Group Classification and Symmetry Reduction of a (1+1)-Dimensional Porous Medium Equation

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies.