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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 541935, 13 pages
http://dx.doi.org/10.1155/2013/541935
Research Article

A New Method Based on the RKHSM for Solving Systems of Nonlinear IDDEs with Proportional Delays

1School of Sports and Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China
2School of Mathematics and Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China

Received 10 November 2012; Revised 1 March 2013; Accepted 25 March 2013

Academic Editor: Zhenya Yan

Copyright © 2013 Changbo He 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

An efficient computational method is given in order to solve the systems of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays. Representation of the solution and an iterative method are established in the reproducing kernel space. Some examples are displayed to demonstrate the computation efficiency of the method.

1. Introduction

In functional-differential equations (FDEs), there is a class of infinite-delay-differential equations (IDDEs) with proportional delays such systems are often encountered in many scientific fields such as electric mechanics, quantum mechanics, and optics. In view of this, developing the research for this class of IDDEs possesses great significance on theory and practice, for this attracts constant interest of researchers. Ones have found that there exist very different mathematical challenges between FDEs with proportional delays and those with constant delays. Some researches on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. In the last few years, there has been a growing interest in studying the existence of solutions of functional differential equations with state dependent delay [17]. Initial-value problem for neutral functional-differential equations with proportional time delays had been studied in [811]; in the literature [11] authors had discussed the existence and uniqueness of analytic solution of linear proportional delays equations.

Ishiwata et al. used the rational approximation method and the collocation method to compute numerical solutions of delay-differential equations with proportional delays in [12, 13]. At [1417], Hu et al. gave the numerical method to compute numerical solutions of neutral delay differential equations. For neutral delay differential equations with proportional delays, Chen and Wang proposed the variational iteration method [18] and the homotopy perturbation method [19]. Recently, time-delay systems become interested in applications like population growth models, transportation, communications, and agricultural models so those systems were widely studied both in a theoretical aspect and in that of related applications [2022].

We consider the following nonlinear infinite delay-differential equation (IDDE) with proportional delay [23]: where , is a given initial value, , and for , , is a continuous function; as , .

Next, the following system of nonlinear infinite-delay-differential equations (IDDEs) with proportional delays will be studied: where , for , , , are continuous bounded function, and , as , . and are reproducing kernel spaces. Equations (2) are obtained through homogenization of initial condition for model in [24]. In this study, existence and a new iterative algorithm are established for the nonlinear infinite-delay-differential equations (IDDEs) with proportional delays in the reproducing kernel spaces.

The paper is organized as follows. In Section 2, some definitions of the reproducing kernel spaces are introduced. In Section 3, main results and the structure of the solution for operator equation are discussed. Existence of the solution to (2) and an iterative method are developed for the kind of problems in the reproducing kernel space. We verify that the approximate solution converges to the exact solution uniformly. In Section 4, some experiments are given to demonstrate the computation efficiency of the algorithm. The conclusion is given in Section 5.

2. Preliminaries

Definition 1 (see [25] (reproducing kernel)). Let be a nonempty abstract set. A function is a reproducing kernel of the Hilbert space if and only if(a), ,(b), , .
The condition (b) is called “the reproducing property”; a Hilbert space which possesses a reproducing kernel is called a Reproducing Kernel Hilbert Space (RKHS).

Next, two reproducing kernel spaces are given.

Definition 2. is absolutely continuous real value functions, .
is a Hilbert space, for ; the inner product and norm in are given by respectively.

Theorem 3. The space is a reproducing kernel space that is, for any and each fixed , there exists , , such that . And the corresponding reproducing kernel can be represented as follows [26]:

Definition 4. is absolutely continuous real-valued function, .
The inner product and norm in can be defined by respectively, where . It has been proved that is a complete reproducing kernel space and its reproducing kernel is as follows [27]:

3. Statements of the Main Results

In this section, the implementation method of obtaining the solution of (2) is proposed in the reproducing kernel space .

Put the differential operator ; then we can convert (2) into the following form: where , and It is clear that is a bounded linear operator. Let , where is dense in the interval , and , where is the conjugate operator of . Define the normal orthogonal system in , which derives from Gram-Schmidt orthogonalization process of ,

Lemma 5. Assume that are dense in ; then is a complete system in and .

Proof. Ones have
Clearly, .
For any , let , , which means that Note that is dense in ; therefore, . It follows that from the existence of .

3.1. Construction the of Iterative Sequence and

Next we construct the iterative sequence and , putting where and is a orthogonal projection operator. Then by (13) it followed that:

Lemma 6. Let be dense on ; if the solution of (2) is unique, then the solution satisfies the form

Proof. Note that and is an orthonormal basis of ; hence according to Lemma 5 we have In the same manner

Take ; define the iterative sequence

3.2. The Boundedness of Sequence and

From we have where ; ; and are bounded functions with respect to , , respectively.

Lemma 7. For , , and , are continuous bounded functions on , we have that and are bounded.

Proof. By the expression of , and the assumptions, we know that and are bounded. In the following, we will discuss the boundedness of .
Since the function which is in is dense in , without loss of generality, we assume that is continuous.
Note that One gets where . Thus, we have So Furthermore, we see that In view of the expression of , we know that are bounded for . It follows that is bounded from the boundedness of , , , , and . In the same way, , , and are bounded.

Lemma 8. Assume that , are continuous bounded functions for , , and as , , then

Proof. Note that (14) and the assumptions, by Lemma 7, , , thus , where , from (19), we have . In the same way, we obtain .

3.3. Construction the of Another Iterative Sequence and

Theorem 9. Let be dense on ; if the solution of (2) exists and unique, then the solution satisfies the form

Proof. Note that and is an orthonormal basis of ; hence we have In the same manner

In the following, a new method of solving (2) is presented. Equations (31) and (32) can be denoted by where and . In fact, and are unknown; we will approximate and by using the known and .

We take , and define the following iterative sequence where Next, lemmas are given.

Lemma 10. The following iterative sequences satisfy respectively.

Proof. If , so, If , (40) (42), we have by (41), . In the same way, we have Similarly,

Theorem 11. The iterative form and the iterative form are the same.

Proof. In Lemma 10, let ; then but thus Equation (46) can be written as In fact, ; then by (51) and (52), we have Equation (53) is the same as (50). We may prove for similarly.

So, by Theorem 11 and Lemma 8, we have the following Theorem.

Theorem 12. Under the conditions of Lemma 8, satisfy , .

Lemma 13. If and , then there exists , such that and .

Proof. It is easy to obtain from the properties in the reproducing kernel space.

By Lemma 13 and Theorem 12, it is easy to obtain the following Lemma 14.

Lemma 14. If , , , and and satisfy the conditions of Lemma 8, then

Theorem 15. Let be dense in , and and satisfy the conditions of Lemma 8, then the n-term approximate solutions and in (35) converge to the exact solution and of (2), respectively, and , , where and are given, respectively, by (36), and (37).

Proof. Firstly, we will prove the convergence of .
By (35), we infer that From the orthogonality of , it follows that From (59) and (60), we know sequence and is increasing. By Theorem 12, and are bounded; hence and are convergent such that This implies that Without loss of generality, assume ; we have Considering the completeness of , there exist and in such that
Secondly, we will prove that and are the solutions of (2).
By Lemma 14 and the proof of (1), we may know that , and respectively, converge uniformly to and . Taking limits in (35), we have Since in the same way, we have it follows that If , then
If , then
From (70) and (71), it is clear that Furthermore, it is easy to see by induction that Since is dense in , for any , there exists subsequence such that Hence, let in (74), and (75); by Lemma 14 and the continuity of and , we have That is, and are the solutions of (2) and where and .

In the proof of the convergence in Theorem 15 we only use , ; thus we obtain the following corollary.

Corollary 16. Suppose and are bounded; then the iterative sequence (35) is convergent to the exact solution of (2).

Theorem 17. Assume and are the solutions of (2), and are the approximate errors of , where and are given by (35). Then the errors , are monotone decreasing in the sense of .

Proof. From (35), and (79), it follows that In the same way, we obtain from (35), (80) Equations (81) and (82) show that the errors and are monotone decreasing in the sense of .

4. Numerical Examples

In order to demonstrate the efficiency of our algorithm for solving (2), we will present two numerical examples in the reproducing kernel space . Let be the number of discrete points in . Denote , . All computations are performed by the Mathematica software package. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Example 18. In this example we consider the problem with exact solution , , and , . Applying the presented method in Section 3, we calculate the approximate solution and in as follows.

 Step 1. By the method of the appendix, the corresponding reproducing kernel functions can be obtained.

 Step 2. Choosing a dense subset in , then we get the orthogonalization coefficients .

Step 3. According to (10), we can get the normal orthogonal systems .

Step 4. Selecting the initial value , we obtain and by (19) developed in the paper.

The graphs of the superimposed image emerge in Figure 1. At the same time, we have computed the approximate solutions and in and also calculated the relative errors and in Table 1. The root mean square errors (RMSE) about with and with are shown in Table 2.

tab1
Table 1: Relative errors in for Example 18.
tab2
Table 2: The RMS errors in for Example 18.
fig1
Figure 1: The left is the superimposed image of with in . The right is the superimposed image of with in .

Example 19. Considering equations The true solutions are , , , . The numerical results are given in Figure 2 and Tables 3 and 4. The figures and tables illustrate that the method given in the paper is efficient.

tab3
Table 3: Relative errors in for Example 19.
tab4
Table 4: The RMS errors in for Example 19.
fig2
Figure 2: The left is the superimposed image of with in . The right is the superimposed image of with in .

5. Conclusion

In this paper, RKHSM has been successfully applied to find the solutions of systems of nonlinear IDDEs with proportional delays. The efficiency and accuracy of the proposed decomposition method were demonstrated by two test problems. It is concluded from above tables and figures that the RKHSM is an accurate and efficient method to solve IDDEs with proportional delays. Moreover, the method is also effective for solving some nonlinear initial-boundary value problems and nonlocal boundary value problems.

Appendix

The Reproducing Kernel Space

is defined as are absolutely continuous real value functions, , . The inner product in is given by where and the norm is denoted by .

Theorem A.1. The space is a reproducing kernel space; that is, for any and each fixed , there exists , , such that . The reproducing kernel can be denoted by

Proof. Applying to the integrations by parts for (A.1), we have