Electronic Journal of Differential Equations (Apr 2015)
Lie group classification and exact solutions of the generalized Kompaneets equations
Abstract
We study generalized Kompaneets equations (GKEs) with one functional parameter, and using the Lie-Ovsiannikov algorithm, we carried out the group classification. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra. Using the direct method, we find the equivalence group. We obtain six non-equivalent (up to transformations from the equivalence group) GKEs that allow wider invariance algebras than the kernel one. We find a number of exact solutions of the non-linear GKE which has the maximal symmetry properties.