This paper introduces a mathematical model of a thermal protection system incorporating the Riemann-Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation. In contrast, the third layer, characterized by its porous nature, is modeled using a fractional-order heat conduction equation. The thermal contact resistances between the layers are taken into account. The external surface is subjected to a boundary condition of the second kind, incorporating an aerothermal heat flux derived from NASA Langley Research Center data, while the internal surface is governed by a Dirichlet boundary condition. Additionally, the temperature-dependent material properties are considered. A significant contribution of this study is the development of a numerical scheme for a three-layer thermal protection system model, in which one layer is porous and described using the Riemann-Liouville fractional derivative. The proposed approach allows for accurate simulation of heat conduction in systems with complex material structures. The influence of the fractional derivative order β on the temperature profile was investigated, showing that variations in β significantly affect the thermal response of the system. Furthermore, a mesh refinement study was conducted to assess the impact of spatial discretization on the numerical results. These findings establish the model as a valuable tool for computer simulations and provide a basis for further development and optimization of mathematical and computational approaches in the analysis of thermal protection systems.
Brociek et al. (Tue,) studied this question.