Lie symmetry analysis of the nonlinear generalized heat equation for varying cross-section geometry

TL;DR

This paper applies Lie symmetry analysis to a nonlinear generalized heat equation with varying cross-section, classifying symmetries based on thermal coefficients and deriving invariant solutions for different cases.

Contribution

It provides a complete classification of symmetries for the generalized heat equation with geometric dependence, including transformations and invariant solutions.

Findings

Symmetry structure splits into two main cases based on the ratio C(u)/K(u).

Additional symmetries appear under specific coefficient relations.

Invariant solutions are derived for physically relevant subclasses.

Abstract

We study the nonlinear generalized heat equation $C(u)u_t=\frac{1}{z^{\nu}}\left(K(u)z^{\nu}u_z\right)_z$, where $C(u)$ and $K(u)$ are temperature-dependent thermal coefficients and $\nu>0$ is a geometric parameter describing the varying cross-section geometry. By applying the classical Lie symmetry method, we derive the determining equations and perform a complete classification of the admitted Lie point symmetries according to the functional dependence between $C(u)$ and $K(u)$. The analysis shows that the symmetry structure splits naturally into two principal cases: $C(u)/K(u)$ non-constant and $C(u)/K(u)=\beta$ constant. In the first case, only the basic symmetries are admitted for arbitrary coefficients, whereas additional generators appear under special compatibility relations. In the second case, the equation can be transformed to a linear radial heat equation by the substitution $v=\int K(u)du$, yielding an extended symmetry algebra. For each case, we construct the infinitesimal generators, commutator tables, one-parameter transformation groups, and corresponding invariant reductions. Invariant and similarity solutions are obtained and then specialized to several physically relevant subclasses, including power-law, exponential-type, and linear constitutive coefficients. The results provide a unified symmetry-based model for the analysis of generalized nonlinear heat equations in non-Cartesian geometries.