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

Authors: Ahmet Yıldırım
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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(2.1)Lu(t)+Nu(t)=g(t), 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 (2.2)un+1(t)=un(t)+∫0tλ(Lun(ξ)+Nu˜n(ξ)−g(ξ))dξ, 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 δu˜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:(3a)dundt=un(un+1−un−1), with
the initial condition (3b)un(0)=n, whose
exact solution can be written as (3.1)un(t)=n1−2t.We apply variational iteration method to the discussed
problem. Using He’s variational iteration method, the correction functional can
be written in the form(3.2)un,m+1(t)=un,m(t)+∫0tλ(s){dun,m(s)ds−(un,m(s))(un+1,m(s)−un−1,m(s))}ds. The stationary conditions (3.3)1+λ=0,λ′=0 follow immediately. This in turn gives (3.4)λ=−1. Substituting this value of the Lagrange
multiplier λ=−1 into the functional (3.2) gives the iteration
formula (3.5)un,m+1(t)=un,m(t)−∫0t{dun,m(s)ds−(un,m(s))(un+1,m(s)−un−1,m(s))}ds. We can start with un,0=n, and we obtain the following successive
approximations: (3.6)un,0(t)=n,un,1(t)=n+2nt,un,2(t)=n+2nt+4nt2,un,3(t)=n+2nt+4nt2+8nt3,un,4(t)=n+2nt+4nt2+8nt3+16nt4.         Hence, the solution series in general gives (3.7)un(t)=n+2nt+4nt2+8nt3+16nt4…,(3.8)un(t)=n(1+2t+4t2+8t3+16t4…). The closed form of the series (3.8) is un(t)=n/(1−2t) which gives exact solution of problem.


## 4. Application to mKDV Lattice Equation

Consider the following discretized mKDV
lattice equation:(16a)dundt=(1−un2)(un+1−un−1), with
the initial condition (16b)un(0)=tanh(k)tanh(kn).We
apply variational iteration method to the discussed problem. Using He’s variational
iteration method, the correction functional can be written in the form(4.1)un,m+1(t)=un,m(t)+∫0tλ(s){dun,m(s)ds−(1−un,m2(s))(un+1,m(s)−un−1,m(s))}ds. The stationary conditions (4.2)1+λ=0,λ′=0 follow immediately. This in turn gives (4.3)λ=−1. Substituting this value of the Lagrange
multiplier λ=−1 into the functional (4.1) gives the iteration
formula (4.4)un,m+1(t)=un,m(t)−∫0t{dun,m(s)ds−(1−un,m2(s))(un+1,m(s)−un−1,m(s))}ds. We can start with un,0=tanh(k)tanh(kn), and we obtain the following successive
approximations:(4.5)un,0(t)=tanh(k)tanh(kn),un,1(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,un,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.The
other components ofun,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 constantk=0.1, and time t=0.5.
nADM-u6VIM-u2Absolute error−25−0,09804197166−0,098043733310.00000176165−15−0,08824837298−0,088257281530,00000890855−5−0,03789706610−0,037886124150,0000109419500,0099009469920,0099337091490,00003276215750,053503102820,053510810230,00000770741150,0918555873270,091847455910,00000813142250,098573655420,098572052190,00000160323Table 2
For constantk=0.1, and time t=1.5.
nADM-u6VIM-u2Absolute error−25−0,09725516662−0,097307607880,00005244126−15−0,083118934180−0,083371566770,00025263259−5−0,01977150813−0,019382852770,0003886553600,028944780180,028901127440,0000436527450,066130631220,066256911010,00012627979150,094355539040,094142089920,00021344912250,098932183370,098892564530,00003961884Tables1 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.



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