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

## TL;DR

This paper applies Lie symmetry analysis to a one-dimensional heat equation with power law diffusivity, deriving explicit solutions expressed through special functions, and classifies solutions into three distinct classes based on symmetry groups.

## Contribution

It introduces a systematic Lie symmetry approach to solve a heat equation with non-constant diffusivity, providing explicit solutions in terms of special functions and classifying solution types.

## Key findings

- Explicit solutions in terms of Bessel, hypergeometric, and Coulomb functions.
- Classification of solutions into three distinct symmetry-based classes.
- Identification of classical point symmetries for the heat equation with power law diffusivity.

## Abstract

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.