# New Family of Iterative Methods for Solving Nonlinear Models

Authors: Faisal Ali; Waqas Aslam; Kashif Ali; Muhammad Adnan Anwar; Akbar Nadeem
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We introduce a new family of iterative methods for solving mathematical models whose 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.


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## Body

## 1. 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, [1–24] 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)xn+1=xn-fxnf′xn,f′xn≠0,n=0,1,2,….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 off(x) 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.


## 2. Iterative Methods

Letα be the simple root of the nonlinear equation:(2)fx=0.We rewrite (2) in the form of the following coupled system using the quadrature formula and fundamental law of calculus:(3)fx≈fγ+∑i=1pwif′γ+τix-γx-γ+∑i=1pwif′′γ+τix-γx-γ22+gx=0,(4)gx=fx-fγ-∑i=1pwif′γ+τix-γx-γ-∑i=1pwif′′γ+τix-γx-γ22,where γ denotes the initial approximation sufficiently close to the exact root α of (2), τi are knots in [0,1], and wi are the weights satisfying(5)∑i=1pwi=1.Equation (3) can be written as follows:(6)x=γ-2fγ+gx2∑i=1pwif′γ+τix-γ+x-γ∑i=1pwif′′γ+τix-γ(7)=c+Nx,where(8)c=γ,(9)Nx=-2fγ+gx2∑i=1pwif′γ+τix-γ+x-γ∑i=1pwif′′γ+τix-γ.Here N(x) is a nonlinear operator which can be decomposed as follows using the decomposition technique mainly due to Daftardar-Gejji and Jafari [5].(10)Nx=N∑i=0∞xi=Nx0+∑i=1∞N∑j=0ixj-N∑j=0i-1xj.The purpose of the above decomposition is to find the solution of (2) in the form of a series:(11)x=∑i=0∞xi.Combining (7), (10), and (11), we get(12)x=∑i=0∞xi=c+Nx0+∑i=1∞N∑j=0ixj-N∑j=0i-1xj.Thus, we have the following iterative scheme:(13)x0=c,x1=Nx0,x2=Nx0+x1-Nx0,⋮xm+1=Nx0+x1+⋯+xm-Nx0+x1+⋯+xm-1,m=1,2,….Therefore, we have(14)x1+x2+⋯+xm+1=Nx0+x1+⋯+xm,m=1,2,….Since x0=c, (11) gives(15)x=c+∑i=1∞xi.From (8) and (13), we have(16)x0=c=γ.From (4), one can easily compute(17)gx0=0.Using (9), (13), and (16), we have(18)x1=Nx0=-2fγ+gx02∑i=1pwif′γ+τix0-γ+x0-γ∑i=1pwif′′γ+τix0-γ.Using condition (5) and (17), the above equation yields(19)x1=-fγf′γ.We note that x is approximated as(20)x=limm→∞⁡Xm,whereXm=x0+x1+⋯+xm.For m=1, in (20) and using (16) and (19), we have(21)x≈X1=x0+x1=γ-fγf′γ.This formulation allows us to propose the following iterative method for solving nonlinear equation (2).Algorithm 1.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(22)xn+1=xn-fxnf′xn,f′xn≠0,n=0,1,2,…,which is the well-known Newton’s method for solving nonlinear (2). Now from (21), we have(23)x0+x1-γ=-fγf′γ.From (4) and using (23), the following can easily be obtained:(24)gx0+x1=fx0+x1-fγ-∑i=1pwif′γ+τix0+x1-γx0+x1-γ-∑i=1pwif′′γ+τix0+x1-γx0+x1-γ22.=fx0+x1-fγ+∑i=1pwif′γ+τix0+x1-γfγf′γ-∑i=1pwif′′γ+τix0+x1-γfγ22f′γ2.Similarly, from (9), by using (14) and (24), we have(25)x1+x2=Nx0+x1=-2fγ+gx0+x12∑i=1pwif′γ+τix0+x1-γ+x0+x1-γ∑i=1pwif′′γ+τix0+x1-γ=-fγf′γ-2fx0+x12∑i=1pwif′γ+τix0+x1-γ+x0+x1-γ∑i=1pwif′′γ+τix0+x1-γ.For m=2, in (20) and by using (13), (14), (16), and (25), we get(26)x≈X2=x0+x1+x2=c+Nx0+x1=γ-fγf′γ-2fx0+x12∑i=1pwif′γ+τix0+x1-γ+x0+x1-γ∑i=1pwif′′γ+τix0+x1-γ.Using above relation, we can suggest the following two-step iterative method for solving nonlinear equation (2).Algorithm 2.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(27)xn+1=yn-2fyn2∑i=1pwif′γ+τiyn-xn+yn-xn∑i=1pwif′′γ+τiyn-xn,yn=xn-fxnf′xnn=0,1,2,….From (26), we obtained(28)x0+x1+x2-γ=fγf′γ-2fx0+x12∑i=1pwif′γ+τix0+x1-γ+x0+x1-γ∑i=1pwif′′γ+τix0+x1-γ.From (4) and by using (28), we have(29)gx0+x1+x2=fx0+x1+x2-fγ-∑i=1pwif′γ+τix0+x1+x2-γx0+x1+x2-γ-∑i=1pwif′′γ+τix0+x1+x2-γx0+x1+x2-γ22.From (9), by using (14) and (29), we obtain(30)x1+x2+x3=Nx0+x1+x2=-2fγ+gx0+x1+x22∑i=1pwif′γ+τix0+x1+x2-γ+x0+x1+x2-γ∑i=1pwif′′γ+τix0+x1+x2-γ=x0+x1+x2-γ-2fx0+x1+x22∑i=1pwif′γ+τix0+x1+x2-γ+x0+x1+x2-γ∑i=1pwif′′γ+τix0+x1+x2-γ.For m=3, in (20), we have(31)x≈X3=x0+x1+x2+x3=c+Nx0+x1+x2=γ-fγf′γ-2fx0+x12∑i=1pwif′γ+τix0+x1-γ+x0+x1-γ∑i=1pwif′′γ+τix0+x1-γ-2fx0+x1+x22∑i=1pwif′γ+τix0+x1+x2-γ+x0+x1+x2-γ∑i=1pwif′′γ+τix0+x1+x2-γ.Using this formulation, we suggest the following three-step iterative method for solving nonlinear equations (2).Algorithm 3.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(32)xn+1=zn-2fzn2∑i=1pwif′γ+τizn-xn+zn-xn∑i=1pwif′′γ+τizn-xn,zn=yn-2fyn2∑i=1pwif′γ+τiyn-xn+yn-xn∑i=1pwif′′γ+τiyn-xn,yn=xn-fxnf′xnn=0,1,2,…According to our knowledge, Algorithms 2 and 3 are new to solve nonlinear equation (2).


### 2.1. Some Special Cases of Algorithm2

Now, we present some special cases of Algorithm2. For p=1,w1=1 and τ1=0, Algorithm 2 reduces to the following iterative method for solving nonlinear equations.Algorithm 4.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme(33)xn+1=yn-2fyn2f′xn+yn-xnf′′xn,yn=xn-fxnf′xnn=0,1,2,….Using p=1,w1=1 and τ1=1, Algorithm 2 reduces to the following iterative method.Algorithm 5.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(34)xn+1=yn-2fyn2f′yn+yn-xnf′′yn,yn=xn-fxnf′xnn=0,1,2,….Now, using p=1, w1=1 and τ1=1/2, Algorithm 2 reduces to the following iterative method.Algorithm 6.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(35)xn+1=yn-2fyn2f′xn+yn/2+yn-xnf′′xn+yn/2,yn=xn-fxnf′xnn=0,1,2,….Now by using p=2, w1=1/4,w2=3/4,τ1=0 and τ2=2/3, Algorithm 2 reduces to the following iterative method for solving f(x)=0.Algorithm 7.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(36)xn+1=yn-8fyn2f′xn+6f′xn+2yn/3+yn-xnf′′xn+3yn-xnf′′xn+2yn/3yn=xn-fxnf′xnn=0,1,2,….To the best of our knowledge, Algorithms 4–7 are new iterative methods for solving nonlinear equation (2).


### 2.2. Some Special Cases of Algorithm3

Now, we present some special cases of Algorithm3. For p=1, w1=1 and τ1=0, Algorithm 3 reduces to the following iterative method for solving nonlinear equations.Algorithm 8.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(37)xn+1=zn-2fzn2f′xn+zn-xnf′′xnzn=yn-2fyn2f′xn+yn-xnf′′xnyn=xn-fxnf′xn,n=0,1,2,….Using p=1, w1=1 and τ1=1, Algorithm 3 reduces to the following iterative method for solving f(x)=0.Algorithm 9.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(38)xn+1=zn-2fzn2f′zn+zn-xnf′′znzn=yn-2fyn2f′yn+yn-xnf′′ynyn=xn-fxnf′xn,n=0,1,2,….Now, using p=1, w1=1 and τ1=1/2, Algorithm 3 reduces to the following iterative method for solving fx=0.Algorithm 10.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(39)xn+1=zn-2fzn2f′zn+xn/2+zn-xnf′′zn+xn/2,zn=yn-2fyn2f′yn+xn/2+yn-xnf′′yn+xn/2,yn=xn-fxnf′xn,n=0,1,2,….Using p=2, w1=1/4, w2=3/4,τ1=0, and τ2=2/3, Algorithm 2 reduces to the following iterative method for solving f(x)=0.Algorithm 11.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(40)xn+1=zn-8fzn2f′xn+6f′xn+2zn/3+zn-xnf′′xn+3zn-xnf′′xn+2zn/3,zn=yn-8fyn2f′xn+6f′xn+2yn/3+yn-xnf′′xn+3yn-xnf′′xn+2yn/3,yn=xn-fxnf′xn,n=0,1,2,….To the best of our knowledge, Algorithms8–11 are new iterative methods for solving nonlinear equation (2).


## 2.1. Some Special Cases of Algorithm2

Now, we present some special cases of Algorithm2. For p=1,w1=1 and τ1=0, Algorithm 2 reduces to the following iterative method for solving nonlinear equations.Algorithm 4.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme(33)xn+1=yn-2fyn2f′xn+yn-xnf′′xn,yn=xn-fxnf′xnn=0,1,2,….Using p=1,w1=1 and τ1=1, Algorithm 2 reduces to the following iterative method.Algorithm 5.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(34)xn+1=yn-2fyn2f′yn+yn-xnf′′yn,yn=xn-fxnf′xnn=0,1,2,….Now, using p=1, w1=1 and τ1=1/2, Algorithm 2 reduces to the following iterative method.Algorithm 6.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(35)xn+1=yn-2fyn2f′xn+yn/2+yn-xnf′′xn+yn/2,yn=xn-fxnf′xnn=0,1,2,….Now by using p=2, w1=1/4,w2=3/4,τ1=0 and τ2=2/3, Algorithm 2 reduces to the following iterative method for solving f(x)=0.Algorithm 7.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(36)xn+1=yn-8fyn2f′xn+6f′xn+2yn/3+yn-xnf′′xn+3yn-xnf′′xn+2yn/3yn=xn-fxnf′xnn=0,1,2,….To the best of our knowledge, Algorithms 4–7 are new iterative methods for solving nonlinear equation (2).


## 2.2. Some Special Cases of Algorithm3

Now, we present some special cases of Algorithm3. For p=1, w1=1 and τ1=0, Algorithm 3 reduces to the following iterative method for solving nonlinear equations.Algorithm 8.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(37)xn+1=zn-2fzn2f′xn+zn-xnf′′xnzn=yn-2fyn2f′xn+yn-xnf′′xnyn=xn-fxnf′xn,n=0,1,2,….Using p=1, w1=1 and τ1=1, Algorithm 3 reduces to the following iterative method for solving f(x)=0.Algorithm 9.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(38)xn+1=zn-2fzn2f′zn+zn-xnf′′znzn=yn-2fyn2f′yn+yn-xnf′′ynyn=xn-fxnf′xn,n=0,1,2,….Now, using p=1, w1=1 and τ1=1/2, Algorithm 3 reduces to the following iterative method for solving fx=0.Algorithm 10.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(39)xn+1=zn-2fzn2f′zn+xn/2+zn-xnf′′zn+xn/2,zn=yn-2fyn2f′yn+xn/2+yn-xnf′′yn+xn/2,yn=xn-fxnf′xn,n=0,1,2,….Using p=2, w1=1/4, w2=3/4,τ1=0, and τ2=2/3, Algorithm 2 reduces to the following iterative method for solving f(x)=0.Algorithm 11.
For a givenx0, compute the approximate solution xn+1 by the following iterative scheme:(40)xn+1=zn-8fzn2f′xn+6f′xn+2zn/3+zn-xnf′′xn+3zn-xnf′′xn+2zn/3,zn=yn-8fyn2f′xn+6f′xn+2yn/3+yn-xnf′′xn+3yn-xnf′′xn+2yn/3,yn=xn-fxnf′xn,n=0,1,2,….To the best of our knowledge, Algorithms8–11 are new iterative methods for solving nonlinear equation (2).


## 3. Convergence Analysis

In this section, convergence criteria of newly suggested methods are studied in the form of the following theorem.Theorem 12.
Assume that the functionf:I⊂R→R on an open interval I has a simple root α∈I. Let f(x) be sufficiently differentiable in the neighborhood of α; 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.Proof.
Letα be a simple zero of f(x). Since f is sufficiently differentiable, the Taylor series expansions of f(xn), f′(xn), and f′′(xn) about α are given by(41)fxn=f′αen+c2en2+c3en3+c4en4+c5en5+Oen6,(42)f′xn=f′α1+2c2en+3c3en2+4c4e43+5c5en4+6c6en5+Oen5,(43)f′′xn=f′α2c2+6c3en+12c4e42+20c5en3+30c6en4+Oen5,where en=xn-α and cj=1/j!fjα/f′(α),j=2,3,….
From (41) and (42), we get(44)fxnf′xn=en-c2en2+2c22-c3en3+-4c23+7c2c3-3c4en4+8c24-20c22c3+10c2c4+6c32-4c5en5+Oen6.Using (44), we have(45)yn=xn-fxnf′xn=α+c2en2-2c22-c3en3+4c23-7c2c3+3c4en4+-8c24+20c22c3-10c2c4-6c32+4c5en5+Oen6.From (45), we get(46)fyn=c2en2-2c22-c3en3+5c23-7c2c3+3c4en4+-12c24+24c22c3-10c2c4-6c32+4c5en5+Oen6.Now, the Taylor’s series expansions of ∑i=1p‍wif′(xn+τi(zn-xn)) and ∑i=1p‍wif′′(xn+τi(zn-xn)) are given by(47)∑i=1pwif′xn+τiyn-xn=1-2∑i=1pwiτic2-c2en+∑i=1pwi2τic22+3τi2c3-6τic3+3c3en2+∑i=1pwi-4τic23-6τi2c2c3-4τi3c4+10τic2c3+12τi2c4-12τic4+4c4en3+Oen4,(48)∑i=1pwif′′xn+τiyn-xn=2c2+∑i=1pwi-6τic3+6c3en+∑i=1pwi6τic2c3+24τi2c4-48τic4+24c4en2+∑i=1pwi-12τic22c3-48τi2c2c4-120τi3c5+48τic2c4+12τic32+360τi2c5-360τic5+120c5en3+Oen4.Using (45), (47), and (48), we obtain the error term for Algorithm 2 as follows:(49)zn=yn-2fyn∑i=1pwif′xn+τiyn-xn+yn-xn∑i=1pwif′′xn+τiyn-xn=α+∑i=1pwi-2c22τi+c22en3+∑i=1pwi-4τi2c23+10τic23+3τi2c2c3-3c23-7τic2c3+2c2c3en4+Oe5.From (49), we have(50)fzn=∑i=1pwi-2c22τi+c22en3+∑i=1pwi-4τi2c23+10τic23+3τi2c2c3-3c23-7τic2c3+2c2c3en4+Oe5.The Taylor’s series expansions of ∑i=1p‍wif′(xn+τi(zn-xn)) and ∑i=1pwif′′(xn+τi(zn-xn)) are given by(51)∑i=1pwif′xn+τizn-xn=1-2∑i=1pwiτic2-c2en+∑i=1pwi3τi2c3-6τic3+3c3en2+∑i=1pwi-4τi2c23+2τic23-4τi3c4+12τi2c4-12τic4+4c4en3+Oen4,∑i=1pwif′′xn+τizn-xn=2+∑i=1pwi-4τic2-2c2en+∑i=1pwi6τi2c3-6τic3en2+∑i=1pwi-8τi2c23+2τic23-8τi3c4+2c23+24τic4-16c4en3+Oen4.Using (49) and (51), the error term for Algorithm 3 is obtained as(52)xn+1=zn-2fzn∑i=1pwif′xn+τizn-xn+zn-xn∑i=1pwif′′xn+τizn-xn=α+∑i=1pwi4τi2c23-4τic23+c23en4+Oen5.Now, we prove the convergence orders of the special cases of Algorithms 2 and 3.The Taylor’s series expansion of2f′(xn)+(yn-xn)f′′(xn) is given by(53)2f′xn+yn-xnf′′xn=2+2c2en+2c22en2+-4c23+10c2c3-4c4en3+8c24-26c22c3+18c2c4+12c32-10c5en4+Oen5.Using (45), (46), and (53), the error term for Algorithm 4 is obtained as(54)zn=yn-2fyn2f′xn+yn-xnf′′xn=α+c22en3+-3c23+2c2c3en4+Oen5.Now, expending 2f′(yn)+(yn-xn)f′′(yn) by Taylor’s series, we get(55)2f′yn+yn-xnf′′yn=2-2c2en+6c22en2+-12c23+6c2c3en3+24c24-18c22c3+18c2c4-12c32en4+Oen5.Using equations (45), (46), and (55), the error term for Algorithm 5 is obtained as(56)zn=yn-2fyn2f′yn+yn-xnf′′yn=α-c22en3+3c23-2c2c3en4+Oen5.The Taylor’s series expansions of f′xn+yn/2 and f′′(xn+yn/2) are given by(57)f′xn+yn2=1+c2en+c22+34c3en2+-2c23+72c2c3+12c4en3+4c24-374c22c3+92c2c4+3c32+516c5en4+Oen5,(58)f′′xn+yn2=2c2+3c3en+3c2c3+6c4en2+-6c22c3+12c2c4+6c32+15c5en3+12c22c3-18c22c4-21c2c32+45c2c5+33c3c4+45c6en4+Oen5.Using (45), (46), (57), and (58), we obtain the error term for Algorithm 6 as follows:(59)zn=yn-2fyn2f′xn+yn/2+yn-xnf′′xn+yn/2=α+c23-34c2c3en4+Oen5.Expanding f′xn+2yn/3 and f′′xn+2yn/3 by Taylor’s series, we have(60)f′xn+2yn3=1+23c2en+43c22+13c3en2+-83c23+4c2c3+427c4en3+163c24-323c22c3+449c2c4+83c32+581c5en4+Oen5,f′′xn+2yn3=2c2+2c3en+4c2c3+83c4en2+-8c22c3+8c32+323c2c4+409c5en3+16c23c3-28c2c32+1003c3c4-323c22c4+803c2c5+809c6en4+Oen5.Using (42), (43), (45), (46), and (60), the error term for Algorithm 7 is obtained as(61)zn=yn-8fyn2f′xn+6f′xn+2yn/3+yn-xnf′′xn+3yn-xnf′′xn+2yn/3=α+c23-12c2c3en4+Oen5.Using (54), we have(62)fzn=c22en3+-3c23+2c2c3en4+Oen5.Using (42), (43), (54), and (62), the error term for Algorithm 8 is obtained as(63)xn+1=zn-2fzn2f′xn+zn-xnf′′xn=α+c23en4+Oen5.Now, from (56), we have(64)fzn=-c22en3+3c23-2c2c3en4+Oen5,f′zn=1-2c23en3+6c24-4c22c3en4+Oen5,f′′zn=2c2-6c22c3en3+18c23c3-12c2c32en4+Oen5.Using (56) and (64), the error term for Algorithm 9 is obtained as(65)xn+1=zn-2fzn2f′zn+zn-xnf′′zn=α+c23en4+Oen5.Using (58), we get(66)fzn=c23-34c2c3en4+-4c24+7c22c3-52c2c4-32c32en5+Oen6.Taylor’s series expansions of f′zn+xn/2 and f′′(zn+xn/2) are given by(67)f′zn+xn2=1+c2en+34c3en2+12c4en3+c24-34c22c3+516c5en4+Oen5,f′′zn+xn2=2c2+3c3en+6c4en2+15c5en3+3c23c3-94c2c32+45c6en4+Oen5.Therefore, using (59), (66), and (67), the error term for Algorithm 10 is obtained as(68)xn+1=zn-2fzn2f′zn+xn/2+zn-xnf′′zn+xn/2=α+-34c23c3+916c2c32en6+Oen7.Using (61), we have(69)fzn=c23-12c2c3en4+-4c24+6c22c3-2518c2c4-c32en5+Oen6.Taylor’s series expansions of f′(xn+2zn/3) and f′′xn+2zn/3 are given by(70)f′xn+2zn3=1+23c2en+13c3en2+427c4en3+43c24-23c22c3+581c5en4+Oen5,f′′xn+2zn3=2c2+2c3en+83c4en2+409c5en3+4c23c3-2c2c32+809c6en4+Oen5.Using (61), (69), and (70), the error term for Algorithm 11 is obtained as(71)xn+1=zn-8fzn2f′xn+6f′xn+2zn/3+zn-xnf′′xn+3zn-xnf′′xn+2zn/3=α+34c25-78c23c3+14c2c32en6+Oen7.This complete the proof.


## 4. 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,(72)1,564,000=1,000,000eλ+435,000λeλ-1,(73)xt=-g2ω2eωt-e-ωt2-sin⁡ωt.We take x0=1.5 for (72) and x0=-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 is, Algorithm 6 (AG 1), Algorithm 7 (AG 2), Algorithm 10 (AG 3), and Algorithm 11 (AG 4) with standard Newton’s method (NM, (1)), Noor et al. methods ([18], Algorithm 2.6) (NR 1), ([18], Algorithm 2.7) (NR 2), ([18], Algorithm 2.12) (NR 3), and ([18], Algorithm 2.13) (NR 4). In Table 3, we present numerical results for various nonlinear equations which also indicate that our methods are more efficient.Table 1
Numerical results for population model (equation (72))
Method n x n f(xn) xn-xn-1 NM 7 0.1009979296857497890 5.933183e-16 1.104193e-18 NR1 6 0.1009979296857497889 2.329650e-34 6.369050e-20 NR2 6 0.1009979296857497889 1.115803e-29 1.138639 e - 17 NR3 4 0.1009979296857497890 5.933183e-16 3.778475e-19 NR4 4 0.1009979296857497890 5.933183e-16 1.398260e-18 AG1 4 0.1009979296857497889 2.632474e-83 1.401588 e - 22 AG2 4 0.1009979296857497889 1.002432e-77 3.228512e-21 AG3 3 0.1009979296857497900 1.406532 e - 12 1.744956e-11 AG4 3 0.1009979296857497900 1.406532 e - 12 3.152777e-12Table 2
Numerical results for motion of a particle (equation (73))
Method n x n f ( x n ) x n - x n - 1 NM 10 -0.3170617745729570950 1.543287e-22 1.557967e-19 NR1 9 -0.3170617745729570950 6.413654e-28 4.163537e-14 NR2 9 -0.3170617745729570950 1.888111e-24 2.182328e-12 NR3 6 -0.3170617745729570950 1.543287e-22 2.116367e-38 NR4 6 -0.3170617745729570950 1.543287e-22 2.709996e-37 AG1 5 -0.3170617745729570950 1.743432e-44 5.976014e-12 AG2 5 -0.3170617745729570950 2.080693e-42 1.938938e-11 AG3 4 -0.3170617745729571000 3.332577e-19 7.533267e-33 AG4 4 -0.3170617745729571000 3.332577e-19 6.625240e-24Table 3 f(x) x _ ∘ Method Number of iterations x [ k ] f ( x _ n ) δ e ^ ( x ^ 2 + 7 x - 30 ) - 1 2.9 NM 7 3.0000000000000000007 9.100000e-18 3.263396e-10 NR1 7 3.0000000000000000007 9.105537e-18 3.263396e-10 NR2 7 3.0000000000000000007 9.105537e-18 3.263396e-10 NR3 5 3.0000000000000000000 0.000000e+00 1.415161e-15 NR4 5 3.0000000000000000000 0.000000e+00 3.366886e-27 AG1 4 3.0000000000000000000 2.921644e-79 3.555566e-21 AG2 4 3.0000000000000000000 8.793703e-51 4.351958e-14 AG3 3 3.0000000000000000000 0.000000e+00 1.645921e-14 AG4 3 3.0000000000000000000 0.000000e+00 3.392389e-14  x ^ 2 - e ^ x + x + 2 1.65 NM 7 2.2041177331716202960 3.305103e-20 1.295280e-10 NR1 7 2.2041177331716202960 1.864247e-22 7.456850e-12 NR2 7 2.2041177331716202960 3.743730e-20 1.033112e-10 NR3 5 2.2041177331716202960 3.305103e-20 1.194680e-24 NR4 5 2.2041177331716202960 3.305103e-20 7.482519e-36 AG1 4 2.2041177331716202960 2.534748e-56 1.035700e-14 AG2 4 2.2041177331716202960 3.405915e-48 1.073168e-12 AG3 3 2.2041177331716203000 1.464912e-17 3.775506e-15 AG4 3 2.2041177331716203000 1.464912e-17 2.401455e-11  ( x + 2 ) e ^ ( - x ) - 1 - 1.35 NM 8 -1.8414056604369606378 2.472561e-19 8.481033e-21 NR1 7 -1.8414056604369606378 1.469668e-22 5.161458e-12 NR2 8 -1.8414056604369606378 1.750316e-40 5.508062e-21 NR3 5 -1.8414056604369606378 2.472561e-19 5.885759e-25 NR4 5 -1.8414056604369606378 2.472561e-19 3.160149e-36 AG1 4 -1.8414056604369606378 1.123854e-58 2.235422e-15 AG2 4 -1.8414056604369606378 2.782588e-49 4.782839 e - 13 AG3 3 -1.8414056604369606000 2.007912e-16 6.603435e-17 AG4 3 -1.8414056604369606000 2.007912e-16 1.073115e-11  2 x ^ 2 - 1 / 2   s i n ⁡ x - 1 / 2  π 0.5 NM 6 0.9977265554138634538 8.696898e-21 7.838761e-13 NR1 5 0.9977265554138634538 1.302709e-21 3.286611e-11 NR2 6 0.9977265554138634538 3.176527e-37 4.745782e-19 NR3 4 0.9977265554138634538 8.696898e-21 5.066551e-25 NR4 4 0.9977265554138634538 8.696898e-21 3.934532e-25 AG1 4 0.9977265554138634538 1.775184e-99 2.198399e-25 AG2 4 0.9977265554138634538 3.133051e-99 2.528325e-25 AG3 3 0.9977265554138634500 1.410674e-17 5.046616e-21 AG4 3 0.9977265554138634500 1.410674e-17 4.537399e-21  8 x ^ 3 - 3 x ^ 2   s i n ⁡ x - 5 0.5 NM 8 0.9716662861175176121 2.044833e-20 2.890826e-20 NR1 7 0.9716662861175176121 1.855259e-30 3.505396e-16 NR2 7 0.9716662861175176121 3.234098e-23 1.380792e-12 NR3 5 0.9716662861175176121 2.044833e-20 5.272663e-24 NR4 5 0.9716662861175176121 2.044833e-20 4.036720e-35 AG1 4 0.9716662861175176121 1.832541e-52 5.721104e-14 AG2 4 0.9716662861175176121 7.018615e-46 2.443339e-12 AG3 3 0.9716662861175176100 3.426161e-17 5.515718e-14 AG4 3 0.9716662861175176100 3.426161e-17 4.503689 e - 11Figure 1
Log of residuals for equation (72).Figure 2
Log of residuals for equation (73).We use Maple18 for all numerical computations and MATLAB 9 to plot the graphs. For the computer programs, we use the following stopping criteria:(74)xn+1-xn+fxn<ϵ,whereϵ=10-15.


## 5. Conclusions

We have introduced a new family of iterative methods (Algorithms2 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.



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