Symmetries of Differential Equations: From Sophus Lie to Computer Algebra

Abstract

The topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it. The whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered. We show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations. The computer algebra packages SODS and SPDE, respectively, which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation, are presented. Furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics.