Conditional correctness of the initial-boundary value problem for the mixed type fourth-order partial differential equation with two degenerate lines
Keywords:
boundary problem, mixed type equation with two degenerate lines, ill-posed problem, a priori estimate, estimate of conditional stability, uniqueness, set of correctness.Abstract
This paper is devoted to investigate conditional correctness of the initial-boundary value problem for a mixed type fourth-order partial differential equation with two degenerate lines. Considered problem is ill-posed in the sense of J. Hadamard. Following to the idea of A.N. Tikhonov about approximate solution of ill-posed problems we proof the theorems of the uniqueness and conditional stability.
References
Lavrent’ev M. M. and Savelyev L. Ya., Theory of operators and ill-posed problems, -2 th ed., Pererab. and add. Novosibirsk:Published by Institute of Mathematics, 2010. p.941
Pyatkov S.G., Solvability of boundary value problems for a second-order equation of mixed type, Nonclassical partial differential equations, Collect. Sci. Works, Novosibirsk, 1988, pp. 77–90.
Pyatkov S.G., On the solvability of a boundary value problem for a parabolic equation with a changing time direction, Dokl. Acad. Sciences of the USSR.1985. T. 285, №6.P.1322-1327.
Krien M.G., Gokhberg I. T. Introduction to the theory of linear non-self-adjoint operators in Hilbert space // Moscow: Nauka 1965, pp. 448.
Pyatkov S.G., Some properties of proper and attached functions of unfamiliar Sturm-Liouville problems // Nonclassical equations of mathematical physics. Collection of scientific papers. Novosibirsk: Publishing House of the Institute of Mathematics. 2005, pp. 240-251.
Fayazov K.S., Khajiev I.O., Fayazova Z.K. “Ill-posed boundary value problem for operator-differential equation of fourth order”, Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: (2018) Volume 1: Issue 2, Article 3.
Khajiev I.O. Estimation of the conditional stability of an ill-posed initial-boundary problem for a high-order mixed type equation. Uzbek Mathematical Journal, 2021, Volume 65, Issue 4, pp. 48-61.
Fayazov K. S, Khudayberganov Y. K.Ill-posed boundary value problem for mixed type system equations with two degenerate lines, Siberian Electronic Mathematical Reports, Vol 17, pp. 647-660.
