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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 634670, 9 pages
http://dx.doi.org/10.1155/2013/634670
Research Article

Convergence of Variational Iteration Method for Second-Order Delay Differential Equations

Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, China

Received 24 October 2012; Revised 17 December 2012; Accepted 17 December 2012

Academic Editor: Changbum Chun

Copyright © 2013 Hongliang Liu 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

This paper employs the variational iteration method to obtain analytical solutions of second-order delay differential equations. The corresponding convergence results are obtained, and an effective technique for choosing a reasonable Lagrange multiplier is designed in the solving process. Moreover, some illustrative examples are given to show the efficiency of this method.

1. Introduction

The second-order delay differential equations often appear in the dynamical system, celestial mechanics, kinematics, and so forth. Some numerical methods for solving second-order delay differential equations have been discussed, which include -method [1], trapezoidal method [2], and Runge-Kutta-Nyström method [3]. The variational iteration method (VIM) was first proposed by He [4, 5] and has been extensively applied due to its flexibility, convenience, and efficiency. So far, the VIM is applied to autonomous ordinary differential systems [6], pantograph equations [7], integral equations [8], delay differential equations [9], fractional differential equations [10], the singular perturbation problems [11], and delay differential-algebraic equations [12]. Rafei et al. [13] and Marinca et al. [14] applied the VIM to oscillations. Tatari and Dehghan [15] consider the VIM for solving second-order initial value problems. For a more comprehensive survey on this method and its applications, the readers refer to the review articles [1619] and the references therein. But the VIM for second-order delay differential equations has not been considered.

The article apply the VIM to second-order delay differential equations to obtain the analytical or approximate analytical solutions. The corresponding convergence results are obtained. Some illustrative examples confirm the theoretical results.

2. Convergence

2.1. The First Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay differential equations where is a differentiable function, is a strictly monotone increasing function and satisfies that and , there exists such that , and is a given continuous mapping and satisfies the Lipschitz condition where are Lipschitz constants; denotes the standard Euclidean norm.

Now the VIM for (1) can read where for ; denotes the restrictive variation, that is, . Thus, we have Using integration by parts to (4), we have From the above formula, the stationary conditions are obtained as Moreover, the general Lagrange multiplier can be readily identified by (7). Thus, the variational iteration formula can be written as

Theorem 1. Suppose that the initial value problems (1) satisfy the condition (2), and , . Then the sequence defined by (9) and (10) with converges to the solution of (1).

Proof. From (1), we have Let , . If , then . From (9) and (11), we have From (10) and (12), we have Using integration by parts, we have Since is bounded, is bounded. Moreover, it follows from (2) and the inequality that where , . Moreover, where is constant. Therefore, we have

2.2. The Second Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay oscillation differential equations where is a differentiable function, is a strictly monotone increasing function and satisfies that and , there exists such that , is a constant, and is a given continuous mapping and satisfies the Lipschitz condition where are Lipschitz constants.

Now the VIM for (20) can read where for ; denotes the restrictive variation, that is, . Thus, we have Using integration by parts to (23), we have From the above formula, the stationary conditions are obtained as Moreover, the general Lagrange multiplier can be readily identified by (26). Thus, the variational iteration formula can be written as

Theorem 2. Suppose that the initial value problems (20) satisfy the condition (21), and , . Then the sequence defined by (28) with converges to the solution of (20).

Proof. The proof process is similar that in Theorem 1.

2.3. The Third Kind of Second-Order Delay Differential Equations

In order to improve the iteration speed, we modify the above iterative formulas and reconstruct the Lagrange multiplier. Consider the initial value problems of second-order delay differential equations where is a differentiable function, is a strictly monotone increasing function and satisfies that and , there exists such that , are bounded functions, and is a given continuous mapping and satisfies the Lipschitz condition where , are Lipschitz constants.

Now the VIM for (2.19) can read where for ; denotes the restrictive variation, that is, . Thus, we have Using integration by parts to (32), we have From the above formula, the stationary conditions are obtained as We suppose that , are the fundamental solutions of (35); the corresponding general solution of (35) is Using the initial conditions of (35), we have Note that is the Wronski determinant of , . We have Using the Liouville formula, we have So can be expressed as Note that Equation (40) can be expressed as Substituting (42) into (31) and (32), we obtain

Theorem 3. Suppose that the initial value problems (2.19) satisfy the condition (30), and , . Then the sequence defined by (43) and (44) with converges to the solution of (2.19).

Proof. The proof process is similar to that in Theorem 1.

3. Illustrative Examples

In this section, some illustrative examples are given to show the efficiency of the VIM for solving second-order delay differential equations.

Example 4. Consider the initial value problem of second-order differential equation with pantograph delay with the exact solution . Using the VIM given in formulas (9) and (10), we construct the correction functional We take as the initial approximation and obtain that The exact and approximate solutions are plotted in Figure 1, which shows that the method gives a very good approximation to the exact solution.

634670.fig.001
Figure 1: Results for Example 4.

Example 5. Consider the second-order delay differential equation Using the VIM given in formulas (28), we construct the correction functional We take as the initial approximation, and obtain that

Example 6. Consider the second-order delay differential equation with the exact solution . From (35), we can solve that . Using the VIM given in formulas (43) and (44), we construct the correction functional We take as the initial approximation and obtain that We use the iterative formulas (9) and (43) for Example 6, respectively. When the iteration number , the corresponding relative errors are showed in Table 1.
Table 1 shows that the iteration speed of the iterative formula (43) for Example 6 is much faster than that of iterative formula (9). This demonstrates that it is important to choose a reasonable Lagrange multiplier.

tab1
Table 1: The errors of the iteration solutions.

4. Conclusion

In this paper, we apply the VIM to obtain the analytical or approximate analytical solutions of second-order delay differential equations. Some illustrative examples show that this method gives a very good approximation to the exact solution. The VIM is a promising method for second-order delay differential equations.

Acknowledgments

The authors would like to thank the projects from NSF of China (11226322, 11271311), the Program for Changjiang Scholars and the Innovative Research Team in the University of China (IRT1179), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China, and the Fund Project of Hunan Province Education Office (11C1120).

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