The numerical solution of the nonlinear hyperbolic-parabolic heat equation

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Abstract

The article discusses a mathematical model and a finite-difference scheme for the heating process of an infinite plate. The disadvantages of using the classical parabolic heat equation for this case and the rationale for using the hyperbolic heat equation are given. The relationship between the hyperbolic thermal conductivity equation and the theory of equations with the retarded argument (delay equation) is shown. The considered mixed equation has 2 parts: parabolic and hyperbolic. Difference schemes use an integrointerpolation method to reduce errors. The problem with a nonlinear thermal conductivity coefficient was chosen as the initial boundary-value problem. The heat source in the parabolic part of the equation is equal to 0, and in the hyperbolic part of the equation sharp heating begins. The initial boundary-value problem with boundary conditions of the third kind in an infinite plate with nonlinear coefficients is formulated and numerically solved. An iterative method for solving the problem is described. A visual graph of the solution results is presented. A theoretical justification for the difference scheme is given. Also we consider the case of the nonlinear mixed equation of the fourth order.

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1. Introduction In theV.N. Khankhasaev’s paper [1-3], which is bound up with the problem of mathematical modeling of the process of switching off the electric arc in the flue gas flow, various mathematical models bound up with the hyperbolic equation of thermal conductivity (obtained by generalization of the Fourier hypothesis [4]) were studied both analytically and numerically. In course of investigations bound up with the transfer processes in the case of high-intensity influence of the gas, the earlier hypotheses presuming the proportionality of the flow density to the vector of the potential gradient, which are based on the known physics laws, lead to an infinite rate of distribution of the perturbations, what contradicts to fundamental laws of nature. The set known physics laws constructed on basis of the given theory includes the following laws:
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About the authors

Vladislav N. Khankhasaev

East Siberia State University of Technology and Management; Buryat State University

Email: hanhvladnick@mail.ru
ORCID iD: 0000-0001-6840-8270
Scopus Author ID: 6504419222
ResearcherId: HKV-0681-2023

Candidate of Physical and Mathematical Sciences, assistant professor of Department of ”Mathematics named after Ts.B. Shoynzhurov” of East Siberia State University of Technology and Management, assistant professor of Department of Fundamental Mathematics of Buryat State University

40V Klyuchevskaya St, Ulan-Ude, 670013, Russian Federation; 24a Smolina St, Ulan-Ude, 670000, Russian Federation

Safron A. Bairov

East Siberia State University of Technology and Management

Author for correspondence.
Email: bairov.sofron@gmail.com
ORCID iD: 0009-0002-6638-5073

Postgraduate student of Department of ”Mathematics named after Ts.B. Shoynzhurov” of East Siberia State University of Technology and Management

40V Klyuchevskaya St, Ulan-Ude, 670013, Russian Federation

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