Aldashev S.A. Well-Posedness of the Dirichlet Problem in a Multidimensional Domain for a Hyperbolic-Parabolic Equation

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https://doi.org/10.15688/mpcm.jvolsu.2019.3.3

 

Serik Aymurzaevich Aldashev
Doctor of Physical and Mathematical Sciences, Professor,
Department of Mathematics and Mathematical Modeling,
Abay Kazakh National Pedagogical University
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https://orcid.org/0000-0002-8223-6900 
Tole bi St., 86, 0500012 Almaty, Republic of 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 nonconducting, then we obtain multidimensional hyperbolic equations. If the medium has a large conductivity, then we get a multidimensional parabolic equation. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to a multidimensional hyperbolic-parabolic equation. It is also known that the vibrations of elastic membranes in space by the Hamiltonian principle can be modeled by multidimensional hyperbolic equations. The study of the process of heat propagation in a medium filled with mass leads to multidimensional parabolic equations. Consequently, by investigating the mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also arrive at multidimensional hyperbolic-parabolic equations. When studying these applications, it becomes necessary of obtaining an explicit representation of the solutions of the investigated problems. In this paper we give a multidimensional domain where the Dirichlet problem for a hyperbolic-parabolic equation is uniquely solvable and an explicit form of its classical solution is obtained.

Key words: multidimensional domain, Dirichlet problem, unique solvability, spherical functions, Bessel function.

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Well-Posedness of the Dirichlet Problem in a Multidimensional Domain for a Hyperbolic-Parabolic Equation by Aldashev S.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 22 No. 3 2019, pp. 31-39

 

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