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

Abstract

In this paper, from the group-theoretic point of view it is investigated such class of the generalized Kompaneets equations (GKEs): $$u_t=\frac{1}{x^2}\cdot {\left[x^4(u_x+f(u))\right]}_x,(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},$$ 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 ${\mathfrak{g}}^{\cap }=⟨{\partial }_t⟩$. 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 $f(u)=u^{\frac{4}{3}}$ 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.

20 pages