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Abstract and Applied Analysis
Volume 2014, Article ID 425648, 9 pages
http://dx.doi.org/10.1155/2014/425648
Research Article

An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
3Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Received 24 March 2014; Revised 2 May 2014; Accepted 3 May 2014; Published 15 May 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 A. H. Bhrawy 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

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials with , and the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.