Symmetries of differential equations: From Sophus Lie to computer algebra. (English) Zbl 0664.35004

Schwarz, Fritz

SIAM Rev. 30, No. 3, 450-481 (1988).

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 SODE 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.

MSC:

35C05	Solutions to PDEs in closed form
35-04	Software, source code, etc. for problems pertaining to partial differential equations
35A30	Geometric theory, characteristics, transformations in context of PDEs

Keywords:

symmetry analysis; computer algebra; analytical calculations; similarity solutions; computer algebra packages; algebraic manipulations

Software:

SPDE; REDUCE