Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 850343, 8 pages

http://dx.doi.org/10.1155/2014/850343

Research Article

Variational Iteration Method for Singular Perturbation Initial Value Problems with Delays

Yongxiang Zhao,1,2 Aiguo Xiao,³ Li Li,² and Chengjian Zhang¹

¹School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, China

²School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404000, China

³School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China

Received 7 October 2013; Revised 21 December 2013; Accepted 21 December 2013; Published 5 March 2014

Academic Editor: Sotiris. K. Ntouyas

Copyright © 2014 Yongxiang Zhao 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 variational iteration method (VIM) is applied to solve singular perturbation initial value problems with delays (SPIVPDs). Some convergence results of VIM for solving SPIVPDs are given. The obtained sequence of iterates is based on the use of general Lagrange multipliers; the multipliers in the functionals can be identified by the variational theory. Moreover, the numerical examples show the efficiency of the method.

1. Introduction

Singular perturbation initial value problems with delays play an important role in the research of various applied sciences, such as control theory, population dynamics, medical science, environment science, biology, and economics [1, 2]. These problems are characterized by a small parameter multiplying the highest derivatives, and the state variables depend not only on the value of the time, but also on the value prior to the time. Because the classical Lipschitz constant and one-sided Lipschitz constant are generally of size , the classical convergence theory, B-convergence theory, and D-convergence theory cannot be directly applied to SPIVPDs.

Starting from pioneering ideas going back to Inokuti-Sekine-Mura method [3], the variational iteration method was first proposed in later 1990s by He [4–6]. By recent years, this method has been extensively applied to various ODEs, integral equations, delay differential equations and fractional differential equations, two-point boundary value problems, oscillations, and stiff ODEs, notably, He [7, 8], Wazwaz [9, 10], Draganescu et al. [11, 12], Saadatmandi and Dehghan [13], Salkuyeh [14], Lu [15], Xu [16], Rafei et al. [17], Darvishi et al. [18], Tatari and Dehghan [19], Mamode [20], Saadati and Dehghan [21], Yu [22], Marinca et al. [23], Yang and Dumitru [24], and Wu [25] to mention only a few. Recently, Zhao and Xiao [26] have applied this method for solving singular perturbation initial value problems. For more comprehensive survey on this method and its applications, the reader is referred to the review articles [26–28] and the references therein.

In this paper, we apply the VIM to SPIVPDs to obtain the analytical or approximate analytical solutions. The convergence results of VIM for solving SPIVPDs are obtained. Some illustrative examples confirm the theoretical results.

In the remaining parts of the text, we denote for simplicity; here, is the singular perturbation parameter. The vectors mean that each component   . denotes the standard Euclidean norm of a vector.

2. Convergence Analysis

2.1. Case 1

Consider the following singular perturbation initial value problem with delays:

where and are the state variables and   is the singular perturbation parameter. , are given continuous mappings which satisfy the following Lipschitz conditions:

where ,     are continuous bounded functions.

According to VIM, we can construct the correction functionals as follows:

where ,   are general Lagrange multipliers, which can be defined optimally via variational theory, and denote the restrictive variation; that is, . Thus, we have

and the stationary conditions are obtained as

Moreover, the general Lagrange multiplier can be readily identified by

Therefore, the variational iteration formulas can be written as

Now, we show that the iterative sequences ,   defined by (7a) and (7b) with ,   converge to the solution of (1).

Theorem 1. Let ,  , . The sequences defined by (7a) and (7b) with ,   converge to the solution of (1).

Proof. Obviously from system (1), we have

Introduce ,  ,  ,  ,  , where ,  . Now from (7a), (7b)-(8a), and (8b) we obtain

Moreover, we can derive

Now, the integration interval is split into two parts

From the Lipschitz conditions (2a) and (2b), we have

where ,  ,  . Therefore,

Moreover, we can derive

Noting that , , , , , are constants. By using Stirling’s formula, we have

thus, , as .

2.2. Case 2

Consider the special case of (1):

where are given continuous mappings which satisfy the Lipschitz conditions (2a) and (2b); the matrices ,   can be decomposed into ,  , respectively, where and :

where ,      are continuous bounded functions.

It is easy to show that the right hand sides of (16) also satisfy the Lipschitz conditions. If the right hand sides of (16) are considered as nonlinear terms, then we can also use the correction functionals constructed in Case 1 and get similar results to Theorem 1. Now, we construct the following correction functionals:

where , , in which ,  ,  , are general Lagrange multipliers and denote the restrictive variations; that is, . Thus, we have

and the stationary conditions are obtained as

Moreover, the general Lagrange multipliers can be readily identified by

Therefore, the variational iteration formula can be written as

The following theorem shows that the sequences ,   defined by (22a) and (22b) with ,   converge to the solution of (16).

Theorem 2. Let ,  ,  . The sequences defined by (22a) and (22b) with converge to the solutions of (16).

Proof. By a similar process to the proof of Theorem 1, we can easily obtain Obviously from system (16) we have

Introduce ,  ,  , where ,  . Now from (22a), (22b)-(23a), and (23b) we obtain

From the Lipschitz conditions (17a) and (17b), we have

similarly, we can derive

where , , , . Noting that ,  ,  ,  ,  ,  ,  , and are constants, we can derive from (26) that , as .

3. Numerical Examples

In this section, some numerical examples are given to show the efficiency of the VIM for solving SPIVPs.

Example 3. Consider SPIVPD (cf. [2]):

where

By using the VIM in Case 1, we construct the following iteration formula:

To get iterate sequence, we start with an initial approximation ,   and let . By means of formulas (29a) and (29b), we have

Figure 1 shows the efficiency of VIM for SPIVPDs.

(a)

(b)

Figure 1: Results for problem 1.

Example 4. Consider SPIVPD (cf. [2]):

where

By using the VIM in Case 2, we construct the following iteration formula:

To get iterate sequence, we start with an initial approximation ,   and let . By means of formulas (33) and (34), we have

Figure 2 shows the efficiency of VIM for SPIVPDs.

(a)

(b)

Figure 2: Results for problem 2.

4. Conclusion

The VIM used in this paper is the variational iteration algorithm I; there are also variational iteration algorithms II and III [29]. In this paper, we apply the VIM to obtain the analytical or approximate analytical solutions of SPIVPDs. The convergence results of VIM for solving SPIVPDs are given. The illustrative examples show the efficiency of the method. When considering the system (16), the choice of correction functionals of Case 1 or Case 2 relies on the practical problems and this choice will result in the difference of the speed of convergence.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by projects NSF of China (11126329, 11271311, and 11201510), Projects Board of Education of Chongqing City (KJ121110), and Key Laboratory for Nonlinear Science and System Structure.

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