Lie symmetry method for a nonlinear heat-diffusion equation
We investigate the nonlinear heat-diffusion equation $C(u)u_t = (K(u)u_x)_x$, where $C(u)$ and $K(u)$ are coefficients that depend on u. By applying the classical Lie symmetry method, we determine the admitted Lie point symmetries and compute the corresponding infinitesimal generators according to the functional relationship between C(u) and K(u). The admitted symmetries are used to reduce the partial differential equation to ordinary differential equations and to construct invariant solutions. Particular cases of physical interest are analyzed in detail, including Storm-type materials and the dependence of power-laws of $C(u)$ and $K(u)$ on u, whose similarity solutions are obtained. Using a Kirchhoff-type transformation, we recover the corresponding results available in the literature for the standard nonlinear diffusion equation. We also discuss the importance of keeping $C(u)$ and $K(u)$ separated in order to preserve their physical meaning.