Non-standard method for solving hyperbolic heat equations
2024
Treilande, Tabita | Iltins, Ilmars
In this article, we build upon the pioneering work of Abraham Temkin (1919-2007), who introduced a novel separation of variables method for non-stationary heat conduction in the 1960s. Our extension applies this method to the hyperbolic heat equation, incorporating a relaxation term. The hyperbolic heat equation, a partial differential equation combining features of both hyperbolic and parabolic equations, finds wide applications across various scientific fields, including physics, engineering, geophysics, medical imaging, and more. Our investigation centres on the application of the Temkin’s method to the hyperbolic heat equation, with the aim to provide insights into its effectiveness in solving direct and inverse problems. The method relies on the observation that for nonstationary heat conduction with dynamic boundary conditions, the influence of initial conditions on the temperature distribution diminishes over time. Consequently, it is reasonable to assume that the temperature distribution is primarily influenced by the time-dependent boundary conditions. By expressing the solution to the given problem as a series, where each term is a product of a derivative of the given boundary condition and an unknown function P dependent on a spatial variable, we obtain a set of ordinary differential equations. These equations lead us to deduce the spatial functions, which are found to be polynomial in nature. While this approach holds promise for formulating an inverse problem to determine the speed of propagation, our current numerical results are inconclusive.
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Editeur Latvia University of Life Sciences and Technologies
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