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Newton-EGmsor Methods for Solution of Second Order Two-Point Nonlinear Boundary Value Problems

Journal of Mathematics and System Science 2012年第3期2卷 185-190页

作者： Jumat Sulaiman Mohd Khatim Hasan Mohamed Othman Samsul Ariffin Abdul Karim School of Science and Technology Universiti Malaysia Sabah Kota Kinabalu 88400 Sabah Malaysia School of Information Technology Universiti Kebangsaan Malaysia UKM Bangi 43600 Selangor Malaysia Department of Communication Technology and Network Universiti Putra Malaysia UPM Serdang 43400 Selangor Malaysia Department of Fundamental and Applied Sciences Universiti Teknologi Petronas Tronoh 31750 Perak Darul Ridzuan Malaysia

The convergence results of block iterative schemes from the EG （Explicit Group） family have been shown to be one of efficient iterative methods in solving any linear systems generated from approximation equations. Apart from block iterative methods, the formulation of the msor （Modified Successive Over-Relaxation） method known as SOR method with red-black ordering strategy by using two accelerated parameters, ω and ω′, has also improved the convergence rate of the standard SOR method. Due to the effectiveness of these iterative methods, the primary goal of this paper is to examine the performance of the EG family without or with accelerated parameters in solving second order two-point nonlinear boundary value problems. In this work, the second order two-point nonlinear boundary value problems need to be discretized by using the second order central difference scheme in constructing a nonlinear finite difference approximation equation. Then this approximation equation leads to a nonlinear system. As well known that to linearize nonlinear systems, the Newton method has been proposed to transform the original system into the form of linear system. In addition to that, the basic formulation and implementation of 2 and 4-point EG iterative methods based on GS （Gauss-Seidel）, SOR and msor approaches, namely EGGS, EGSOR and EGmsor respectively are also presented. Then, combinations between the EG family and Newton scheme are indicated as EGGS-Newton, EGSOR-Newton and EGmsor-Newton methods respectively. For comparison purpose, several numerical experiments of three problems are conducted in examining the effectiveness of tested methods. Finally, it can be concluded that the 4-point EGmsor-Newton method is more superior in accelerating the convergence rate compared with the tested methods.

关键词： Explicit group msor iteration second order scheme two-point nonlinear boundary value problem.

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