Lie symmetry method for a nonlinear heat-diffusion equation

TL;DR

This paper applies Lie symmetry analysis to a nonlinear heat-diffusion equation with variable coefficients, deriving symmetries, reducing the PDE to ODEs, and constructing invariant solutions for specific physical cases.

Contribution

It systematically determines Lie point symmetries for the nonlinear heat equation with variable coefficients and finds similarity solutions for physically relevant cases.

Findings

Identified Lie symmetries depending on the relationship between C(u) and K(u)

Reduced PDEs to ODEs using symmetry methods

Constructed explicit similarity solutions for Storm-type and power-law cases

Abstract

We investigate the nonlinear heat-diffusion equation \( C(u)\,\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\!\left( K(u)\,\frac{\partial u}{\partial x} \right) \), 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 power-law dependence of \( C(u) \) and \( K(u) \) on \( u \). For these cases, similarity solutions are obtained.