Practical Guide to the Symbolic Computation of Symmetries of Differential Equations

TL;DR

This paper presents a computational approach and free software tools for finding symmetries of nonlinear differential equations, aiding in their analytic solutions.

Contribution

It introduces algorithms implemented in Maxima/WxMaxima for automatically computing and solving symmetry-determining equations for differential equations.

Findings

Automated computation of symmetry-determining equations.

Algorithms that solve these equations automatically.

Implementation in freely available Maxima/WxMaxima software.

Abstract

Symmetries play an critical role in finding analytic solutions to nonlinear differential equations. A symmetry is a mapping of the solutions of the differential equation into the solutions and have been studied extensively for over a century. We present a computational approach to finding symmetries and computer algebra programs to compute the usually very large system of determining partial differential equations. We also provide computer algebra algorithm that at least automatically solves most of these equations and in simple cases provides a complete solution. The algorithms are programmed in maxima/wxmaxima that is freely available.