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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, Volume 19, Number 1, Pages 136–154 (Mi vsgtu1335)

Differential Equations and Mathematical Physics

Inverse problem for a nonlinear partial differential equation of the eighth order

T. K. Yuldashev

M. F. Reshetnev Siberian State Aerospace University, Krasnoyarsk, 660014, Russian Federation

Abstract: We study the questions of solvability of the inverse problem for a nonlinear partial differential equation of the eighth order, left-hand side of which is the superposition of pseudoparabolic and pseudohyperbolic operators of the fourth order. The applicability of the Fourier method of separation of variables is proved in study of mixed and inverse problems for a nonlinear partial differential equation of the eighth order. Using the method of separation of variables, the mixed problem is reduced to the study of the countable system of nonlinear Volterra integral equations of the second kind. Use the given additional conditions led us to study of nonlinear Volterra integral equation of the first kind with respect to the second unknown function (with respect to restore function). With the help of nonclassical integral transform the one-value restore of the second unknown function is reduced to study of the unique solvability of nonlinear Volterra integral equation of the second kind. As a result is obtained a system of two nonlinear Volterra integral equations of the second kind with respect to two unknown functions. This system is one-value solved by the method of successive approximations. Further the stability of solutions of the mixed and inverse problems is studied with respect to initial value and additional given functions.

Keywords: inverse problem, nonlinear partial differential equation, equation of the eighth order, superposition of two operators, correctness of solution

DOI: https://doi.org/10.14498/vsgtu1335

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Bibliographic databases:

UDC: 517.957
MSC: 35K70, 35R30
Original article submitted 24/VII/2014
revision submitted – 15/X/2014

Citation: T. K. Yuldashev, “Inverse problem for a nonlinear partial differential equation of the eighth order”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 136–154

Citation in format AMSBIB
\Bibitem{Yul15} \by T.~K.~Yuldashev \paper Inverse problem for a nonlinear partial differential equation of~the eighth order \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2015 \vol 19 \issue 1 \pages 136--154 \mathnet{http://mi.mathnet.ru/vsgtu1335} \crossref{https://doi.org/10.14498/vsgtu1335} \zmath{https://zbmath.org/?q=an:06968953} \elib{http://elibrary.ru/item.asp?id=23681747} 

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This publication is cited in the following articles:
1. T. K. Yuldashev, “Ustoichivost i differentsiruemost po malomu parametru smeshannoi zadachi dlya nelineinogo uravneniya v chastnykh proizvodnykh vosmogo poryadka”, Zhurnal SVMO, 18:1 (2016), 82–93
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