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Abstract and Applied Analysis
Volume 2012, Article ID 415431, 22 pages
Research Article

On the Exact Analytical and Numerical Solutions of Nano Boundary-Layer Fluid Flows

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11757, Egypt
3Department of Mathematics, Faculty of Science, Tabuk University, Tabuk 71491, Saudi Arabia

Received 31 May 2012; Revised 24 July 2012; Accepted 25 July 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Emad H. Aly and Abdelhalim Ebaid. 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.


The nonlinear boundary value problem describing the nanoboundary-layer flow with linear Navier boundary condition is investigated theoretically and numerically in this paper. The -expansion method is applied to search for the all possible exact solutions, and its results are then validated by the Chebyshev pseudospectral differentiation matrix (ChPDM) approach which has been recently introduced and successfully used. This numerical technique is firstly applied and, on comparing with the other recent work, it is found that the results are very accurate and effective to deal with the current problem. It is then used to examine and validate the present analytical analysis. Although the -expansion method has been used widely to solve nonlinear wave equations, its application for nonlinear boundary value problems has not been discussed yet, and the present paper may be the first to address this point. It is clarified that the exact solutions obtained via the -expansion method cannot be obtained by using some of the other methods. In addition, the domain of the physical parameters involved in the current boundary value problem is also discussed. Furthermore, the convex, vicinity of zero, and asymptotic solutions are deduced.