Lie symmetry analysis and exact solutions of the one-dimensional heat equation with power law diffusivity

A heat equation with non-constant diffusivity depending as a power law on the spatial variable is analysed using Lie's method to identify classical point symmetries. It is shown that the group invariant solutions of a four-dimensional symmetry subgroup can be decomposed into three different classes. These admit explicit solutions which can either be expressed in terms of Bessel functions, confluent hypergeometric functions or Coulomb wave functions.