Group classification of linear evolution equations

TL;DR

This paper solves the group classification problem for (1+1)-dimensional linear evolution equations of arbitrary order, providing a complete classification and exact solutions using Lie reduction techniques.

Contribution

It offers the first complete group classification for arbitrary-order linear evolution equations and demonstrates their semi-normalization, enabling algebraic classification methods.

Findings

Complete group classification for arbitrary r > 2

Identification of semi-normalized classes for algebraic methods

Exact solutions obtained via Lie reduction

Abstract

The group classification problem for the class of (1+1)-dimensional linear $r$th order evolution equations is solved for arbitrary values of $r>2$. It is shown that a related maximally gauged class of homogeneous linear evolution equations is uniformly semi-normalized with respect to linear superposition of solutions and hence the complete group classification can be obtained using the algebraic method. We also compute exact solutions for equations from the class under consideration using Lie reduction and its specific generalizations for linear equations.