BOUNDARY VALUE PROBLEMS FOR MIXED-TYPE DIFFERENTIAL EQUATIONS OF THE FIRST AND SECOND ORDER WITH RESPECT TO THE TIME VARIABLE
This work is devoted to the study of boundary value problems for mixed-type differential equations
of the first and second order with respect to the time variable. Boundary value problems for mixed-
type equations arise in various fields of natural sciences, including laser physics, plasma modeling,
and mathematical biology. In this paper, we establish theorems on the uniqueness and conditional
stability of the solution to the problem under consideration within a set of well-posedness. An a
priori estimate of the solution is obtained using the method of logarithmic convexity and spectral
decomposition.
1. Bers, L., Mathematical Problems of Subsonic and Transonic Gas Dynamics. Moscow: Foreign Literature
Publishing House, 1961. - 206 p.
2. Kuzmin A.G., Nonclassical Equations of Mixed Type and Their Applications in Gas Dynamics. Leningrad:
Leningrad State University Publishing, 1990. - 280 p.
3. Frankl F.I., Selected Works on Gas Dynamics. Moscow: Nauka, 1973. - 711 p.
4. He Kan Cher., The Singular Tricomi Problem for Equations of Mixed Type with Two Degeneracy Lines:
PhD Thesis in Physical and Mathematical Sciences. Novosibirsk, 1976.
5. Sabitov, K.B., Karamova, A.A., Spectral Properties of Solutions to the Tricomi Problem for a
Mixed-Type Equation with Two Type-Changing Lines and Their Applications. Izvestiya RAN. Seriya
Matematicheskaya, 2001. Vol. 65, No. 4, pp. 133-150.
6. Fayazov, K.S., An Ill-posed Boundary Value Problem for a Second-Order Equation of Mixed Type. Uzbek
Mathematical Journal, 1995, No. 2, pp. 89-93.
7. 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, Vol. 1, Issue 2, Article 3.
8. Fayazov K.S., Khudaybergenov Y.K., Ill-posed Boundary Value Problem for a System of Mixed-Type
Equations with Two Degeneracy Lines. Siberian Electronic Mathematical Reports, 2020, Vol. 17, pp.
647-660.
9. Lavrent‘ev M.M.,Saveliev L.Y., Theory of operators and ill-posed problems., Publishing House of the
Institute of Mathematics, Novosibirsk, 2010.
10. Krein M. G., Gohberg I. T., Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert
Space. Moscow: Nauka, 1965, 448 pages.
11. Pyatkov S. G., Some Properties of Eigenfunctions and Associated Functions of Indefinite Sturm-Liouville
Problems. In: Nonclassical Equations of Mathematical Physics: Collected Scientific Works. Novosibirsk:
Institute of Mathematics Publishing, 2005, pp. 240-251.
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