New Family of Iterative Methods for Solving Nonlinear Models

We introduce a new family of iterativemethods for solvingmathematicalmodelswhose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.


Introduction
Solving nonlinear equations is one of the important problems in mathematical sciences, especially in numerical analysis.There is a vast literature available to find the solution of nonlinear equations; see, for example,  and references therein.The construction of numerical methods is usually based on diverse techniques such as Taylor series, quadrature formulas, homotopy perturbation, and decomposition.One of the most powerful and well-known techniques for finding the solution of nonlinear equations is Newton's method which converges quadratically [19].
+1 =   −  (  )   (  ) ,   (  ) ̸ = 0,  = 0, 1, 2, . . . .(1) To improve the efficiency, several modified higher order methods have been presented in the literature by using different techniques [1-6, 9-12, 14-20, 22-24].Recently, some useful methods have been introduced in [7,8,13,21].Abbasbandy [1] used Adomian decomposition method (ADM) [2] to find the simple root of nonlinear equations.It is worth mentioning here that the involvement of higher order derivatives of Adomian polynomials is a major weakness of ADM.This weakness was eradicated by Daftardar-Gejji and Jafari [5], when they introduced a new decomposition technique.The said decomposition technique is quite simple as compared to the ADM as it does not need the higher order derivatives of functions.This technique has extensively been used to develop some useful algorithms for solving nonlinear equations [4,5,10,18].In 2015, Noor et al. [18], using the same decomposition technique along with the idea of coupled system, proposed two fourth-order iterative methods ( [18], Algorithms 2.12 and 2.13) with efficiency index of 1.2600 each.In this work, using the quadrature formula along with the fundamental law of calculus, truncating the series of () at quadratic level and decomposition technique of [5], we construct a family of new iterative methods for solving nonlinear equations.As special cases, we propose two fourth-order and two sixth-order methods.The number of evaluations per iteration for the third-order methods is 4 and 5; the fourth-order methods 5 and 6; and sixth-order methods 8 and 9; thus the efficiency indices of our methods are 1.3161, 1.2457, 1.3195, 1.2600, 1.2510, and 1.2203.The convergence criteria of newly constructed families are also presented.In order to demonstrate the validity and better efficiency of our proposed methods, we solve the nonlinear equations arising in the population model and in the motion of a particle on an inclined plane [18].We also present the graphical analysis for the endorsement of numerical results.

Iterative Methods
Let  be the simple root of the nonlinear equation: We rewrite (2) in the form of the following coupled system using the quadrature formula and fundamental law of calculus: where  denotes the initial approximation sufficiently close to the exact root  of (2),   are knots in [0, 1], and   are the weights satisfying Equation ( 3) can be written as follows: =  +  () , where Here () is a nonlinear operator which can be decomposed as follows using the decomposition technique mainly due to Daftardar-Gejji and Jafari [5].
The purpose of the above decomposition is to find the solution of (2) in the form of a series: Combining ( 7), (10), and (11), we get Thus, we have the following iterative scheme: Therefore, we have Since  0 = , (11) gives From ( 8) and ( 13), we have From ( 4), one can easily compute Using ( 9), (13), and ( 16), we have Using condition ( 5) and ( 17), the above equation yields We note that  is approximated as For  = 1, in (20) and using ( 16) and ( 19), we have This formulation allows us to propose the following iterative method for solving nonlinear equation (2).
Algorithm 3.For a given  0 , compute the approximate solution  +1 by the following iterative scheme: According to our knowledge, Algorithms 2 and 3 are new to solve nonlinear equation (2).
2.1.Some Special Cases of Algorithm 2. Now, we present some special cases of Algorithm 2. For  = 1,  1 = 1 and  1 = 0, Algorithm 2 reduces to the following iterative method for solving nonlinear equations.
Algorithm 4. For a given  0 , compute the approximate solution  +1 by the following iterative scheme Using  = 1,  1 = 1 and  1 = 1, Algorithm 2 reduces to the following iterative method.
Algorithm 7.For a given  0 , compute the approximate solution  +1 by the following iterative scheme: To the best of our knowledge, Algorithms 4-7 are new iterative methods for solving nonlinear equation (2).
2.2.Some Special Cases of Algorithm 3. Now, we present some special cases of Algorithm 3.For  = 1,  1 = 1 and  1 = 0, Algorithm 3 reduces to the following iterative method for solving nonlinear equations.
Algorithm 11.For a given  0 , compute the approximate solution  +1 by the following iterative scheme: To the best of our knowledge, Algorithms 8-11 are new iterative methods for solving nonlinear equation (2).

Convergence Analysis
In this section, convergence criteria of newly suggested methods are studied in the form of the following theorem.Proof.Let  be a simple zero of ().Since  is sufficiently differentiable, the Taylor series expansions of (  ),   (  ), and   (  ) about  are given by where   =   − and   = (1/!)( () ()/  ()),  = 2, 3, . . . .From (41) and (42), we get Using (44), we have From (45), we get Using ( 45), (47), and (48), we obtain the error term for Algorithm 2 as follows: From (49), we have Using ( 49) and ( 51), the error term for Algorithm 3 is obtained as Now, we prove the convergence orders of the special cases of Algorithms 2 and 3.

Numerical Examples
In this section, we demonstrate the validity and efficiency of our proposed iterative schemes by considering the nonlinear equations obtained from population model and the motion of a particle on an inclined plane, that is, 1,564,000 = 1,000,000  + 435,000  (  − 1) , (72) We take  0 = 1.5 for (72) and  0 = −1.2 for (73) as initial guess for computer program.Tables 1 and 2 and Figures 1 and 2 give the comparison of our newly proposed iterative methods, that

Conclusions
We have introduced a new family of iterative methods (Algorithms 2 and 3), based on decomposition technique, for solving nonlinear equations using coupled system of equations.Several new iterative methods have been established as special cases of newly established family.We have explored the convergence criteria of our new methods and investigated for convergence order and efficiency index.We present the comparative study both numerically and graphically (Tables 1 and 2 and Figures 1 and 2) of our newly constructed methods with some known methods by considering two real life models.In Table 3, we present numerical results by considering various nonlinear equations.

Theorem 12 .
Assume that the function  :  ⊂ R → R on an open interval  has a simple root  ∈ .Let () be sufficiently differentiable in the neighborhood of ; then the convergence orders of the methods defined by Algorithms 2-11 are 3, 4, 3, 3, 4, 4, 4, 4, 6 and 6, respectively.