Mathematical Model of Heat Conduction for a Semi-Infinite Body, Taking into Account Memory Effects and Spatial Correlations

Abstract

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed.