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, [1–19]. 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
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