# The numerical solution of the nonlinear hyperbolic-parabolic heat equation

**Journal**: Discrete and Continuous Models and Applied Computational Science (Vol.32, No. 1)

**Publication Date**: 2024-04-01

**Authors** : Vladislav Khankhasaev; Safron Bairov;

**Page** : 112-121

**Keywords** : hyperbolic-parabolic equation; delay equations; initial boundary-value problem; finite difference schemes; equations of the high order;

### 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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Last modified: 2024-07-19 21:41:46