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Numerov's Method for a Class of Nonlinear Multipoint Boundary Value Problems
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.
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 ). 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 ), 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 , 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  made use of the Chebyshev series for approximating solutions of nonlinear first-order multipoint boundary value problems, and the work in  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 , 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  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  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 ). 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
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.
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 .
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:
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 .
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).
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