Lie group classification and invariant exact solutions of the generalized Kompaneets equations

TL;DR

This paper classifies symmetries of a generalized Kompaneets equation using Lie group methods, identifies equations with enhanced symmetry properties, and constructs exact solutions based on these symmetries.

Contribution

It provides a comprehensive Lie group classification of the generalized Kompaneets equations and finds invariant solutions, highlighting the equation with $f(u)=u^{4/3}$ as maximally symmetric.

Findings

Six non-equivalent equations with extended symmetries

The equation with $f(u)=u^{4/3}$ has a 3D symmetry algebra

All group-invariant exact solutions are derived

Abstract

In this paper, from the group-theoretic point of view it is investigated such class of the generalized Kompaneets equations (GKEs): $$u_t=\frac1{x^2}\cdot\left[x^4(u_x+f(u))\right]_x, \ (t,x) \in \mathbb{R}_{+} \times \mathbb{R}_{+},$$ where $u=u(t,x)$, $u_t=\frac{\partial u}{\partial t}$, $u_x=\frac{\partial u}{\partial x}$, $u_{xx}=\frac{\partial^2 u}{\partial x^2}$; $f(u)$ is an arbitrary smooth function of the variable $u$. Using the Lie--Ovsiannikov algorithm, the group classification of the class under study is carried out. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra $\mathfrak{g}^\cap=\langle \partial_t \rangle$. Using the direct method, the equivalence group $G^\sim$ of the class is found. It is obtained six non-equivalent (up to the equivalence transformations from the group $G^\sim$) GKEs that allow wider invariance algebras than $\mathfrak{g}^\cap$. It is shown that, among the non-linear equations from the class, the GKE with the function $f(u)=u^{\frac43}$ has the maximal symmetry properties, namely, it admits a three-dimensional maximal Lie invariance algebra. Using the obtained operators, it is found all possible non-equivalent group-invariant exact solutions of the GKE under consider.