of the solution of an integrodifferential equation is studied.
Key words:
Boussinesq type integrodifferential equation, fourthorder equation, degenerate kernel, integral conditions, one valued solvability.
DOI:
https://doi.org/10.23671/VNC.2019.2.32118
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UDC:
517.968
MSC: 35A02, 35M10, 35S05 Received: 17.11.2016
Citation:
T. K. Yuldashev, “A coefficient determination in nonlocal problem for Boussinesq type integrodifferential equation with degenerate kernel”, Vladikavkaz. Mat. Zh., 21:2 (2019), 67–84
Citation in format AMSBIB
\Bibitem{Yul19}
\by T.~K.~Yuldashev
\paper A coefficient determination in nonlocal problem for Boussinesq type integrodifferential equation with degenerate kernel
\jour Vladikavkaz. Mat. Zh.
\yr 2019
\vol 21
\issue 2
\pages 6784
\mathnet{http://mi.mathnet.ru/vmj694}
\crossref{https://doi.org/10.23671/VNC.2019.2.32118}
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http://mi.mathnet.ru/eng/vmj694 http://mi.mathnet.ru/eng/vmj/v21/i2/p67
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