# Iteration Methods with an Auxiliary Function for Nonlinear Equations

Authors: Faisal Ali; Waqas Aslam; Imran Khalid; Akbar Nadeem
Journal: Journal of Mathematics
 (2020)

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## Abstract

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary functiongx to solve the nonlinear equation fx=0. In this paper, we introduce the expansion of gx with the inclusion of weights wi such that ∑i=1pwi=1 and knots τi∈0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function gx, and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.


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## Body

## 1. Introduction

Most of the problems in science and engineering involves nonlinear equation of the formfx=0, where f:D⊆R⟶R is a sufficiently smooth function in the neighborhood of a simple zero α∈D. Many physical problems related to diverse areas such as biological applications in population dynamics and genetics where impulses arise naturally, motion of a particle on an inclined plane and projectile motion in physics, Van der Waals problem, and continuous stirred tank reactor equation in chemistry etc., can be modelled by nonlinear equations. Consequently, many numerical methods based on different techniques have been developed for solving nonlinear equations, see for example [1–12] and references therein. The following quadratically convergent Newton’s method [13], which needs 2 function evaluations per iteration, is considered as the fundamental tool for obtaining the numerical solution of nonlinear equations:(1)xn+1=xn−fxnf′xn,n=0,1,2,3,…,f′xn≠0.Definition 1.
(see [12]).
Letp be the convergence order of the iterative method and q is the number of functional evaluations per iteration required by the method, then the efficiency index (IE) of the method is defined as(2)IE=p1/q.
Obviously, the efficiency index (IE) of Newton’s method is 1.4142.
An appropriate selection of an initial guess makes Newton’s method very efficient, whereas this method works ordinarily for an inappropriate initial guess. In second century BC, an ancient Chinese, Ying Buzu Shu, introduced a more useful method, which gives accurate results in some cases even where Newton’s method does not work, for which two initial approximations are chosen. Usingx1 and x2 as initial approximations, the next approximation can be obtained as(3)x3=x2fx1−x1fx2fx1−fx2.
During the last two decades, ancient Chinese algorithms have been a scorching topic for many researchers [14–16]. In 2016, another Chinese He [17] introduced two modifications (Bubbfil Algorithms 1 and 2) of the above method with fast convergence.
Several modifications of Newton’s method have also been introduced, e.g., [5, 12, 18, 19]. Weerakon and Fernando [20] constructed the following useful cubically convergent method:(4)xn+1=xn−2fxnf′xn+f′yn,yn=xn−fxnf′xn.
The midpoint rule [19, 21],(5)xn+1=xn−fxnf′xn−fxn/2f′xn,with an order of convergence 3, is also a well-known technique. There exist several other third- and fourth-order methods involving diverse techniques for nonlinear equations, e.g., [5, 18, 22, 23]. Abbasbandy [1] used the Adomian decomposition method (ADM) [2] to find the simple root of nonlinear equations. But employing ADM involves higher order derivatives of Adomian polynomials which is a major weakness of ADM. But this is not the case in the decomposition technique due to Cherruault [24, 25]. The technique of Cherruault has extensively been used to develop some useful algorithms for solving nonlinear equations [3, 9, 26]. Recently, Shah et al. [27], using the same technique and an auxiliary function gx along with the expansion of the original equation fx=0, have introduced a family of iterative methods for nonlinear equations.
In the present paper, making use of the decomposition technique of [24, 25], together with the expansions of both fx and gx, we construct a new family of swiftly convergent iterative methods. In order to exhibit the efficiency of proposed methods, we present numerical as well as graphical analysis by considering three mathematical models from different branches of science, i.e., physics, mathematics, and chemistry.


## 2. Construction of Methods

Assume the nonlinear equation:(6)fx=0,with a simple zero α and the initial guess γ sufficiently close to α. We take gx as an auxiliary function, such that gx is sufficiently smooth and gα≠0,(7)fxgx=0.Using the quadrature formula and the fundamental law of calculus, the following coupled system can be formed:(8)fγgγ+x−γfγKg′,x,γ+gγQf′,x,γ+hx=0,(9)hx=fxgx−fγgγ−x−γfγKg′,x,γ+gγQf′,x,γ,where Kg′,x,γ and Qf′,x,γ represents ∑i=1pwig′γ+τix−γ and ∑i=1pwif′γ+τix−γ respectively; wi and τi are weights and knots, respectively, such that ∑i=1pwi=1 and τi∈0,1.We write equation (8) in terms of nonlinear operator Nx as follows:(10)x=γ−fγgγ+hxfγKg′,x,γ+gγQf′,x,γ=c+Nx,where(11)c=γ,(12)Nx=−fγgγ+hxfγKg′,x,γ+gγQf′,x,γ.We construct a sequence of higher order iterative methods by decomposingNx in the following way [24, 25]:(13)Nx=N∑i=0∞xi=Nx0+∑i=1∞N∑j=0ixj−N∑j=0i−1xj.Our purpose is to seek the solution in the form of the following series:(14)x=∑i=0∞xi.Combining equations (10), (13), and (14), we get(15)∑i=0∞xi=c+Nx0+∑i=1∞N∑j=0ixj−N∑j=0i−1xj.Thus, we have the following iterative scheme:(16)x0=c,x1=Nx0,x2=Nx0+x1−Nx0,⋮xn+1=Nx0+x1+⋯+xn−Nx0+x1+⋯+xn−1,n=1,2,….Thus,(17)x1+x2+⋯+xn+1=Nx0+x1+⋯+xn,n=1,2,3,…,x=c+∑i=1∞xi.From equations (11), (12), and (16), we have(18)x0=c=γ,x1=Nx0=−fγgγ+hx0fγKg′,x,γ+gγQf′,x,γ.Using equation (9), we get(19)x1=−fγgγfγg′γ+gγf′γ.We note thatx is approximated as(20)Xm=x0+x1+⋯+xm,where(21)limm⟶∞Xm=x.Form=1, we have(22)x≈X1=x0+x1=γ−fγgγfγg′γ+gγf′γ.Thus, the following recurrence relation can be formed by using equation (22).Algorithm 1.
(28).(23)xn+1=xn−fxngxnfxng′xn+f′xngxn,n=0,1,2,…,i.e., corresponding to the initial approximation x0, n+1st approximation can be determined. It is notable that the above algorithm is actually Newton’s method applied to fxgx.
Now, form=2, equation (20) gives(24)x≈X2=x0+x1+x2.
Using equation (12), we have(25)Nx0+x1=−fγgγ+hx0+x1fγKg′,x0+x1,γ+gγQf′,x0+x1,γ.
Using equations (9), (16), (22), and (24), we get(26)x≈X2=x0+x1+x2,=γ−fγgγfγg′γ+gγf′γ−fx0+x1gx0+x1fγKg′,x0+x1,γ+gγQf′,x0+x1,γ.
Thus, the following recurrence relation can be formed using equation (26).Algorithm 2.
(27)yn=xn−fxngxnfxng′xn+gxnf′xn,n=0,1,2,…,xn+1=yn−fyngynfxn∑i=1pwig′xn+τiyn−xn+gxn∑i=1pwif′xn+τiyn−xn,which is a new predictor-corrector method, and corresponding to initial approximation x0, n+1st approximation can be determined.
Now, form=3, equation (20) gives(28)x≈X3=x0+x1+x2+x3.
Using equation (12), we have(29)Nx0+x1+x2=−fγgγ+hx0+x1+x2fγKg′,x0+x1+x2,γ+gγQf′,x0+x1+x2,γ.
Using equations (9), (16), (26), and (28), we get(30)x≈X3=x0+x1+x2+x3=γ−fγgγfγg′γ+gγf′γ−fx0+x1gx0+x1fγKg′,x0+x1,γ+gγQf′,x0+x1,γ−fx0+x1+x2gx0+x1+x2fγKg′,x0+x1+x2,γ+gγQf′,x0+x1+x2,γ.
Thus, the following recurrence relation can be formed using equation (30).Algorithm 3.
(31)yn=xn−fxngxnfxng′xn+gxnf′xn,n=0,1,2,…,zn=yn−fyngynfxn∑i=1pwig′xn+τiyn−xn+gxn∑i=1pwif′xn+τiyn−xn,xn+1=zn−fzngznfxn∑i=1pwig′xn+τizn−xn+gxn∑i=1pwif′xn+τizn−xn,which is also a new three-step iterative technique.


### 2.1. Some Special Cases of Algorithm2

Algorithm 4.
Takingp=1,w1=1, and τ1=0, Algorithm 2 reduces to the following iterative method:(32)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′xn+gxnf′xn,n=0,1,2,….
Shah and Noor [28] have established the above formula.Algorithm 5.
Takingp=1,w1=1, and τ1=1, Algorithm 2 reduces to the following iterative method.(33)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′yn+gxnf′yn,n=0,1,2,….Algorithm 6.
Takingp=1,w1=1, and τ1=1/2, Algorithm 2 reduces to the following iterative method:(34)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′yn+xn/2+gxnf′yn+xn/2,n=0,1,2,….Algorithm 7.
Takingp=2,w1=1/4,w2=3/4,τ1=0, and τ2=2/3, Algorithm 2 reduces to the following iterative method:(35)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−4fyngynfxng′xn+gxnf′xn+3fxng′xn+2yn/3+3gxnf′xn+2yn/3,n=0,1,2,….Algorithm 8.
Takingp=2,w1=1/2,w2=1/2,τ1=0, and τ2=1/2, Algorithm 2 reduces to the following iterative method:(36)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−2fyngynfxng′xn+gxnf′xn+fxng′xn+yn/2+gxnf′xn+yn/2,n=0,1,2,….
To the best of our knowledge, Algorithms5–8 are new methods having convergence orders 3,3,4, and 4, respectively.


### 2.2. Some Special Cases of Algorithm3

Algorithm 9.
Takingp=1,w1=1, and τ1=0, Algorithm 3 reduces to the following iterative method:(37)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′xn+gxnf′xn,xn+1=zn−fzngznfxng′xn+gxnf′xn,n=0,1,2,….Algorithm 10.
Takingp=1,w1=1, and τ1=1, Algorithm 3 reduces to the following iterative method:(38)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′yn+gxnf′yn,xn+1=zn−fzngznfxng′zn+gxnf′zn,n=0,1,2,….Algorithm 11.
Takingp=1,w1=1, and τ1=1/2, Algorithm 3 reduces to the following iterative method:(39)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′yn+xn/2+gxnf′yn+xn/2,xn+1=zn−fzngznfxng′zn+xn/2+gxnf′zn+xn/2,n=0,1,2,….Algorithm 12.
Takingp=2,w1=1/4,w2=3/4,τ1=0, and τ2=2/3, Algorithm 3 reduces to the following iterative method:(40)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−4fyngynfxng′xn+gxnf′xn+3fxng′xn+2yn/3+3gxnf′xn+2yn/3,xn+1=zn−4fzngznfxng′xn+gxnf′xn+3fxng′xn+2zn/3+3gxnf′xn+2zn/3,n=0,1,2,….Algorithm 13.
Takingp=2,w1=1/2,w2=1/2,τ1=0, and τ2=1/2, Algorithm 3 reduces to the following iterative method:(41)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−2fyngynfxng′xn+gxnf′xn+fxng′xn+yn/2+gxnf′xn+yn/2,xn+1=zn−2fzngznfxng′xn+gxnf′xn+fxng′xn+zn/2+gxnf′xn+zn/2,n=0,1,2,….
To the best of our knowledge, Algorithms10–13 are new iterative methods for solving nonlinear equation (6) with convergence orders 3,4,4, and 4, respectively.
Obviously, different iterative methods can be obtained from Algorithms2 and 3 by using distinct values of the auxiliary function gx. Suitable selection of auxiliary function gx plays a significant role for the better performance of these methods. To demonstrate the efficiency of our proposed algorithms, we choose gx=Ix. Thus, corresponding to this choice, algorithm 10+i, i=4,5,6,…,13, is the special case of algorithm i.Algorithm 14.
(42)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′xn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm14 is 3, and the total number of evaluations per iteration is 3. Thus, IE=1.4422.Algorithm 15.
(43)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′yn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm15 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 16.
(44)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′xn+yn/2+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm16 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 17.
(45)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−4ynfyn3xnf′xn+2yn/3+xnf′xn+4fxn,n=0,1,2,….
The convergence order of the method described in Algorithm17 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 18.
(46)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−2ynfynxnf′xn+yn/2+xnf′xn+2fxn,n=0,1,2,….
The convergence order of the method described in Algorithm18 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 19.
(47)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′xn+fxn,xn+1=zn−znfznxnf′xn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm19 is 4, and the total number of evaluations per iteration is 4. Thus, IE=1.4142.Algorithm 20.
(48)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′yn+fxn,xn+1=zn−znfznxnf′zn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm20 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 21.
(49)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′xn+yn/2+fxn,xn+1=zn−znfznxnf′xn+zn/2+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm21 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 22.
(50)yn=xn−xnfxnxnf′xn+fxn,zn=yn−4ynfyn3xnf′xn+2yn/3+xnf′xn+4fxn,xn+1=zn−4znfzn3xnf′xn+2zn/3+xnf′xn+4fxn,n=0,1,2,….
The convergence order of the method described in Algorithm22 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 23.
(51)yn=xn−xnfxnxnf′xn+fxn,zn=yn−2ynfynxnf′xn+yn/2+xnf′xn+2fxn,xn+1=zn−2znfznxnf′xn+zn/2+xnf′xn+2fxn,n=0,1,2,….
The convergence order of the method described in Algorithm23 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.


## 2.1. Some Special Cases of Algorithm2

Algorithm 4.
Takingp=1,w1=1, and τ1=0, Algorithm 2 reduces to the following iterative method:(32)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′xn+gxnf′xn,n=0,1,2,….
Shah and Noor [28] have established the above formula.Algorithm 5.
Takingp=1,w1=1, and τ1=1, Algorithm 2 reduces to the following iterative method.(33)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′yn+gxnf′yn,n=0,1,2,….Algorithm 6.
Takingp=1,w1=1, and τ1=1/2, Algorithm 2 reduces to the following iterative method:(34)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−fyngynfxng′yn+xn/2+gxnf′yn+xn/2,n=0,1,2,….Algorithm 7.
Takingp=2,w1=1/4,w2=3/4,τ1=0, and τ2=2/3, Algorithm 2 reduces to the following iterative method:(35)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−4fyngynfxng′xn+gxnf′xn+3fxng′xn+2yn/3+3gxnf′xn+2yn/3,n=0,1,2,….Algorithm 8.
Takingp=2,w1=1/2,w2=1/2,τ1=0, and τ2=1/2, Algorithm 2 reduces to the following iterative method:(36)yn=xn−fxngxnfxng′xn+gxnf′xn,xn+1=yn−2fyngynfxng′xn+gxnf′xn+fxng′xn+yn/2+gxnf′xn+yn/2,n=0,1,2,….
To the best of our knowledge, Algorithms5–8 are new methods having convergence orders 3,3,4, and 4, respectively.


## 2.2. Some Special Cases of Algorithm3

Algorithm 9.
Takingp=1,w1=1, and τ1=0, Algorithm 3 reduces to the following iterative method:(37)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′xn+gxnf′xn,xn+1=zn−fzngznfxng′xn+gxnf′xn,n=0,1,2,….Algorithm 10.
Takingp=1,w1=1, and τ1=1, Algorithm 3 reduces to the following iterative method:(38)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′yn+gxnf′yn,xn+1=zn−fzngznfxng′zn+gxnf′zn,n=0,1,2,….Algorithm 11.
Takingp=1,w1=1, and τ1=1/2, Algorithm 3 reduces to the following iterative method:(39)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−fyngynfxng′yn+xn/2+gxnf′yn+xn/2,xn+1=zn−fzngznfxng′zn+xn/2+gxnf′zn+xn/2,n=0,1,2,….Algorithm 12.
Takingp=2,w1=1/4,w2=3/4,τ1=0, and τ2=2/3, Algorithm 3 reduces to the following iterative method:(40)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−4fyngynfxng′xn+gxnf′xn+3fxng′xn+2yn/3+3gxnf′xn+2yn/3,xn+1=zn−4fzngznfxng′xn+gxnf′xn+3fxng′xn+2zn/3+3gxnf′xn+2zn/3,n=0,1,2,….Algorithm 13.
Takingp=2,w1=1/2,w2=1/2,τ1=0, and τ2=1/2, Algorithm 3 reduces to the following iterative method:(41)yn=xn−fxngxnfxng′xn+gxnf′xn,zn=yn−2fyngynfxng′xn+gxnf′xn+fxng′xn+yn/2+gxnf′xn+yn/2,xn+1=zn−2fzngznfxng′xn+gxnf′xn+fxng′xn+zn/2+gxnf′xn+zn/2,n=0,1,2,….
To the best of our knowledge, Algorithms10–13 are new iterative methods for solving nonlinear equation (6) with convergence orders 3,4,4, and 4, respectively.
Obviously, different iterative methods can be obtained from Algorithms2 and 3 by using distinct values of the auxiliary function gx. Suitable selection of auxiliary function gx plays a significant role for the better performance of these methods. To demonstrate the efficiency of our proposed algorithms, we choose gx=Ix. Thus, corresponding to this choice, algorithm 10+i, i=4,5,6,…,13, is the special case of algorithm i.Algorithm 14.
(42)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′xn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm14 is 3, and the total number of evaluations per iteration is 3. Thus, IE=1.4422.Algorithm 15.
(43)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′yn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm15 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 16.
(44)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−ynfynxnf′xn+yn/2+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm16 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 17.
(45)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−4ynfyn3xnf′xn+2yn/3+xnf′xn+4fxn,n=0,1,2,….
The convergence order of the method described in Algorithm17 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 18.
(46)yn=xn−xnfxnxnf′xn+fxn,xn+1=yn−2ynfynxnf′xn+yn/2+xnf′xn+2fxn,n=0,1,2,….
The convergence order of the method described in Algorithm18 is 3, and the total number of evaluations per iteration is 4. Thus, IE=1.3161.Algorithm 19.
(47)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′xn+fxn,xn+1=zn−znfznxnf′xn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm19 is 4, and the total number of evaluations per iteration is 4. Thus, IE=1.4142.Algorithm 20.
(48)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′yn+fxn,xn+1=zn−znfznxnf′zn+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm20 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 21.
(49)yn=xn−xnfxnxnf′xn+fxn,zn=yn−ynfynxnf′xn+yn/2+fxn,xn+1=zn−znfznxnf′xn+zn/2+fxn,n=0,1,2,….
The convergence order of the method described in Algorithm21 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 22.
(50)yn=xn−xnfxnxnf′xn+fxn,zn=yn−4ynfyn3xnf′xn+2yn/3+xnf′xn+4fxn,xn+1=zn−4znfzn3xnf′xn+2zn/3+xnf′xn+4fxn,n=0,1,2,….
The convergence order of the method described in Algorithm22 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.Algorithm 23.
(51)yn=xn−xnfxnxnf′xn+fxn,zn=yn−2ynfynxnf′xn+yn/2+xnf′xn+2fxn,xn+1=zn−2znfznxnf′xn+zn/2+xnf′xn+2fxn,n=0,1,2,….
The convergence order of the method described in Algorithm23 is 4, and the total number of evaluations per iteration is 6. Thus, IE=1.2599.


## 3. Convergence Analysis

In this section, convergence criteria of proposed algorithms are studied in the form of the following theorem.Theorem 1.
Assume that the functionf:I⊂ℝ⟶ℝ on an open interval I has a simple root α∈I. Let fx be sufficiently differentiable in the neighborhood of α, then the convergence orders of the methods defined by Algorithms 2 and 3 are three and four, respectively.Proof.
Letα be a simple zero of fx. Since f is sufficiently differentiable, the Taylor series of fxn and f′xn about α are given by(52)fxn=f′αen+c2en2+c3en3+c4en4+c5en5+Oen6,f′xn=f′α1+2c2en+3c3en2+4c4e43+5c5en4+6c6en5+Oen6,where en=xn−α and cj=1/j!fjα/f′α, j=2,3,…
Expandingfxngxn,f′xngxn and fxng′xn by Taylor series, we get(53)fxngxn=f′αgαen+c2gα+g′αen2+12g″α+c2g′α+c3gαen3+Oen4,(54)f′xngxn=f′αgα+2c2gα+g′αen+12g″α+2c2g′α+3c3gαen2+16g‴α+c2g″α+3c3g′α+4c4gαen3+Oen4,(55)fxng′xn=f′αg′αen+c2g′α+g″αen2+12g‴α+c2g″α+c3g′αen3+Oen4.
From equations (53), (54), and (55), we get(56)fxngxnf′xngxn+fxng′xn=en−c2+g′αgαen2+2c22−2c3+2c2g′αgα−g″αgα+2g′α2gα2en3+Oen4.
Using equation (56), we have(57)yn=xn−fxngxnf′xngxn+fxng′xn=α+c2+g′αgαen2+2c3−2c22−2c2g′αgα+g″αgα−2g′α2gα2en3+Oen4.
The Taylor series offyn and gyn are given as(58)fyn=f′αc2+g′αgαen2+2c3−2c22−2c2g′αgα+g″αgα−2g′α2gα2en3+Oen4,gyn=gα+Ag′α+A2g″α2!+A3g‴α3!+⋯,where(59)A=c2+g′αgαen2+2c3−2c22−2c2g′αgα−2g′α2gα2+g″αgαen3+Oen4.
Letvn=xn+τiyn−xn. Taylor’s expansions of gvn,g′vn and f′vn are given by(60)gvn=gα+g′α1−τien+g′α2τigα+g′ατic2+g″ατi22−g″ατi+g″α2en2+2τic3−c22g′α+τic21−τig″α+g‴α61−3τi+3τi2−τi3+τigα2g′αg″α−2c2g′α2−2g′α3gα−τig′αg″αen3+Oen4,g′vn=g′α+g″α1−τien+τig″αc2+g′αgα+g‴α21−τi2en2++Oen3,f′vn=f′α1+2c21−τien+2τic2+g′αgαc2+3c3τi2−2τi+1en2+2τig′αgα3c3−2c22−3τic3−2c2g′αgα+2c2τi5c3−2c22−3τic3+g″αgα+4c41−3τi+3τi2−τi3en3+Oen4.
Now, expandingfyngyn,fxng′vn and f′vngxn by Taylor series, we get(61)fyngyn=f′αg′α+gαc2en2+g″α−2g′αc2+2gαc3−2g′α2gα−2gαc22en3+Oen4,(62)fxng′vn=f′αg′αen+g″α1−τi+c2g′αen2+g′αc3+g″ατigα+c2g″α+g‴α21−τi2en3+Oen4,(63)f′vngxn=f′αgα+g′α−2gατic2+2gαc2en+gα2τic22+3τi2c3−6τic3+3c3+2c2g′α+g″α2en2+gα4c4−4c23τi−6c2c3τi2−4c4τi3+10c2c3τi+12c4τi2−12c4τi+g′α3c3−3c3τi2−2c22τi−2c2τig′α2gα+c21+τig″α+16g‴αen3+Oen4.
Using equations (61)–(63), we obtain(64)fyngynfxng′vn+f′vngxn=g′αgα+c2en2+2c2g′αgατi−3−4g′α2gα2+g″αgα+2c22τi−2+2c3en3+Oen4.
Using equation (64) and the fact ∑i=1pwi=1, the error term for Algorithm 2 is as follows:(65)zn=yn−fyngynfxng′vn+f′vngxn=α+2c2g′αgα2−τi+2g′α2gα2+2c221−τien3+Oen4.
The last expression shows that the convergence order of Algorithm2 is 3.
The Taylor series expansions offzn and gzn are given by(66)fzn=f′α2c2g′αgα2−τi+2g′α2gα2+2c221−τien3+Oen4,gzn=gα+Bg′α+B2g″α2!+B3g‴α3!+⋯,where(67)B=4c2g′αgα+2g′α2gα2−2g′αc2τigα+2c22−2c22τien3+Oen4.
Letun=xn+τizn−xn. Thus, Taylor’s expansions of gun and g′un are given by(68)gun=gα+g′α1−τien+g″α21−τi2en2+2c22τig′α1−τi+2c2τig′α2gα2−τi+2τig′α3gα2+g‴α61−3τi+3τi2−τi3en3+Oen4,g′un=g′α+g″α1−τien+g‴α21−τi2en2+2c2τig′αg″αgα2−τi+2τig′α2g″αgα2+2c22τi1−τig″α++g″″α61−3τi+3τi2−τi3en3+Oen4.
Now, expandingfzngzn,fxng′un and f′ungxn by Taylor series, we get(69)fzngzn=f′α2g′α2gα−2g′ατic2+4g′αc2−2gατic22+2gαc22en3+Oen4,(70)fxng′un=f′αg′αen+g″α1−τi+c2g′αen2+c3g′α+c2g″α1−τi+g‴α21−τi2en3+Oen4,(71)f′ungxn=f′αgα+g′α−2gατic2+2gαc2en+3c3τi2−2τi+1gα+2c21−τig′α+g″α2en2+Oen3.
Using equations (69), (70), and (71), we obtain(72)fzngznfxng′un+f′ungxn=2c2g′αgα2−τi+2g′α2gα2+2c221−τien3+Oe4.
Thus, by using equation (72) and the fact ∑i=1pwi=1, the error term for Algorithm 3 is as follows:(73)xn+1=zn−fzngznfxng′un+f′ungxn,en+1=α+Oen4.
This completes the proof.


## 4. Applications in Science and Engineering

In this section, we consider three scientific and engineering models related to mathematics, physics, and chemistry which include motion of a particle on an inclined plane, Van der Waal’s equation of state, and continuous stirred tank rector equation. We make use of Maple software for computational work and MATLAB for graphical analysis. We apply the methods proposed in Algorithm15 (AG1), Algorithm 16 (AG2), Algorithm 20 (AG3), and Algorithm 21 (AG4) along with Weerakon and Fernando method (WF, equation (3)), midpoint method (MP, equation (4)), Shah et al.’s [27] methods (Algorithm 11 (SH1), Algorithm 13 (SH2), and Algorithm 14 (SN3)), Noor et al.’s [9] method (Algorithm 2.13 (NR1)), Shah and Noor’s [28] method (Algorithm 2.6 (NR2)), and basic Newton’s method (NM, equation (1)) to show that the methods proposed in the present article work more efficiently (see the following Tables 1–3 and Figures 1–3).Table 1
Numerical result for motion of a particle.
Methodnxnfxnxn−x∗NM6–0.31706177457295709508.245568e − 223.926144e − 11WF5−0.31706177457295709501.612296e − 711.751504e − 24MP5−0.31706177457295709508.062909e − 781.477384e − 26SH15−0.31706177457295709504.280457e − 661.081215e − 22SH25−0.31706177457295709501.543287e − 221.310848e − 22SN35−0.31706177457295709501.543287e − 221.441913e − 22NR14−0.31706177457295710003.332577e − 191.042056e − 20NR24−0.31706177457295710003.332577e − 191.119981e − 13AG15−0.31706177457295709502.670763e − 561.882972e − 19AG25−0.31706177457295709505.041170e − 433.979192e − 15AG34−0.31706177457295710003.332577e − 191.782496e − 16AG44−0.31706177457295710003.332577e − 199.411615e − 12Table 2
Numerical results for Van der Waals equation of state.
Methodnxnfxnxn−x∗NM91.72000000000000000000.000000e + 004.291037e − 14WF61.72000000000000000001.436781e − 343.172435e − 12MP61.72000000000000000001.617053e − 403.506693e − 14SH161.72000000000000000004.269024e − 374.705826e − 13SH261.72000000000000000000.000000e + 006.576319e − 13SN361.72000000000000000000.000000e + 007.745611e − 13NR161.72000000000000000000.000000e + 005.067884e − 14NR261.72000000000000000000.000000e + 001.095066e − 23AG151.72000000000000000006.680166e − 424.588240e − 14AG261.72000000000000000008.041535e − 412.743100e − 14AG341.72000000000000000000.000000e + 001.631356e − 22AG451.72000000000000000000.000000e + 003.924480e – 15Table 3
Numerical result for continuous stirred tank reactor (CSTR).
Methodnxnfxnxn−x∗NM8−1.45000000000000000000.000000e + 004.758488e − 16WF6−1.45000000000000000005.336686e − 582.952929e − 20MP5−1.45000000000000000001.329498e − 304.323476e − 11SH16−1.45000000000000000007.836919e − 592.155427e − 20SH25−1.45000000000000000000.000000e + 005.727534e − 24SN35−1.45000000000000000000.000000e + 008.031736e − 26NR15−1.45000000000000000000.000000e + 001.017168e − 27NR219−2.85000000007986480001.339461e − 201.822556e − 10AG15−1.45000000000000000007.313531e − 609.512503e − 21AG24−1.45000000000000000004.569485e − 721.234708e − 24AG34−1.45000000000000000000.000000e + 001.944442e − 17AG43−1.45000000000000000000.000000e + 001.828241e − 13Figure 1
Log of residuals for equation (75).Figure 2
Log of residuals for equation (78).Figure 3
Log of residuals for equation (80).We use the following stopping criteria for computer programmes:(74)xn+1−xn+fxn<ε,whereε=10−15.(1)
Motion of particle on an inclined plane [9].We solve the nonlinear model formed due to the motion of a particle on an inclined plane whose angle of inclinationθ changes at a constant rate dθ/dt=ω<0.(75)xt=−g2ω2eωt−e−ωt2−sinωt,with(76)x0=−0.45.(2)
Van der Waals equation of state [29].Now, we consider the governing nonlinear equation for the special case of well-known Van der Waals equation of state, i.e.,(77)P+a1n2V2V−na2=nRT.Particularly, we solve the above equation in the following form:(78)x3−5.22x2+9.0825x−5.2675=0,with(79)x0=1.65.(3)
Continuos stirred tank reactor (CSTR) [30].We consider the nonlinear model of continuous stirred tank reactor, i.e.,(80)x4+11.50x3+47.49x2+83.06325x+51.23266875=0,with(81)x0=−1.75.


## 5. Conclusions

In this paper, introducing the weights and knots in the expansion of the auxiliary functiongx, i.e., different from usual cases, we have developed a new family of iterative methods for nonlinear equations, i.e., Algorithms 15, 16, 20, and 21. We have compared our methods with some existing methods both numerically and graphically. The results obtained clearly reveal that the proposed methods are more rapidly convergent methods. These methods, besides giving some more efficient new techniques, are the generalized shape of some well-known methods.



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