Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 518406, 13 pages
http://dx.doi.org/10.1155/2015/518406

Research Article

Reproducing Kernel Algorithm for the Analytical-Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems

Omar Abu Arqub

Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan

Received 8 December 2014; Accepted 2 May 2015

Academic Editor: Francesco Tornabene

Copyright © 2015 Omar Abu Arqub. 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 reproducing kernel algorithm is described in order to obtain the efficient analytical-numerical solutions to nonlinear systems of two point, second-order periodic boundary value problems with finitely many singularities. The analytical-numerical solutions are obtained in the form of an infinite convergent series for appropriate periodic boundary conditions in the space , whilst two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed approach. The main characteristic feature of the utilized algorithm is that the global approximation can be established on the whole solution domain, in contrast with other numerical methods like onestep and multistep methods, and the convergence is uniform. Two numerical experiments are carried out to verify the mathematical results, whereas the theoretical statements for the solutions are supported by the results of numerical experiments. Our results reveal that the present algorithm is a very effective and straightforward way of formulating the analytical-numerical solutions for such nonlinear periodic singular systems.

1. Introduction

Mathematical models of classical applications from physics, chemistry, and mechanics take the form of systems of singular periodic boundary value problems (BVPs) of second order which are a combination of singular differential system and periodic boundary conditions. Commonly, the singularity typically occurring at endpoints or in the form of a set of finite cardinality of the interval of integration. Periodic BVPs for systems of ordinary differential equations with singularities appear also in numerous applications which are of interest in modern applied mathematics. To name but a few, computations of self-similar blow-up solutions of nonlinear partial differential equations lead to the computation of problems from this class [1, 2], the density profile equation in hydrodynamics may be reduced to a system of singular periodic BVP [3, 4], the investigation of problems in the theory of shallow membrane caps is associated with such systems [5], and in ecology, in the computation of avalanche run-up, this problem class is translated into a system of singular periodic BVP [6, 7].

Most scientific problems and phenomenons in different fields of sciences and engineering occur nonlinearly. To set the scene, we know that except a limited number of these problems and phenomenons, most of them do not have analytical solutions. So these nonlinear equations should be solved using numerical methods or other analytical methods. Anyhow, when applied to the systems of singular periodic BVPs, standard numerical methods designed for regular BVPs suffer from a loss of accuracy or may even fail to converge [8–10], because of the singularity, whilst analytical methods commonly used to solve nonlinear differential equations are very restricted and numerical techniques involving discretization of the variables on the other hand give rise to rounding off errors. As a result, there are some restrictions to solve these singular periodic systems; firstly, we encountered with the nonlinearity of systems; secondly, these systems are singular BVPs with periodic boundary values.

Investigation about systems of singular periodic BVPs numerically is scarce and missing. In this study, the reproducing kernel Hilbert space (RKHS) method has been successfully applied as a numerical solver for such systems. The present technique has the following characteristics; first, it is of global nature in terms of the solution obtained as well as its ability to solve other mathematical and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for nonlinear cases; third, in the proposed technique, it is possible to pick any point in the given domain and as well the numerical solutions and their derivatives will be applicable; fourth, the approach does not require discretization of the variables, it is not effected by computation round off errors, and one is not faced with necessity of large computer memory and time; fifth, the proposed approach does not resort to more advanced mathematical tools; that is, the algorithm is simple to understand, implement, and should be thus easily accepted in the mathematical and engineering application’s fields. More precisely, we provide the analytical-numerical solutions for the following differential singular system:

subject to the following periodic boundary conditions:

where , are unknown functions to be determined, are continuous terms in as , , , and are depending on the system discussed, and , are two reproducing kernel spaces. Here, the two functions may take the values or for some which make (1) be singular at , whilst are continuous real-valued functions on , in which . Through this paper, we assume that (1) and (2) have a unique solution on .

A number of theoretical results for the solutions of various types of systems of singular differential equations have been developed over the last couple of decades. The reader is asked to refer to [9–15] in order to know more details about these analyses, including their kinds and history, their modifications and conditions for use, their scientific applications, their importance and characteristics, and their relationship including the differences.

Reproducing kernel theory has important application in numerical analysis, computational mathematics, image processing, machine learning, finance, and probability and statistics [16–19]. Recently, a lot of research work has been devoted to the applications of the reproducing kernel theory representative in the RKHS method, which provides the analytical-numerical solutions for linear and nonlinear problems, for wide classes of stochastic and deterministic problems involving operator equations, differential equations, fuzzy differential equations, integral equations, and integrodifferential equations. The RKHS method was successfully used by many authors to investigate several scientific applications side by side with their theories. The reader is kindly requested to go through [20–38] in order to know more details about RKHS method, including its history, its modification for use, its scientific applications, its kernel functions, and its characteristics.

The rest of the paper is organized as follows. In the next section, several inner product spaces are constructed in order to apply the method. In Section 3, the analytical-numerical solutions and theoretical basis of the method are introduced. In Section 4, an iterative method for the analytical-numerical solutions is described and the -truncation numerical solutions are proved to converge to the analytical solutions. In Section 5, we derive the error estimation and the error bound in order to capture the behavior of the numerical solutions. In order to verify the mathematical simulation of the proposed method, two nonlinear numerical examples are presented in Section 6. Some concluding remarks are presented in Section 7. This paper ends in Appendices, with two parts about the kernel function of the space .

2. Building Several Inner Product Spaces

In functional analysis, the RKHS is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. In this section, we utilize the reproducing kernel concept in order to construct the RKHS’s and . After that, two reproducing kernels functions and are building in order to formulate and utilize the analytical-numerical solutions via RKHS technique. Throughout this paper is the set of complex numbers, , and .

Prior to discussing the applicability of the RKHS method on solving singular periodic differential systems and their associated numerical algorithm, it is necessary to present an appropriate brief introduction to preliminary topics from the reproducing kernel theory.

Definition 1 (see [16]). Let be a Hilbert space of a function on a set . A function is a reproducing kernel of if the following conditions are satisfied. Firstly, for each . Secondly, for each and each .

The second condition in Definition 1 is called “the reproducing property” which means that the value of the function at the point is reproduced by the inner product of with . Indeed, a Hilbert space of functions on a nonempty abstract set is called a RKHSs, if there exists a reproducing kernel of .

It is worth mentioning that the reproducing kernel function of a Hilbert space is unique, and the existence of is due to the Riesz representation theorem, where completely determines the space . Moreover, every sequence of functions , which converges strongly to a function in , converges also in the pointwise sense. This convergence is uniform on every subset on in which is bounded. In this occasion, these spaces have wide applications including complex analysis, harmonic analysis, quantum mechanics, statistics, and machine learning. For the theoretical background of the reproducing kernel theory and its applications, we refer the reader to [16–38].

Definition 2. The inner product space is defined as are absolutely continuous real-valued functions on , , and , . On the other hand, the inner product and the norm of are defined, respectively, by

and , where .

The Hilbert space is called a reproducing kernel if for each fixed and any , there exist and such that . Henceforth and not to conflict unless stated otherwise, we denote simply by .

Theorem 3. The Hilbert space is a complete reproducing kernel and its reproducing kernel function can be written as

where and , , are unknown coefficients of .

Proof. The proof and the coefficients of the reproducing kernel function are given in Appendices A and B, respectively.

Definition 4 (see [20]). The inner product space is defined as is absolutely continuous real-valued function on and . On the other hand, the inner product and the norm of are defined, respectively, by

and , where .

Theorem 5 (see [20]). The Hilbert space is a complete reproducing kernel and its reproducing kernel function can be written as

where and , , are unknown coefficients of and are given as , , , and .

The spaces and are complete Hilbert with some special properties. So, all the properties of the Hilbert space will be held. Further, this space possesses some special and better properties which could make some problems be solved easier. For instance, many problems studied in space, which is a complete Hilbert space, require large amount of integral computations and such computations may be very difficult in some cases. Thus, the numerical integrals have to be calculated in the cost of losing some accuracy. However, the properties of and require no more integral computation for some functions, instead of computing some values of a function at some nodes. In fact, this simplification of integral computation not only improves the computational speed, but also improves the computational accuracy.

3. Formulation of the Analytical-Numerical Solutions

In this section, formulation of the differential linear operator and implementation method are presented in the space . Meanwhile, we construct an orthogonal function system based on the Gram-Schmidt orthogonalization process in order to obtain the analytical-numerical solutions. For the remaining sections, the lowercase letter whenever used means for each .

To deal with (1) and (2) in more realistic form via the RKHS approach, multiplying both sides of (1) by and define the differential operator as

such that

As a result, (1) and (2) can be converted into the equivalent form as follows:

subject to the periodic boundary conditions

where .

Theorem 6. The operator is bounded and linear.

Proof. For boundedness, we need to prove , where . From the definition of the inner product and the norm of , we have . By the Schwarz inequality and the reproducing properties , , and of , we get

where . Thus, or , where . The linearity part is obvious. The proof is complete.

Next, some theoretic basis of the RKHS method is introduced. Initially, we construct an orthogonal function system of ; to do so, put and , where is dense on and is the adjoint operator of . In other words, , .

Algorithm 7. The orthonormal function system of the space can be derived from the Gram-Schmidt orthogonalization process of as follows:
Step 1. For and do the following:

 If , then set

 If , then set

 If , then set

Output: the orthogonalization coefficients of the orthonormal system .
Step 2. For set

Output: the orthonormal function system .
Step 3. Stop.
It is easy to see that = = = . Thus, can be written in the form , where indicates that the operator applies to the function of .

Theorem 8. If is dense on , then is a complete function system of the space .

Proof. For each fixed , let , . In other words, one has . Note that is dense on ; therefore . It follows that from the existence of . So, the proof of the theorem is complete.

Lemma 9. If , then , , and .

Proof. For the first part, note that , where are absolutely continuous on . If those are integrated again from to , the result is itself as . Again, integrated from to , yield that = . So, or . By using Holder’s inequality and (3), one can note the following inequalities:

Thus, . For the second part, since , this means that . Thus, one can find . In the third part, clearly, , which yield that . Hence, one can write .

4. Iterative Algorithm for the Analytical-Numerical Solutions

In this section, an iterative algorithm of obtaining the analytical-numerical solutions is represented in the reproducing kernel space . The numerical solution is obtained by taking finitely many terms in this series representation form. Also, the numerical solutions and their derivatives are proved to converge uniformly to the analytical solution and their derivatives, respectively.

The internal structure of the following theorem is to give the representation form of the analytical solutions. After that, the convergence of the numerical solutions to the analytical solutions will be proved.

Theorem 10. For each , the series are convergent in the sense of the norm of . On the other hand, if is dense on , then the analytical solutions of (9) and (10) could be represented by

Proof. Let be solutions of (9) and (10) in . Since , are the Fourier series expansion about normal orthogonal system , and is the Hilbert space, then the series are convergent in the sense of . On the other hand, using (15), yields that

Therefore, the form of (17) is the analytical solutions of (9) and (10). So, the proof of the theorem is complete.

Let be the normal orthogonal system derived from the Gram-Schmidt orthogonalization process of . Then according to (17), the analytical solution of (9) and (10) can be denoted by

where . In fact, , , are unknown; we will approximate using known . For numerical computations, define initial functions , put , and set the -term numerical approximations to by

where the coefficients of the normal orthogonal system are given as

According to Lemma 9, it is clear that, for any , the analytical-numerical solutions of (9) and (10) satisfy

where , , and . As a result, if as , then the numerical solutions and are converged uniformly to the analytical solutions and , , respectively.

Lemma 11. If , as , and is continuous in with respect to for and , then as .

Proof. Firstly, we will prove that in the sense of . Since

By reproducing property of , we have and . Thus, . From the symmetry of , it follows that as . Hence, as soon as . On the other hand, by Lemma 9, for any it holds that as . Therefore, in the sense of as and . Thus, by means of the continuation of it is obtained that as .

Lemma 12. One has as .

Proof. The proof of is obtained by using the mathematical induction as follows: if , then = = = . Using the orthogonality of , yields that

Now, if , then . Again, if , then = . Thus, . It is easy to see that . But on the other aspect as well, from (22) converge uniformly to . It follows that, on taking limits in (20), . Therefore, , where is an orthogonal projector from to Span. Thus,

Theorem 13. If are bounded and is dense on , then the -term numerical solutions in the iterative formula (20) converges to the analytical solutions of (9) and (10) in the space and , where are given by (21).

Proof. The proof consists of the following steps: firstly, we will prove that the sequences in (20) are increasing in the sense of . By Theorem 8, is the complete orthonormal system in . Hence, we have

Therefore, are increasing.
Secondly, we will prove the convergence of . From (20), we have . From the orthogonality of , it follows that = = . Since the sequences are increasing. Due to the condition that are bounded, are convergent as . Then, there exists constants such that . It implies that , . On the other hand, since it follows for that

Furthermore, . Consequently, as , we have . Considering the completeness of , there exists such that as in the sense of .
Thirdly, we will prove that are the analytical solutions of (9) and (10). Since is dense on , for any , there exists a subsequence , such that as . From Lemma 12, It is clear that . Hence, let ; by Lemma 11 and the continuity of , we have . That is, satisfies (9). Also, since , clearly, satisfies the periodic boundary conditions of (10). In other words, are the analytical solutions of (9) and (10), where and are given by (21). The proof is complete.

5. Error Estimation and Error Bound

When solving practical problems, it is necessary to take into account all the errors of the measurements. Moreover, in accordance with the technical progress and the degree of complexity of the problem, it becomes necessary to improve the technique of measurement of quantities. Considerable errors of measurement become inadmissible in solving complicated mathematical, physical, and engineering problems. The reliability of the numerical result will depend on an error estimate or bound, therefore the analysis of error and the sources of error in numerical methods is also a critically important part of the study of numerical technique. In this section, we derive error bounds for the present method and problem in order to capture behavior of the solutions.

In the next theorem, we show that the error of the approximate solutions is decreasing, while the next lemma is presented in order to prove the recent theorem.

Theorem 14. Let , where and are given by (19) and (20), respectively. Then, the sequences are decreasing in the sense of the norm of and as .

Proof. Based on the previous results, it is obvious that

Clearly, , and consequently are decreasing in the sense of . On the other aspect as well, by Theorem 10, are convergent, so, or as . This completes the proof.

Lemma 15. Let be the analytical solutions of (9) and (10), and are the numerical solutions of . Suppose that is the subset of , where is the dense subset in as . Then, , for and .

Proof. Set the projective operator . Then, we have = = = = .

Theorem 16. Let be the analytical solutions of (9) and (10), and are the numerical solutions of . Suppose that is the subset of , where is the dense subset in as . Then, , where are the product of the maximum of determinate function and the sup of convergent basis about the variable in .

Proof. Since and for every given , there is always satisfying and . On the other hand, Lemma 15 and imply that . So, we obtain

By applying the reproducing kernel properties and to (30), we conclude

whilst on the other aspect as well,

Here, we take the norm of for the variable and the function is derived on in . So, we have . Hence, we obtain

So, the proof of the theorem is complete.

6. Numerical Algorithm and Numerical Outcomes

In this final section, we consider two nonlinear examples in order to illustrate the performance of the RKHS algorithm in finding the numerical solutions for systems of singular periodic BVPs and justify the accuracy and applicability of the method. These examples have been solved by the presented algorithm while the results obtained are compared with the analytical solutions of each example by computing the absolute and the relative errors and are found to be in good agreement with each other. In the process of computation, all the symbolic and numerical computations performed by using Maple 13 software package.

An algorithm is a precisely defined sequence of steps for performing a specified task. The aim of the next algorithm is to implement a procedure to solve periodic singular differential systems in numeric form in terms of their grid nodes based on the use of RKHS method.

Algorithm 17. To find the numerical solutions of for (9) and (10), we do the following steps:
Input. The endpoints of , the integers , and the kernel functions and .
Output. Numerical solutions of .
Step 1. Fixed in and set ;

 If , set ;

 else set ;

 For do the following: Set ; Set ;

Output: the orthogonal function system .
Step 2. For and , do Algorithm 7 for and ;
Output: the orthogonalization coefficients .
Step 3. For and , do the following:

 Set ;

Output: the orthonormal function system .
Step 4. Set ;
Set ;
Set ;
Output: the numerical solutions of .
Step 5. Stop.

Using RKHS method, take , , with the reproducing kernel functions and on in which Algorithms 7 and 17 are used throughout the computations; some graphical results and tabulated data are presented and discussed quantitatively at some selected grid points on to illustrate the numerical solutions for the following periodic singular differential systems.

Example 18. Consider the nonlinear differential system:

subject to the periodic boundary conditions:

where in which and are chosen such that the analytical solutions are and .

Example 19. Consider the nonlinear differential system:

subject to the periodic boundary conditions:

where in which and are chosen such that the analytical solutions are and .

Results from numerical analysis are an approximation, in general, which can be made as accurate as desired. Because a computer has a finite word length, only a fixed number of digits are stored and used during computations. Next, the agreement between the analytical-numerical solutions is investigated for Examples 18 and 19 at various in by computing the absolute errors and the relative errors of numerically approximating their analytical solutions for the corresponding equivalent system as shown in Tables 1, 2, 3, and 4, respectively.

Table 1: Numerical results of the first dependent variable for Example 18 at various .

Node
Analytical solution
Numerical solution
Absolute error
Relative error

Table 2: Numerical results of the second dependent variable for Example 18 at various .

Node
Analytical solution
Numerical solution
Absolute error
Relative error

Table 3: Numerical results of the first dependent variable for Example 19 at various .

Node
Analytical solution
Numerical solution
Absolute error
Relative error

Table 4: Numerical results of the second dependent variable for Example 19 at various .

Node
Analytical solution
Numerical solution
Absolute error
Relative error

Anyhow, it is clear from the tables that, the numerical solutions are in close agreement with the analytical solutions, while the accuracy is advanced by using only few tens of the RKHS iterations. Indeed, we can conclude that higher accuracy can be achieved by computing further RKHS iterations. As a computational conclusion, it is to be noted from the tables that the two dependent solutions are relatively of the same order of errors on average for the absolute and the relative error, respectively, for the two examples.

As we mentioned earlier, it is possible to pick any point in and as well the numerical solutions and all their numerical derivatives up to order two will be applicable. Next, the numerical values of the absolute errors for the first and the second derivatives of the numerical solutions of Example 18 have been plotted in Figures 1 and 2, respectively, at various in . As the plots show, while the value of approaches to the boundary of , the numerical values for both derivatives approach smoothly to the -axis. It is observed that the increase in the number of node results in a reduction in the absolute errors and correspondingly an improvement in the accuracy of the obtained solutions. This goes in agreement with the known fact, the error is decreasing, where more accurate solutions are achieved using an increase in the number of nodes. On the other hand, the cost to be paid while going in this direction is the rapid increase in the number of iterations required for convergence.

Figure 1: The numerical values of the absolute error function for the first derivative of Example 18: blue: the first dependent variable and red: the second dependent variable.

Figure 2: The numerical values of the absolute error function for the second derivative of Example 18: blue: the first dependent variable and red: the second dependent variable.

7. Conclusions

The applications of the RKHS algorithm were extended successfully for solving nonlinear systems of singular periodic BVPs. In this approach, reproducing kernel spaces are constructed, in which the given periodic boundary conditions of the systems can be involved. The analytical-numerical solutions were calculated in the form of a convergent series in the space with easily computable components; in the meantime the -term numerical solutions are obtained and are proved to converge to the analytical solutions. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions; whilst the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in the space . Further, an error estimation and error bound based on the reproducing kernel theory are proposed in order to capture the behavior of the numerical solutions. Tabulated data, graphical results, and numerical comparisons with the analytical solutions are presented and discussed quantitatively to illustrate the numerical solutions. The basic ideas of this iterative novel approach can be widely employed to solve other strongly nonlinear singular systems.

Appendices

Here, we provide a detailed proof of Theorem 3 and the expansion formulas for the unknown coefficients and , , of the reproducing kernel function in the space .

A. Proof of Theorem 3

The proof of the completeness and reproducing property of is similar to the proof in [21]; let us find out the expression form of . Through several integrations by parts, we obtain

Thus, from (3) one can write

Since , it follows that and . Again, since , it yields that , . Hence,

On the other hand, if , , , , , and , then (A.3) implies that . Now, for any , if satisfies

then . Obviously, is the reproducing kernel function of . Next, we give the expression form of . The auxiliary formula of (A.4) is given by , and its auxiliary values are with multiplicity . So, let the expression form of be as defined in (4). But on the other aspect as well, for (A.4), let satisfy the equation , . Integrating from to with respect to and letting , we have the jump degree of at given by . Through the last descriptions the unknown coefficients and , , of (4) can be obtained. This completes the proof.

B. Coefficients of the Reproducing Kernel Function

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to express his gratitude to the unknown referees for carefully reading the paper and their helpful comments.

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