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Mathematical Problems in Engineering
Volume 2006, Article ID 68753, 11 pages
http://dx.doi.org/10.1155/MPE/2006/68753

Approximate transformations for van der Pol-type equations

1Department of High Performance Computing, Faculty of Natural Sciences, Ufa State Aviation Technical University, 12 K. Marx Street, Ufa 450025, Russia
2International Institute for Symmetry Analysis and Mathematical Modeling, North West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
3Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modeling, North West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa

Received 13 December 2004; Revised 20 September 2005; Accepted 16 July 2006

Copyright © 2006 R. K. Gazizov and C. M. Khalique. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Classification of van der Pol-type equations with respect to admitted approximate transformation groups transforming a small parameter is given. It is shown that approximate symmetries transforming the small parameter as well as the usual approximate symmetries can be used for approximate integration (e.g., by method of successive reduction of order) of ordinary differential equations with a small parameter.