Analytical and Computational Approaches for Bi-Stable Reaction and p-Laplacian Diffusion Flame Dynamics in Porous Media

In this paper, we present a mathematical approach for studying the changes in pressure and temperature variables in flames. This conception extends beyond the traditional second-order Laplacian diffusion model by considering the p-Laplacian operator and a bi-stable reaction term, thereby providing a more generalized framework for flame diffusion analysis. Given the structure of our equations, we provide the boundedness and uniqueness of the solutions in a weak sense from both analytical and numerical approaches. We further reformulate the governing equations in the context of traveling wave solutions, applying singular geometric perturbation theory to derive the analytical expressions of these profiles. This theoretical development is complemented by numerical assessments, which not only validate our theoretical predictions, but also optimize the traveling wave speed to minimize the error between numerical and analytical solutions. Additionally, we explore self-similar structured solutions. The paper then concludes with a perspective on future research, with emphasis being placed on the need for experimental validation in laboratory settings. Such empirical studies could test the robustness of our model and allow for refinement based on actual measurements, thereby broadening the applicability and accuracy of our findings in practical scenarios.