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Manuscript Title: LIE, a PC program for Lie analysis of differential equations.
Authors: A.K. Head
Program title: LIE version 4.1
Catalogue identifier: ACPB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 77(1993)241
Programming language: Mumath.
Computer: IBM PC.
Operating system: MSDOS or PCDOS version 2 or later.
RAM: 256K words
Word size: 16
Keywords: General purpose, Algebras, Computer algebra, Differential equations, Exact solutions, Symbolic computations, Lie symmetries, Mumath.
Classification: 4.2, 4.3, 5.

Nature of problem:
Differential equations are onmipresent 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 solution. So provision is made for convenient user intervention to give help although in practice this is rarely necessary.

The main restriction is on the size of the problem as MUMATH (and therefore LIE) is limited to using 256 Kilobytes of memory for program and workspace combined. A typical large problem for LIE would be the 3-dimensional Navier-Stokes equations of fluid dynamics. There are also restrictions on the type of functions that can occur in the differential equations as these must fall within the mathematical knowledge of the program. These are detailed in LIE.DOC but have not proved restrictive in practice.

Unusual features:
A self-contained stand-alone program that makes Lie analysis available for all models of the most widespread and numerous computer type. Automatic symbolic solution of the Lie determining equations by a heuristic procedure that covers a wide range of cases. It is not necessary to obtain an underlying computer algebra system to run the program.

Running time:
11 seconds for test run of 1-dimensional heat equation HEAT1.DAT given in file LIE.DOC. 20 minutes for 3-dimensional Navier-Stokes equations of fluid dynamics NAVSTOKE.DAT. These times are for a 386SX CPU with 33 Mhz clock and 32 Kilobyte cache.

[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, Comput. Phys. Commun. 66(1991)319.