# A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

Authors: Ramandeep Behl; Eulalia Martínez
Journal: Complexity
 (2020)

---

## Abstract

In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter,θ∈ℝ, and for θ=−1, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.


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## Body

## 1. Introduction

Solution of nonlinear equations of the following form:(1)Fx=0,where F:D⊆ℝn⟶ℝn, is a more challenging and broader problem as compared to the scalar equations. Several issues like neurophysiology, kinematics syntheses, transport theory, chemical equilibrium, reactor and steering combustion, and economics modeling problems can be phrased in the terms of (1) and details can be found in [1–9].There are different ways to construct iterative schemes for nonlinear systems. We center in the natural way methods constructed for scalar equations and then extend to the multidimensional case. Some researchers like Cordero et al. [10], Abad et al. [11], Cordero et al. [12], and Wang and Zhang [13] adopted this way to construct schemes for nonlinear systems. There is no doubt that some scalar schemes can be extended to a system of nonlinear equations with the help of some transformations or substitutions. But, the most important and crucial task is to retain the same convergence order and lower computational cost.Therefore, researchers also tried some other approaches and procedures in order to construct new and higher-order iterative methods for systems of nonlinear equations. Recently, Sharma et al. [14] proposed fourth-order and six-order iterative methods based on weighted-Newton iteration. Very recently, Artidiello et al. [15] provided fourth-order methods based on the weight function approach. Some researchers have also used the approaches like quadrature formulae, Adomian polynomial, divided difference approach for constructing iterative schemes to solve nonlinear systems. For the details of the other approaches, one can refer some standard text books [16, 17, 18].In this paper, our main objective is to construct a new high-order family of iterative methods for a system of nonlinear equations. The development of our scheme is based on the scalar scheme proposed by Kou et al. [19]. First of all, we extend their scheme for a system of nonlinear equations retaining the same convergence order. Then, we increase the convergence order up to 6 by adding one additional substep to the same scheme. The efficiency of the suggested iterative techniques is demonstrated on several applied science problems and academic numerical examples. We found that our proposed methods perform better than the existing ones in terms of residual error, difference between two consecutive iterations, and asymptotic error constants.


## 2. Higher-Order Scheme for Multidimensional Case

Here, we consider a higher-order scheme for simple zeros on univariate function proposed by Kou et al. [19], which is defined as follows:(2)yi=xi−fxif′xi,xi+1=xi−θfxi+fyif′xi−1−θfxi2f′xifxi−fyi,where θ∈ℝ is a free disposable parameter. Expression (2) has at least cubic convergence for every θ except θ≠−1. It has maximum fourth-order convergence for θ=−1. It is a linear combination of well-known Potra-Pták and Newton–Steffensen methods. More details can be found in Kou et al.’s study [19]. We can easily obtain Potra-Pták [20] and Newton–Steffensen [21], for θ=1 and θ=0, respectively, in scheme (2). We can rewrite scheme (2) in the following way:(3)yi=xi−fxif′xi,xi+1=xi−θfxi+fyif′xi−1−θ1yi,xi;fxi−yifxif′xi2,or(4)yi=xi−fxif′xi,xi+1=xi−θfxi+fyif′xi−1−θfxiyi,xi;f,where yi,xi;f is the well-known divided difference of first order. But, scheme (2) does not work for a system of nonlinear equations. So, our foremost aim is to extend this scheme for a system of nonlinear equations and retain the same convergence order. Therefore, we rewrite expression (4) in the following way:(5)yj=xj−F′xj−1Fxj,xj+1=xj−θF′xj−1Fxj+Fyj−1−θyj,xj;F−1Fxj,where θ is a free disposable parameter and yj,xj;F is a finite difference of order one.Now, by adding one extra substep, we have following higher-order scheme:(6)yj=xj−F′xj−1Fxj,zj=xj−θF′xj−1Fxj+Fyj−1−θyj,xj;F−1Fxj,xj+1=zj−yj,zj;F−1Fzj.In Theorem1, we demonstrate that the minimum convergence orders of methods (5) and (6) are three and five, respectively. In the proof of this result, we use the tools and procedure introduced in [10] which we recall briefly.LetF:D⊆ℝn⟶ℝn be sufficiently differentiable in D. The qth derivative of F at u∈ℝn, q≥1, is the q-linear function Fqu:ℝn×⋯×ℝn⟶ℝn such that Fquv1,…,vq∈ℝn. It is easy to observe that(1)
Fquv1,…,vq−1,⋅∈ℒℝn.(2)
Fquvσ1,…,vσq=Fquv1,…,vq, for all permutation σ of 1,2,…,q.From the above properties, we can use the following notation:(a)
Fquv1,…,vq=Fquv1…vq(b)
Fquvq−1Fpvp=FquFpuvq+p−1On the other hand, forξ+h∈ℝn lying in a neighborhood of a solution ξ of Fx=0, we can apply Taylor’s expansion, and assuming that the Jacobian matrix F′ξ is nonsingular, we have(7)Fξ+h=F′ξh+∑q=2p−1Cqhq+Ohp,where Cq=1/q!F′ξ−1Fqξ, q≥2. We observe that Cqhq∈ℝn since Fqξ∈ℒℝn×⋯×ℝn,ℝn and F′ξ−1∈ℒℝn.In addition, we can expressF′ as(8)F′ξ+h=F′ξI+∑q=2p−2qCqhq−1+Ohp−1,where I is the identity matrix. Therefore, qCqhq−1∈ℒℝn. From (8), we obtain(9)F′ξ+h−1=I+X2h+X3h2+X4h4+⋯F′ξ−1+Ohp,where(10)X2=−2C2,X3=4C22−3C3,X4=−8C23+6C2C3+6C3C2−4C4,⋮We denoteej=xj−ξ as the error in the jth-iteration in the multidimensional case. The equation ej+1=Mejp+Oejp+1, where M is a p-linear function, M∈ℒℝn×⋯×ℝn,ℝn, is called the error equation and p is the order of convergence. Let us observe that ejp is ej,ej,…,ej.Theorem 1.
LetF:D⊆ℝn⟶ℝn be a sufficiently differentiable function in an open neighborhood D of its zero ξ. Let us consider that F′x is continuous and nonsingular in the neighborhood of ξ. In addition, we assume the initial guess x0 is close enough to ξ for the guaranteed convergence. Then, the iterative schemes (5) and (6) have minimum third and fifth order of convergence, respectively.Proof.
Letej=xj−ξ be the error of the jth iteration. Developing Fxj and F′xj in a neighborhood of ξ, we write(11)Fxj=F′ξej+C2ej2+C3ej3+C4ej4+C5ej5+C6ej6+Oej7and(12)F′xj=F′ξI+2C2ej+3C3ej2+4C4ej3+5C5ej4+6C6ej5+Oej6,where I is the identity matrix of size n×n and Ci=1/k!F′ξ−1Fkξ, k≥2.
With the help of expression (12), we have(13)F′xj−1=I−2C2ej+Ω0ej2+Ω1ej3+Ω2ej4+Ω3ej5+Ω4ej6+Oej7F′ξ−1,where Ωi=ΩiC2,C3,…,C6, given by(14)Ω0=4C22−3C3,Ω1=−8C23−6C2C3−6C3C2+4C4,Ω2=8C2C4+9C32+8C4C2−12C22C3−12C2C3C2−12C3C22+16C24−5C5,Ω3=10C2C5+12C3C4+12C4C3+10C5C2−16C22C4−18C2C32−16C2C4C2−18C3C2C3−18C32C2−16C4C22+24C23C3+24C22C3C2+24C2C3C22+24C3C23−32C25−6C6,⋮
From expressions (11) and (2), we yield(15)F′xj−1Fxj=ej+Λ0ej2+Λ1ej3+Λ2ej4+Λ3ej5+Λ4ej6+Oej7,where Λj=ΛjC2,C3,…,C6, i.e.,(16)Λ0=−C2,Λ1=2C22−2C3,Λ2=−4C23−4C2C3−3C3C2+3C4,Λ3=6C2C4+6C32+4C4C2−8C22C3−6C2C3C2−6C3C22+8C24−4C5,Λ4=8C2C5+9C3C4+8C4C3+5C5C2−12C22C4−12C2C32−8C2C4C2−12C3C2C3−9C32C2−8C4C22+16C23C3+12C22C3C2+12C2C3C22+12C3C23−16C25−5C6,⋮
By inserting expression (15) in the first substep of (5), we obtain(17)yj−ξ=−Λ0ej2−Λ1ej3−Λ2ej4−Λ3ej5−Λ4ej6+Oej7,which further produces(18)Fyj=F′ξ−Λ0ej2−Λ1ej3+C2Λ02−Λ2ej4+2C2Λ0Λ1−Λ3ej5+−C3Λ03−C2Λ12+2Λ0Λ2+Λ4ej6+Oej7,and(19)yj,xj;F−1=1−C2ej+C2Λ0+C22−C3ej2+−2C22Λ0+C22C3+Λ1+C3Λ0−C23−C4ej3+3C23Λ0+C22−3C3+Λ02−2Λ1+C2−4C3Λ0+2C4+Λ2+C4Λ0+C3Λ1−Λ02+C24+C32−C5ej4+Oej5.
By using expressions (11)–(19) in scheme (5), we have(20)ej+1=Γ0ej2+Γ1ej3+Γ2ej4+Γ3ej5+Γ4ej6+Oej7,where Γi=ΓiΛ0,Λ1,…,Λ4,C2,C3,…,C6,θ, i.e.,(21)Γ0=θC2+Λ0,Γ1=−C2θ+1Λ0+θ2C3+Λ1−2C22θ,⋮
By using the values ofΛ0=−C2 and Λ1=2C22−2C3 from (15) in Γ0 and Γ1, we have(22)Γ0=0,Γ1=θ+1C22,and adopting (22) in (21), we yield(23)ej+1=zj−ξ=θ+1C22ej3+Γ2ej4+Γ3ej5+Γ4ej6+Oej7.
From expression (23), it clear that scheme (5) has the least third-order convergence for every θ.
Now, expanding theFzj in a neighborhood of ξ, we have(24)Fzj=F′ξθ+1C22ej3+Γ2ej4+Γ3ej5+C25θ+12+Γ4ej6+Oej7,which further produces, with the help of expression (18),(25)zj,yj;F−1=1−C22ej2−θ−1C23+2C3C2ej3+C24−C3C22−Γ2C2+Λ2C2ej4+Oej5.
By using equations (24) and (25) in the final substep of (6), we obtain(26)ej+1=θ+1C24ej5+Γ¯4ej6+Oej7.It is clear from expression (26) that we have minimum fifth-order convergence for every θ∈ℝ−−1. In addition, we have maximum sixth order of convergence for θ=−1. Hence, schemes (5) and (6) have minimum third and fifth order convergence, respectively, for every θ≠−1∈ℝ.


## 3. Numerical Experiments

Here, we checked the efficiency and effectiveness of our methods on 4 real life problems. Some of them have been taken from the paper of R. Behl et al. (see [22]). Additionally, we also consider a large system of equations of 150×150 to verify the theoretical results. The details of all numerical problems can be seen in examples (1–5). Furthermore, we also mentioned the starting points and solutions of the considered problems in the examples. Now, we employ our fourth and sixth-order schemes (5) and (6), called OS14 and OS26, respectively, for θ=−1 in order to verify the computational performance of them with the existing methods.Now, we compare (2) with sixth-order methods suggested by Mona et al. [23] and Lotfi et al. [24]. We consider methods (27) forλ=1,β=2,p=1,andq=3/2 and (5), respectively, called MS and LS. In addition, we also compared our scheme with sixth-order family of iterative methods proposed by Sharma and Arora [25]. From them, we choose method (13), called SS. Finally, we compare (2) with a sixth-order family of iterative method that is recently designed by Abbasbandy et al. [26] and Hueso et al. [27]. From them, we choose their best expressions (8) and (14, 15) fort1=−9/4ands2=9/8, respectively, denoted by AS and HS.In Tables1–5, we depicted the iteration indexes k, residual error of the corresponding function Fxj, error between the two consecutive iterations xj+1−xj, approximate computational order of convergence ρ=logxj+1−xj/xj−xj−1/logxj−xj−1/xj−1−xj−2,i=2,3,…, xj+1−xj/xj−xj−1p (where pp is either 4 or 6), and η is the last calculated value.Table 1
Convergence behavior of different methods on differential equation (27).
Methods
j
F
x
j
x
j
+
1
−
x
j
ρ
∗
x
j
+
1
−
x
j
/
x
j
−
x
j
−
1
6
η
MS
1
1.4
−
4
5.0
−
4
3.431307196
+
16
2
6.5
−
24
2.1
−
23
1.270833506
−
3
3
9.3
−
121
2.9
−
120
4.9974
3.431307196
+
16
LS
1
1.3
−
3
4.7
−
3
1.069128721
+
12
2
9.6
−
20
8.2
−
20
7.809583095
−
6
3
2.1
−
103
3.3
−
103
4.9773
1.069128721
+
12
AS
1
1.3
−
3
4.6
−
3
6.184712903
−
7
2
1.5
−
21
4.9
−
21
5.332292596
−
7
3
2.7
−
129
8.7
−
129
5.9964
6.184712903
−
7
HS
1
2.2
−
4
4.2
−
4
1.536785689
+
17
2
9.0
−
24
3.5
−
23
6.562626596
−
3
3
8.4
−
119
2.9
−
118
4.9846
1.536785689
+
17
SS
1
6.3
−
4
2.4
−
3
9.186167408
−
7
2
4.5
−
23
1.5
−
22
8.028226594
−
7
3
3.5
−
138
1.1
−
137
5.9970
9.186167408
−
7
OS
1
4
1
4.7
−
3
1.0
−
2
1761209601
−
4
2
7.5
−
13
2.1
−
12
1.806107555
−
4
3
1.1
−
51
3.3
−
51
4.0011
1.761209601
−
4
OS
2
6
1
5.4
−
5
1.7
−
4
9.855399505
−
8
2
7.4
−
31
2.1
−
30
9.979976566
−
8
3
2.9
−
186
8.1
−
186
6.0002
9.855399505
−
8
The lowest residual errors, lowest difference between two consecutive iterations, and lowest asymptotic error constants belong to our schemeOS26. So, we can say that OS26 has better convergence behavior than other mentioned methods.Table 2
Convergence behavior of different methods on Fisher’s equation (32).
Methods
j
F
x
j
x
j
+
1
−
x
j
ρ
∗
x
j
+
1
−
x
j
/
x
j
−
x
j
−
1
6
η
MS
1
2.7
−
3
5.5
−
4
2.350947999
+
16
2
9.5
−
21
1.5
−
21
5.507288873
−
2
3
1.6
−
108
2.5
−
109
4.9964
2.350947999
+
16
LS
1
4.4
−
3
1.0
−
3
7.024389352
+
12
2
4.6
−
20
7.4
−
21
5.815894917
−
3
3
2.2
−
107
1.2
−
108
5.1204
7.024389352
+
12
AS
1
6.9
−
3
1.5
−
3
9.022505886
−
5
2
4.2
−
21
6.5
−
22
7.021836209
−
5
3
4.5
−
131
7.0
−
132
5.9941
9.022505886
−
5
HS
1
6.0
−
3
1.3
−
3
4.016099735
+
14
2
1.3
−
18
2.0
−
19
5.236602433
−
2
3
1.8
−
97
2.8
−
98
4.9940
4.016099735
+
14
SS
1
4.8
−
3
1.0
−
3
5.610610203
−
5
2
2.9
−
22
4.5
−
23
4.391559871
−
5
3
3.0
−
138
4.7
−
139
5.9945
5.610610203
−
5
OS
1
4
1
4.8
−
2
1.1
−
2
1.088252704
−
3
2
9.3
−
11
1.6
−
11
9.250513914
−
4
3
4.1
−
46
6.4
−
47
3.9920
1.088252704
−
3
OS
2
6
1
9.9
−
4
2.2
−
4
5.569289669
−
6
2
3.5
−
27
5.5
−
28
4.830577428
−
6
3
9.9
−
169
1.6
−
169
5.9974
5.569289669
−
6
From the table, it is clear that there is big difference of residual errors, among our methodOS26 and other existing methods. In addition, OS26 has the lowest difference between two consecutive iterations and asymptotic error constants.Table 3
Convergence behavior of different methods on 2D Bratu problem (34).
Methods
j
F
x
j
x
j
+
1
−
x
j
ρ
∗
x
j
+
1
−
x
j
/
x
j
−
x
j
−
1
6
η
MS
1
7.5
−
15
4.1
−
14
9.148606524
+
66
2
1.2
−
80
6.4
−
80
1.375781477
+
1
3
1.2
−
409
6.3
−
409
4.9997
9.148606524
+
66
LS
1
6.4
−
16
2.5
−
15
2.808772449
+
1
2
2.9
−
87
5.3
−
87
1.447342836
−
13
3
9.1
−
445
3.3
−
444
4.9853
2.808772449
+
1
AS
1
4.4
−
15
2.4
−
14
1.973434769
−
12
2
6.9
−
95
3.5
−
94
1.428095547
−
12
3
7.9
−
574
3.9
−
573
5.9994
1.973434769
−
12
HS
1
2.1
−
13
1.2
−
12
3.495510769
+
1
2
2.1
−
71
1.2
−
70
7.368055345
−
11
3
1.7
−
361
9.3
−
361
4.9997
3.495510769
+
1
SS
1
4.4
−
15
2.4
−
14
1.433541371
−
12
2
7.1
−
95
3.6
−
94
1.433541371
−
12
3
9.2
−
574
4.5
−
573
5.9994
1.433541371
−
12
OS
1
4
1
7.5
−
11
4.1
−
10
7.171191212
−
9
2
3.8
−
47
2.0
−
46
6.870113499
−
9
3
2.1
−
192
1.1
−
191
3.9995
7.171191212
−
9
OS
2
6
1
2.3
−
18
1.3
−
17
8.564522080
−
16
2
8.2
−
118
4.7
−
117
8.429675254
−
16
3
1.6
−
714
9.0
−
714
5.9999
8.564522080
−
16
The schemeOS26 has the lowest residual error and asymptotic error constant as compared to the other mentioned methods. In addition, lowest difference between two consecutive iterations also belongs to OS26.Table 4
Convergence behavior of different methods on Hammerstein integral equation (38).
Methods
j
F
x
j
x
j
+
1
−
x
j
ρ
∗
x
j
+
1
−
x
j
/
x
j
−
x
j
−
1
6
η
MS
1
3.9
−
6
8.2
−
7
2.773373608
+
30
2
5.8
−
36
1.2
−
36
3.943508142
3
4.6
−
185
9.9
−
186
4.9992
2.773373608
+
30
LS
1
8.6
−
6
1.8
−
6
1.232404905
+
31
2
8.4
−
37
1.7
−
37
4.445532921
−
3
3
1.4
−
189
3.0
−
190
4.9224
1.232404905
+
31
AS
1
1.1
−
5
2.4
−
6
7.072478176
−
6
2
5.4
−
39
1.2
−
39
6.596956919
−
6
3
8.0
−
239
1.7
−
239
5.9991
7.072478176
−
6
HS
1
3.0
−
5
6.5
−
6
9.220736175
+
25
2
6.9
−
31
1.5
−
31
1.994799598
3
4.4
−
159
9.4
−
160
4.9991
9.220736175
+
25
SS
1
9.4
−
6
2.0
−
6
5.324312398
−
6
2
1.5
−
39
3.2
−
40
4.964844066
−
6
3
2.7
−
242
5.7
−
243
5.9991
5.324312398
−
6
OS
1
4
1
9.0
−
4
1.9
−
4
3.268514485
−
4
2
2.0
−
18
4.2
−
19
3.089659514
−
4
3
4.8
−
77
1.0
−
77
3.9983
3.268514485
−
4
OS
2
6
1
7.6
−
7
1.6
−
7
3.836358838
−
7
2
3.2
−
47
6.9
−
48
3.761775905
−
7
3
2.0
−
289
4.2
−
290
5.9998
3.836358838
−
7
From this table, it is clear that there is a big difference of residual errors amongOS26 and other existing methods. In addition, OS26 has the lowest difference between two consecutive iterations and asymptotic error constants.Table 5
Convergence behavior of different methods on example 5.
Methods
j
F
x
j
x
j
+
1
−
x
j
ρ
∗
x
j
+
1
−
x
j
/
x
j
−
x
j
−
1
6
η
MS
1
3.4
−
6
1.5
−
8
1.549161749
+
39
2
6.6
−
45
2.8
−
47
3.031863687
3
1.9
−
238
8.2
−
241
5.0000
1.549161749
+
39
LS
1
1.4
−
6
6.1
−
9
1.946645858
−
8
2
2.3
−
55
1.0
−
57
1.946645858
−
8
3
4.5
−
348
1.9
−
350
6.0000
1.946645858
−
8
AS
1
1.4
−
6
5.9
−
9
1.814507829
−
8
2
1.8
−
55
7.8
−
58
1.814507829
−
8
3
9.7
−
349
4.1
−
351
6.0000
1.814507829
−
8
HS
1
4.6
−
6
2.0
−
8
3.301595366
+
38
2
3.5
−
44
1.5
−
46
2.485682422
3
8.4
−
235
3.6
−
237
5.0000
3.301595366
+
38
SS
1
7.7
−
7
3.3
−
9
1.007050020
−
8
2
3.1
−
57
1.3
−
59
1.007050020
−
8
3
1.3
−
359
5.4
−
362
6.0000
1.007050020
−
8
OS
1
4
1
4.3
−
4
1.8
−
6
3.734033747
−
6
2
9.9
−
27
4.2
−
29
3.734034356
−
6
3
2.8
−
117
1.2
−
119
4.0000
3.734033747
−
6
OS
2
6
1
6.5
−
8
2.8
−
10
8.150704909
−
10
2
8.5
−
65
3.6
−
67
8.150704909
−
10
3
4.3
−
406
1.8
−
408
6.0000
8.150704909
−
10
Our schemeOS26 has minimum residual error constant among other mentioned methods. It also has the lowest x4−x3 and asymptotic error constants as compared to other existing methods.During the current numerical experiments with programming language Mathematica (Version 9), all computations have been performed with multiple precision arithmetic with 1000 digits of mantissa, which minimize round-off errors. The meaning ofa±b is a×10±b in all tables. We adopted the command “AbsoluteTiming” in order to calculate the CPU time. We run our programs three times and depicted the average CPU time in Table 6. Also, one can observe the times used for each iterative method and each problem in Figure 1, where we want to point out that for big size problems, the method OS26 uses the minimum time, so it is being very competitive. The configuration of the used computer is given as follows:Processor: Intel(R) Core(TM) i7-4790 CPU@ 3.60 GHzMade of: HPRAM: 8:00 GBSystem type: 64-bit-Operating System, x64-based processorTable 6
CPU time taken by different methods on examples 1–5.
Methods
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Total time
Average time
MS
0.141097
1.658164
636.148568
49.370628
1418.257989
2105.576446
421.1152892
LS
0.109075
1.031724
323.672644
24.845433
973.952124
1323.611
264.7222
AS
0.586411
1.590115
634.605760
49.738892
1492.151852
2178.67303
435.734606
HS
0.646454
2.371663
611.849108
50.585468
2207.404740
2872.857433
574.571866
SS
1.262873
1.570112
376.072834
148.621206
1018.385809
1545.912834
309.1825668
OS
1
4
0.260183
0.609427
251.443292
25.600944
779.653091
1057.566937
211.5133874
OS
2
6
0.283189
0.764537
364.446516
25.026559
723.418750
1113.939551
222.7879102
Our schemeOS14 has the lowest average CPU time. Since, OS14 has convergence order fourth and others have sixth. Then, it is quite obvious that OS14 has lower convergence behavior as compared to other mentioned methods in terms of residual error, asymptotic error constant, etc. However, it is better than the others in regard of CPU timing. On the other hand, the second lowest average CPU time belongs to our sixth-order convergent scheme OS26 and also illustrates better convergence performance as compared to other mentioned methods.Figure 1
CPU times for each example with the corresponding method.Example 1.
Let us consider a boundary value problem described in the book of Ortega and Rheinbolt [16], which is defined as follows:(27)y″=12y3+3y′−32−x+12,y0=0,y1=1.
In addition, we consider the following partition of the interval0,1, which is given by(28)x0=0<x1<x2<x3<…<xn,wherei=0,1,…,n−1,xi+1=xi+h,h=1n.
Moreover, we also consider the notation,yi=yxi, for i=0,1,…,n. We discretize problem (27) with the help of following numerical approximation for first and second derivatives(29)yj′≈yj+1−yj−12h,yj″≈yj−1−2yj+yj+1h2,j=1,2,…,n−1.
Hence, we yield a nonlinear system ofn−1×n−1, which is given as follows:(30)yj+1−2yj+yj−1−h22yj3−32−xjh2−3yj+1−yj−12h−1h2=0,j=1,2,…,n−1.
Let us consider initial approximationyi0=1.8,…,1.8,1.8T. In particular, we will solve this problem for n=7 so that we can obtain a 6×6 system of nonlinear equations. The solution of this problem is(31)yxi=0.076543…,0.165873…,0.271521…,0.398454…,0.553886…,0.748687…T.
The numerical results appeared in Table1. Moreover, we represent in Figure 2 the difference between two consecutive iterations but in logarithmic scale and taking absolute value, logxj+1−xj, so the biggest value corresponds to the minimum distance error, by the properties of logarithmic function, that is, the OS26 iterative method presents the best results.Figure 2
Difference between consecutive iterations in logarithmic scale for example 1.Example 2.
Consider another typical nonlinear problem, that is, Fisher’s equation [28] with homogeneous Neumann’s boundary conditions and D the diffusion coefficient:(32)ut=Duxx+u1−u,ux,0=1.5+0.5cosπx,0≤x≤1,ux0,t=0,∀t≥0,ux1,t=0,∀t≥0.
Again using a finite difference discretization, equation (32) reduces to a system of nonlinear equations. Consider ui,j=uxi,tj to be its approximate solution at the grid points of the mesh. Let M and N be the number of steps in x and t directions, and h and k be the respective step size. Apply central difference to uxxxi,tj=ui+1,j−2ui,j+ui−1,j/h2, backward difference for utxi,tj=ui,j−ui,j−1/k, and forward difference for uxxi,tj=ui+1,j−ui,j/h,t∈0,1. For the solution of the system, we consider M=5 and N=5 which reduces to a nonlinear system of size 25, with the initial vector u0xi,tj=1+i+j/M2T,i,j=1,2,…,M convergence towards the following solution:(33)uxi,tj=1.4606…,1.2959…,1.2166…,1.1688…,1.1355…1.3988…,1.2783…,1.2113…,1.1672…,1.1349…1.3081…,1.2509…,1.2028…,1.1645…,1.1341…1.2260…,1.2245…,1.1944…,1.1619…,1.1333…1.1812…,1.2089…,1.1893…,1.1602…,1.1327…,and the numerical results of comparing the behavior different methods are shown in Table 2.Example 3.
The 2D Bratu problem [29, 30] is defined in Ω=x,t:0≤x≤1,0≤t≤1, as(34)uxx+utt+Ceu=0,onwith boundary conditionsu=0on∂Ω.
The approximated solution of a nonlinear partial differential equation can be found using finite difference discretization which reduces to solving a system of nonlinear equations. Letui,j=uxi,tj be its approximate solution at the grid points of the mesh. Let M and N be the number of steps in x and t directions and h and k be the respective step size. To solve the given PDE, apply central difference to uxx and utt, i.e., uxxxi,tj=ui+1,j−2ui,j+ui−1,j/h2,C=0.1,t∈0,1. We look for the solution of the system for M = 11 and N = 11 of size 100, with the initial sinusoidal approximation u0xi,tj=0.1sinπihsinπjk; by using different methods, one can see the obtained result in Table 3. The solution of this problem can be seen in Figure 3.Figure 3
Approximated solution for Bratu equation withC=0.1.Example 4.
In this example, we consider the following Hammerstein integral equation (see [pp. 19, 20]) Ortega to check the effectiveness and applicability of our proposed methods, and compared with other existing methods, the nonlinear integral equation is given by(35)xs=1+15∫01Fs,txt3dt,where xs∈C0,1,s,t∈0,1 and the kernel F can be written as(36)Fs,t=1−st,ift≤s,s1−t,ifs≤t.
To transform the above equation into a finite-dimensional problem by using Gauss Legendre quadrature formula given as∫01ftdt≃∑j=1nwjftj, where the abscissas tj and the weights wj are determined for n=8. Denoting the approximations of xti by xi, one gets the system of nonlinear equations:(37)5xi−5−∑j=18aijxj3=0,wherei=1,2,…,8,aij=wjtj1−ti,ifj≤i,wjti1−tj,ifi<j.where the abscissas tj and the weights wj are known and depicted in Table 7 when n=8.
The different methods converge towards the solutionxti=1.00209…,1.00990…,1.01972…,1.02643…,1.02643…,1.01972…,1.00990…,1.00209…T, and different conditions of convergence can be checked in Table 1. The initial guess x0=1/2,…,1/2T.Table 7
Abscissas and weights of Gauss Legendre quadrature formula fort=8.
j
t
j
w
j
1
0.01985507175123188415821957…
0.05061426814518812957626567…
2
0.10166676129318663020422303…
0.11119051722668723527217800…
3
0.23723379504183550709113047…
0.15685332293894364366898110…
4
0.40828267875217509753026193…
0.18134189168918099148257522…
5
0.59171732124782490246973807…
0.18134189168918099148257522…
6
0.76276620495816449290886952…
0.15685332293894364366898110…
7
0.89833323870681336979577696…
0.11119051722668723527217800…
8
0.98014492824876811584178043…
0.05061426814518812957626567…Example 5.
Finally, let us consider the following system of nonlinear equations (chosen from Grau-Sánchez et al. [31]):(38)Fx1,x2,…,xn=∑j=1,j≠inxj−e−xi,1≤i≤n.
We choosen=150 in order to check the theoretical results mentioned with a large size system. We take the initial guess x0=1/2,…,1/2T for this problem. The obtained solution of this problem is(39)ξ=0.5671433…,0.5671433…,0.5671433……,0.5671433…150timesT.
The obtained results can be observed in Table5.


## 4. Concluding Remarks

In this paper, we proposed a new high-order and efficient family of iterative techniques for a system of nonlinear equations. The suggested methods are the extension of the study by Kou et al. [19] for multidimensional case. In addition, our scheme has minimum 5th-order convergence by adding additional substep and for particular value θ=−1, it reached the maximum 6th order. From the obtained numerical results in Tables 1–5, it is confirmed that the lowest residual error, difference between two consecutive iterations and asymptotic error constant, belongs to our method OS26. Finally, our schemes OS16 and OS26, respectively, also consume the first and second lowest average CPU time as compared to the other existing methods in examples 1–5. In the future work, we will try to propose higher-order version of the same or new methods with lower computational cost and that employ more accurate solution.



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