Applying He ’ s Variational Iteration Method for Solving Differential-Difference Equation

We extend He’s variational iteration method VIM to find the approximate solutions for nonlinear differential-difference equation. Simple but typical examples are applied to illustrate the validity and great potential of the generalized variational iteration method in solving nonlinear differentialdifference equation. The results reveal that the method is very effective and simple. We find the extended method for nonlinear differential-difference equation is of good accuracy.


Introduction
In recent years, some promising approximate analytical solutions are proposed, such as expfunction method 1 , homotopy perturbation method 2-11 , and variational iteration method VIM 12-25 .The variational iteration method is the most effective and convenient one for both weakly and strongly nonlinear equations.This method has been shown to effectively, easily, and accurately solve a large class of nonlinear problems with component converging rapidly to accurate solutions.
Differential-difference equations DDEs have been the focus of many nonlinear studies.DDEs describe many important phenomena and dynamical processes in many different fields, such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains, and so on.DDEs play important role in the study of modern physics and also play a crucial role in numerical simulations of nonlinear partial differential equations NLPDEs , queueing problems, and discretization in solid state and quantum physics.At the same time, finding exact solutions of DDEs is extremely important in mathematical physics.
On the other hand, in order to find directly exact solutions to DDEs, some methods 16-26 for solving nonlinear differential equations are applied to DDEs.For example, Dehghan and Shakeri 16 have extended successfully multilinear variable separation approach to special DDEs.Baldwin et al. 26 ,Wang et al. 27 have applied homotopy analysis method HAM to DDEs.Dai and Zhang 28 have given a Jacobian elliptic function expansion method to solve the doubly periodic traveling wave solutions and kink-type tanh solitary solutions to some DDEs.There also have been some methods for nonlinear DDEs, such as Backlund transformation 29, 30 , Hirota method 31, 32 , Darboux transformation 33 , and Adomian decomposition method 34 .

He's variational iteration method
Now, to illustrate the basic concept of He's variational iteration method, we consider the following general nonlinear differential equation given in the form where L is a linear operator, N is a nonlinear operator, and g t is a known analytical function.We can construct a correction functional according to the variational method as where λ is a general Lagrange multiplier, which can be identified optimally via variational theory, the subscript n denotes the nth approximation, and u n is considered as a restricted variation, namely δ u n 0.
In the following example, we will illustrate the usefulness and effectiveness of the proposed technique.

Application to Volterra equation
Consider the following Volterra equation: with the initial condition whose exact solution can be written as We apply variational iteration method to the discussed problem.Using He's variational iteration method, the correction functional can be written in the form The stationary conditions 1 λ 0, λ 0 3.3 follow immediately.This in turn gives λ −1.

3.4
Substituting this value of the Lagrange multiplier λ −1 into the functional 3.2 gives the iteration formula We can start with u n,0 n, and we obtain the following successive approximations:

3.8
The closed form of the series 3.8 is u n t n/ 1 − 2t which gives exact solution of problem.

Application to mKDV lattice equation
Consider the following discretized mKDV lattice equation: with the initial condition We apply variational iteration method to the discussed problem.Using He's variational iteration method, the correction functional can be written in the form

4.3
Substituting this value of the Lagrange multiplier λ −1 into the functional 4.1 gives the iteration formula We can start with u n,0 tanh k tanh kn , and we obtain the following successive approximations: The other components of u n,m t can be generated in a similar way.Generally speaking, it is possible to calculate more components via some calculation software such as Maple to improve the accuracy of the approximate solutions.In order to verify numerically whether the proposed methodology leads to high accuracy, we evaluate the numerical solutions using only second-order approximation and compared it with Adomian decomposition solution ADM using six-term approximation 34 .Tables 1 and 2 show the absolute errors between ADM-u 6 and numerical solution VIM-u 2 of 16a with initial condition 16b .
Tables 1 and 2 show that the numerical approximate solution has a high degree of accuracy.As we know, the more terms added to the approximate solution, the more accurate it will be.Although we only considered second-order approximation, it achieves a high level of accuracy.

Conclusion
In this paper, by the variational iteration method, firstly, we obtain the exact solution of Volterra equation.Secondly, we obtain the approximate solution of mKDV lattice equation.The method is extremely simple, easy to use, and is very accurate for solving nonlinear differential-difference equation.Also, the method is a powerful tool to search for solutions of various linear/nonlinear problems.This variational iteration method will become a much more interesting method to solve nonlinear DDEs in science and engineering.

Table 1 :
For constant k 0.1, and time t 0.5.

Table 2 :
For constant k 0.1, and time t 1.5.