# Numerov's Method for a Class of Nonlinear Multipoint Boundary Value Problems

Authors: Yuan-Ming Wang; Wen-Jia Wu; Massimo Scalia
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The purpose of this paper is to give a numerical treatment for a class of nonlinear multipoint boundary value problems. The multipoint boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition. The problems are discretized by the fourth-order Numerov's method. The existence and uniqueness of the numerical solution are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the method are proved. An accelerated monotone iterative algorithm with the quadratic rate of convergence is developed for solving the resulting nonlinear discrete problems. Some applications and numerical results are given to demonstrate the
high efficiency of the approach.


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## Body

## 1. Introduction

Multipoint boundary value problems arise in various fields of applied science. An often discussed problem is the following nonlinear second-order multipoint boundary value problem:(1.1)-u′′(x)=f(x,u(x)),0<x<1,u(0)=∑i=1pαiu(ξi),u(1)=∑i=1pβiu(ηi),
where f(x,u) is a continuous function of its arguments and for each i, αi, βi∈[0,∞) and ξi, ηi∈(0,1). An application of this problem appears in the design of a large-size bridge with multipoint supports, where u(x) denotes the displacement of the bridge from the unloaded position (e.g., see [1]). For other applications of problem (1.1), we see [2–4] and the references therein. It is allowed in (1.1) that αi=0 or βi=0 for some or all i. This implies that the boundary condition in (1.1) includes various commonly discussed multipoint boundary conditions. In particular, the boundary condition in (1.1) is reduced to
(1.1a)u(0)=0,u(1)=∑i=1pβiu(ηi),
if αi=0 for all i (see [5–14]), to the form (1.1b)u(0)=∑i=1pαiu(ξi),u(1)=0,
if βi=0 for all i (see [15]), to the four-point boundary condition (1.1c)u(0)=αu(ξ),u(1)=βu(η),
if p=1 and ξ1=ξ, η1=η (see [11, 15–17]), and to the two-point boundary condition (1.1d)u(0)=u(1)=0,
if αi=0 and βi=0 for all i. Condition 1.1c includes the three-point boundary condition when ξ=η (see [16, 17]).The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated in [18, 19] by Il’in and Moiseev. In [20], Gupta studied a three-point boundary value problem for nonlinear second-order ordinary differential equations. Since then, more general nonlinear second-order multipoint boundary value problems in the form (1.1) have been studied. Most of the discussions were concerned with the existence and multiplicity of solutions by using different methods. Applying the fixed point index theorem in cones, the works in [5–14] showed the existence of one or more solutions to the problem (1.1)-1.1a, while the works in [15–17] were devoted to the existence of solutions for the three- or four-point boundary value problem (1.1)–1.1c. For the problem (1.1) with the more general multipoint boundary conditions, some existence results were obtained in [21, 22] by using the fixed point index theory or the topological degree theory. Based on the method of upper and lower solutions, the authors of [17, 23] obtained some sufficient conditions so that (1.1) or its some special form has at least one solution. Additional works that deal with the existence problem of nonlinear second-order multipoint boundary value problems can be found in [24–29].On the other hand, there are also some works that are devoted to numerical methods for the solutions of multipoint boundary value problems. The work in [30] made use of the Chebyshev series for approximating solutions of nonlinear first-order multipoint boundary value problems, and the work in [31] showed how an adaptive finite difference technique can be developed to produce efficient approximations to the solutions of nonlinear multipoint boundary value problems for first-order systems of equations. Another method for computing the solutions of nonlinear first-order multipoint boundary value problems was described in [32], where a multiple shooting technique was developed. Some other works for the computational methods of first-order multipoint boundary value problems can be seen in [33–35]. In [36–38] the authors gave several constructive methods for the solutions of multipoint discrete boundary value problems, including the method of adjoints, the invariant embedding method, and the shooting-type method. In the case of second-order multipoint boundary value problems, there are only a few computational algorithms in the literature. The paper [39] set up a reproducing kernel Hilbert space method for the solution of a second-order three-point boundary value problem. Based upon the shooting technique, a numerical method was developed in [1] for approximating solutions and fold bifurcation solutions of a class of second-order multipoint boundary value problems.As we know, Numerov's method is one of the well-known difference methods to solve the second-order ordinary differential equation-u′′=f(x,u). Because Numerov's method possesses the fourth-order accuracy and a compact property, it has attracted considerable attention and has been extensively applied in practical computations (cf. [40–51]). Although many theoretical investigations have focused on Numerov's method for two-point boundary conditions such as 1.1d (cf. [40, 41, 43, 44, 47–51]), there is relatively little discussion on the analysis of Numerov's method applied to fully multipoint boundary conditions in (1.1). The study presented in this paper is aimed at filling in such a gap by considering Numerov's method for the numerical solution of the multipoint boundary value problem (1.1) with the more general boundary conditions, including the boundary conditions 1.1a, 1.1b, and 1.1c. It is not difficult to give a Numerov's difference approximation to (1.1) in the same manner as that for two-point boundary value problems. However, a lack of explicit information about the boundary value of the solution in the multipoint boundary conditions prevents us from using the standard analysis process of treating two-point boundary value problems, and so we here develop a different approach for the analysis of Numerov's difference approximation to (1.1). Our specific goals are (1) to establish the existence and uniqueness of the numerical solution, (2) to show the convergence of the numerical solution to the analytic solution with the fourth-order accuracy, and (3) to develop an efficient computational algorithm for solving the resulting nonlinear discrete problems. To achieve the above goals, we use the method of upper and lower solutions and its associated monotone iterations. It should be mentioned that the proposed fourth-order Numerov's discretization methodology may be straightforwardly extended to the following nonhomogeneous multipoint boundary condition: (1.2)u(0)=∑i=1pαiu(ξi)+λ1,u(1)=∑i=1pβiu(ηi)+λ2,
where λ1 and λ2 are two prescribed constants.The outline of the paper is as follows. In Section2, we discretize (1.1) into a finite difference system by Numerov's technique. In Section 3, we deal with the existence and uniqueness of the numerical solution by using the method of upper and lower solutions. The convergence of the numerical solution and the fourth-order accuracy of the method are proved in Section 4. Section 5 is devoted to an accelerated monotone iterative algorithm for solving the resulting nonlinear discrete problem. Using an upper solution and a lower solution as initial iterations, the iterative algorithm yields two sequences that converge monotonically from above and below, respectively, to a unique solution of the resulting nonlinear discrete problem. It is shown that the rate of convergence for the sum of the two produced sequences is quadratic (the error metric is the sum of the infinity norm of the error between the mth-iteration of the upper solution and the true solution with the infinity norm of the error between the mth-iteration of the lower solution and the true solution) and under an additional requirement, the quadratic rate of convergence is attained for one of these two sequences. In Section 6, we give some applications to three model problems and present some numerical results demonstrating the monotone and rapid convergence of the iterative sequences and the fourth-order accuracy of the method. We also compare our method with the standard finite difference method and show its advantages. The final section contains some concluding remarks.


## 2. Numerov's Method

Leth=1/L be the mesh size, and let xi=ih(0≤i≤L) be the mesh points in [0,1]. Assume that for all 1≤i≤p, the points ξi and ηi in the boundary condition of (1.1) serve as mesh points. This assumption is always satisfied by a proper choice of mesh size h. For convenience, we use the following notations: (2.1)Sα[u(ξ)]=∑i=1pαiu(ξi),Sβ[u(η)]=∑i=1pβiu(ηi)
and introduce the finite difference operators δh2 and 𝒫h as follows: (2.2)δh2u(xi)=u(xi-1)-2u(xi)+u(xi+1),1≤i≤L-1,Phu(xi)=h212(u(xi-1)+10u(xi)+u(xi+1)),1≤i≤L-1.
Using the following Numerov's formula (cf. [52, 53]):(2.3)δh2u(xi)=Phu′′(xi)+O(h6),1≤i≤L-1,
we have from (1.1) and (2.1) that (2.4)-δh2u(xi)=Phf(xi,u(xi))+O(h6),1≤i≤L-1,u(0)=Sα[u(ξ)],u(1)=Sβ[u(η)].
After dropping the 𝒪(h6) term, we derive a Numerov's difference approximation to (1.1) as follows: (2.5)-δh2uh(xi)=Phf(xi,uh(xi)),1≤i≤L-1,uh(0)=Sα[uh(ξ)],uh(1)=Sβ[uh(η)],
where uh(xi) represents the approximation of u(xi).For two constantsM̲ and M¯ satisfying M¯≥M̲>-π2, we define (2.6)h(M̲,M¯)={12M¯,M̲>-8,M¯>0,1,M̲>-8,M¯≤0,min⁡{12M¯,12π2(1+M̲π2)},M̲≤-8,M¯>0,12π2(1+M̲π2),M̲≤-8,M¯≤0.
A fundamental and useful property of the operators δh2 and 𝒫h is stated below.Lemma 2.1 (See Lemma 3.1 of [50]).
LetM̲, M¯, and Mi be some constants satisfying
(2.7)-π2<M̲≤Mi≤M¯,0≤i≤L.
If
(2.8)-δh2uh(xi)+Ph(Miuh(xi))≥0,1≤i≤L-1,uh(0)≥0,uh(1)≥0,
and h<h(M̲,M¯), then uh(xi)≥0 for all 0≤i≤L.The following results are also useful for our forthcoming discussions. Their proofs will be given in the appendix.Lemma 2.2.
Assume(2.9)σ≡max⁡{∑i=1pαi,∑i=1pβi}<1.
Let M̲, M¯, and Mi be the given constants such that
(2.10)-8(1-σ)<M̲≤Mi≤M¯,0≤i≤L.
If
(2.11)-δh2uh(xi)+Ph(Miuh(xi))≥0,1≤i≤L-1,uh(0)≥Sα[uh(ξ)],uh(1)≥Sβ[uh(η)],
and h<h(M̲,M¯), then uh(xi)≥0 for all 0≤i≤L.Lemma 2.3.
Let the condition (2.9) be satisfied, and let M̲, M¯, and Mi be the given constants satisfying (2.10). Assume that the functions uh(xi) and g(xi) satisfy
(2.12)-δh2uh(xi)+Ph(Miuh(xi))=g(xi),1≤i≤L-1,uh(0)=Sα[uh(ξ)],uh(1)=Sβ[uh(η)].
Then when h<h(M̲,M¯),
(2.13)‖uh‖∞≤‖g‖∞(8(1-σ)+min⁡(M̲,0))h2,
where ∥uh∥∞=max1≤i≤L-1|uh(xi)| denotes discrete infinity norm for any mesh function uh(xi).Remark 2.4.
It is clear from Lemma2.1 that if σ=0 then the condition (2.10) in Lemma 2.2 can be replaced by the weaker condition (2.7). Lemmas 2.1 and 2.2 guarantee that the linear problems based on (2.8) and (2.11) with the inequality relation “≥” replaced by the equality relation “=” are well posed.


## 3. The Existence and Uniqueness of the Solution

To investigate the existence and uniqueness of the solution of (2.5), we use the method of upper and lower solutions. The definition of the upper and lower solutions is given as follows.Definition 3.1.
A functionũh(xi) is called an upper solution of (2.5) if
(3.1)-δh2ũh(xi)≥Phf(xi,ũh(xi)),1≤i≤L-1,ũh(0)≥Sα[ũh(ξ)],ũh(1)≥Sβ[ũh(η)].
Similarly, a function ûh(xi) is called a lower solution of (2.5) if it satisfies the above inequalities in the reversed order. A pair of upper and lower solutions ũh(xi) and ûh(xi) are said to be ordered if ũh(xi)≥ûh(xi) for all 0≤i≤L.It is clear that every solution of (2.5) is an upper solution as well as a lower solution. For a given pair of ordered upper and lower solutions ũh(xi) and ûh(xi), we set (3.2)〈ûh,ũh〉={uh;ûh(xi)≤uh(xi)≤ũh(xi)(0≤i≤L)},[ûh(xi),ũh(xi)]={ui∈R;ûh(xi)≤ui≤ũh(xi)}
and make the following basic hypotheses:  (H1)
For each0≤i≤L, there exists a constant Mi such that Mi>-π2 and(3.3)f(xi,vi)-f(xi,vi′)≥-Mi(vi-vi′)
whenever ûh(xi)≤vi′≤vi≤ũh(xi); (H2)
h<h(M̲,M¯), where M¯=max0≤i≤LMi and M̲=min0≤i≤LMi.The existence of the constantMi in (H1) is trivial if f(xi,u) is a C1-function of u∈[ûh(xi),ũh(xi)]. In fact, Mi may be taken as any nonnegative constant satisfying (3.4)Mi≥max⁡{-fu(xi,u);u∈[ûh(xi),ũh(xi)]}.Theorem 3.2.
Letũh(xi) and ûh(xi) be a pair of ordered upper and lower solutions of (2.5), and let hypotheses (H1) and (H2) be satisfied. Then system (2.5) has a maximal solution u¯h(xi) and a minimal solution u̲h(xi) in 〈ûh,ũh〉. Here, the maximal property of u¯h(xi) means that for any solution uh(xi) of (2.5) in 〈ûh,ũh〉, one hase uh(xi)≤u¯h(xi) for all 0≤i≤L. The minimal property of u̲h(xi) is similarly understood.Proof.
The proof is constructive. Using the initial iterationsu¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=ûh(xi) we construct two sequences {u¯h(m)(xi)} and {u̲h(m)(xi)}, respectively, from the following iterative scheme:
(3.5)-δh2uh(m)(xi)+Ph(Miuh(m)(xi))=Ph(Miuh(m-1)(xi)+f(xi,uh(m-1)(xi))),1≤i≤L-1,uh(m)(0)=Sα[uh(m-1)(ξ)],uh(m)(1)=Sβ[uh(m-1)(η)],
where Mi is the constant in (H1). By Lemma 2.1, these two sequences are well defined. We shall first prove that for all m=0,1,…,
(3.6)u̲h(m)(xi)≤u̲h(m+1)(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L.
Let w¯h(0)(xi)=u¯h(0)(xi)-u¯h(1)(xi). Then by (3.1) and (3.5),
(3.7)-δh2w¯h(0)(xi)+Ph(Miw¯h(0)(xi))≥0,1≤i≤L-1,w¯h(0)(0)≥0,w¯h(0)(1)≥0.
It follows from Lemma 2.1 that w¯h(0)(xi)≥0, that is, u¯h(0)(xi)≥u¯h(1)(xi) for all 0≤i≤L. A similar argument using the property of a lower solution gives u̲h(1)(xi)≥u̲h(0)(xi) for all 0≤i≤L. Let wh(1)(xi)=u¯h(1)(xi)-u̲h(1)(xi). We have from (3.3) and (3.5) that
(3.8)-δh2wh(1)(xi)+Ph(Miwh(1)(xi))≥0,1≤i≤L-1,wh(1)(0)≥0,wh(1)(1)≥0.
Again by Lemma 2.1, wh(1)(xi)≥0, that is, u¯h(1)(xi)≥u̲h(1)(xi) for all 0≤i≤L. This proves (3.6) for m=0. Finally, an induction argument leads to the desired result (3.6) for all m=0,1,….
In view of (3.6), the limits
(3.9)lim⁡m→∞u¯h(m)(xi)=u¯h(xi),lim⁡m→∞u̲h(m)(xi)=u̲h(xi),0≤i≤L
exist and satisfy
(3.10)u̲h(m)(xi)≤u̲h(m+1)(xi)≤u̲h(xi)≤u¯h(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L,m=0,1,….
Letting m→∞ in (3.5) shows that both u¯h(xi) and u̲h(xi) are solutions of (2.5).
Now, ifuh(xi) is a solution of (2.5) in 〈ûh,ũh〉, then the pair uh(xi) and ûh(xi) are also a pair of ordered upper and lower solutions of (2.5). The above arguments imply that u̲h(xi)≤uh(xi) for all 0≤i≤L. Similarly, we have uh(xi)≤u¯h(xi) for all 0≤i≤L. This shows that u¯h(xi) and u̲h(xi) are the maximal and the minimal solutions of (2.5) in 〈ûh,ũh〉, respectively. The proof is completed.Theorem3.2 shows that the system (2.5) has a maximal solution u¯h(xi) and a minimal solution u̲h(xi) in 〈ûh,ũh〉. If u¯h(xi)=u̲h(xi) for all 0≤i≤L, then u¯h(xi) or u̲h(xi) is a unique solution of (2.5) in 〈ûh,ũh〉. In general, these two solutions do not coincide. Consider, for example, the case
(3.11)∑i=1pαi=∑i=1pβi=1.
If there exist two different constants c¯ and c̲ such that f(x,c¯)=f(x,c̲)=0 for all x∈(0,1) then both c¯ and c̲ are solutions of (2.5). Hence to show the uniqueness of a solution it is necessary to impose some additional conditions on αi, βi and f. Assume that there exists a constant M̲u such that (3.12)f(xi,vi)-f(xi,vi′)≤-M̲u(vi-vi′),0≤i≤L
whenever ûh(xi)≤vi′≤vi≤ũh(xi). This condition is trivially satisfied if f(xi,u) is a C1-function of u∈[ûh(xi),ũh(xi)] for all 0≤i≤L. In fact, M̲u may be taken as (3.13)M̲u=min⁡0≤i≤Lmin⁡{-fu(xi,u);u∈[ûh(xi),ũh(xi)]}.
The following theorem gives a sufficient condition for the uniqueness of a solution.Theorem 3.3.
Let the conditions in Theorem3.2 hold. If, in addition, the conditions (2.9) and (3.12) hold and either
(3.14)-8(1-σ)<M̲u≤0orM̲u>0,h<12M̲u,
then the system (2.5) has a unique solution uh*(xi) in 〈ûh,ũh〉. Moreover, the relation (3.10) holds with u¯h(xi)=u̲h(xi)=uh*(xi) for all 0≤i≤L.Proof.
It suffices to showu¯h(xi)=u̲h(xi) for all 0≤i≤L, where u¯h(xi) and u̲h(xi) are the limits in (3.9). Let wh(xi)=u¯h(xi)-u̲h(xi). Then wh(xi)≥0, and by (2.5),
(3.15)-δh2wh(xi)=Ph(f(xi,u¯h(xi))-f(xi,u̲h(xi))),1≤i≤L-1,wh(0)=Sα[wh(ξ)],wh(1)=Sβ[wh(η)].
Therefore, we have from (3.12) that
(3.16)-δh2wh(xi)+Ph(M̲uwh(xi))≤0,1≤i≤L-1,wh(0)=Sα[wh(ξ)],wh(1)=Sβ[wh(η)].
By Lemma 2.2, wh(xi)≤0 for all 0≤i≤L. This proves u¯h(xi)=u̲h(xi) for all 0≤i≤L.
To give another sufficient condition, we assume that there exists a nonnegative constantM¯u* such that
(3.17)|f(xi,vi)-f(xi,vi′)|≤M¯u*|vi-vi′|,0≤i≤L
whenever ûh(xi)≤vi′≤vi≤ũh(xi). If f(xi,u) is a C1-function of u∈[ûh(xi),ũh(xi)] for all 0≤i≤L, the above condition is clearly satisfied by
(3.18)M¯u*=max⁡0≤i≤Lmax⁡{|fu(xi,u)|;u∈[ûh(xi),ũh(xi)]}.Theorem 3.4.
Let the conditions in Theorem3.2 hold. If, in addition, the conditions (2.9) and (3.17) hold and
(3.19)M¯u*<8(1-σ),
then the conclusions of Theorem 3.3 are also valid.Proof.
Applying Lemma2.3 with Mi=0 to (3.15) leads to
(3.20)‖wh‖∞≤‖g‖∞8(1-σ)h2,
where g(xi)=𝒫h(f(xi,u¯h(xi))-f(xi,u̲h(xi))). By (3.17), we obtain ∥g∥∞≤h2M¯u*∥wh∥∞. Consequently,
(3.21)‖wh‖∞≤M¯u*‖wh‖∞8(1-σ).
This together with (3.19) implies wh(xi)=0, that is, u¯h(xi)=u̲h(xi) for all 0≤i≤L.It is seen from the proofs of Theorems3.2–3.4 that the iterative scheme (3.5) not only leads to the existence and uniqueness of the solution of (2.5) but also provides a monotone iterative algorithm for computing the solution. However, the rate of convergence of the iterative scheme (3.5) is only of linear order because it is of Picard type. A more efficient monotone iterative algorithm with the quadratic rate of convergence will be developed in Section 5.


## 4. Convergence of Numerov's Method

In this section, we deal with the convergence of the numerical solution and show the fourth-order accuracy of Numerov's scheme (2.5). Throughout this section, we assume that the function f(x,u) and the solution u(x) of (1.1) are sufficiently smooth.Letu(xi) be the value of the solution of (1.1) at the mesh point xi, and let uh(xi) be the solution of (2.5). We consider the error eh(xi)=u(xi)-uh(xi). In fact, we have from (2.4) and (2.5) that (4.1)-δh2eh(xi)=Ph(f(xi,u(xi))-f(xi,uh(xi)))+O(h6),1≤i≤L-1,eh(0)=Sα[eh(ξ)],eh(1)=Sβ[eh(η)].Theorem 4.1.
Let the condition (2.9) hold, and let [u*,i,ui*] be an interval in R such that u(xi),uh(xi)∈[u*,i,ui*]. Assume that
(4.2)max⁡0≤i≤Lmax⁡{fu(xi,u);u∈[u*,i,ui*]}<8(1-σ).
Then for sufficiently small h,
(4.3)‖u-uh‖∞≤C*h4,
where C* is a positive constant independent of h.Proof.
Applying the mean value theorem to the first equality of (4.1), we have
(4.4)-δh2eh(xi)+Ph(Mieh(xi))=O(h6),1≤i≤L-1,eh(0)=Sα[eh(ξ)],eh(1)=Sβ[eh(η)],
where Mi=-fu(xi,θi) and θi∈[u*,i,ui*]. Let M¯=miniMi and M̲=miniMi. Then by (4.2), -8(1-σ)<M̲≤Mi≤M¯. We, therefore, obtain from Lemma 2.3 that when h<h(M̲,M¯),
(4.5)‖eh‖∞≤C1‖O(h6)‖∞h2,
where C1 is a positive constant independent of h. Finally, the error estimate (4.3) follows from ∥𝒪(h6)∥∞≤C2h6 for some positive constant independent of h.Theorem4.1 shows that Numerov's scheme (2.5) possesses the fourth-order accuracy under the conditions of the theorem.


## 5. An Accelerated Monotone Iterative Algorithm

The iterative scheme (3.5) gives an algorithm for solving the system (2.5). However, as already mentioned in Section 3, its rate of convergence is only of linear order because it is of Picard type. To raise the rate of convergence while maintaining the monotone convergence of the sequence, we propose an accelerated monotone iterative algorithm. An advantage of this algorithm is that its rate of convergence for the sum of the two produced sequences is quadratic (in the sense mentioned in Section 1) with only the usual differentiability requirement on the function f(·,u). If the function fu(·,u) possesses a monotone property in u, this algorithm is reduced to Newton's method, and one of the two produced sequences converges quadratically.


### 5.1. Monotone Iterative Algorithm

Letũh(xi) and ûh(xi) be a pair of ordered upper and lower solutions of (2.5) and assume that f(·,u) is a C1-function of u∈〈ûh,ũh〉. It follows from Theorems 3.2–3.4 that (2.5) has a unique solution uh*(xi) in 〈ûh,ũh〉 under the conditions of the theorems. To compute this solution, we use the following iterative scheme: (5.1)-δh2uh(m)(xi)+Ph(Mi(m-1)uh(m)(xi))=Ph(Mi(m-1)uh(m-1)(xi)+f(xi,uh(m-1)(xi))),1≤i≤L-1,uh(m)(0)=Sα[uh(m)(ξ)],uh(m)(1)=Sβ[uh(m)(η)],
where uh(0)(xi) is either ũh(xi) or ûh(xi), and for each i, (5.2)Mi(m)=max⁡{-fu(xi,u);u∈[u̲h(m)(xi),u¯h(m)(xi)]}.
The functions u¯h(m)(xi) and u̲h(m)(xi) in the definition of Mi(m) are obtained from (5.1) with uh(0)(xi)=ũh(xi) and uh(0)(xi)=ûh(xi), respectively. It is clear from (5.2) that (5.3)f(xi,vi)-f(xi,vi′)≥-Mi(m)(vi-vi′),0≤i≤L,
whenever u̲h(m)(xi)≤vi′≤vi≤u¯h(m)(xi). Moreover, (5.4)Mi(m)={-fu(xi,u¯h(m)(xi)),iffu(xi,u)ismonotonenonincreasinginu∈[u̲h(m)(xi),u¯h(m)(xi)],-fu(xi,u̲h(m)(xi)),iffu(xi,u)ismonotonenondecreasinginu∈[u̲h(m)(xi),u¯h(m)(xi)].
Hence, if fu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, then the iterative scheme (5.1) for {u¯h(m)(xi)}/{u̲h(m)(xi)} is reduced to Newton's form: (5.5)-δh2uh(m)(xi)-Ph(fu(xi,uh(m-1)(xi))uh(m)(xi))=-Ph(fu(xi,uh(m-1)(xi))uh(m-1)(xi)-f(xi,uh(m-1)(xi))),1≤i≤L-1,uh(m)(0)=Sα[uh(m)(ξ)],uh(m)(1)=Sβ[uh(m)(η)].To show that the sequences given by (5.1) are well-defined and monotone for an arbitrary C1-function f(·,u), we let M̲u be given by (3.13) and let (5.6)M¯u=max⁡0≤i≤Lmax⁡{-fu(xi,u);u∈[ûh(xi),ũh(xi)]}.Lemma 5.1.
Let the condition (2.9) hold, and let ũh(xi) and ûh(xi) be a pair of ordered upper and lower solutions of (2.5). Assume that M̲u>-8(1-σ) and h<h(M̲u,M¯u). Then the sequences {u¯h(m)(xi)}, {u̲h(m)(xi)}, and {Mi(m)} given by (5.1) and (5.2) with u¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=ûh(xi) are all well defined and possess the monotone property
(5.7)u̲h(m)(xi)≤u̲h(m+1)(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L,m=0,1,….Proof.
SinceMi(0)=max{-fu(xi,u);ui∈[ûh(xi),ũh(xi)]}, -8(1-σ)<M̲u≤Mi(0)≤M¯u and h<h(M̲u,M¯u), we have from Lemma 2.2 that the first iterations u¯h(1)(xi) and u̲h(1)(xi) are well defined. Let w¯h(0)(xi)=u¯h(0)(xi)-u¯h(1)(xi). Then, by (3.1) and (5.1),
(5.8)-δh2w¯h(0)(xi)+Ph(Mi(0)w¯h(0)(xi))≥0,1≤i≤L-1,w¯h(0)(0)≥Sα[w¯h(0)(ξ)],w¯h(0)(1)≥Sβ[w¯h(0)(η)].
We have from Lemma 2.2 that w¯h(0)(xi)≥0, that is, u¯h(0)(xi)≥u¯h(1)(xi) for every 0≤i≤L. Similarly by the property of a lower solution, u̲h(1)(xi)≥u̲h(0)(xi) for every 0≤i≤L. Let wh(1)(xi)=u¯h(1)(xi)-u̲h(1)(xi). Then by (5.1) and (5.3),
(5.9)-δh2wh(1)(xi)+Ph(Mi(0)wh(1)(xi))≥0,1≤i≤L-1,wh(1)(0)=Sα[wh(1)(ξ)],wh(1)(1)=Sβ[wh(1)(η)].
It follows from Lemma 2.2 that wh(1)(xi)≥0, that is, u¯h(1)(xi)≥u̲h(1)(xi) for every 0≤i≤L. This proves the monotone property (5.7) for m=0.
Assume, by induction, that there exists some integerm0≥0 such that for all 0≤m≤m0, the iterations u¯h(m)(xi),  u¯h(m+1)(xi),  u̲h(m)(xi), and u̲h(m+1)(xi) are well-defined and satisfy (5.7). Then Mi(m0+1) is well defined and -8(1-σ)<M̲u≤Mi(m0+1)≤M¯u. Since h<h(M̲u,M¯u), we have from Lemma 2.2 that the iterations u¯h(m0+2)(xi) and u̲h(m0+2)(xi) exist uniquely. Let w¯h(m0+1)(xi)=u¯h(m0+1)(xi)-u¯h(m0+2)(xi). Since
(5.10)Mi(m0+1)w¯h(m0+1)(xi)=(Mi(m0+1)-Mi(m0))u¯h(m0+1)(xi)+Mi(m0)u¯h(m0+1)(xi)-Mi(m0+1)u¯h(m0+2)(xi),
the iterative scheme (5.1) implies that
(5.11)-δh2w¯h(m0+1)(xi)+Ph(Mi(m0+1)w¯h(m0+1)(xi))=Ph(Mi(m0)(u¯h(m0)(xi)-u¯h(m0+1)(xi))+f(xi,u¯h(m0)(xi))-f(xi,u¯h(m0+1)(xi))),1≤i≤L-1,w¯h(m0+1)(0)=Sα[w¯h(m0+1)(ξ)],w¯h(m0+1)(1)=Sβ[w¯h(m0+1)(η)].
Using the relation (5.3) yields
(5.12)-δh2w¯h(m0+1)(xi)+Ph(Mi(m0+1)w¯h(m0+1)(xi))≥0,1≤i≤L-1,w¯h(m0+1)(0)=Sα[w¯h(m0+1)(ξ)],w¯h(m0+1)(1)=Sβ[w¯h(m0+1)(η)].
By Lemma 2.2, w¯h(m0+1)(xi)≥0, that is, u¯h(m0+1)(xi)≥u¯h(m0+2)(xi) for every 0≤i≤L. Similarly we have u̲h(m0+2)(xi)≥u̲h(m0+1)(xi) for every 0≤i≤L. Let wh(m0+2)(xi)=u¯h(m0+2)(xi)-u̲h(m0+2)(xi). Then by (5.1) and (5.3), wh(m0+2)(xi) satisfies (5.12) with w¯h(m0+1)(xi) replaced by wh(m0+2)(xi). Therefore, by Lemma 2.2, wh(m0+2)(xi)≥0, that is, u¯h(m0+2)(xi)≥u̲h(m0+2)(xi) for every 0≤i≤L. This shows that the monotone property (5.7) is also true for m=m0+1. Finally, the conclusion of the lemma follows from the principle of induction.We next show monotone convergence of the sequences{u¯h(m)(xi)} and {u̲h(m)(xi)}.Theorem 5.2.
Let the hypothesis in Lemma5.1 hold. Then the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} given by (5.1) converge monotonically to the unique solution uh*(xi) of (2.5) in 〈ûh,ũh〉, respectively. Moreover,
(5.13)u̲h(m)(xi)≤u̲h(m+1)(xi)≤uh*(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L,m=0,1,….Proof.
It follows from the monotone property (5.7) that the limits
(5.14)lim⁡m→∞u¯h(m)(xi)=u¯h(xi),lim⁡m→∞u̲h(m)(xi)=u̲h(xi),0≤i≤L
exist and they satisfy (3.10). Since the sequence {Mi(m)} is monotone nonincreasing and is bounded from below by M̲u given in (3.13), it converges as m→∞. Letting m→∞ in (5.1) shows that both u¯h(xi) and u̲h(xi) are solutions of (2.5) in 〈ûh,ũh〉. Since M̲u>-8(1-σ) and h<h(M̲u,M¯u), the condition (3.14) of Theorem 3.3 is satisfied. Thus by Theorem 3.3, u¯h(xi)=u̲h(xi)(≡uh*(xi)) and uh*(xi) is the unique solution of (2.5) in 〈ûh,ũh〉. The monotone property (5.13) follows from (3.10).Whenfu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the iterative scheme (5.1) for {u¯h(m)(xi)}/{u̲h(m)(xi)} is reduced to Newton iteration (5.5). As a consequence of Theorem 5.2, we have the following conclusion.Corollary 5.3.
Let the hypothesis in Lemma5.1 be satisfied. If fu(xi,u) is monotone nonincreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u¯h(m)(xi)} given by (5.5) with u¯h(0)(xi)=ũh(xi) converges monotonically from above to the unique solution uh*(xi) of (2.5) in 〈ûh,ũh〉. Otherwise, if fu(xi,u) is monotone nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u̲h(m)(xi)} given by (5.5) with u̲h(0)(xi)=ûh(xi) converges monotonically from below to uh*(xi).


### 5.2. Rate of Convergence

We now show the quadratic rate of convergence of the sequences given by (5.1). Assume that there exists a nonnegative constant M¯* such that (5.15)|fu(xi,vi)-fu(xi,vi′)|≤M¯*|vi-vi′|∀vi,vi′∈[ûh(xi),ũh(xi)],0≤i≤L.
Clearly, this assumption is satisfied if f(·,u) is a C2-function of u.Theorem 5.4.
Let the hypotheses in Lemma5.1 and (5.15) hold. Also let {u¯h(m)(xi)} and {u̲h(m)(xi)} be the sequences given by (5.1) and let uh*(xi) be the unique solution of (2.5) in 〈ûh,ũh〉. Then there exists a constant ρ, independent of m, such that
(5.16)‖u¯h(m)-uh*‖∞+‖u̲h(m)-uh*‖∞≤ρ(‖u¯h(m-1)-uh*‖∞+‖u̲h(m-1)-uh*‖∞)2,m=1,2,….Proof.
Letw¯h(m)(xi)=u¯h(m)(xi)-uh*(xi). Subtracting (2.5) from (5.1) gives
(5.17)-δh2w¯h(m)(xi)+Ph(Mi(m-1)w¯h(m)(xi))=Ph(Mi(m-1)w¯h(m-1)(xi)+f(xi,u¯h(m-1)(xi))-f(xi,uh*(xi))),1≤i≤L-1,w¯h(m)(0)=Sα[w¯h(m)(ξ)],w¯h(m)(1)=Sβ[w¯h(m)(η)].
By the intermediate value theorem,
(5.18)Mi(m-1)=-fu(xi,θi(m-1)),
where θi(m-1)∈[u̲h(m-1)(xi),u¯h(m-1)(xi)], and by the mean value theorem,
(5.19)f(xi,u¯h(m-1)(xi))-f(xi,uh*(xi))=fu(xi,γi(m-1))w¯h(m-1)(xi),
where γi(m-1)∈[uh*(xi),u¯h(m-1)(xi)]. Let
(5.20)gi(m-1)=(fu(xi,γi(m-1))-fu(xi,θi(m-1)))w¯h(m-1)(xi).
Then we have from (5.17) that
(5.21)-δh2w¯h(m)(xi)+Ph(Mi(m-1)w¯h(m)(xi))=Phgi(m-1),1≤i≤L-1,w¯h(m)(0)=Sα[w¯h(m)(ξ)],w¯h(m)(1)=Sβ[w¯h(m)(η)].
Since -8(1-σ)<M̲u≤Mi(m-1)≤M¯u and h<h(M̲u,M¯u), it follows from Lemma 2.3 that there exists a constant ρ1, independent of m, such that
(5.22)‖w¯h(m)‖∞≤ρ1‖Phgi(m-1)‖∞h2.
To estimate gi(m-1), we observe from (5.15) that
(5.23)|gi(m-1)|≤M¯*|γi(m-1)-θi(m-1)|⋅|w¯h(m-1)(xi)|.
Since both γi(m-1) and θi(m-1) are in [u̲h(m-1)(xi),u¯h(m-1)(xi)], the above estimate implies that
(5.24)|gi(m-1)|≤M¯*|u¯h(m-1)(xi)-u̲h(m-1)(xi)|⋅|w¯h(m-1)(xi)|.
Using this estimate in (5.22), we obtain
(5.25)‖w¯h(m)‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖w¯h(m-1)‖∞,
or
(5.26)‖u¯h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖u¯h(m-1)-uh*‖∞.
Similarly, we have
(5.27)‖u̲h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖u̲h(m-1)-uh*‖∞.
Addition of (5.26) and (5.27) gives
(5.28)‖u¯h(m)-uh*‖∞+‖u̲h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞(‖u¯h(m-1)-uh*‖∞+‖u̲h(m-1)-uh*‖∞).
Then the estimate (5.16) follows immediately.Theorem5.4 gives a quadratic convergence for the sum of the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} in the sense of (5.16). If fu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u¯h(m)(xi)}/{u̲h(m)(xi)} has the quadratic convergence. This result is stated as follows.Theorem 5.5.
Let the conditions in Theorem5.4 hold. Then there exists a constant ρ, independent of m, such that
(5.29)‖u¯h(m)-uh*‖∞≤ρ‖u¯h(m-1)-uh*‖∞2,m=1,2,…
if fu(xi,u) is monotone nonincreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L and
(5.30)‖u̲h(m)-uh*‖∞≤ρ‖u̲h(m-1)-uh*‖∞2,m=1,2,…
if fu(xi,u) is monotone nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L.Proof.
Consider the monotone nonincreasing case. In this case, the sequence{u¯h(m)(xi)} is given by (5.5) with u¯h(0)(xi)=ũh(xi). This implies that θi(m-1)=u¯h(m-1)(xi), where θi(m-1) is the intermediate value in (5.18). Since γi(m-1) in (5.19) is in [uh*(xi),u¯h(m-1)(xi)], we see that
(5.31)|γi(m-1)-θi(m-1)|≤|u¯h(m-1)(xi)-uh*(xi)|.
Thus, (5.24) is now reduced to
(5.32)|gi(m-1)|≤M¯*|w¯h(m-1)(xi)|2.
The argument in the proof of Theorem 5.4 shows that (5.29) holds with ρ=ρ1M¯*, where ρ1 is the constant in (5.22). The proof of (5.30) is similar.


## 5.1. Monotone Iterative Algorithm

Letũh(xi) and ûh(xi) be a pair of ordered upper and lower solutions of (2.5) and assume that f(·,u) is a C1-function of u∈〈ûh,ũh〉. It follows from Theorems 3.2–3.4 that (2.5) has a unique solution uh*(xi) in 〈ûh,ũh〉 under the conditions of the theorems. To compute this solution, we use the following iterative scheme: (5.1)-δh2uh(m)(xi)+Ph(Mi(m-1)uh(m)(xi))=Ph(Mi(m-1)uh(m-1)(xi)+f(xi,uh(m-1)(xi))),1≤i≤L-1,uh(m)(0)=Sα[uh(m)(ξ)],uh(m)(1)=Sβ[uh(m)(η)],
where uh(0)(xi) is either ũh(xi) or ûh(xi), and for each i, (5.2)Mi(m)=max⁡{-fu(xi,u);u∈[u̲h(m)(xi),u¯h(m)(xi)]}.
The functions u¯h(m)(xi) and u̲h(m)(xi) in the definition of Mi(m) are obtained from (5.1) with uh(0)(xi)=ũh(xi) and uh(0)(xi)=ûh(xi), respectively. It is clear from (5.2) that (5.3)f(xi,vi)-f(xi,vi′)≥-Mi(m)(vi-vi′),0≤i≤L,
whenever u̲h(m)(xi)≤vi′≤vi≤u¯h(m)(xi). Moreover, (5.4)Mi(m)={-fu(xi,u¯h(m)(xi)),iffu(xi,u)ismonotonenonincreasinginu∈[u̲h(m)(xi),u¯h(m)(xi)],-fu(xi,u̲h(m)(xi)),iffu(xi,u)ismonotonenondecreasinginu∈[u̲h(m)(xi),u¯h(m)(xi)].
Hence, if fu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, then the iterative scheme (5.1) for {u¯h(m)(xi)}/{u̲h(m)(xi)} is reduced to Newton's form: (5.5)-δh2uh(m)(xi)-Ph(fu(xi,uh(m-1)(xi))uh(m)(xi))=-Ph(fu(xi,uh(m-1)(xi))uh(m-1)(xi)-f(xi,uh(m-1)(xi))),1≤i≤L-1,uh(m)(0)=Sα[uh(m)(ξ)],uh(m)(1)=Sβ[uh(m)(η)].To show that the sequences given by (5.1) are well-defined and monotone for an arbitrary C1-function f(·,u), we let M̲u be given by (3.13) and let (5.6)M¯u=max⁡0≤i≤Lmax⁡{-fu(xi,u);u∈[ûh(xi),ũh(xi)]}.Lemma 5.1.
Let the condition (2.9) hold, and let ũh(xi) and ûh(xi) be a pair of ordered upper and lower solutions of (2.5). Assume that M̲u>-8(1-σ) and h<h(M̲u,M¯u). Then the sequences {u¯h(m)(xi)}, {u̲h(m)(xi)}, and {Mi(m)} given by (5.1) and (5.2) with u¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=ûh(xi) are all well defined and possess the monotone property
(5.7)u̲h(m)(xi)≤u̲h(m+1)(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L,m=0,1,….Proof.
SinceMi(0)=max{-fu(xi,u);ui∈[ûh(xi),ũh(xi)]}, -8(1-σ)<M̲u≤Mi(0)≤M¯u and h<h(M̲u,M¯u), we have from Lemma 2.2 that the first iterations u¯h(1)(xi) and u̲h(1)(xi) are well defined. Let w¯h(0)(xi)=u¯h(0)(xi)-u¯h(1)(xi). Then, by (3.1) and (5.1),
(5.8)-δh2w¯h(0)(xi)+Ph(Mi(0)w¯h(0)(xi))≥0,1≤i≤L-1,w¯h(0)(0)≥Sα[w¯h(0)(ξ)],w¯h(0)(1)≥Sβ[w¯h(0)(η)].
We have from Lemma 2.2 that w¯h(0)(xi)≥0, that is, u¯h(0)(xi)≥u¯h(1)(xi) for every 0≤i≤L. Similarly by the property of a lower solution, u̲h(1)(xi)≥u̲h(0)(xi) for every 0≤i≤L. Let wh(1)(xi)=u¯h(1)(xi)-u̲h(1)(xi). Then by (5.1) and (5.3),
(5.9)-δh2wh(1)(xi)+Ph(Mi(0)wh(1)(xi))≥0,1≤i≤L-1,wh(1)(0)=Sα[wh(1)(ξ)],wh(1)(1)=Sβ[wh(1)(η)].
It follows from Lemma 2.2 that wh(1)(xi)≥0, that is, u¯h(1)(xi)≥u̲h(1)(xi) for every 0≤i≤L. This proves the monotone property (5.7) for m=0.
Assume, by induction, that there exists some integerm0≥0 such that for all 0≤m≤m0, the iterations u¯h(m)(xi),  u¯h(m+1)(xi),  u̲h(m)(xi), and u̲h(m+1)(xi) are well-defined and satisfy (5.7). Then Mi(m0+1) is well defined and -8(1-σ)<M̲u≤Mi(m0+1)≤M¯u. Since h<h(M̲u,M¯u), we have from Lemma 2.2 that the iterations u¯h(m0+2)(xi) and u̲h(m0+2)(xi) exist uniquely. Let w¯h(m0+1)(xi)=u¯h(m0+1)(xi)-u¯h(m0+2)(xi). Since
(5.10)Mi(m0+1)w¯h(m0+1)(xi)=(Mi(m0+1)-Mi(m0))u¯h(m0+1)(xi)+Mi(m0)u¯h(m0+1)(xi)-Mi(m0+1)u¯h(m0+2)(xi),
the iterative scheme (5.1) implies that
(5.11)-δh2w¯h(m0+1)(xi)+Ph(Mi(m0+1)w¯h(m0+1)(xi))=Ph(Mi(m0)(u¯h(m0)(xi)-u¯h(m0+1)(xi))+f(xi,u¯h(m0)(xi))-f(xi,u¯h(m0+1)(xi))),1≤i≤L-1,w¯h(m0+1)(0)=Sα[w¯h(m0+1)(ξ)],w¯h(m0+1)(1)=Sβ[w¯h(m0+1)(η)].
Using the relation (5.3) yields
(5.12)-δh2w¯h(m0+1)(xi)+Ph(Mi(m0+1)w¯h(m0+1)(xi))≥0,1≤i≤L-1,w¯h(m0+1)(0)=Sα[w¯h(m0+1)(ξ)],w¯h(m0+1)(1)=Sβ[w¯h(m0+1)(η)].
By Lemma 2.2, w¯h(m0+1)(xi)≥0, that is, u¯h(m0+1)(xi)≥u¯h(m0+2)(xi) for every 0≤i≤L. Similarly we have u̲h(m0+2)(xi)≥u̲h(m0+1)(xi) for every 0≤i≤L. Let wh(m0+2)(xi)=u¯h(m0+2)(xi)-u̲h(m0+2)(xi). Then by (5.1) and (5.3), wh(m0+2)(xi) satisfies (5.12) with w¯h(m0+1)(xi) replaced by wh(m0+2)(xi). Therefore, by Lemma 2.2, wh(m0+2)(xi)≥0, that is, u¯h(m0+2)(xi)≥u̲h(m0+2)(xi) for every 0≤i≤L. This shows that the monotone property (5.7) is also true for m=m0+1. Finally, the conclusion of the lemma follows from the principle of induction.We next show monotone convergence of the sequences{u¯h(m)(xi)} and {u̲h(m)(xi)}.Theorem 5.2.
Let the hypothesis in Lemma5.1 hold. Then the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} given by (5.1) converge monotonically to the unique solution uh*(xi) of (2.5) in 〈ûh,ũh〉, respectively. Moreover,
(5.13)u̲h(m)(xi)≤u̲h(m+1)(xi)≤uh*(xi)≤u¯h(m+1)(xi)≤u¯h(m)(xi),0≤i≤L,m=0,1,….Proof.
It follows from the monotone property (5.7) that the limits
(5.14)lim⁡m→∞u¯h(m)(xi)=u¯h(xi),lim⁡m→∞u̲h(m)(xi)=u̲h(xi),0≤i≤L
exist and they satisfy (3.10). Since the sequence {Mi(m)} is monotone nonincreasing and is bounded from below by M̲u given in (3.13), it converges as m→∞. Letting m→∞ in (5.1) shows that both u¯h(xi) and u̲h(xi) are solutions of (2.5) in 〈ûh,ũh〉. Since M̲u>-8(1-σ) and h<h(M̲u,M¯u), the condition (3.14) of Theorem 3.3 is satisfied. Thus by Theorem 3.3, u¯h(xi)=u̲h(xi)(≡uh*(xi)) and uh*(xi) is the unique solution of (2.5) in 〈ûh,ũh〉. The monotone property (5.13) follows from (3.10).Whenfu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the iterative scheme (5.1) for {u¯h(m)(xi)}/{u̲h(m)(xi)} is reduced to Newton iteration (5.5). As a consequence of Theorem 5.2, we have the following conclusion.Corollary 5.3.
Let the hypothesis in Lemma5.1 be satisfied. If fu(xi,u) is monotone nonincreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u¯h(m)(xi)} given by (5.5) with u¯h(0)(xi)=ũh(xi) converges monotonically from above to the unique solution uh*(xi) of (2.5) in 〈ûh,ũh〉. Otherwise, if fu(xi,u) is monotone nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u̲h(m)(xi)} given by (5.5) with u̲h(0)(xi)=ûh(xi) converges monotonically from below to uh*(xi).


## 5.2. Rate of Convergence

We now show the quadratic rate of convergence of the sequences given by (5.1). Assume that there exists a nonnegative constant M¯* such that (5.15)|fu(xi,vi)-fu(xi,vi′)|≤M¯*|vi-vi′|∀vi,vi′∈[ûh(xi),ũh(xi)],0≤i≤L.
Clearly, this assumption is satisfied if f(·,u) is a C2-function of u.Theorem 5.4.
Let the hypotheses in Lemma5.1 and (5.15) hold. Also let {u¯h(m)(xi)} and {u̲h(m)(xi)} be the sequences given by (5.1) and let uh*(xi) be the unique solution of (2.5) in 〈ûh,ũh〉. Then there exists a constant ρ, independent of m, such that
(5.16)‖u¯h(m)-uh*‖∞+‖u̲h(m)-uh*‖∞≤ρ(‖u¯h(m-1)-uh*‖∞+‖u̲h(m-1)-uh*‖∞)2,m=1,2,….Proof.
Letw¯h(m)(xi)=u¯h(m)(xi)-uh*(xi). Subtracting (2.5) from (5.1) gives
(5.17)-δh2w¯h(m)(xi)+Ph(Mi(m-1)w¯h(m)(xi))=Ph(Mi(m-1)w¯h(m-1)(xi)+f(xi,u¯h(m-1)(xi))-f(xi,uh*(xi))),1≤i≤L-1,w¯h(m)(0)=Sα[w¯h(m)(ξ)],w¯h(m)(1)=Sβ[w¯h(m)(η)].
By the intermediate value theorem,
(5.18)Mi(m-1)=-fu(xi,θi(m-1)),
where θi(m-1)∈[u̲h(m-1)(xi),u¯h(m-1)(xi)], and by the mean value theorem,
(5.19)f(xi,u¯h(m-1)(xi))-f(xi,uh*(xi))=fu(xi,γi(m-1))w¯h(m-1)(xi),
where γi(m-1)∈[uh*(xi),u¯h(m-1)(xi)]. Let
(5.20)gi(m-1)=(fu(xi,γi(m-1))-fu(xi,θi(m-1)))w¯h(m-1)(xi).
Then we have from (5.17) that
(5.21)-δh2w¯h(m)(xi)+Ph(Mi(m-1)w¯h(m)(xi))=Phgi(m-1),1≤i≤L-1,w¯h(m)(0)=Sα[w¯h(m)(ξ)],w¯h(m)(1)=Sβ[w¯h(m)(η)].
Since -8(1-σ)<M̲u≤Mi(m-1)≤M¯u and h<h(M̲u,M¯u), it follows from Lemma 2.3 that there exists a constant ρ1, independent of m, such that
(5.22)‖w¯h(m)‖∞≤ρ1‖Phgi(m-1)‖∞h2.
To estimate gi(m-1), we observe from (5.15) that
(5.23)|gi(m-1)|≤M¯*|γi(m-1)-θi(m-1)|⋅|w¯h(m-1)(xi)|.
Since both γi(m-1) and θi(m-1) are in [u̲h(m-1)(xi),u¯h(m-1)(xi)], the above estimate implies that
(5.24)|gi(m-1)|≤M¯*|u¯h(m-1)(xi)-u̲h(m-1)(xi)|⋅|w¯h(m-1)(xi)|.
Using this estimate in (5.22), we obtain
(5.25)‖w¯h(m)‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖w¯h(m-1)‖∞,
or
(5.26)‖u¯h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖u¯h(m-1)-uh*‖∞.
Similarly, we have
(5.27)‖u̲h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞‖u̲h(m-1)-uh*‖∞.
Addition of (5.26) and (5.27) gives
(5.28)‖u¯h(m)-uh*‖∞+‖u̲h(m)-uh*‖∞≤ρ1M¯*‖u¯h(m-1)-u̲h(m-1)‖∞(‖u¯h(m-1)-uh*‖∞+‖u̲h(m-1)-uh*‖∞).
Then the estimate (5.16) follows immediately.Theorem5.4 gives a quadratic convergence for the sum of the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} in the sense of (5.16). If fu(xi,u) is monotone nonincreasing/nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L, the sequence {u¯h(m)(xi)}/{u̲h(m)(xi)} has the quadratic convergence. This result is stated as follows.Theorem 5.5.
Let the conditions in Theorem5.4 hold. Then there exists a constant ρ, independent of m, such that
(5.29)‖u¯h(m)-uh*‖∞≤ρ‖u¯h(m-1)-uh*‖∞2,m=1,2,…
if fu(xi,u) is monotone nonincreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L and
(5.30)‖u̲h(m)-uh*‖∞≤ρ‖u̲h(m-1)-uh*‖∞2,m=1,2,…
if fu(xi,u) is monotone nondecreasing in u∈[u̲h(m)(xi),u¯h(m)(xi)] for all 0≤i≤L.Proof.
Consider the monotone nonincreasing case. In this case, the sequence{u¯h(m)(xi)} is given by (5.5) with u¯h(0)(xi)=ũh(xi). This implies that θi(m-1)=u¯h(m-1)(xi), where θi(m-1) is the intermediate value in (5.18). Since γi(m-1) in (5.19) is in [uh*(xi),u¯h(m-1)(xi)], we see that
(5.31)|γi(m-1)-θi(m-1)|≤|u¯h(m-1)(xi)-uh*(xi)|.
Thus, (5.24) is now reduced to
(5.32)|gi(m-1)|≤M¯*|w¯h(m-1)(xi)|2.
The argument in the proof of Theorem 5.4 shows that (5.29) holds with ρ=ρ1M¯*, where ρ1 is the constant in (5.22). The proof of (5.30) is similar.


## 6. Applications and Numerical Results

In this section, we give some applications of the results in the previous sections to three model problems. We present some numerical results to demonstrate the monotone and rapid convergence of the sequence from (5.1) and to show the fourth-order accuracy of Numerov's scheme (2.5), as predicted in the analysis.In order to implement the monotone iterative algorithm (5.1), it is necessary to find a pair of ordered upper and lower solutions of (2.5). The construction of this pair depends mainly on the function f(·,u), and much discussion on the subject can be found in [54] for continuous problems. To demonstrate some techniques for the construction of ordered upper and lower solutions of (2.5), we assume that f(x,0)≥0 for all x∈[0,1] and there exists a nonnegative constant C such that (6.1)f(x,C)≤0,x∈[0,1].
Then -δh2C=0≥𝒫hf(xi,C) for all 1≤i≤L-1. This implies that ũh(xi)≡C and ûh(xi)≡0 are a pair of ordered upper and lower solutions of (2.5) if, in addition, the condition (2.9) holds. On the other hand, assume that there exist nonnegative constants a, b with a<8(1-σ) such that (6.2)f(x,u)≤au+bforx∈[0,1],u≥0,
where σ<1 is defined by (2.9). We have from Lemma 2.2 that the solution ũh(xi) of the linear problem (6.3)-δh2ũh(xi)-aPhũh(xi)=h2b,1≤i≤L-1,ũh(0)=Sα[ũh(ξ)],ũh(1)=Sβ[ũh(η)]
exists uniquely and is nonnegative. Clearly by (6.2), this solution is a nonnegative upper solution of (2.5).As applications of the above construction of upper and lower solutions, we next consider three specific examples. In each of these examples, the analytic solutionu(x) of (1.1) is explicitly known, against which we can compare the numerical solution uh*(xi) of the scheme (2.5) to demonstrate the fourth-order accuracy of the scheme. The order of accuracy is calculated by (6.4)error∞(h)=∥u-uh*∥∞,order∞(h)=log⁡2(error∞(h)error∞(h/2)).
All computations are carried out by using a MATLAB subroutine on a Pentium 4 computer with 2G memory, and the termination criterion of iterations for (5.1) is given by (6.5)‖u¯h(m)-u̲h(m)‖∞<10-14.Example 6.1.
Consider the four-point boundary value problem:(6.6)-u′′(x)=θu(x)(1-u(x))+q(x),0<x<1,u(0)=19u(12),u(1)=18u(14),
where θ is a positive constant and q(x) is a nonnegative continuous function. Clearly, problem (6.6) is a special case of (1.1) with
(6.7)f(x,u)=θu(1-u)+q(x).
To obtain an explicit analytic solution of (6.6), we choose
(6.8)q(x)=8+(π22)sin(2πx)-θz(x)(1-z(x)),z(x)=4x(1-x)+1+sin(2πx)8.
Then the function u(x)=z(x) is a solution of (6.6). Moreover, q(x)≥0 in [0,1] if θ≤32-2π2.For problem (6.6), the corresponding Numerov scheme (2.5) is now reduced to (6.9)-δh2uh(xi)=Phf(xi,uh(xi)),1≤i≤L-1,uh(0)=19uh(12),uh(1)=18uh(14).
To find a pair of ordered upper and lower solutions of (6.9), we observe from (6.7) that f(x,0)=q(x)≥0 for all x∈[0,1], and, therefore, ûh(xi)≡0 is a lower solution. Since q(x)≤14, we have from (6.7) that the condition (6.2) is satisfied for the present function f with a=θ and b=14. Therefore, the solution ũh(xi) of (6.3) (corresponding to (6.9)) with a=θ and b=14 is a nonnegative upper solution if θ<7. This implies that ũh(xi) and ûh(xi)≡0 are a pair of ordered upper and lower solutions of (6.9).Letθ=π/2. Using u¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=0, we compute the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} from the iterative scheme (5.1) for (6.9) and various values of h. In all the numerical computations, the basic feature of monotone convergence of the sequences was observed. Let h=1/32. In Figure 1, we present some numerical results of these sequences at xi=0.5, where the solid line denotes the sequence {u¯h(m)(xi)} and the dashed-dotted line stands for the sequence {u̲h(m)(xi)}. As described in Theorem 5.2, the sequences converge to the same limit as m→∞, and their common limit uh*(xi) is the unique solution of (6.9) in 〈0,ũh〉. Besides, these sequences converge rapidly (in five iterations). More numerical results of uh*(xi) at various xi are explicitly given in Table 1. We also list the values of the analytic solution u(xi). Clearly, the numerical solution uh*(xi) meets the analytic solution u(xi) closely.Table 1
Solutionsuh*(xi) and u(xi) of Example 6.1.
xiuh*(xi)u(xi)1/160.407210723513250.407210429045641/80.650888888302900.650888347648323/160.849860646000070.849859941563911/41.0000007621589215/161.099860647989911.099859941563913/81.150888894375341.150888347648327/161.157210736887651.157210429045641/21.125000026236031.12500000000000Figure 1
The monotone convergence of({u¯h(m)(xi)},{u̲h(m)(xi)}) at xi=0.5 for Example 6.1.To further demonstrate the accuracy of the numerical solutionuh*(xi), we list the maximum error error∞(h) and the order order∞(h) in the first three columns of Table 2 for various values of h. The data demonstrate that the numerical solution uh*(xi) has the fourth-order accuracy. This coincides with the analysis very well.Table 2
The accuracy of the numerical solutionuh*(xi) of Example 6.1.
Scheme (6.9)SFD schemeherror∞(h)order∞(h)error∞(h)order∞(h)1/43.43422969288754e-034.104510401942242.82813777940529e-022.121792306231501/81.99640473104834e-044.026474937587266.49796599949681e-032.030667914584311/161.22506422453039e-054.006621718132061.59032351665434e-032.007659855976181/327.62158923750533e-074.001655592580193.95475554192615e-042.001914274721461/644.75802997002006e-084.000409162223329.87377889736241e-052.000478519459601/1282.97292546136418e-094.000245051907672.46762611546547e-052.000119610947921/2561.85776283245787e-104.033193250685316.16855384505399e-062.000029779424241/5121.13469234008790e-111.54210662950405e-06For comparison, we also solve (6.6) by the standard finite difference (SFD) method. This method leads to a difference scheme in the form (6.9) with 𝒫h=ℐ (an identical operator). Thus, a similar iterative scheme as (5.1) can be used in actual computations. The corresponding maximum error error∞(h) and the order order∞(h) are listed in the last two columns of Table 2. We see that the standard finite difference method possesses only the second-order accuracy.Example 6.2.
Our second example is for the following five-point boundary value problem:(6.10)-u′′(x)=θu(x)1+u(x)+q(x),0<x<1,u(0)=36u(14)+14u(12),u(1)=14u(12)+36u(34),
where θ is a positive constant and q(x) is a nonnegative continuous function. The corresponding Numerov's scheme (2.5) for this example is given by
(6.11)-δh2uh(xi)=Phf(xi,uh(xi)),1≤i≤L-1,uh(0)=36uh(14)+14uh(12),uh(1)=14uh(12)+36uh(34),
where
(6.12)f(xi,uh(xi))=θuh(xi)1+uh(xi)+q(xi),0≤i≤L.
Let
(6.13)q(x)=(κ2-θ1+sin(κx+π/6))sin(κx+π6),κ=2π3.
Then q(x)≥0 in [0,1] if θ≤3κ2/2, and u(x)=sin(κx+π/6) is a solution of (6.10). Clearly, ûh(xi)≡0 is a lower solution of (6.11). On the other hand, the condition (6.2) is satisfied for the present problem with a=θ and b=κ2. Therefore, the solution ũh(xi) of (6.3) (corresponding to (6.11)) with a=θ and b=κ2 is a nonnegative upper solution of (6.11) if θ<2(9-23)/3.
Letθ=κ. Using u¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=0, we compute the sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} from the iterative scheme (5.1) for (6.11). Let h=1/32. Some numerical results of these sequences at xi=0.5 are plotted in Figure 2, where the solid line denotes the sequence {u¯h(m)(xi)} and the dashed-dotted line stands for the sequence {u̲h(m)(xi)}. We see that the sequences possess the monotone convergence given in Theorem 5.2 and converge rapidly (in five iterations) to the unique solution uh*(xi) of (6.11) in 〈0,ũh〉. The maximum error error∞(h) and the order order∞(h) of the numerical solution uh*(xi) by the scheme (6.11) and the SFD scheme are presented in Table 3. The numerical results clearly indicate that the proposed scheme (6.11) is more efficient than the SFD scheme.Table 3
The accuracy of the numerical solutionuh*(xi) of Example 6.2.
Scheme (6.11)SFD schemeherror∞(h)order∞(h)error∞(h)order∞(h)1/43.64212838788403e-044.011564004866062.66151346177448e-022.013410455030111/82.25815711794031e-054.002928327870376.59222051766362e-032.003388857769901/161.40848640284297e-064.000734779104841.64418842865244e-032.000849881432791/328.79855768243232e-084.000189464284744.10805033531858e-042.000212644569911/645.49837642083162e-093.999482149845241.02686121950857e-042.000053172936301/1283.43771899835588e-103.996845423516722.56705843380001e-052.000013268346571/2562.15327755626049e-114.037359609172186.41758706221296e-062.000003336713531/5121.31139543668724e-121.60439305485482e-06Figure 2
The monotone convergence of({u¯h(m)(xi)},{u̲h(m)(xi)}) at xi=0.5 for Example 6.2.Example 6.3.
Our last example is given by(6.14)-u′′(x)=θ(q4(x)-u4(x)),0<x<1,u(0)=28u(18)+312u(14)+14u(12),u(1)=312u(14)+14u(12)+312u(34),
where θ is a positive constant and q(x) is a continuous function. For this example, the corresponding Numerov scheme (2.5) is reduced to
(6.15)-δh2uh(xi)=Phf(xi,uh(xi)),1≤i≤L-1,uh(0)=28uh(18)+312uh(14)+14uh(12),uh(1)=312uh(14)+14uh(12)+312uh(34),
where
(6.16)f(xi,uh(xi))=θ(q4(xi)-uh4(xi)),0≤i≤L.
To accommodate the analytical solution of u(x)=sin(κx+π/6) where κ=2π/3, we let
(6.17)q(x)=(κ2θsin(κx+π6)+sin4(κx+π6))1/4.
As in the previous examples, ûh(xi)≡0 is a lower solution of (6.15) and the solution ũh(xi) of (6.3) (corresponding to (6.15)) with a=0 and b=κ2+θ is a nonnegative upper solution.
Letθ=π2/2. We compute the corresponding sequences {u¯h(m)(xi)} and {u̲h(m)(xi)} from the iterative scheme (5.1) with the initial iterations u¯h(0)(xi)=ũh(xi) and u̲h(0)(xi)=0. Let h=1/32. Figure 3 shows the monotone and rapid convergence of these sequences at xi=0.5, where the solid line denotes the sequence {u¯h(m)(xi)} and the dashed-dotted line stands for the sequence {u̲h(m)(xi)} as before. The data in Table 4 show the maximum error error∞(h) and the order order∞(h) of the numerical solution uh*(xi) by the scheme (6.15) and the SFD scheme for various values of h. The fourth-order accuracy of the numerical solution uh*(xi) by the present Numerov scheme is demonstrated in this table.Table 4
Table1 The accuracy of the numerical solution uh*(xi) of Example 6.3.
Scheme (6.15)SFD schemeherror∞(h)order∞(h)error∞(h)order∞(h)1/84.16044850615194e-064.002629451519171.20005715017424e-031.990119559075841/162.59554536974349e-074.000716513501123.02076017246300e-041.997565963637401/321.62141038373420e-084.000192999646887.56465233968662e-051.999393725591531/641.01324593160257e-094.000178954763071.89195798938613e-051.999845712260861/1286.33200158972613e-114.005921169753554.73040083492915e-061.999953141074271/2563.94129173741931e-124.069771742784091.18263862036727e-061.999990438281071/5122.34701147405758e-132.95661614635456e-07Figure 3
The monotone convergence of({u¯h(m)(xi)},{u̲h(m)(xi)}) at xi=0.5 for Example 6.3


## 7. Conclusions

In this paper, we have given a numerical treatment for a class of nonlinear multipoint boundary value problems by the fourth-order Numerov method. The existence and uniqueness of the numerical solution and the convergence of the method have been discussed. An accelerated monotone iterative algorithm with the quadratic rate of convergence has been developed for solving the resulting nonlinear discrete problem. The proposed Numerov method is more attractive due to its fourth-order accuracy, compared to the standard finite difference method.In this work, we have generalized the method of upper and lower solutions to nonlinear multipoint boundary value problems. We have also developed a technique for designing and analyzing compact and monotone finite difference schemes with high accuracy. They are very useful for accurate numerical simulations of many other nonlinear problems, such as those related to integrodifferential equations (e.g., [55, 56]) and those in information modeling (e.g., [57–60]).



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