Computer algebra solving of second order ODEs using symmetry methods

Abstract

An update of the ODEtools Maple package, for the analytical solving of 1 st and 2 nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant steps of the symmetry scheme. The package also includes commands for testing the returned results, and for classifying 1 st and 2 nd order ODEs. PROGRAM SUMMARYTitle of the software package: ODEtools. Catalogue number: (supplied by Elsevier)Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Licensing provisions: noneOperating systems under which the program has been tested: UNIX systems, Macintosh, DOS (AT 386, 486 and Pentium based) systems, DEC VMS, IBM CMS.Programming language used: Maple V Release 3/4. Memory required to execute with typical data: 16 Megabytes.No. of lines in distributed program, excluding On-Line Help, etc.: 10159.Keywords: 1 st /2 nd order ordinary differential equations, symmetry methods, symbolic computation. Nature of mathematical problemAnalytical solving of 1 st and 2 nd order ordinary differential equations using symmetry methods, and the inverse problem; that is: given a set of point and/or dynamical symmetries, to find the most general invariant 1 st or 2 nd order ODE. Methods of solutionComputer algebra implementation of Lie group symmetry methods. Restrictions concerning the complexity of the problemBesides the inherent restrictions of the method (there is as yet no general scheme for solving the associated PDE for the coefficients of the infinitesimal symmetry generator), the present implementation does not work with systems of ODEs nor with ODEs of differential order higher than two. Typical running timeThis depends strongly on the ODE to be solved. For the case of first order ODEs, it usually takes from a few seconds to 1 or 2 minutes. In the tests we ran with the first 500 1 st order ODEs from Kamke's book [1], the average times were: 8 sec. for a solved ODE and 15 sec. for an unsolved one, using a Pentium 200 with 64 Mb. RAM, on a Windows 95 platform. In the case of second order ODEs, the average times for the non-linear 2 nd order examples of Kamke's Book were 35 seconds for a solved ODE and 50 seconds for an unsolved one. The tests were run using the Maple version under development, but almost equivalent results are obtained using the available Maple R4 and R3 (the code presented in this work runs in all these versions). Unusual features of the programThe ODE-solver here presented is an implementation of all the steps of the symmetry method solving scheme; that is, the command receives an ODE, and when successful it directly returns a closed form solution for the undetermined function. Also, this solver permits the user to optionally participate in the solving process by giving advice concerning the functional form for the coefficients of the infinitesimal symmetry generator (infinitesimals). Many of the intermediate steps of the symmetry scheme are available as ...

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