Abstract:
We study the numerical solution to the nonlinear heat conduction problem for a plate with a nonlinear heat source (thermal conductivity coefficient and internal heat source are exponential functions of temperature). In particular, for the nonlinear problem, the phenomena of self-similarity, inertia, and heat localization were found, which also manifest themselves in solutions of linear hyperbolic heat equations. With a self-similar change in temperature in some ranges of spatial and temporal variables, similarity (self-similarity) of temperature curves is observed. When heat is localized in a certain range of spatial variable, the temperature does not change over time. The inertia of heat is revealed in the finite speed of its propagation, despite the solution of the parabolic heat equation. The listed phenomena are also observed in solutions of linear hyperbolic heat equations, the derivation of which takes into account the time dependence of the heat flow in the formula of Fourier’s law, leading to a finite rate of heat propagation. In nonlinear problems, a similar effect manifests itself due to the dependence of the physical properties and heat source on temperature, leading to a similar delay in heat flow.
Keywords:nonlinear boundary value problem, dependence of physical properties and heat source on temperature, numerical solution, n, inertia and heat localization, self-similar temperature change, thermal explosion, hyperbolic heat equations
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