An Application of He's Variational Iteration Method for Solving Duffing-Van Der Pol Equation

In this paper, we apply He's variational iteration method (VIM) and the Adomian decomposition method (ADM) to approximate the solution of Duffing-Van Der Pol equation (DVP). In VIM, a correction functional is constructed by a general Lagrange multiplier which can be identified via a variational theory. The VIM yields an approximate solution in the form of a quickly convergent series. Comparisons of the two series solutions with the classical Runge-Kutta order four RK45 method show that the VIM is a powerful method for the solution of nonlinear equations. The convergent of He's variational iteration method to this equation is also considered.


Introduction
Chaotic systems have received a flurry of research effort in the past few decades. Such systems which are nonlinear by nature, can occur in various natural and manmade systems, and are characterized by a great sensitivity to initial conditions [16].
The Duffing ‫ـ‬ Van der Pol equation provides an important mathematical model for dynamic systems having a single unstable fixed point, along with a single stable limit cycle and is governed by the non-linear differential equation where, the over dot represents the derivative with respect to time,  and  which are positive coefficients. It generates the limit cycle for small values of μ, developing into relaxation oscillations when μ becomes large which can be evaluated through the Lindsted's perturbation method [6]. Examples of such phenomena arise in all of the natural and engineering sciences [17,18], in many physical problems, as well [8,11]. Most scientific problems in solid mechanic are inherently non-linear. Except a limited number of these problems, most of them do not have analytical solution. Some of them are solved by using numerical techniques and some are done so the analytical perturbation method [19].
Recently introduced variational iteration method by He [7,[12][13][14][15] which gives rapidly convergent successive approximations of the exact solution if such a solution exists, has proved successful in deriving analytical and approximate solutions of linear and nonlinear differential equations. This method is preferable over numerical methods as it is free from rounding off errors and neither requires large computer power/memory. He [13,14,22] has applied this method for solving analytical solutions of autonomous ordinary differential equation, non-linear partial differential equations with variable coefficients and integro -differential equations. The variational iteration method was successfully applied to Burger's and coupled Burger's equations [1], to Schruodinger-KdV, generalized KdV and shallow water equations [2], to linear Helmholtz partial differential equation [9], to seventh order Sawada -Kotera equation [10], to Van der Pol-Duffing Oscillators [21], Linear and nonlinear wave equations, KdV, K(2,2), Burgers, and cubic Boussinesq equations have been solved by Wazwaz [23,24] by using the variational iteration method.
In the present paper we employ VIM method for solving Duffing ‫ـ‬ Van der Pol equation. Further, we compare the result with the given solutions by using Adomian Decomposition Method [3,4,20] and we prove the convergence of the method.

Adomian Decomposition Method for Solving Duffing -Van der Pol Equation
To solve eq. (1), ADM is employed. We rewrite it in the following form 0 ) is either a linear or a nonlinear operator. The principle of the ADM is based on the decomposition of the non-linear operator A in the following form: Where, L+R is linear, N non-linear, L invertible with L -1 as inverse defined by As usual in ADM the solutions of Eq. (3) can be considered to be as the sum of the following infinite series Where n A are called Adomian Polynomials [3,4,5].

Computing Adomian Polynomials for Equation (1)
Computing Adomian Polynomial by Algorithm (2.1) yields to Now, substituting (8) in (7) and so on … The five terms of the approximations to the solutions are considered as for the convergence of the method, we refer the reader to [5] in which the problem of convergence has been discussed briefly .

He's Variational Iteration Method for Solving Duffing -Van der Pol Equation
To explain the basic idea of He's variational iteration method (VIM), we consider a general nonlinear oscillator with specified initial conditions (2) as follows (more general form can be considered without the loss of generality) 0 ) , , where, f andare continuous nonlinear operators with respect to their arguments, g and ) t ( u is an unknown variable. We first consider Eq. (10) as …(11) and for Eq.(1) we have: denotes the linear operator with respect to u and N is a non-linear operator with respect to u .We then construct a correction functional for Eq.(11) as [12] where, ) ( 0 t u is the initial guess and the subscript n denotes the n-th iteration,

Numerical Results
The numerical approach for (9) and (20) is computed by using Matlab. We consider the following four cases Case 1: , and the initial conditions 1 ) , and the initial conditions

Convergence Analysis:
The He's variational iteration formula makes a recurrence sequence   ) (t u n . Obviously, the limit of the sequence will be the solution of eq.(10) if the sequence is convergent. In this section, we give a new proof of convergence of He's variational Where L is as noted in (12).

Proof
Suppose n u and 1 + n u satisfy the variational iteration formula (19). Applying …(33) On the other hand, using the specified initial conditions and the definition of the initial guess, we have

Conclusions
In this work, we have given a new proof of convergence of He's Variational Iteration Method by presenting a new formulation of He's method. We have compared this method with ADM and RK45, and we can conclude that the main property of this method is in its flexibility and ability to solve Duffing -Van Der Pol accurately and conveniently without decomposing the non-linear terms, which are very complex. This technique gives an accurate and easy computable solution by means of a truncated series whose convergence is fast.