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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 198943, 12 pages
http://dx.doi.org/10.1155/2010/198943
Research Article

Some New Iterative Methods for Nonlinear Equations

1Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad 44000, Pakistan
2Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received 1 October 2010; Accepted 20 December 2010

Academic Editor: Carlo Cattani

Copyright © 2010 Muhammad Aslam Noor 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 suggest and analyze some new iterative methods for solving the nonlinear equations using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

1. Introduction

It is well known that a wide class of problem which arises in several branches of pure and applied science can be studied in the general framework of the nonlinear equations . Due to their importance, several numerical methods have been suggested and analyzed under certain conditions. These numerical methods have been constructed using different techniques such as Taylor series, homotopy perturbation method and its variant forms, quadrature formula, variational iteration method, and decomposition method; see, for example, [119]. To implement the decomposition method, one has to calculate the so-called Adomian polynomial, which is itself a difficult problem. Other technique have also their limitations. To overcome these difficulties, several other techniques have been suggested and analyzed for solving the nonlinear equations. One of the decompositions is due to Daftardar-Gejji and Jafari [6]. In this paper, we use this decomposition method to construct some new iterative methods. To apply this technique, we first use the new series representation of the nonlinear function, which is obtained by using the quadrature formula and the fundamental theorem of calculus. We rewrite the nonlinear equation as a coupled system of nonlinear equations. Applying the decomposition of Daftardar-Gejji and Jafari [6], we are able to construct some new iterative methods for solving the nonlinear equations. Our method of construction of these iterative methods is very simple as compared with other methods. We also prove convergence of the proposed methods, which is of order four. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. Our results can be considered as an important improvement and refinement of the previously results.

2. Iterative Methods

Consider the nonlinear equation Using the quadrature formula and the fundamental theorem of calculus, (2.1) can be written as where is an initial guess sufficiently close to α, which is a simple root of (2.1). We can rewrite the nonlinear equation (2.1) as a coupled system From (2.3), we have where It is clear that the operator is nonlinear. We now construct a sequence of higher-order iterative methods by using the decomposition technique, which is mainly due to Daftardar-Gejji and Jafari [6]. This decomposition of the nonlinear function is quite different from that of Adomian decomposition. In this method, one does not have to calculate the so-called the Adomian polynomial, which is another advantage of this decomposition. The main idea of this technique is to look for a solution having the series form The nonlinear operator can be decomposed as Combining (2.5), (2.8), and (2.9), we have Thus, we have the following iterative scheme: Then, It can be shown that the series converges absolutely and uniformly to a unique solution of (2.6). see [6].

From (2.7) and (2.12), we have From (2.4), (2.8) and using the idea of Yun [19], we obtain Note that is approximated by where .

For ,

For , This formulation allows us to suggest the following one-step iterative method for solving the nonlinear equation (2.1).

Algorithm 2.1. For a given , compute the approximate solution by the following iterative scheme: It is a well-known Newton method for solving nonlinear equations (2.1), which has second-order convergence.
From (2.1), we have From (2.4), (2.8) and using the idea of Yun [19], we have For , Using this relation, we can suggest the following two-step iterative method for solving nonlinear equation (2.1).

Algorithm 2.2. For a given , compute the approximate solution by the iterative following scheme: From (2.22), we obtain From (2.4), (2.8) and using the idea of Yun [19], we get For , Using this formulation, we can suggest the following three-step iterative method for solving nonlinear equation (2.1).

Algorithm 2.3. For a given , compute the approximate solution by the iterative following scheme.

3. Convergence Analysis

In this section, we consider the convergence criteria of the iterative methods developed in Section 2. In a similar way, one can consider the convergence of other algorithms.

Theorem 3.1. Let be a simple zero of sufficiently differentiable function for an open interval I. If is sufficiently close to , then the iterative methods defined by Algorithm 2.3 has fourth-order convergence.

Proof. Let α be a simple zero of . Then, by expanding and in Taylor’s Series about , we have where and .
From (3.1) and (3.2), we have From (3.3), we get Expanding ,, in Taylor’s Series about α and using (3.4), we have From (3.2), (3.6), and (3.7), we have From (3.5) and (3.8), we obtain From (3.4) and (3.9), we have Expanding ,, in Taylor’s Series about α and using (3.10), we obtain From (3.2), (3.12), and (3.13), we have From (3.11) and (3.14), we obtain From (3.10) and (3.15), we have Thus, we have Error equation (3.17) shows that the Algorithm 2.3 is fourth-order convergent.

4. Numerical Results

We now present some examples to illustrate the performance of the newly developed two-step and three-step iterative methods in this paper. We compare Newton method (NM), method of M. A. Noor et al. [9] (NNT), method of Chun [3] (CM), Algorithm 2.2 (NR1), and the Algorithm 2.3 (NR2) introduced in this paper. We used . The following stopping criteria is used for computer programs:(i), (ii).

The computational order of convergence approximated by means of

We consider the following nonlinear equations as test problems which are the same as M. Aslam Noor and K. Inayat Noor [10].

5. Conclusion

In this paper, we have considered one-step, two-step, and three-step iterative methods for solving nonlinear equations by using a different decomposition technique. Our method of derivation of the iterative methods is very simple as compared with the Adomian decomposition methods. From the Table 1, it is obvious that three-step method introduced in this paper performs better than the fourth-order method of Chun [3]. Using the technique and idea of this paper, one can suggest and analyze higher-order multistep iterative methods for solving nonlinear equations as well as system of nonlinear equations. It is an open problem to extend the technique and ideas of this paper for solving the obstacle problems associated with the variational inequalities and related problems see [2023] and the references therein. This is another direction for future research.

tab1
Table 1: Numerical examples.

Acknowledgments

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, CIIT, for providing excellent research facilities. This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia and Research Grant no. KSU.VPP.108.

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