Investigation of Doubly Nonlinear Parabolic Equation

Abstract

We study the properties of solutions for a porous medium equation (PME) in non-divergent form with a source term. The PME is a fundamental model in various physical and biological processes, including fluid flow through porous media, heat transfer, and population dynamics. Unlike the classical heat equation, the PME exhibits nonlinear diffusion, leading to rich mathematical structures and solution behaviours. Our main focus is obtaining exact solutions using the separable variable method under certain parameter constraints. These solutions provide explicit representations of the evolving profile of the medium and provide insight into the dynamics of the equation. Additionally, we construct a self-similar Barenblatt-type solution, a fundamental tool for analysing long-time asymptotics and the spreading behaviour of solutions. Self-similar solutions provide insights into the scaling properties of the PME and the influence of the source term on solution evolution. Moreover, we have constructed a numerical scheme, calculated numerical results and based on numerical solutions shown graphs in some particular cases.