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Abstract Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package — InSyDE. The package is designed, apart from materializing the algorithms presented, to provide a set of tools to allow the user to analyse the intermediary steps of the process. Computer programs in physics Program Title: InSyDE — Invariants and Symmetries of (racional second order ordinary) Differential Equations. Program Files doi: http://dx.doi.org/10.17632/4ytft6zgk7.1 Licensing provisions: GPLv3 Programming language: Maple 17. Nature of problem Search for first integrals of rational 2ODEs. Solution method: The method of solution is based on an algorithm described in this paper. Restrictions: If the rational 2ODE that is being analysed presents a very high degree in (x , y , z), then the method may not work well. Unusual features: Our implementation can find first integrals in many cases where the rational 2ODE under study cannot be reduced by other powerful solvers. Besides that, the package presents some useful research commands. [ABSTRACT FROM AUTHOR]

Abstract: Various extensions of the tanh-function method and their implementations for finding explicit travelling wave solutions to nonlinear partial differential equations (PDEs) have been reported in the literature. However, some solutions are often missed by these packages. In this paper, a new algorithm and its implementation called TWS for solving single nonlinear PDEs are presented. TWS is implemented in Maple 10. It turns out that, for PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases. Program summary: Program title:TWS Catalogue identifier:AEAM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAM_v1_0.html Program obtainable from:CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:1250 No. of bytes in distributed program, including test data, etc.:78 101 Distribution format:tar.gz Programming language:Maple 10 Computer:A laptop with 1.6 GHz Pentium CPU Operating system:Windows XP Professional RAM:760 Mbytes Classification:5 Nature of problem:Finding the travelling wave solutions to single nonlinear PDEs. Solution method:Based on tanh-function method. Restrictions:The current version of this package can only deal with single autonomous PDEs or ODEs, not systems of PDEs or ODEs. However, the PDEs can have any finite number of independent space variables in addition to time t. Unusual features:For PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases. Additional comments:It is easy to use. Running time:Less than 20 seconds for most cases, between 20 to 100 seconds for some cases, over 100 seconds for few cases. References: [1] E.S. Cheb-Terrab, K. von Bulow, Comput. Phys. Comm. 90 (1995) 102. [2] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299 (2002) 179. [3] E. Fan, Phys. Lett. 277 (2000) 212. [4] W. Malfliet, Amer. J. Phys. 60 (1992) 650. [5] W. Malfliet, W. Hereman, Phys. Scripta 54 (1996) 563. [6] E.J. Parkes, B.R. Duffy, Comput. Phys. Comm. 98 (1996) 288. [Copyright &y& Elsevier]