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Manuscript Title: Computer algebra solving of first order ODEs using symmetry methods. | ||

Authors: E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P. da Mota | ||

Program title: ODEtools | ||

Catalogue identifier: ADFP_v1_0Distribution format: tar.gz | ||

Journal reference: Comput. Phys. Commun. 101(1997)254 | ||

Programming language: Maple. | ||

Computer: Macintosh. | ||

Operating system: UNIX, DOS, VMS, CMS. | ||

RAM: 8M words | ||

Keywords: Computer algebra, First order ordinary, Differential equations, Symmetry methods, Symbolic computation. | ||

Classification: 5. | ||

Nature of problem:Analytical solving of first order ordinary differential equations. | ||

Solution method:Lie group symmetry methods. | ||

Restrictions:Besides 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 higher order ODEs. | ||

Unusual features:The first order ODE solver here presented is a Maple implementation of all the steps of the symmetry method solving scheme; i.e., when successful it returns a closed solution for the given ODE, not only the symmetry generator. Also, this solver permits the user to (optionally) participate in the solving process by giving an advice (HINT option) concerning the functional form for the coefficients of the infinitesimal symmetry generator (infinitesimals). This is relevant since the solver here presented was designed to tackle arbitrary first order ODEs, so its defaults may not be the best choice for all the cases. All the intermediate steps of the symmetry method solving scheme are available as user-level commands too. For instance, using the package's commands, it is possible to obtain the infinitesimals, the related canonical coordinates, the most general first order ODE invariant under a symmetry group, or even the symmetries of a given solution. The package also includes a command for classifying ODEs (according to Kamke's book [1] popping up Help pages based on Kamke's advice for solving them, facilitating the study of a given ODE and the use of the package with pedagogical purposes. | ||

Running time:This depends strongly on the ODE to be solved, usually taking from a few seconds to a few minutes. In the tests we ran (with 467 first order ODEs from Kamke's book [1], see sec. 4), the average times were: 8 sec. for a solved ODE and 75 sec. for an unsolved one, using a Pentium 90 with 64 Mb. RAM on a Windows 95 platform. | ||

References: | ||

[1] | E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen (Chelsea, New York, 1959). |

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