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

Abstract

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 differential-difference 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.

1. Introduction

In recent years, some promising approximate analytical solutions are proposed, such as exp-function 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].

2. 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 𝐿𝑢(𝑡)+𝑁𝑢(𝑡)=𝑔(𝑡), (2.1) 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 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀𝜆(𝐿𝑢_𝑛(𝜉)+𝑁𝑢^{0𝑥002𝑑𝑐}_𝑛(𝜉)−𝑔(𝜉))𝑑𝜉, (2.2) where 𝜆 is a general Lagrange multiplier, which can be identified optimally via variational theory, the subscript 𝑛 denotes the 𝑛th approximation, and ^0𝑥0𝑢^02𝑑𝑐_𝑛 is considered as a restricted variation, namely 𝛿𝑢^{0𝑥002𝑑𝑐}_𝑛=0.

In the following example, we will illustrate the usefulness and effectiveness of the proposed technique.

3. Application to Volterra Equation

Consider the following Volterra equation: 𝑑𝑢_𝑛𝑑𝑡=𝑢_𝑛(𝑢_𝑛+1−𝑢_𝑛−1), (3a) with the initial condition 𝑢_𝑛(0)=𝑛, (3b) whose exact solution can be written as 𝑢_𝑛𝑛(𝑡)=1−2𝑡. (3.1)

We apply variational iteration method to the discussed problem. Using He’s variational iteration method, the correction functional can be written in the form 𝑢_𝑛,𝑚+1(𝑡)=𝑢_𝑛,𝑚(𝑡)+^𝑡₀𝜆(𝑠){𝑑𝑢_𝑛,𝑚(𝑠)𝑑𝑠−(𝑢_𝑛,𝑚(𝑠))(𝑢_𝑛+1,𝑚(𝑠)−𝑢_{𝑛−1,𝑚}(𝑠))}𝑑𝑠. (3.2) The stationary conditions 𝜆1+𝜆=0,_′=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 𝑢_𝑛,𝑚+1(𝑡)=𝑢_𝑛,𝑚(𝑡)−^𝑡₀{𝑑𝑢_𝑛,𝑚(𝑠)𝑑𝑠−(𝑢_𝑛,𝑚(𝑠))(𝑢_𝑛+1,𝑚(𝑠)−𝑢_{𝑛−1,𝑚}(𝑠))}𝑑𝑠. (3.5) We can start with 𝑢_𝑛,0=𝑛, and we obtain the following successive approximations: 𝑢_𝑛,0𝑢(𝑡)=𝑛,_𝑛,1𝑢(𝑡)=𝑛+2𝑛𝑡,_𝑛,2(𝑡)=𝑛+2𝑛𝑡+4𝑛𝑡²,𝑢_𝑛,3(𝑡)=𝑛+2𝑛𝑡+4𝑛𝑡²+8𝑛𝑡³,𝑢_𝑛,4(𝑡)=𝑛+2𝑛𝑡+4𝑛𝑡²+8𝑛𝑡³+16𝑛𝑡⁴. (3.6) Hence, the solution series in general gives 𝑢_𝑛(𝑡)=𝑛+2𝑛𝑡+4𝑛𝑡²+8𝑛𝑡³+16𝑛𝑡⁴𝑢…,_𝑛(𝑡)=𝑛(1+2𝑡+4𝑡²+8𝑡³+16𝑡⁴…). (3.7)(3.8) The closed form of the series (3.8) is 𝑢_𝑛(𝑡)=𝑛/(1−2𝑡) which gives exact solution of problem.

4. Application to mKDV Lattice Equation

Consider the following discretized mKDV lattice equation: 𝑑𝑢_𝑛𝑑𝑡=(1−𝑢²_𝑛)(𝑢_𝑛+1−𝑢_𝑛−1), (16a) with the initial condition 𝑢_𝑛(0)=tanh(𝑘)tanh(𝑘𝑛). (16b)

We apply variational iteration method to the discussed problem. Using He’s variational iteration method, the correction functional can be written in the form 𝑢_𝑛,𝑚+1(𝑡)=𝑢_𝑛,𝑚(𝑡)+^𝑡₀𝜆(𝑠){𝑑𝑢_𝑛,𝑚(𝑠)𝑑𝑠−(1−𝑢²_𝑛,𝑚(𝑠))(𝑢_𝑛+1,𝑚(𝑠)−𝑢_{𝑛−1,𝑚}(𝑠))}𝑑𝑠. (4.1) The stationary conditions 𝜆1+𝜆=0,_′=0 (4.2) follow immediately. This in turn gives 𝜆=−1. (4.3) Substituting this value of the Lagrange multiplier 𝜆=−1 into the functional (4.1) gives the iteration formula 𝑢_𝑛,𝑚+1(𝑡)=𝑢_𝑛,𝑚(𝑡)−^𝑡₀{𝑑𝑢_𝑛,𝑚(𝑠)𝑑𝑠−(1−𝑢²_𝑛,𝑚(𝑠))(𝑢_𝑛+1,𝑚(𝑠)−𝑢_{𝑛−1,𝑚}(𝑠))}𝑑𝑠. (4.4) We can start with 𝑢_𝑛,0=tanh(𝑘)tanh(𝑘𝑛), and we obtain the following successive approximations:𝑢_𝑛,0𝑢(𝑡)=tanh(𝑘)tanh(𝑘𝑛),_𝑛,1(𝑡)=tanh(𝑘)tanh(𝑘𝑛)+[tanh(𝑘)(tanh(𝑘(𝑛+1))−tanh(𝑘(𝑛−1)))−tanh²(𝑘)tanh²𝑢(𝑘𝑛)(tanh(𝑘)(tanh(𝑘(𝑛+1))−tanh(𝑘(𝑛−1))))]𝑡,_𝑛,2(𝑡)=tanh(𝑘)tanh(𝑘𝑛)+[tanh(𝑘)(tanh(𝑘(𝑛+1))−tanh(𝑘(𝑛−1)))−tanh²(𝑘)tanh²(𝑘𝑛)(tanh(𝑘)(tanh(𝑘(𝑛+1))−tanh(𝑘(𝑛−1))))]𝑡+[tanh(𝑘)tanh(𝑘(𝑛+2))−2tanh(𝑘)tanh(𝑘𝑛)−tanh²(𝑘)tanh²(𝑘(𝑛+1))(tanh(𝑘)tanh(𝑘(𝑛+2)))−tanh(𝑘)tanh(𝑘𝑛)+tanh(𝑘)tanh(𝑘(𝑛−2))+tanh²(𝑘)tanh²(𝑘(𝑛−1))(tanh(𝑘)tanh(𝑘𝑛))−tanh(𝑘)tanh(𝑘(𝑛−2))−2tanh(𝑘)tanh(𝑘𝑛)tanh(𝑘)tanh(𝑘(𝑛+1))−tanh(𝑘)tanh(𝑘(𝑛−1))tanh(𝑘)tanh(𝑘(𝑛+1))−tanh(𝑘)tanh(𝑘(𝑛−1))−tanh²(𝑘)tanh²(𝑘𝑛)tanh(𝑘)tanh(𝑘(𝑛+1))−tanh(𝑘)tanh(𝑘(𝑛−1))−tanh²(𝑘)tanh²(𝑘𝑛)(tanh(𝑘)tanh(𝑘(𝑛+2))−2tanh(𝑘)tanh(𝑘𝑛)−tanh²(𝑘)tanh²(𝑘(𝑛+1)))×(tanh(𝑘)tanh(𝑘(𝑛+2))−tanh(𝑘)tanh(𝑘𝑛))+tanh(𝑘)tanh(𝑘(𝑛−2))+tanh²(𝑘)tanh²(𝑘(𝑛−1))tanh(𝑘)tanh(𝑘𝑛)−tanh(𝑘)tanh(𝑘(𝑛−2))]0.5𝑡². (4.5)

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-𝑢₆ and numerical solution (VIM-𝑢₂) of (16𝑎) with initial condition (16𝑏).

Table 1 

For constant 𝑘=0.1, and time 𝑡=0.5.

Table 2 

For constant 𝑘=0.1, and time 𝑡=1.5.

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.

5. 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.

Acknowledgment

The author thanks The Scientific and Technological Research Council of Turkey (TÜB İ TAK) for their financial support.