# A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

Authors: Minqiang Xu; Jing Niu; Li Guo
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.


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## Body

## 1. Introduction

Nonlinear systems of second-order BVPs are widely used in applied physics, mechanical engineering, biology, etc. In this article, we mainly focus on numerical solutions for such problems. In general, these systems can be described by(1)u″+a1xu′+a2xv″+a3xv′+ℱu,v=f,x∈a,b,v″+b1xv′+b2xu″+b3xu′+Gu,v=g,x∈a,b,ua=0,ub=0,va=0,vb=0,in which ℱ and G represent the nonlinear terms, aix,bix∈Ca,b for i=1,2,3, and f,g. are given functions defined in a,b.Concerning the existence and uniqueness of solution of problem (1), we refer to [1–3]. Nevertheless, to the best of our knowledge, research studies on numerical methods for nonlinear systems of second-order BVPs were seldom reported. Recently, several techniques for approximate solutions of such systems have been presented. Arqub and Abo-Hammour [4] suggested a continuous genetic algorithm to deal with second-order linear systems. In [5, 6], the authors combined homotopy perturbation methods and iterative RKM to solve problem (1). In [7–9], Peng et al. proposed an alternative iterative method called symplectic method to solve nonlinear BVPs. Furthermore, methods such as sinc-collocation methods and Jacobi matrix methods were also considered in [10, 11].In this work, QNM [12] and simplified reproducing kernel method (SRKM) [13] are combined to design a numerical method for solving nonlinear systems (1). It is worth noting that QNM acts directly on operator equations to linearize nonlinear problems by Fréchet derivative. Here we extend QNM to solve nonlinear systems (1). Many improved versions of RKM have been developed to solve linear second-order BVPs [14–21]. However, the high-order RKM has been seldom discussed. Motivated by designing RKM of high-order accuracy, we establish a reproducing kernel space with polynomial form by applying the well-known Lobatto orthogonal polynomials. NDM converges fast and LRKM avoids the Schmidt orthogonalization process and uses very few points. Therefore, QDM-LRKM is efficient and simple to implement.The article is built up as follows. Several fundamental definitions aboutFréchet derivative and Hilbert spaces are recalled in Section 2. The QNM-LRKM is introduced in Section 3. Some numerical examples validate the robustness and effectiveness in Section 4. Finally, a summary is provided in the last section.


## 2. Preliminaries

To simplify our discussion and demonstrate the idea clearly, we rewrite system (1) into a system with homogenous boundary conditions and the domain defined on −1,1. Actually, by a scaling from a,b to −1,1, that is, x=b−a/2t+b+a/2, we can derive a system defined on −1,1 as follows:(2)u^″+a^1tu^′+a^2tv^″+a^3tv^′+ℱ^u^;v^=f^t;t∈−1,1,v^″+b^1tv^′+b^2tu^″+b^3tu′+G^u^;v^=g^t;t∈−1,1,u^−1=u^1=0,v^−1=v^1=0,where a^1t=b−a/2a1xt, a^2t=a2xt, a^3t=b−a/2a3xt, b^1t=b−a/2b1xt, b^2t=b2xt, b^3t=b−a/2b3xt, ℱ^u^;v^=b−a2/4ℱu^;v^, G^u^;v^=b−a2/4Gu^;v^, f^t=b−a2/4fxt, and g^t=b−a2/4gxt.


### 2.1. Several Reproducing Kernel Spaces and Hilbert Spaces

Denote byPn the space of polynomial functions with degrees no more than n. Let Pn and Ln be the well-known Legendre and Lobatto polynomials, respectively. To be more precise,(3)P0t=1,P1t=t,Pnt=2n−1ntPn−1t−n−1nPn−2t,n=2,3,…,satisfying the orthogonality condition on −1,1, which means(4)Pm,Pn=∫−11PmtPntdt=22n+1,m≠n,0,m=n.Furthermore, Lobatto polynomials are defined by(5)L0t=1−t2,L1t=1+t2,Lnt=∫−1tPn−1sds=22n−1Pn−Pn−2,n=2,3,….Clearly, Lobatto polynomials satisfy(6)Ln−1=Ln1=0,n=2,3,….LetL2−1,1=u∫−11u2xdx<∞ and Wn−1,−1 be the standard linear inner product space of polynomials with degrees not exceeding n, namely,(7)u,vWn=∫−11uvdx,∀u,v∈Wn−1,1,and the norm(8)uWn−1,1=u,uWn,∀u∈Wn−1,1.From Theorem 3.7 of [22], we obtain that the inner space Wn−1,1 is a Hilbert space as Wn−1,1 is a finite dimension closed subspace of L2−1,1.Definition 1.
(9)
W
n
0
−
1,1
=
u
u
∈
W
n
−
1,1
,
u
−
1
=
u
1
=
0
.
Leteii=2n be the standard orthonormal basis functions obtained by the Gram–Schmidt process of Lii=2n.Lemma 1.
W
n
0
−
1,1 is a reproducing kernel space with kernel function(10)Ks,t=∑i=2neiteis.Definition 2.
(11)
W
n
−
1,1
=
W
n
−
1,1
⊕
W
n
−
1,1
=
u
1
,
u
2
T
u
1
,
u
2
∈
W
n
−
1,1
.
We equip the following inner product and norm forWn−1,1:(12)U,VWn=∑i=12ui,viWn−1,1,U,V∈Wn−1,1,UWn2=u1Wn−1,12+u2Wn−1,12,U∈Wn−1,1.
Under such definitions, we obtain thatWn−1,1 is a Hilbert space. Similarly, we can define a subspace Wn0−1,1 of Wn−1,1,(13)Wn0−1,1=Wn0−1,1⊕Wn0−1,1.


### 2.2. Fréchet Derivative

LetX and Y be Banach spaces with norms ⋅X and ⋅Y, respectively. ℱ denotes an operator from X to Y.Definition 3.
(see [23]). We say that the linear operator A is the Fréchet derivative of ℱ at u0∈X if(14)limh⟶0ℱu0+h−ℱu0−AhYhX=0,and A is denoted by ℱ′u0.
Letℱ:X1×X2⟶Y, A1:X1×X2⟶Y, and A2:X1×X2⟶Y, where X1 and X2 are also Banach spaces with norms ⋅X1 and ⋅X2.Definition 4.
We say that the linear operatorA1 is the partial Fréchet derivative of ℱ on u at u0,v0∈X1×X2 if(15)limh⟶0ℱu0+h,v0−ℱu0,v0−A1h,v0YhX1=0,and A1 is denoted by ℱu′u0,v0.
Similarly, we can define the partialFréchet derivative of ℱ at u0,v0∈X1×X2 with respect to v by ℱv′u0,v0.


## 2.1. Several Reproducing Kernel Spaces and Hilbert Spaces

Denote byPn the space of polynomial functions with degrees no more than n. Let Pn and Ln be the well-known Legendre and Lobatto polynomials, respectively. To be more precise,(3)P0t=1,P1t=t,Pnt=2n−1ntPn−1t−n−1nPn−2t,n=2,3,…,satisfying the orthogonality condition on −1,1, which means(4)Pm,Pn=∫−11PmtPntdt=22n+1,m≠n,0,m=n.Furthermore, Lobatto polynomials are defined by(5)L0t=1−t2,L1t=1+t2,Lnt=∫−1tPn−1sds=22n−1Pn−Pn−2,n=2,3,….Clearly, Lobatto polynomials satisfy(6)Ln−1=Ln1=0,n=2,3,….LetL2−1,1=u∫−11u2xdx<∞ and Wn−1,−1 be the standard linear inner product space of polynomials with degrees not exceeding n, namely,(7)u,vWn=∫−11uvdx,∀u,v∈Wn−1,1,and the norm(8)uWn−1,1=u,uWn,∀u∈Wn−1,1.From Theorem 3.7 of [22], we obtain that the inner space Wn−1,1 is a Hilbert space as Wn−1,1 is a finite dimension closed subspace of L2−1,1.Definition 1.
(9)
W
n
0
−
1,1
=
u
u
∈
W
n
−
1,1
,
u
−
1
=
u
1
=
0
.
Leteii=2n be the standard orthonormal basis functions obtained by the Gram–Schmidt process of Lii=2n.Lemma 1.
W
n
0
−
1,1 is a reproducing kernel space with kernel function(10)Ks,t=∑i=2neiteis.Definition 2.
(11)
W
n
−
1,1
=
W
n
−
1,1
⊕
W
n
−
1,1
=
u
1
,
u
2
T
u
1
,
u
2
∈
W
n
−
1,1
.
We equip the following inner product and norm forWn−1,1:(12)U,VWn=∑i=12ui,viWn−1,1,U,V∈Wn−1,1,UWn2=u1Wn−1,12+u2Wn−1,12,U∈Wn−1,1.
Under such definitions, we obtain thatWn−1,1 is a Hilbert space. Similarly, we can define a subspace Wn0−1,1 of Wn−1,1,(13)Wn0−1,1=Wn0−1,1⊕Wn0−1,1.


## 2.2. Fréchet Derivative

LetX and Y be Banach spaces with norms ⋅X and ⋅Y, respectively. ℱ denotes an operator from X to Y.Definition 3.
(see [23]). We say that the linear operator A is the Fréchet derivative of ℱ at u0∈X if(14)limh⟶0ℱu0+h−ℱu0−AhYhX=0,and A is denoted by ℱ′u0.
Letℱ:X1×X2⟶Y, A1:X1×X2⟶Y, and A2:X1×X2⟶Y, where X1 and X2 are also Banach spaces with norms ⋅X1 and ⋅X2.Definition 4.
We say that the linear operatorA1 is the partial Fréchet derivative of ℱ on u at u0,v0∈X1×X2 if(15)limh⟶0ℱu0+h,v0−ℱu0,v0−A1h,v0YhX1=0,and A1 is denoted by ℱu′u0,v0.
Similarly, we can define the partialFréchet derivative of ℱ at u0,v0∈X1×X2 with respect to v by ℱv′u0,v0.


## 3. QNM-SRKM for Solving Equation (2)

In this section, we present the QNM-SRKM to solve the nonlinear system (2).


### 3.1. Analysis of QNM

By equation (1), we define ℒij:Wn0−1,1⟶L2−1,1 as(16)ℒ11:u^⟼u^″+a^1u^′,ℒ12:v^⟼a^2v^″+a^3v^′,ℒ21:u^⟼b^2u^″+b^3u^′,ℒ22:v^⟼v″+b^1v^′.Let(17)ℒ=ℒ11ℒ12ℒ21ℒ22,NU=ℱ^u^,v^G^u^,v^.One can correspondingly rewrite equation (2) into the following operator form:(18)ℒU+NU=F,U−1=0,U1=0,where U=u^,v^T and F=f^,g^T∈L2−1,1.Theorem 1.
TheFréchet derivative of N at U0 is(19)N′U0:U⟼ℱuu0,v0ℱvu0,v0Guu0,v0Gvu0,v0U.To showFréchet derivative of the nonlinear operator more clearly, we introduce two examples.Example 1.
LetNU=xv+u2u+x2v2, then N′U0U=2u0xxℐxℐ2x2v0xU, where ℐ is identity operator.Example 2.
LetNU=u″+xeu+v3v″+u2+xsinv, then N′U0U=u″v″+xeu03v022u02xcosv0U.
Now, we will introduce the QNM to linearize the operator equation (18).
The Quasi-Newton’s Scheme: FindingUkk=1∞ such that(20)ℒUk+1+N′UkUk+1−Uk+NUk=F,U−1=0,U1=0,k=0,1,….


### 3.2. Analysis of LRKM

By improving the RKM, an effective and simple algorithm is derived to solve the linear system (20). For convenience, we rewrite (20) into the following form:(21)ℒU=F,U−1=0,U1=0.Lemma 2.
ℒ is linear and bounded.Proof. Obviously, ℒ11,ℒ12,ℒ21, and ℒ22 are linear operators. For ∀U∈Wn0, we have(22)ℒUL22=∑i=12ℒi1u+ℒi2vL22≤∑i=12ℒi1⋅uWn+ℒi2⋅vWn2≤2∑i=12ℒi12+ℒi22⋅uWn2+vWn2≤2∑i=12ℒi12+ℒi22⋅UWn2.The boundedness ofℒiji,j=12 results in the boundedness of the operator ℒ.Let(23)Ψi1=ℒ∗Rx,xi,0T,Ψi2=ℒ∗0,Rx,xiT,wherein xii=1n are different points in −1,1, ℒ∗ is the conjugate operator of ℒ, and R is the reproducing kernel of L2−1,1.Lemma 3.
Ψ
i
1
x
=
ℒ
11
K
x
,
x
i
ℒ
12
K
x
,
x
i
,
Ψ
i
2
x
=
ℒ
21
K
x
,
x
i
ℒ
22
K
x
,
x
i
,
i
=
1,2
,
…
,
n.Proof. From the reproducibility of R, we immediately deduce that(24)Ψi1x=ℒ∗Rx,xi0=ℒ11∗ℒ21∗ℒ12∗ℒ22∗Rx,xi0=ℒ11∗Rx,xiℒ12∗Rx,xi=ℒ11Rx,xiℒ12Rx,xi.Similarly, we haveΨi2x=ℒ21Kx,xiℒ22Kx,xi,i=1,2,…,n.Lemma 4.
Ψ
11
,
Ψ
21
,
…
,
Ψ
n
1
,
Ψ
12
,
Ψ
22
,
…
,
Ψ
n
2 are linearly independent in Wn0−1,1.Considering the numerical implement, we apply the SRKM proposed in [12, 13] to solve equation (21). We first denote a finite-dimensional subspace S2n by(25)S2n≜spanΨi21n∪Ψi21n¯.LetU^n≜P2n+4U^, where P2n:W∞⟶S2n is the projection operator. Then, U^n converges to U^ in the sense of ⋅Wn−1,1, which means that(26)U^−U^nWn⟶0,asn⟶∞.AsU^n∈Wn−1,1, there exits undetermined coefficients ai1andai2i=1,2…n such that(27)U^nx=∑i=1nai1Ψi1x+∑i=1nai2Ψi2x.The LRKM reads as follows: FindingU^∈Wn0−1,1 such that(28)ℒ11u^xi+ℒ12v^xi=f1xi,i=1,2,…,n,ℒ21u^xi+ℒ22v^xi=f2xi,i=1,2,…,n.OnceU^n is obtained, one can get an approximate solution Un of U by applying scaling from −1,1 to a,b:(29)U^nx=U^2b−ax−b+ab−a.Next, we will discuss the convergence of LRKM.Theorem 2.
u
^
n
x and v^nx uniformly converge to u^x and v^x on −1,1 as n⟶∞, respectively.Proof
(30)
u
^
n
−
u
^
=
u
^
n
⋅
−
u
^
⋅
,
K
⋅
,
x
≤
u
^
n
−
u
^
W
2
3
⋅
K
x
,
x
W
n
≤
u
^
n
−
u
^
W
n
2
+
v
^
n
−
v
^
W
n
2
K
W
n
=
U
^
n
−
U
^
W
n
⋅
K
W
n
.From the continuity ofK, we have KWn≤M. Notice that U^n−U^Wn⟶0 as n⟶∞, and we conclude that(31)u^nx−u^x≤MU^n−U^⟶0.Analogously,v^nx−v^x≤MU^n−U^⟶0. The proof is completed.


## 3.1. Analysis of QNM

By equation (1), we define ℒij:Wn0−1,1⟶L2−1,1 as(16)ℒ11:u^⟼u^″+a^1u^′,ℒ12:v^⟼a^2v^″+a^3v^′,ℒ21:u^⟼b^2u^″+b^3u^′,ℒ22:v^⟼v″+b^1v^′.Let(17)ℒ=ℒ11ℒ12ℒ21ℒ22,NU=ℱ^u^,v^G^u^,v^.One can correspondingly rewrite equation (2) into the following operator form:(18)ℒU+NU=F,U−1=0,U1=0,where U=u^,v^T and F=f^,g^T∈L2−1,1.Theorem 1.
TheFréchet derivative of N at U0 is(19)N′U0:U⟼ℱuu0,v0ℱvu0,v0Guu0,v0Gvu0,v0U.To showFréchet derivative of the nonlinear operator more clearly, we introduce two examples.Example 1.
LetNU=xv+u2u+x2v2, then N′U0U=2u0xxℐxℐ2x2v0xU, where ℐ is identity operator.Example 2.
LetNU=u″+xeu+v3v″+u2+xsinv, then N′U0U=u″v″+xeu03v022u02xcosv0U.
Now, we will introduce the QNM to linearize the operator equation (18).
The Quasi-Newton’s Scheme: FindingUkk=1∞ such that(20)ℒUk+1+N′UkUk+1−Uk+NUk=F,U−1=0,U1=0,k=0,1,….


## 3.2. Analysis of LRKM

By improving the RKM, an effective and simple algorithm is derived to solve the linear system (20). For convenience, we rewrite (20) into the following form:(21)ℒU=F,U−1=0,U1=0.Lemma 2.
ℒ is linear and bounded.Proof. Obviously, ℒ11,ℒ12,ℒ21, and ℒ22 are linear operators. For ∀U∈Wn0, we have(22)ℒUL22=∑i=12ℒi1u+ℒi2vL22≤∑i=12ℒi1⋅uWn+ℒi2⋅vWn2≤2∑i=12ℒi12+ℒi22⋅uWn2+vWn2≤2∑i=12ℒi12+ℒi22⋅UWn2.The boundedness ofℒiji,j=12 results in the boundedness of the operator ℒ.Let(23)Ψi1=ℒ∗Rx,xi,0T,Ψi2=ℒ∗0,Rx,xiT,wherein xii=1n are different points in −1,1, ℒ∗ is the conjugate operator of ℒ, and R is the reproducing kernel of L2−1,1.Lemma 3.
Ψ
i
1
x
=
ℒ
11
K
x
,
x
i
ℒ
12
K
x
,
x
i
,
Ψ
i
2
x
=
ℒ
21
K
x
,
x
i
ℒ
22
K
x
,
x
i
,
i
=
1,2
,
…
,
n.Proof. From the reproducibility of R, we immediately deduce that(24)Ψi1x=ℒ∗Rx,xi0=ℒ11∗ℒ21∗ℒ12∗ℒ22∗Rx,xi0=ℒ11∗Rx,xiℒ12∗Rx,xi=ℒ11Rx,xiℒ12Rx,xi.Similarly, we haveΨi2x=ℒ21Kx,xiℒ22Kx,xi,i=1,2,…,n.Lemma 4.
Ψ
11
,
Ψ
21
,
…
,
Ψ
n
1
,
Ψ
12
,
Ψ
22
,
…
,
Ψ
n
2 are linearly independent in Wn0−1,1.Considering the numerical implement, we apply the SRKM proposed in [12, 13] to solve equation (21). We first denote a finite-dimensional subspace S2n by(25)S2n≜spanΨi21n∪Ψi21n¯.LetU^n≜P2n+4U^, where P2n:W∞⟶S2n is the projection operator. Then, U^n converges to U^ in the sense of ⋅Wn−1,1, which means that(26)U^−U^nWn⟶0,asn⟶∞.AsU^n∈Wn−1,1, there exits undetermined coefficients ai1andai2i=1,2…n such that(27)U^nx=∑i=1nai1Ψi1x+∑i=1nai2Ψi2x.The LRKM reads as follows: FindingU^∈Wn0−1,1 such that(28)ℒ11u^xi+ℒ12v^xi=f1xi,i=1,2,…,n,ℒ21u^xi+ℒ22v^xi=f2xi,i=1,2,…,n.OnceU^n is obtained, one can get an approximate solution Un of U by applying scaling from −1,1 to a,b:(29)U^nx=U^2b−ax−b+ab−a.Next, we will discuss the convergence of LRKM.Theorem 2.
u
^
n
x and v^nx uniformly converge to u^x and v^x on −1,1 as n⟶∞, respectively.Proof
(30)
u
^
n
−
u
^
=
u
^
n
⋅
−
u
^
⋅
,
K
⋅
,
x
≤
u
^
n
−
u
^
W
2
3
⋅
K
x
,
x
W
n
≤
u
^
n
−
u
^
W
n
2
+
v
^
n
−
v
^
W
n
2
K
W
n
=
U
^
n
−
U
^
W
n
⋅
K
W
n
.From the continuity ofK, we have KWn≤M. Notice that U^n−U^Wn⟶0 as n⟶∞, and we conclude that(31)u^nx−u^x≤MU^n−U^⟶0.Analogously,v^nx−v^x≤MU^n−U^⟶0. The proof is completed.


## 4. Numerical Experiments

Example 3.
Let us consider a linear problem suggested in [5, 10]:(32)u″x+u′x+xux+v′x+2xvx=fx,x∈0,1,2u′x+x2ux+v″x+vx=gx,x∈0,1,u0=u1=0,v0=v1=0.
Here,fx=πcosπx−21+xcosx+4x−2x2−4sinx+2xsinπx,gx=1−π2sinπx+2−x3+x2+x−3sinx−4xcosx. The exact solution is ux=21−xsinx,vx=sinπx. Applying the LRKM, absolute errors are presented in Table 1 and Figure 1. Besides, we compare our method with RKM [5] and the sinc-collocation method [10] in Table 2. Numerical results show that LRKM has high accuracy in solving the linear equation, but uses the least number of points.Table 1
Absolute errors for Example3.
n
u
−
u
n
C
u
′
−
u
n
′
C
v
−
v
n
C
v
′
−
v
n
′
C
5
3.32
×
10
−
4
2.09
×
10
−
3
1.51
×
10
−
3
1.44
×
10
−
2
7
1.80
×
10
−
6
1.96
×
10
−
5
1.85
×
10
−
5
2.18
×
10
−
4
9
9.44
×
10
−
9
1.20
×
10
−
7
1.13
×
10
−
7
1.89
×
10
−
6
11
4.21
×
10
−
10
7.28
×
10
−
9
1.28
×
10
−
9
2.86
×
10
−
8Figure 1
Absolute erroru−u9 (a) and v−v9 (b) for Example 3.
(a)
(b)Table 2
Comparison with RKM [5] and the sinc-collocation method [10] for Example 3.
x
i
Method [5] n=21
Method [10] N=10
Our methodn=9
u
−
u
n
C
v
−
v
n
C
u
−
u
n
C
v
−
v
n
C
u
−
u
n
C
v
−
v
n
C
0.08
8.0
×
10
−
4
1.9
×
10
−
3
3.0
×
10
−
4
2.0
×
10
−
3
4.03
×
10
−
10
5.86
×
10
−
8
0.24
1.9
×
10
−
3
5.1
×
10
−
3
8.5
×
10
−
5
9.8
×
10
−
4
6.57
×
10
−
9
6.68
×
10
−
10
0.40
2.4
×
10
−
3
6.9
×
10
−
3
3.5
×
10
−
4
1.1
×
10
−
3
5.56
×
10
−
9
3.59
×
10
−
8
0.56
2.4
×
10
−
3
6.9
×
10
−
3
2.6
×
10
−
4
1.4
×
10
−
3
8.16
×
10
−
9
3.62
×
10
−
8
0.72
1.8
×
10
−
3
5.2
×
10
−
3
2.0
×
10
−
4
5.5
×
10
−
5
1.13
×
10
−
9
3.68
×
10
−
8
0.88
8.0
×
10
−
4
2.4
×
10
−
3
2.6
×
10
−
4
7.7
×
10
−
4
2.37
×
10
−
9
2.37
×
10
−
9
0.96
2.0
×
10
−
4
8.0
×
10
−
4
2.6
×
10
−
3
8.3
×
10
−
4
2.24
×
10
−
9
3.98
×
10
−
8Example 4.
Let us consider a nonlinear system proposed in [5, 10]:(33)u″x+xux+2xvx+xu2x=f,x∈0,1,v″x+vx+x2ux+sinxv2x=g,x∈0,1,u0=u1=0,v0=v1=0,where f=2xsinπx+x5−2x4+x2−2 and g=x31−x+1−π2sinπx+sinxsin2πx. The exact solution is u=x−x2;v=sinπx. Selecting initial functions u0, v0 as the polynomial that satisfies the boundary conditions and applying the technique of QNM-LRKM three times, we depict numerical results of uxi and vxi in Table 3 and Figure 2. Numerical comparison with homotopy analysis and collocation method for problem 2 is shown in Table 4. Numerical results indicate that our method uses fewer nodes but obtains the more accurate numerical solution. We take n=11 in LRKM (28) and different iteration steps k, and absolute errors are shown in Table 5. Numerical results indicate that the convergence order of NDM is nearly 2.Table 3
Absolute errors for Example4.
n
u
−
u
n
C
u
′
−
u
n
′
C
v
−
v
n
C
v
′
−
v
n
′
C
5
2.92
×
10
−
5
2.55
×
10
−
4
1.50
×
10
−
3
1.46
×
10
−
2
7
3.62
×
10
−
7
1.95
×
10
−
5
1.88
×
10
−
5
2.12
×
10
−
4
9
1.84
×
10
−
9
9.84
×
10
−
8
1.14
×
10
−
7
1.88
×
10
−
6
11
2.67
×
10
−
10
2.43
×
10
−
9
2.73
×
10
−
9
4.42
×
10
−
8Figure 2
Absolute erroru−u9 (a) and v−v9 (b) for Example 4.
(a)
(b)Table 4
Comparison with homotopy method and collocation method for Example4.
x
i
Method [10] N=10
Method [5] n=21
Our methodn=9
u
−
u
n
C
v
−
v
n
C
u
−
u
n
C
v
−
v
n
C
u
−
u
n
C
v
−
v
n
C
0.08
1.4
×
10
−
4
2.4
×
10
−
4
5.0
×
10
−
4
2.0
×
10
−
3
3.76
×
10
−
10
5.82
×
10
−
8
0.24
4.4
×
10
−
5
2.3
×
10
−
3
1.4
×
10
−
3
5.6
×
10
−
3
1.09
×
10
−
9
1.36
×
10
−
9
0.40
6.7
×
10
−
5
8.9
×
10
−
4
2.1
×
10
−
3
7.9
×
10
−
3
1.59
×
10
−
9
3.65
×
10
−
8
0.56
9.3
×
10
−
5
1.4
×
10
−
3
2.2
×
10
−
3
8.2
×
10
−
3
1.82
×
10
−
9
3.62
×
10
−
8
0.72
4.9
×
10
−
5
3.1
×
10
−
3
1.8
×
10
−
3
6.5
×
10
−
3
1.70
×
10
−
9
7.29
×
10
−
8
0.88
8.6
×
10
−
5
1.6
×
10
−
3
9.0
×
10
−
4
3.1
×
10
−
3
8.80
×
10
−
10
2.21
×
10
−
9
0.96
7.1
×
10
−
5
9.8
×
10
−
4
3.0
×
10
−
4
1.0
×
10
−
3
3.35
×
10
−
10
4.00
×
10
−
8Table 5
Convergence order of NDM for Example4.
k
u
−
u
k
C
ε
k
+
1
/
ε
k
2
v
−
v
k
C
ε
k
+
1
/
ε
k
2
1
1.57
×
10
−
2
1.27
×
10
−
2
2
4.08
×
10
−
5
0.160
1.61
×
10
−
5
0.0998
3
2.67
×
10
−
10
0.165
2.52
×
10
−
11
0.0972


## 5. Conclusions

This paper presents an efficient numerical method for solving second-order nonlinear system of BVPs. The algorithm is easy to implement. Numerical results verify that the QNM-LRKM is a reliable numerical technique of high-order accuracy. Precisely, NDM ensures the speed of iteration, while LRKM guarantees the accuracy of the algorithm.



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*Source: 6280372-2020-03-09.xml*