GeM software package for computation of symmetries and conservation laws of differential equations

Abstract

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results. Program summary Title of program: GeM Catalogue identifier: ADYK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS Solaris Operating systems under which the program has been tested: Windows 2000, Windows XP, Linux, Solaris Programming language used: Maple 9.5 Memory required to execute with typical data: below 100 Megabytes No. of lines in distributed program, including test data, etc.: 4939 No. of bytes in distributed program, including test data, etc.: 166 906 Distribution format: tar.gz Nature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations. Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions. Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully. Typical running time: 1–20 seconds for problems that do not involve classification; 5–1000 seconds for problems that involve classification, depending on complexity.