Mathematical Problems in EngineeringVolume 2012 (2012), Article ID 316852, 29 pagesdoi:10.1155/2012/316852

Research Article

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

Yuan-Ming Wang,1,2 Wen-Jia Wu,¹ and Massimo Scalia³

¹Department of Mathematics, East China Normal University, Shanghai 200241, China²Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China³Department of Mathematics, University of Rome “La Sapienza”, Piazzale le Aldo Moro 2, 00185 Rome, Italy

Received 28 March 2011; Revised 16 May 2011; Accepted 13 June 2011

Academic Editor: Shengyong Chen

Copyright © 2012 Yuan-Ming Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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: where is a continuous function of its arguments and for each , , and , . An application of this problem appears in the design of a large-size bridge with multipoint supports, where 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 or for some or all . 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 if for all (see [5–14]), to the form if for all (see [15]), to the four-point boundary condition if and , (see [11, 15–17]), and to the two-point boundary condition if and for all . Condition includes the three-point boundary condition when (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)-, while the works in [15–17] were devoted to the existence of solutions for the three- or four-point boundary value problem (1.1)–. 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 . 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 (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 , , and . 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: where and are two prescribed constants.

The outline of the paper is as follows. In Section 2, 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 th-iteration of the upper solution and the true solution with the infinity norm of the error between the th-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

Let be the mesh size, and let be the mesh points in . Assume that for all , the points and in the boundary condition of (1.1) serve as mesh points. This assumption is always satisfied by a proper choice of mesh size . For convenience, we use the following notations: and introduce the finite difference operators and as follows: Using the following Numerov's formula (cf. [52, 53]): we have from (1.1) and (2.1) that After dropping the term, we derive a Numerov's difference approximation to (1.1) as follows: where represents the approximation of .

For two constants and satisfying , we define A fundamental and useful property of the operators and is stated below.

Lemma 2.1 (See Lemma 3.1 of [50]). Let , , and be some constants satisfying If and , then for all .

The following results are also useful for our forthcoming discussions. Their proofs will be given in the appendix.

Lemma 2.2. Assume Let , , and be the given constants such that If and , then for all .

Lemma 2.3. Let the condition (2.9) be satisfied, and let , , and be the given constants satisfying (2.10). Assume that the functions and satisfy Then when , where denotes discrete infinity norm for any mesh function .

Remark 2.4. It is clear from Lemma 2.1 that if 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 function is called an upper solution of (2.5) if Similarly, a function 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 and are said to be ordered if for all .

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 and , we set and make the following basic hypotheses: (H₁)For each , there exists a constant such that and whenever ; (H₂), where and .

The existence of the constant in is trivial if is a -function of . In fact, may be taken as any nonnegative constant satisfying

Theorem 3.2. Let and be a pair of ordered upper and lower solutions of (2.5), and let hypotheses and be satisfied. Then system (2.5) has a maximal solution and a minimal solution in . Here, the maximal property of means that for any solution of (2.5) in , one hase for all . The minimal property of is similarly understood.

Proof. The proof is constructive. Using the initial iterations and we construct two sequences and , respectively, from the following iterative scheme: where is the constant in . By Lemma 2.1, these two sequences are well defined. We shall first prove that for all , Let . Then by (3.1) and (3.5), It follows from Lemma 2.1 that , that is, for all . A similar argument using the property of a lower solution gives for all . Let . We have from (3.3) and (3.5) that Again by Lemma 2.1, , that is, for all . This proves (3.6) for . Finally, an induction argument leads to the desired result (3.6) for all .In view of (3.6), the limits exist and satisfy Letting in (3.5) shows that both and are solutions of (2.5). Now, if is a solution of (2.5) in , then the pair and are also a pair of ordered upper and lower solutions of (2.5). The above arguments imply that for all . Similarly, we have for all . This shows that and are the maximal and the minimal solutions of (2.5) in , respectively. The proof is completed.

Theorem 3.2 shows that the system (2.5) has a maximal solution and a minimal solution in . If for all , then or is a unique solution of (2.5) in . In general, these two solutions do not coincide. Consider, for example, the case If there exist two different constants and such that for all then both and are solutions of (2.5). Hence to show the uniqueness of a solution it is necessary to impose some additional conditions on , and . Assume that there exists a constant such that whenever . This condition is trivially satisfied if is a -function of for all . In fact, may be taken as The following theorem gives a sufficient condition for the uniqueness of a solution.

Theorem 3.3. Let the conditions in Theorem 3.2 hold. If, in addition, the conditions (2.9) and (3.12) hold and either then the system (2.5) has a unique solution in . Moreover, the relation (3.10) holds with for all .

Proof. It suffices to show for all , where and are the limits in (3.9). Let . Then , and by (2.5), Therefore, we have from (3.12) that By Lemma 2.2, for all . This proves for all . To give another sufficient condition, we assume that there exists a nonnegative constant such that whenever . If is a -function of for all , the above condition is clearly satisfied by

Theorem 3.4. Let the conditions in Theorem 3.2 hold. If, in addition, the conditions (2.9) and (3.17) hold and then the conclusions of Theorem 3.3 are also valid.

Proof. Applying Lemma 2.3 with to (3.15) leads to where . By (3.17), we obtain . Consequently, This together with (3.19) implies , that is, for all .

It is seen from the proofs of Theorems 3.2–3.4 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 and the solution of (1.1) are sufficiently smooth.

Let be the value of the solution of (1.1) at the mesh point , and let be the solution of (2.5). We consider the error . In fact, we have from (2.4) and (2.5) that

Theorem 4.1. Let the condition (2.9) hold, and let be an interval in such that . Assume that Then for sufficiently small , where is a positive constant independent of .

Proof. Applying the mean value theorem to the first equality of (4.1), we have where and . Let and . Then by (4.2), . We, therefore, obtain from Lemma 2.3 that when , where is a positive constant independent of . Finally, the error estimate (4.3) follows from for some positive constant independent of .

Theorem 4.1 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 . If the function possesses a monotone property in , this algorithm is reduced to Newton's method, and one of the two produced sequences converges quadratically.

5.1. Monotone Iterative Algorithm

Let and be a pair of ordered upper and lower solutions of (2.5) and assume that is a -function of . It follows from Theorems 3.2–3.4 that (2.5) has a unique solution in under the conditions of the theorems. To compute this solution, we use the following iterative scheme: where is either or , and for each , The functions and in the definition of are obtained from (5.1) with and , respectively. It is clear from (5.2) that whenever . Moreover, Hence, if is monotone nonincreasing/nondecreasing in for all , then the iterative scheme (5.1) for / is reduced to Newton's form:

To show that the sequences given by (5.1) are well-defined and monotone for an arbitrary -function , we let be given by (3.13) and let

Lemma 5.1. Let the condition (2.9) hold, and let and be a pair of ordered upper and lower solutions of (2.5). Assume that and . Then the sequences , , and given by (5.1) and (5.2) with and are all well defined and possess the monotone property

Proof. Since , and , we have from Lemma 2.2 that the first iterations and are well defined. Let . Then, by (3.1) and (5.1), We have from Lemma 2.2 that , that is, for every . Similarly by the property of a lower solution, for every . Let . Then by (5.1) and (5.3), It follows from Lemma 2.2 that , that is, for every . This proves the monotone property (5.7) for . Assume, by induction, that there exists some integer such that for all , the iterations , , , and are well-defined and satisfy (5.7). Then is well defined and . Since , we have from Lemma 2.2 that the iterations and exist uniquely. Let . Since the iterative scheme (5.1) implies that Using the relation (5.3) yields By Lemma 2.2, , that is, for every . Similarly we have for every . Let . Then by (5.1) and (5.3), satisfies (5.12) with replaced by . Therefore, by Lemma 2.2, , that is, for every . This shows that the monotone property (5.7) is also true for . Finally, the conclusion of the lemma follows from the principle of induction.

We next show monotone convergence of the sequences and .

Theorem 5.2. Let the hypothesis in Lemma 5.1 hold. Then the sequences and given by (5.1) converge monotonically to the unique solution of (2.5) in , respectively. Moreover,

Proof. It follows from the monotone property (5.7) that the limits exist and they satisfy (3.10). Since the sequence is monotone nonincreasing and is bounded from below by given in (3.13), it converges as . Letting in (5.1) shows that both and are solutions of (2.5) in . Since and , the condition (3.14) of Theorem 3.3 is satisfied. Thus by Theorem 3.3, and is the unique solution of (2.5) in . The monotone property (5.13) follows from (3.10).

When is monotone nonincreasing/nondecreasing in for all , the iterative scheme (5.1) for 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 Lemma 5.1 be satisfied. If is monotone nonincreasing in for all , the sequence given by (5.5) with converges monotonically from above to the unique solution of (2.5) in . Otherwise, if is monotone nondecreasing in for all , the sequence given by (5.5) with converges monotonically from below to .

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 such that Clearly, this assumption is satisfied if is a -function of .

Theorem 5.4. Let the hypotheses in Lemma 5.1 and (5.15) hold. Also let and be the sequences given by (5.1) and let be the unique solution of (2.5) in . Then there exists a constant , independent of , such that

Proof. Let . Subtracting (2.5) from (5.1) gives By the intermediate value theorem, where , and by the mean value theorem, where . Let Then we have from (5.17) that Since and , it follows from Lemma 2.3 that there exists a constant , independent of , such that To estimate , we observe from (5.15) that Since both and are in , the above estimate implies that Using this estimate in (5.22), we obtain or Similarly, we have Addition of (5.26) and (5.27) gives Then the estimate (5.16) follows immediately.

Theorem 5.4 gives a quadratic convergence for the sum of the sequences and in the sense of (5.16). If is monotone nonincreasing/nondecreasing in for all , the sequence has the quadratic convergence. This result is stated as follows.

Theorem 5.5. Let the conditions in Theorem 5.4 hold. Then there exists a constant , independent of , such that if is monotone nonincreasing in for all and if is monotone nondecreasing in for all .

Proof. Consider the monotone nonincreasing case. In this case, the sequence is given by (5.5) with . This implies that , where is the intermediate value in (5.18). Since in (5.19) is in , we see that Thus, (5.24) is now reduced to The argument in the proof of Theorem 5.4 shows that (5.29) holds with , where 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 , 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 for all and there exists a nonnegative constant such that Then for all . This implies that and 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 , with such that where is defined by (2.9). We have from Lemma 2.2 that the solution of the linear problem 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 solution of (1.1) is explicitly known, against which we can compare the numerical solution of the scheme (2.5) to demonstrate the fourth-order accuracy of the scheme. The order of accuracy is calculated by 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

Example 6.1. Consider the four-point boundary value problem: where is a positive constant and is a nonnegative continuous function. Clearly, problem (6.6) is a special case of (1.1) with To obtain an explicit analytic solution of (6.6), we choose Then the function is a solution of (6.6). Moreover, in if .

For problem (6.6), the corresponding Numerov scheme (2.5) is now reduced to To find a pair of ordered upper and lower solutions of (6.9), we observe from (6.7) that for all , and, therefore, is a lower solution. Since , we have from (6.7) that the condition (6.2) is satisfied for the present function with and . Therefore, the solution of (6.3) (corresponding to (6.9)) with and is a nonnegative upper solution if . This implies that and are a pair of ordered upper and lower solutions of (6.9).

Let . Using and , we compute the sequences and from the iterative scheme (5.1) for (6.9) and various values of . In all the numerical computations, the basic feature of monotone convergence of the sequences was observed. Let . In Figure 1, we present some numerical results of these sequences at , where the solid line denotes the sequence and the dashed-dotted line stands for the sequence . As described in Theorem 5.2, the sequences converge to the same limit as , and their common limit is the unique solution of (6.9) in . Besides, these sequences converge rapidly (in five iterations). More numerical results of at various are explicitly given in Table 1. We also list the values of the analytic solution . Clearly, the numerical solution meets the analytic solution closely.

Table 1: Solutions and of Example 6.1.

0.40721072351325
0.40721042904564

0.65088888830290
0.65088834764832

0.84986064600007
0.84985994156391

1.00000076215892
1

1.09986064798991
1.09985994156391

1.15088889437534
1.15088834764832

1.15721073688765
1.15721042904564

1.12500002623603
1.12500000000000

Figure 1: The monotone convergence of at for Example 6.1.

To further demonstrate the accuracy of the numerical solution , we list the maximum error and the order in the first three columns of Table 2 for various values of . The data demonstrate that the numerical solution has the fourth-order accuracy. This coincides with the analysis very well.

Table 2: The accuracy of the numerical solution of Example 6.1.

Scheme (6.9)
SFD scheme

4.10451040194224

2.12179230623150

4.02647493758726

2.03066791458431

4.00662171813206

2.00765985597618

4.00165559258019

2.00191427472146

4.00040916222332

2.00047851945960

4.00024505190767

2.00011961094792

4.03319325068531

2.00002977942424

For 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 (an identical operator). Thus, a similar iterative scheme as (5.1) can be used in actual computations. The corresponding maximum error and the order 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: where is a positive constant and is a nonnegative continuous function. The corresponding Numerov's scheme (2.5) for this example is given by where Let Then in if , and is a solution of (6.10). Clearly, is a lower solution of (6.11). On the other hand, the condition (6.2) is satisfied for the present problem with and . Therefore, the solution of (6.3) (corresponding to (6.11)) with and is a nonnegative upper solution of (6.11) if . Let . Using and , we compute the sequences and from the iterative scheme (5.1) for (6.11). Let . Some numerical results of these sequences at are plotted in Figure 2, where the solid line denotes the sequence and the dashed-dotted line stands for the sequence . We see that the sequences possess the monotone convergence given in Theorem 5.2 and converge rapidly (in five iterations) to the unique solution of (6.11) in . The maximum error and the order of the numerical solution 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 solution of Example 6.2.

Scheme (6.11)
SFD scheme

4.01156400486606

2.01341045503011

4.00292832787037

2.00338885776990

4.00073477910484

2.00084988143279

4.00018946428474

2.00021264456991

3.99948214984524

2.00005317293630

3.99684542351672

2.00001326834657

4.03735960917218

2.00000333671353

Figure 2: The monotone convergence of at for Example 6.2.

Example 6.3. Our last example is given by where is a positive constant and is a continuous function. For this example, the corresponding Numerov scheme (2.5) is reduced to where To accommodate the analytical solution of where , we let As in the previous examples, is a lower solution of (6.15) and the solution of (6.3) (corresponding to (6.15)) with and is a nonnegative upper solution. Let . We compute the corresponding sequences and from the iterative scheme (5.1) with the initial iterations and . Let . Figure 3 shows the monotone and rapid convergence of these sequences at , where the solid line denotes the sequence and the dashed-dotted line stands for the sequence as before. The data in Table 4 show the maximum error and the order of the numerical solution by the scheme (6.15) and the SFD scheme for various values of . The fourth-order accuracy of the numerical solution by the present Numerov scheme is demonstrated in this table.

Table 4: Table 1 The accuracy of the numerical solution of Example 6.3.

Scheme (6.15)
SFD scheme

4.00262945151917

1.99011955907584

4.00071651350112

1.99756596363740

4.00019299964688

1.99939372559153

4.00017895476307

1.99984571226086

4.00592116975355

1.99995314107427

4.06977174278409

1.99999043828107

Figure 3: The monotone convergence of at 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]).

Appendix

A. Proofs of Lemmas 2.2 and 2.3

Lemmas 2.2 and 2.3 are the special cases of Lemmas 2.2 and 2.3 in [61]. We include their proofs here in order to make the paper self-contained. Define Let , , and be the th-order matrices with where if and if .

Lemma A.1. Let the condition (2.9) be satisfied. Then the inverse , and

Proof. It can be checked by Corollary 3.20 on Page 91 of [62] that the inverse . Let and let . Then . It is known that the inverse exists, and its elements are given by A simple calculation shows that and for each . This implies Thus the estimate (A.3) follows immediately.

Proof of Lemma 2.2. Define the following th-order matrices or vectors: Using the matrices and defined by (A.2), we have from (2.11) that Since and , it is easy to check that . Thus by the boundary condition in (2.11), where is the th-order matrix defined by (A.2). This leads to Let . To prove for all , it suffices to show that the inverse of exists and is nonnegative. Case 1 (). In this case, the matrix satisfies the condition of Corollary 3.20 on Page 91 of [62], and, therefore, its inverse exists and is positive. Case 2 (). For this case, we define The matrices and can be similarly defined. Let . We know from Case 1 that exists and is positive. Since we need only to prove that the inverse exists and is nonnegative. By Theorem 3 on Page 298 of [63], this is true if Since which implies , we have from Lemma A.1 that It is clear that , and . Thus, we have The estimate (A.12) follows from .

Proof of Lemma 2.3. Using the same notation as before, the system (2.12) can be written as where . Case 1 (). Since the inverse exists and is positive, we have . This shows Thus, by (A.15), which implies the desired estimate (2.13). Case 2 (). It follows from (A.11) that By (A.13) and (A.14), This together with (A.15) leads to the estimate (2.13).

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and suggestions which improved the presentation of the paper. This work was supported in part by the National Natural Science Foundation of China No. 10571059, E-Institutes of Shanghai Municipal Education Commission No. E03004, the Natural Science Foundation of Shanghai No. 10ZR1409300, and Shanghai Leading Academic Discipline Project No. B407.

References

Y. Zou, Q. Hu, and R. Zhang, “On numerical studies of multi-point boundary value problem and its fold bifurcation,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 527–537, 2007.

J. Adem and M. Moshinsky, “Self-adjointness of a certain type of vectorial boundary value problems,” Boletín de la Sociedad Matemática Mexicana, vol. 7, pp. 1–17, 1950.

M. S. Berger and L. E. Fraenkel, “Nonlinear desingularization in certain free-boundary problems,” Communications in Mathematical Physics, vol. 77, no. 2, pp. 149–172, 1980.

M. Greguš, F. Neuman, and F. M. Arscott, “Three-point boundary value problems in differential equations,” Journal of the London Mathematical Society, vol. 2, pp. 429–436, 1971.

C.-Z. Bai and J.-X. Fang, “Existence of multiple positive solutions for nonlinear m-point boundary-value problems,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 297–305, 2003.

P. W. Eloe and Y. Gao, “The method of quasilinearization and a three-point boundary value problem,” Journal of the Korean Mathematical Society, vol. 39, no. 2, pp. 319–330, 2002.

W. Feng and J. R. L. Webb, “Solvability of m-point boundary value problems with nonlinear growth,” Journal of Mathematical Analysis and Applications, vol. 212, no. 2, pp. 467–480, 1997.

Y. Guo, W. Shan, and W. Ge, “Positive solutions for second-order m-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 151, no. 2, pp. 415–424, 2003.

C. P. Gupta, “A Dirichlet type multi-point boundary value problem for second order ordinary differential equations,” Nonlinear Analysis, vol. 26, no. 5, pp. 925–931, 1996.

R. Ma, “Positive solutions of a nonlinear m-point boundary value problem,” Computers & Mathematics with Applications, vol. 42, no. 6-7, pp. 755–765, 2001.

B. Liu, “Solvability of multi-point boundary value problem at resonance (II),” Applied Mathematics and Computation, vol. 136, no. 2-3, pp. 353–377, 2003.

X. Liu, J. Qiu, and Y. Guo, “Three positive solutions for second-order m-point boundary value problems,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 733–742, 2004.

R. Ma, “Multiplicity results for a three-point boundary value problem at resonance,” Nonlinear Analysis, vol. 53, no. 6, pp. 777–789, 2003.

R. Ma, “Positive solutions for nonhomogeneous m-point boundary value problems,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 689–698, 2004.

B. Liu and J. Yu, “Solvability of multi-point boundary value problem at resonance (III),” Applied Mathematics and Computation, vol. 129, no. 1, pp. 119–143, 2002.

W. Feng and J. R. L. Webb, “Solvability of three point boundary value problems at resonance,” Nonlinear Analysis, vol. 30, no. 6, pp. 3227–3238, 1997.

Z. Zhang and J. Wang, “Positive solutions to a second order three-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 237–249, 2003.

V. A. Il'in and E. I. Moiseev, “Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects,” Differential Equations, vol. 23, no. 7, pp. 803–810, 1987.

V. A. Il'in and E. I. Moiseev, “Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator,” Differential Equations, vol. 23, no. 8, pp. 979–987, 1987.

C. P. Gupta, “Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 540–551, 1992.

C. P. Gupta, “A generalized multi-point boundary value problem for second order ordinary differential equations,” Applied Mathematics and Computation, vol. 89, no. 1–3, pp. 133–146, 1998.

B. Liu, “Solvability of multi-point boundary value problem at resonance. Part IV,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 275–299, 2003.

Z. Zhang and J. Wang, “The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 147, no. 1, pp. 41–52, 2002.

C. P. Gupta, S. K. Ntouyas, and P. Ch. Tsamatos, “On an m-point boundary-value problem for second-order ordinary differential equations,” Nonlinear Analysis, vol. 23, no. 11, pp. 1427–1436, 1994.

W. Feng, “On a m-point nonlinear boundary value problem,” Nonlinear Analysis, vol. 30, pp. 5369–5374, 1997.

R. Ma, “Existence theorems for a second order m-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 211, no. 2, pp. 545–555, 1997.

S. A. Marano, “A remark on a second-order three-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 183, no. 3, pp. 518–522, 1994.

C. P. Gupta, “A second order m-point boundary value problem at resonance,” Nonlinear Analysis, vol. 24, no. 10, pp. 1483–1489, 1995.

C. P. Gupta, “Existence theorems for a second order m-point boundary value problem at resonance,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 4, pp. 705–710, 1995.

M. Urabe, “Numerical solution of multi-point boundary value problems in Chebyshev series theory of the method,” Numerische Mathematik, vol. 9, pp. 341–366, 1967.

M. Lentini and V. Pereyra, “A variable order finite difference method for nonlinear multipoint boundary value problems,” Mathematics of Computation, vol. 28, pp. 981–1003, 1974.

F. R. de Hoog and R. M. M. Mattheij, “An algorithm for solving multipoint boundary value problems,” Computing, vol. 38, no. 3, pp. 219–234, 1987.

R. P. Agarwal, “The numerical solution of multipoint boundary value problems,” Journal of Computational and Applied Mathematics, vol. 5, no. 1, pp. 17–24, 1979.

T. Ojika and W. Welsh, “A numerical method for multipoint boundary value problems with application to a restricted three-body problem,” International Journal of Computer Mathematics, vol. 8, no. 4, pp. 329–344, 1980.

W. Welsh and T. Ojika, “Multipoint boundary value problems with discontinuities. I. Algorithms and applications,” Journal of Computational and Applied Mathematics, vol. 6, no. 2, pp. 133–143, 1980.

R. P. Agarwal, “On multi-point boundary value problems for discrete equations,” Journal of Mathematical Analysis and Applications, vol. 96, pp. 520–534, 1983.

R. P. Agarwal and R. A. Usmani, “The formulation of invariant imbedding method to solve multipoint discrete boundary value problems,” Applied Mathematics Letters, vol. 4, no. 4, pp. 17–22, 1991.

R. C. Gupta and R. P. Agarwal, “A new shooting method for multipoint discrete boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 112, no. 1, pp. 210–220, 1985.

F. Geng, “Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2095–2102, 2009.

R. P. Agarwal, “On Nomerov's method for solving two point boundary value problems,” Utilitas Mathematica, vol. 28, pp. 159–174, 1985.

R. P. Agarwal, Difference Equations and Inequalities, vol. 155 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.

L. K. Bieniasz, “Use of the Numerov method to improve the accuracy of the spatial discretization in finite-difference electrochemical kinetic simulations,” Journal of Computational Chemistry, vol. 26, pp. 633–644, 2002.

M. M. Chawla and P. N. Shivakumar, “Numerov's method for non-linear two-point boundary value problems,” International Journal of Computer Mathematics, vol. 17, pp. 167–176, 1985.

L. Collatz, The Numerical Treatment of Differential Equations, Springer, Berlin, Germany, 3rd edition, 1960.

Y. Fang and X. Wu, “A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions,” Applied Numerical Mathematics, vol. 58, no. 3, pp. 341–351, 2008.

T. E. Simos, “A Numerov-type method for the numerical solution of the radial Schrödinger equation,” Applied Numerical Mathematics, vol. 7, no. 2, pp. 201–206, 1991.

Y.-M. Wang, “Monotone methods for a boundary value problem of second-order discrete equation,” Computers & Mathematics with Applications, vol. 36, no. 6, pp. 77–92, 1998.

Y.-M. Wang, “Numerov's method for strongly nonlinear two-point boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 759–763, 2003.

Y.-M. Wang, “The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems,” Applied Numerical Mathematics, vol. 57, no. 3, pp. 253–269, 2007.

Y.-M. Wang and B.-Y. Guo, “On Numerov scheme for nonlinear two-points boundary value problem,” Journal of Computational Mathematics, vol. 16, no. 4, pp. 345–366, 1998.

Y.-M. Wang and R. P. Agarwal, “Monotone methods for solving a boundary value problem of second order discrete system,” Mathematical Problems in Engineering, vol. 5, no. 4, pp. 291–315, 1999.

B. V. Numerov, “A method of extrapolation of perturbations,” Monthly Notices of the Royal Astronomical Society, vol. 84, pp. 592–601, 1924.

R. P. Agarwal and Y.-M. Wang, “The recent developments of Numerov's method,” Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 561–592, 2001.

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.

C. Cattani, “Shannon wavelets for the solution of integrodifferential equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 408418, 22 pages, 2010.

C. Cattani and A. Kudreyko, “Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4164–4171, 2010.

M. Li, S. C. Lim, and S.-Y. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011.

M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368–1372, 2010.

S.-Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation for vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages, 2011.

S.-Y. Chen and Q. Guan, “Parametric shape representation by a deformable NURBS model for cardiac functional measurements,” IEEE Transactions on Biomedical Engineering, vol. 58, pp. 480–487, 2011.

Y.-M. Wang, W.-J. Wu, and R. P. Agarwal, “A fourth-order compact finite difference method for nonlinear higher-order multi-point boundary value problems,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3226–3245, 2011.

R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2nd edition, 2000.

L. Collatz, Funktionalanalysis und Numerische Mathematik, Springer, Berlin, Germany, 1964.