Abstract
Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.
Highlights
A general and powerful theoretical approach to deal with ordinary as well as partial differential equations is provided by the study of their continuous symmetries, i.e., transformations mapping the set of solutions of differential equations into itself
Even if there exist many good packages working in different Computer Algebra Systems (CAS) and suitable for doing automatically the computation of Lie symmetries of differential equations, the opinion of the author is that this program, besides offering the possibility of being freely used without requiring the access to a commercial CAS, contains procedures for investigating problems in different areas of modern group analysis; in addition, it is the first program where the approach to approximate symmetries proposed in [60] can be automatically exploited
When the infinitesimals ξ i and ηα are assumed to be independent of p, it is possible to project the symmetries on the space Z ≡ X × U of the independent and dependent variables, so obtaining a transformation changing an element of the class of differential equations to another element in the same class
Summary
A general and powerful theoretical approach to deal with ordinary as well as partial differential equations is provided by the study of their continuous symmetries, i.e., transformations mapping the set of solutions of differential equations into itself. The origin of this package dates back to 1994 when the author developed some routines useful to manage the lengthy expressions needed to determine the Lie point symmetries of differential equations; this set of procedures constantly grew through the years providing new capabilities, and constitutes an extensively tested package able to perform almost automatically much of the work. Even if there exist many good packages working in different CAS and suitable for doing automatically the computation of Lie symmetries of differential equations, the opinion of the author is that this program, besides offering the possibility of being freely used without requiring the access to a commercial CAS, contains procedures for investigating problems in different areas of modern group analysis (classical point, contact and variational symmetries, group-classification problems, Q-conditional symmetries, Lie remarkable equations); in addition, it is the first program where the approach to approximate symmetries proposed in [60] can be automatically exploited.
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