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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 650208, 9 pages
http://dx.doi.org/10.1155/2013/650208
Research Article

A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-Bénard Convection

Department of Mathematics, Guru Nanak Dev University, Amritsar 143005, India

Received 5 June 2013; Accepted 3 July 2013

Academic Editors: D. Dürr and A. Qadir

Copyright © 2013 Jitender Singh. 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 simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type is considered where components of are known at one of the boundaries and components of are specified at the other boundary. The map is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of Ha (2001) to the BVPs arising in the higher order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-Bénard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role.