Decomposition Technique and a Family of Efficient Schemes for Nonlinear Equations

Various problems of pure and applied sciences can be studied in the unified framework of nonlinear equations. In this paper, a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique.The convergence of the new methods is proven. For the implementation and performance of the new methods, some examples are solved and the results are compared with some existing methods.


Introduction
The conceptualization and creation of diverse iterative methods for finding efficient and precisely the approximate solution of nonlinear equation, () = 0, are a fundamental task in numerical analysis and related areas.This topic has attracted the attention of many researchers, when Abbasbandy [1] initiated the analysis of these methods.Many researchers collate and updated the state of the art of iterative methods.The advantage of multipoint methods is that they do not use higher-order derivatives and have great practical importance because they overcome the theoretical limitations of one-point methods regarding their convergence order and computational efficiency.In this work, a new decomposition technique [2] that is quite different from Adomian decomposition method [3] is applied to some classes of iterative methods for solving the nonlinear equations.We would like to point out that in the implementation of the Adomian decomposition method one has to calculate the derivatives of the so-called Adomian polynomials, which is the drawback of the technique of Adomian method.To overcome the disadvantage, the decomposition of Bhalekar and Daftardar-Gejji [2] is used in this work.The said decomposition technique [2] does not occupy the high-order differentials of the function and is very simple as compared to the Adomian method.Chun et al. [4] has suggested that the nonlinear equations can be written as a coupled system of equations.He et al. [5,6] have used the idea of coupled system of equations with the decomposition technique of Bhalekar and Daftardar-Gejji [2] to develop several iterative methods to solve the nonlinear equations.In this work, decomposition method [2] sophisticatedly combined with the coupled system of equations is used to develop new family of iterative methods.It is shown that the proposed family of iterative methods contains several well-known iterative techniques as special case.The convergence of the suggested class of method is also proven.Some numerical examples are illustrated to exhibit the efficiency and performance of proposed iterative methods.

Iterative Methods
In this section, we are going to present some new iterative methods by the help of quadrature method and basic fundamental law of calculus.Consider the following nonlinear equation: By assuming  as a simple zero of (1) and  as initial guess in the neighbor of , let () be the auxiliary function such that  ()  () = 0. (2) Now we use fundamental law of calculus and quadrature method on () and Taylor's series on (); then (2) becomes We rewrite (3) as system of coupled equations and then, by combining it with (2), we get represent the knots such that   ∈ [0, 1] and  are weights which are taken in such a way that they satisfy the consistency condition such that Now, rearrange (4) as Now we let where Equation (9) shows that () is a nonlinear function.Now, by using the idea of Daftardar-Gejji and Jafari [7], we develop a new iterative scheme.This decomposition technique is developed to solve the nonlinear functions and, by using this decomposition technique, we suggested the iterative method to calculate the root of () = 0.The Daftardar-Gejji and Jafari [7] decomposition technique is easier to use than Adomian decomposition method [3] to give the series solution.
Algorithm 2. The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Some Special Case of Algorithm 2. Now we have to consider value of Algorithm 2. Take  = 1,  1 = 1, and  1 = 0 in Algorithm 2 and it reduces to the following algorithm.
Algorithm 3. The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Take  = 1,  1 = 1, and  1 = 1 in Algorithm 2 and it reduces to the following algorithm.
Algorithm 4. The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Take  = 1,  1 = 1, and  1 = 1/2 in Algorithm 2 and it reduces to the following algorithm.
Algorithm 7. The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 7 is suggested by Shah and Noor [11].
Algorithm 8.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithms 3-8 give main iterative schemes which are helpful to generate several higher-order iterative methods for different values of auxiliary function (  ).So proper selection of auxiliary function manages to help us to diversify the problem for best possible implementation for obtaining the solution of nonlinear equation ( 1).So we consider auxiliary function () =  (−) to obtain the result by putting it in Algorithms 3-8.
After putting the value of () =  (−) in Algorithms 3-8, we get Algorithms 9-14.Algorithm 9.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 10.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 11.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 12.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 13.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Algorithm 14.The iterative scheme which will compute the approximate root  +1 for initial guess  0 is given as Shah and Noor [11] suggest that Algorithm 14 has cubic order of convergence.

Convergence Analysis
This section helps us to find out the convergence of Algorithm 2 by using Taylor's series.(47)

Conclusion
In this paper, the coupled system of equations with the new decomposition technique has been used to develop a family of iterative methods for solving nonlinear equations, which includes several well-known and new methods.Technique of derivation of the iterative methods is very simple as compared to the Adomian decomposition method.This is another aspect of the simplicity.The convergence analysis of the new  iterative methods has been proven.We have solved some examples and the methods are compared, which are exhibited in Tables 1 and 2 and Figures 1 and 2.