Review
New applications of variational iteration method

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Abstract

The variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled Schrodinger-KdV, generalized KdV and shallow water equations. The exact and numerical solutions obtained by variational iteration method are compared with that obtained using Adomian decomposition method. The comparison shows that the two solutions obtained are excellent agreement. The method is made, showing that the former is more effective than the latter. In this paper, He’s variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomials.

Introduction

In various fields of science and engineering, nonlinear evalution equations, as well as their analytic and numerical solutions, are fundamentally important. One of the most attractive and surprising wave phenomena is the creation of solitary waves or solitons. It was approximately two centuries ago that an adequate theory for solitary waves was developed, in the form of modified wave equation known as the KdV equation [1], [2], [3].

It is well known that nonlinear phenomena are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and chemical physics. Calculating exact and numerical solutions, and in particular, traveling wave solutions, of nonlinear equations in mathematical physics plays an important role in soliton theory [3], [4]. Many explicit exact methods have been introduced in the literature [5], [6]. Many authors mainly had paid attention to study solutions of nonlinear equations by using various methods, such as Backlund transformation [7], [8], Darboux transformation [8], Hirota’s bilinear method [7], the tanh method [9], the sine-cosine method [10], [13], the homogeneous balance method [11], [12], and the Riccati expansion method with constant coefficients [14], [15]. Recently, the Adomian decomposition method is used to study the coupled Schrodinger-KdV, generalized KdV equations [16], [17] and shallow water equations [18].

The variational iteration method was first proposed by He [19], [20], [21], [22] and was successfully applied to autonomous ordinary differential equations in [23], to nonlinear polycrystalline solids in [24], and other fields. The combination of a perturbation method, variational iteration method, method of variation of constants and averaging method to establish an approximate solution of one degree of freedom to weakly nonlinear system in [25]. The variational iteration method has many merits and has much advantages over the Adomian method [26].

The motivation of this paper is to extended the analysis of the variational iteration method proposed by He [19], [20], [21], [22], He [27], [28] and [29] to solve three different types of nonlinear equations namely, coupled Schrodinger-KdV, generalized KdV and shallow water equations and to compare with that obtained by other works [16], [17], [18].

Section snippets

Variational iteration method

To illustrate the basic concepts of variational iteration method, we consider the following differential equationLu+Nu=g(x),where L is a linear operator, N a nonlinear operator, and g(x) an inhomogeneous term.

According to the variational iteration method, we can construct a correct functional as followsun+1(x)=un(x)+0xλ{Lun(τ)+Nũn(τ)g(τ)}dτ,where λ is a general Lagrangian multiplier [19], [20], [22], which can be identified optimally via the variational theory, the subscript ndenotes the n th

Coupled Schrodinger-KdV equations

We consider first the coupled Schrodinger-KdV equations [17] iut(uxx+uv)=0,vt+6uvx+vxxx(|u|2)x=0,with an initial conditionu(x,0)=62eiαxk2sech2(kx),v(x,0)=α+16k2316k2tanh2(kx),where α and k are arbitrary constants.

To solve Eqs. (3) by means of variational iteration method, we construct a correctional functional which readsun+1(x,t)=un(x,t)+0tλ1(unt+i(unxx+unv˜n))dτ,vn+1(x,t)=vn(x,t)+0tλ2(vnt+6unv˜nx+vnxxxũ2)dτ,where δunv˜nx,δunv˜n and ũ2 are considered as restricted variations. Its

Conclusions

In this paper, the variational iteration method has been successfully used to study three different models of special interest in mathematics and physics, namely, Schrodinger-KdV, generalized KdV and shallow water equations. The solution obtained by means of the variational iteration method is an infinite power series for appropriate initial condition, which can be, in turn, expressed in a closed form. The exact solutions are compared with that found by others by using Adomian decomposition

Acknowledgement

The authors would like to thank the referees for their comments and discussions.

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      Citation Excerpt :

      The variational iteration approach proposed by He [9] for nonlinear differential equations has been employed to address complex problems involving seepage flow with fractional derivatives and a nonlinear oscillator [8]. Various researchers handled many real life application’s including nonlinear heat transfer and porous media problems [6], the Korteweg–de Vries equation [21], Burgers equations [22], Riccati equations [2], Jaulent–Miodek equations [6], Helmholtz equation [18], KdV equations [24], evolution equations [25], Boussinesq equations [26], logarithmic Schrödinger equations [27], Lane–Emden problems [7], dispersive water wave phenomena [4], a SIR epidemic model [20] and others [3,10,15,19] using the VIM. Other mathematical developments concerning this method can be found in [11,12,20,25,27,29] and references therein.

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