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The Scientific World Journal
Volume 2013, Article ID 540705, 6 pages
http://dx.doi.org/10.1155/2013/540705
Research Article

Cotton-Type and Joint Invariants for Linear Elliptic Systems

1Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
2School of Natural Sciences (SNS), National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan

Received 9 October 2013; Accepted 26 November 2013

Academic Editors: D. Baleanu, H. Jafari, and C. M. Khalique

Copyright © 2013 A. Aslam and F. M. Mahomed. 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

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.