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 Izv. IMI UdGU, 2018, Volume 52, Pages 116–130 (Mi iimi365)

The initial value problem for the quasi-linear partial integro-differential equation of higher order with a degenerate kernel

T. K. Yuldashev

Reshetnev Siberian State University of Science and Technology, Krasnoyarsky Rabochy Av., 31, Krasnoyarsk, 660014, Russia

Abstract: High-order partial differential equations are of great interest when it comes to physical applications. Many problems of gas dynamics, elasticity theory and the theory of plates and shells are reduced to the consideration of high-order partial differential equations. This paper studies the one-valued solvability of the initial value problem for a nonlinear partial integro-differential equation of an arbitrary order with a degenerate kernel. The expression of higher-order partial differential equations as a superposition of first-order partial differential operators has allowed us to apply methods for solving first-order partial differential equations. First-order partial differential equations can be locally solved by the methods of the theory of ordinary differential equations, reducing them to a characteristic system. The existence and uniqueness of the solution to this problem is proved by the method of successive approximation. An estimate of convergence of the iterative Picard process is obtained. The stability of the solution from the second argument of the initial value problem is shown.

Keywords: initial value problem, characteristic, derivative along the direction, degenerate kernel, superposition of partial differential operators, existence and uniqueness of the solution.

DOI: https://doi.org/10.20537/2226-3594-2018-52-09

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Bibliographic databases:

UDC: 517.956.35
MSC: 35A30, 35C15, 35G55, 35L30, 35M10, 35S05

Citation: T. K. Yuldashev, “The initial value problem for the quasi-linear partial integro-differential equation of higher order with a degenerate kernel”, Izv. IMI UdGU, 52 (2018), 116–130

Citation in format AMSBIB
\Bibitem{Yul18} \by T.~K.~Yuldashev \paper The initial value problem for the quasi-linear partial integro-differential equation of higher order with a degenerate kernel \jour Izv. IMI UdGU \yr 2018 \vol 52 \pages 116--130 \mathnet{http://mi.mathnet.ru/iimi365} \crossref{https://doi.org/10.20537/2226-3594-2018-52-09} \elib{http://elibrary.ru/item.asp?id=36508460}