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Manuscript Title: Computer algebra solving of second 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_v2_0Distribution format: tar.gz | ||

Journal reference: Comput. Phys. Commun. 108(1998)90 | ||

Programming language: Maple. | ||

Computer: Pentium 200. | ||

Operating system: UNIX, Macintosh, DOS, DEC VMS, IBM CMS. | ||

RAM: 16M words | ||

Word size: 32 | ||

Keywords: Computer algebra, First/second order, Ordinary differential, Equations, Symmetry methods, Symbolic computation. | ||

Classification: 5. | ||

Nature of problem:Analytical solving of first and second 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 first or second order ODE. | ||

Solution method:Computer algebra implementation of 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 ODEs of differential order higher than two. | ||

Unusual features:The 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 user-level commands too. Using the package's commands, it is then possible to obtain the infinitesimals, the related canonical coordinates, the finite form of the related group transformation equations, etc. Routines for testing the returned results, especially when they come in implicit form, are also provided. Special efforts were put in commands for solving the inverse problem too; that is, commands returning the most general first or second order ODE simultaneously invariant under given symmetries. One of the striking new features of the package - related to second order ODEs - is its ability to deal with dynamical symmetries, both in finding them and in using them in the integration procedures. Finally, the package also includes a command for classifying ODEs, optionally 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. 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 first 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 second 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). | ||

References: | ||

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

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