Nonlinear diffusion equations (NDEs) are fundamental mathematical models describing a vast array of phenomena across science, engineering, and biology. Due to their inherent nonlinearities, obtaining exact or even approximate solutions for these equations poses significant challenges. This paper provides a comprehensive review of various established and emerging methodologies employed to solve NDEs, drawing insights from both analytical and numerical approaches. We explore methods such as the Differential Transform Method (DTM), Generalized Integral Transform Technique (GITT), Lie Symmetry Method, and Residual Power Series Method (RPSM) for analytical and semi-analytical solutions. For numerical approaches, we delve into the Differential Quadrature Method (DQM), Finite Difference Method (FDM), Finite Element Method (FEM), Collocation Methods, and the Method of Lines. The review highlights the applicability of these methods to diverse NDE types, including those with reaction terms, convection, and delays, emphasizing their strengths, limitations, and the critical importance of error analysis and stability considerations.
Azizul et al. (Wed,) studied this question.