Revisiting the Group Classification of the General Nonlinear Heat Equation u t = ( K ( u ) u x ) x

Group classification is a powerful tool for identifying and selecting the free elements—functions or parameters—in partial differential equations (PDEs) that maximize the symmetry properties of the model. In this paper, we revisit the group classification of the general nonlinear heat (or diffusion) equation u t = K ( u ) u x x , where K ( u ) is a non-constant function of the dependent variable. We present the group classification framework, derive the determining equations for the coefficients of the infinitesimal generators of the admitted symmetry groups, and systematically solve for admissible forms of K ( u ) . Our analysis recovers the classical results of Ovsyannikov and Bluman and provides additional clarity and detailed derivations. The classification yields multiple cases, each corresponding to a specific form of K ( u ) , and reveals the dimension of the associated Lie symmetry algebra.