Aldashev S.A. The Tricomi Problem for a Class of Multidimensional Mixed Hyperbolic-Parabolic Equations

Serik A. Aldashev
Doctor of Physical and Mathematical Sciences, Professor,
Institute of Mathematics and Mathematical Modeling
of the Ministry of Education and Science of the Republic of Kazakhstan
This email address is being protected from spambots. You need JavaScript enabled to view it. 
Pushkin St, 125, 050010 Almaty, Kazakhstan

Abstract. It is known that in the mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate multidimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) is reduced to degenerate multidimensional hyperbolic-parabolic equations. It is also known that the oscillations of elastic membranes in space can be modeled according to the Hamilton principle by degenerate multidimensional hyperbolic equations. The study of the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Therefore, by studying the mathematical modeling of the heat propagation process in oscillating elastic membranes, we also arrive at degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it becomes necessary to obtain an explicit representation of the solutions to the problems under study. Boundary value problems for hyperbolicparabolic equations on the plane are well studied, and their multidimensional analogues are little studied. The Tricomi problem for these equations was previously investigated. As far as we know, this problem has not been studied in space. In this paper, the Tricomi problem is shown to be ambiguously solvable for a class of multidimensional mixed hyperbolic-parabolic equations. 

Key words: the Tricomi problem, multidimensional equation, solvability, spherical functions, mixed hyperbolic-parabolic equations.

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The Tricomi Problem for a Class of Multidimensional Mixed Hyperbolic-Parabolic Equations by Aldashev S.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in EnglishMathematical Physics and Computer Simulation. Vol. 25 No. 2 2022, pp. 5-16

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