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Manuscript Title: Symmetries and first order ODE patterns.
Authors: E.S. Cheb-Terrab, A.D. Roche
Program title: Extension to ODEtools package
Catalogue identifier: ADIP_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 113(1998)239
Programming language: Maple.
Operating system: UNIX, Macintosh, Windows (95/NT).
RAM: 16M words
Keywords: Computer algebra, First order differential, Equations, Symmetry methods, Invariant ode patterns, Symbolic computation.
Classification: 5.

Nature of problem:
Analytical solving of first order ordinary differential equations using symmetry methods.

Solution method:
Matching ODEs to the patterns of a pre-determined set of invariant ODE families.

The computational scheme presented works when the input ODE has a symmetry of one of the forms considered in this work. This set of symmetry patterns can be extended, but there are symmetry patterns for which the ideas of this work cannot be applied.

Unusual features:
The computational scheme being presented is able to find symmetries by matching a given ODE to the patterns of a varied set of invariant ODE families. When such a symmetry exists, the routines can explicitly determine it, without solving any differential equations, and use it to return a closed form solution without requiring further participation from the user. The invariant ODE families that are covered include, as particular cases, more than 70 per cent of Kamke's first order examples, as well as many subfamilies for which there is no standard classification. Many of these subfamilies cannot be solved by using other methods nor by ODE-solvers of other computer algebra systems. The combination of this 'symmetry & pattern matching' approach with the standard classification methods of the ODEtools package succeeds in solving 94 per cent of Kamke's first order examples.

Running time:
The methods being presented were implemented in the framework of the ODEtools Maple package. On the average, over Kamke's [1] first order non-trivial examples (see sec. 5), the ODE-solver of ODEtools is now spending approx 5 sec. per ODE when successful, and approx 20 sec. when unsuccessful. When considering only ODEs of first degree in y', these timings drop by 50 per cent. The timings of this paper were obtained using Maple R5 on a Pentium 200 - 128 Mb. of RAM - running Windows95.

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