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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 303960, 30 pages
http://dx.doi.org/10.1155/2012/303960
Review Article

Linearization: Geometric, Complex, and Conditional

Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Islamabad, Pakistan

Received 21 September 2012; Accepted 25 November 2012

Academic Editor: Fazal M. Mahomed

Copyright © 2012 Asghar Qadir. 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

Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of) differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second-order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it), and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory), are reviewed. It is relevant to mention that much of the work is not linearization but uses the base of linearization.