Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 316852, 29 pages
http://dx.doi.org/10.1155/2012/316852
Research Article

Numerov's Method for a Class of Nonlinear Multipoint Boundary Value Problems

Yuan-Ming Wang,1,2 Wen-Jia Wu,1 and Massimo Scalia3

1Department of Mathematics, East China Normal University, Shanghai 200241, China
2Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China
3Department of Mathematics, University of Rome “La Sapienza”, Piazzale le Aldo Moro 2, 00185 Rome, Italy

Received 28 March 2011; Revised 16 May 2011; Accepted 13 June 2011

Academic Editor: Shengyong Chen

Copyright © 2012 Yuan-Ming Wang 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 purpose of this paper is to give a numerical treatment for a class of nonlinear multipoint boundary value problems. The multipoint boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition. The problems are discretized by the fourth-order Numerov's method. The existence and uniqueness of the numerical solution are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the method are proved. An accelerated monotone iterative algorithm with the quadratic rate of convergence is developed for solving the resulting nonlinear discrete problems. Some applications and numerical results are given to demonstrate the high efficiency of the approach.