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Manuscript Title: LIE, a PC program for Lie analysis of differential equations.
Authors: A.K. Head
Program title: LIE version 4.5
Catalogue identifier: ACPB_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 96(1996)311
Programming language: Mumath.
Computer: IBM PC.
Operating system: MSDOS or PCDOS version 2 or later.
RAM: 256K words
Word size: 16
Keywords: Computer algebra, Differential equations, Symbolic computations, Lie symmetries, Mumath, General purpose, Lie algebra, Exact solutions.
Classification: 4.2, 4.3, 5.

Nature of problem:
Differential equations are omnipresent in most departments of science. Explicit exact solutions are of great value when they exist and Lie analysis is the only systematic method of finding them. The procedure is well-known [1-5] but involves a large amount of symbolic calculation that is better done by computer.

Solution method:
The method of solution is well-known [1-5] except in one respect. The Lie determining equations are a set, often a large set, of partial differential equations. They are linear, homogeneous, over-determined and usually redundant. It is required to find their explicit general solution. There is no deterministic procedure known that is certain to solve these equations. This program uses a heuristic procedure, described in the file MORELIE.DOC, that works in most cases but it can sometimes fail to find the complete solution. So provision is made for convenient user intervention to give help although in practice this is rarely necessary.

Reasons for new version:
The previous version was for classical Lie analysis, finding the point symmetries of well-posed differential equations. This is now extended to contact, Lie-Backlund and non-classical symmetries. Memory utilization has been improved and it can analyse the equations of Magneto-Hydrodynamics, a set of 9 partial differential equations in 12 variables (data file MHD.DAT).

Unusual features:
A self-contained stand-alone program that makes Lie analysis available for the most widespread and numerous computer type without the need to obtain an underlying computer algebra system.

Running time:
For current PCs, the program run time is usually less than the time taken to prepare the input data file.

References:
[1] P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, N.Y., 1986).
[2] G.W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer, N.Y., 1989).
[3] J.M. Hill, Solution of Differential Equations by means of One- parameter Groups (Pitman, London, 1982).
[4] F. Schwarz, SIAM Review, 30(1988)450.
[5] B. Champagne, W. Hereman and P. Winternitz, Comp. Phys. Commun. 66(1991)319.