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International Journal of Differential Equations
Volume 2013 (2013), Article ID 865464, 8 pages
http://dx.doi.org/10.1155/2013/865464
Research Article

An Improvement of the Differential Transformation Method and Its Application for Boundary Layer Flow of a Nanofluid

1Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
2Department of Mathematics, Faculty of Science, Ain Shams University, Egypt

Received 4 April 2013; Accepted 22 April 2013

Academic Editor: Jürgen Geiser

Copyright © 2013 Abdelhalim Ebaid et al. 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

The main feature of the boundary layer flow problems of nanofluids or classical fluids is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such a kind of problems. In order to solve these difficulties, the authors usually resort to either Padé approximants or the commercial numerical codes. However, an intensive work is needed to perform the calculations using Padé technique. Due to the importance of the nanofluids flow as a growing field of research and the difficulties caused by using Padé approximants to solve such problems, a suggestion is proposed in this paper to map the semi-infinite domain into a finite one by the help of a transformation. Accordingly, the differential equations governing the fluid flow are transformed into singular differential equations with classical boundary conditions which can be directly solved by using the differential transformation method. The numerical results obtained by using the proposed technique are compared with the available exact solutions, where excellent accuracy is found. The main advantage of the present technique is the complete avoidance of using Padé approximants to treat the infinity boundary conditions.