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

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.


Introduction
The second-order delay differential equations often appear in the dynamical system, celestial mechanics, kinematics, and so forth. Some numerical methods for solving secondorder 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 [16][17][18][19] 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.

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 secondorder delay differential equations where 0 , 1 are Lipschitz constants.

Illustrative Examples
In this section, some illustrative examples are given to show the efficiency of the VIM for solving second-order delay differential equations.
We take 0 ( ) = 2 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.
We take 0 ( ) = 4 as the initial approximation, and obtain that  Example 6. Consider the second-order delay differential equation with the exact solution ( ) = 2 . From (35), we can solve that ( , ) = − + 2 / . Using the VIM given in formulas (43) We take 0 ( ) = 2 as the initial approximation and obtain that We use the iterative formulas (9) and (43) for Example 6, respectively. When the iteration number = 2, 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.

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.