Elsevier

Chaos, Solitons & Fractals

Volume 39, Issue 3, 15 February 2009, Pages 1297-1303
Chaos, Solitons & Fractals

Variational iteration method for solving non-linear partial differential equations

https://doi.org/10.1016/j.chaos.2007.06.025Get rights and content

Abstract

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.

Introduction

Differential equations are widely used to describe physical problems. In most cases, these problems may be too complicated to solve exactly. Alternatively, the numerical methods can provide approximate solutions rather than the exact solutions. In fact there are many of methods to solve these problems numerically such as: finite difference methods, multi-grid methods, perturbation methods, but most of these methods are of low accuracy and are of high expensive. In this paper, the variational iteration method has been shown to solve effectively, easily and accurately a large class of linear and non-linear problems with approximations converging rapidly to accurate solutions. In 1978, Inokuti et al. [1] proposed a general Lagrange multiplier method to solve analytically non-linear problems, at first the proposed method used to solve problems in quantum mechanics. The main feature of the proposed method is: the solution of a mathematical problem with linearization assumption is used as initial approximation (trial-function), then a more highly precise approximation at some special point can be obtained. More details of this method is illustrated in the following problems.

Problem 1 Duffing equation

This equation is used widely by many perturbation techniques to verify their effectiveness, here we shall use Duffing equation with non-linearity of seventh order to illustrate the general evaluation process of the proposed method. Considering:d2udt2+u+εu7=0,u(0)=1,u(0)=0.

The correction functional for this problem takes the form:un+1(t)=un(t)+0tλd2un(τ)dτ2+un(τ)+εu˜n7(τ)dτ,where u˜n is a restricted variation [2], [3], [4], [5], [6], [7], i.e. δu˜n=0. the last term of the right is called “correction”, λ is a general Lagrange multiplier [1], which can be identified optimally via variational theory [1], [2], [3].

The stationary conditions for this problem takes the form:λ(τ)+λ(τ)=0,λ(τ)|τ=t=0,1-λ(τ)|τ=t=0.The multiplier, therefore, can be identified as λ = sin(τt), and the following variational iteration formula can be obtained:un+1(t)=un(t)+0tsin(τ-t)d2un(τ)dτ2+un(τ)+εun7(τ)dτ.Starting with the trial function: u0 = cos αt, where α  0 is a non-zero unknown constant. By the variational iteration formulae (4) we have:u1(t)=cosαt+0tsin(τ-t)-α2+1+35ε64cosατ+21ε64cos3ατ+7ε64cos5ατ+ε64cos7ατdτ.The constant α can be identified by various methods such as least square method. To simplify the integral, we set:α=1+3564ε.Therefore, we have:u1(t)=cosαt+0tsin(τ-t)21ε64cos3ατ+7ε64cos5ατ+ε64cos7ατdτ=cosατ+21ε64(9α2-1)(cos3αt-cost)+7ε64(25α2-1)(cos5αt-cost)+ε64(49α2-1)(cos7αt-cost),with α defined by Eq. (6), which is more accurate than the results can be obtained by perturbations.

Problem 2

In this problem the proposed technique can be also extended to the following partial differential equation (PDE) [4], [5], [6], [7], [8]:2u+uy2=2y+x4,with the boundary and initial conditions:u(0,y)=0,ux(0,y)=a,u(x,0)=ax,uy(x,0)=x2.

The correction variational functional in x- and y-directions can be expressed, respectively, as follows:un+1(x,y)=un(x,y)+0xλ12un(ξ,y)ξ2+2u˜n(ξ,y)y2+u˜n(ξ,y)y2-2y-ξ4dξ,un+1(x,y)=un(x,y)+0yλ22u˜n(x,η)x2+2un(x,η)η2+u˜n(x,η)η2-2η-x4dηwhere u˜n is a restricted variation. The stationary conditions for this problem takes the form:λ1(ξ)=0,λ1(ξ)|ξ=x=0,1-λ1(ξ)|ξ=x=0,λ2(η)=0,λ2(η)|η=y=0,1-λ2(η)|η=y=0.The Lagrange multipliers can be identified as:λ1=ξ-x,λ2=η-y.Therefore, the iteration formulae in x- and y-directions are as follows:un+1(x,y)=un(x,y)+0x(ξ-x)2un(ξ,y)ξ2+2un(ξ,y)y2+un(ξ,y)y2-2y-ξ4dξ,un+1(x,y)=un(x,y)+0y(η-y)2un(x,η)x2+2un(x,η)η2+un(x,η)η2-2η-x4dη.Now we start with the arbitrary initial approximation: u0 = A + Bx, where A and B are constants to be determined, by using the initial conditions (9a), thus we have:u0(x,y)=ax.By the variational iteration formula in x-direction (14), we can have:u1(x,y)=ax+0x(ξ-x)0+0+0-2y-ξ4dξ=ax+yx2+x630,which is the first approximation, in the same way we can have the following second approximation:u2(x,y)=ax+yx2=x(yx+a),which is an exact solution. The approximation can also be obtained by y-direction or alternate use of x- and y-directions iteration formula.

Problem 3

Again the proposed technique can be also extended to PDEs with mixed derivative terms. Consider the following problem:2u+uxy=2(x+y),with the boundary and initial conditions:

u(0,y)=0,ux(0,y)=a,u(x,0)=ax,uy(x,0)=x2.The correction variational functional in x- and y-directions can be expressed, respectively, as follows:un+1(x,y)=un(x,y)+0xλ12un(ξ,y)ξ2+2u˜n(ξ,y)y2+2u˜n(ξ,y)ξy-2ξ-2ydξ,un+1(x,y)=un(x,y)+0yλ22u˜n(x,η)x2+2un(x,η)η2+2u˜n(x,η)xη-2x-2ηdη,where u˜n is a restricted variation.

As in Eq. (13) the Lagrange multipliers can be easily identified:λ1=ξ-x,λ2=η-y.The iteration formulae in x- and y-directions can be, therefore expressed, respectively, as follows:un+1(x,y)=un(x,y)+0x(ξ-x)2un(ξ,y)ξ2+2un(ξ,y)y2+2un(ξ,y)ξy-2ξ-2ydξ,un+1(x,y)=un(x,y)+0y(η-y)2un(x,η)x2+2un(x,η)η2+2un(x,η)xη-2x-2ηdη.Now we start with the arbitrary initial approximation: u0 = A + Bx, where A and B are constants to be determined, by using the initial conditions (20a), thus we can obtain:u0(x,y)=ax.By the variational iteration formula in x-direction (24), we have:u1(x,y)=ax+0x(ξ-x)0+0+0-2ξ-2ydξ=x(yx+a)+x33.which is the first approximation, in the same way we can obtain the following second approximation:u2(x,y)=x(yx+a),which is an exact solution. The approximation can also be obtained by y- direction or alternate use of x- and y-directions iteration formulae.

Problem 4 Combined KdV–MKdV equation [9], [10]

The KdV and MKdV equations are most popular solution equations. The non-linear terms of KdV and MKdV equations often simultaneously exist in practical problems such as fluid physics, physics and quantum field theory, and form the following so-called combined KdV–MKdV equation:ut+puux+qu2ux+uxxx=0.

This equation may describe the wave propagation of the bound particle, sound wave and thermal pulse. In this example we shall solve this equation:ut+puux+qu2ux+uxxx=0,u(0,t)=-p2q+6c2qsechp2-4qc24qc2t,u(x,0)=-p2q+6c2qsechc2x.Its correction variational functional in x- and t-directions can be expressed, respectively, as follows:un+1(x,t)=un(x,t)+0xλ1u˜n(ξ,t)t+pu˜nu˜n(ξ,t)ξ+qu˜n2u˜n(ξ,t)ξ+3un(ξ,t)ξ3dξ,un+1(x,t)=un(x,t)+0tλ2un(x,τ)τ+pu˜nu˜n(x,τ)x+qu˜n2u˜n(x,τ)x+3u˜n(x,τ)x3dτ,where u˜n is a restricted variation. The stationary conditions for this problem takes the form:λ1(ξ)=0,1+λ1(ξ)|ξ=x=0,λ1(ξ)|ξ=x=0,λ1(ξ)|ξ=x=0,λ2(τ)=0,1+λ2(τ)|τ=t=0.The Lagrange multipliers can be easily identified:λ1=-12(ξ-x)2,λ2=-1.The iteration formulae in x- and t-directions can be written as:un+1(x,t)=un(x,t)-120x(ξ-x)2un(ξ,t)t+punun(ξ,t)ξ+qun2un(ξ,t)ξ+3un(ξ,t)ξ3dξun+1(x,t)=un(x,t)-0tun(x,τ)τ+punun(x,τ)x+qun2un(x,τ)x+3un(x,τ)x3dτNow we start with the arbitrary initial approximation: u0 = At + B, where A and B are functions to be determined, by the iteration formula in x-direction (35), we have:u1(x,t)=At+B-120x(ξ2-2xξ+x2)(A+0)dξ=At+B-A20x(ξ2-2xξ+x2)dξ=At+B-A2ξ33-xξ2+x2ξ0x=At+B-Ax36.Imposing the boundary conditions at t = 0 and x = 0, we have:-p2q+6c2qsechc2x=B-Ax36,-p2q+6c2qsechp2-4qc24qc2t=At+B.Solving Eqs. (38), (39) for A and B, we have:A=66t+x36c2qsechp2-4qc24qc2t-sechc2x,B=-p2q+6c2qsechp2-4qc24qc2t-6t6c2/q6t+x3sechp2-4qc24qc2t-sechc2x.Substituting A and B into Eq. (37) we have the following solution:u1(x,t)=-p2q+6c2qsechp2-4qc24qc2t-x36t+x36c2qsechp2-4qc24qc2t-sechc2x,which is an approximate solution satisfies the given boundary conditions.

Problem 5 Camassa–Holm equation [9], [11]

The Camassa–Holm equationut+2pux-uxxt+3uux=2uxuxx+uuxxx.Appeared first in a physical context as describing the shallow water approximation in invisible hydrodynamics. The variable u(x,t) represents the fluid velocity in the horizontal direction and p is a constant. Much of the interest on this equation come from the facts that it is completely integrable and possess peaked solution [9], [11]. The proposed method can be used to solve this equation:

ut+2pux-uxxt+3uux=2uxuxx+uuxxx,u(0,t)=-p+a1edt,u(1,t)=-p+a1e1+dt,u(x,0)=-p+a1ex,u(x,1)=-p+a1ex+d.Its correction variational functional in x- and t-directions can be expressed, respectively, as follows:un+1(x,t)=un(x,t)+0xλ1u˜n(ξ,t)t+2pun(ξ,t)ξ-3u˜n(ξ,t)ξ2t+3u˜n·u˜n(ξ,t)ξ-2u˜n(ξ,t)ξ·2u˜n(ξ,t)ξ2-u˜n·3u˜n(ξ,t)ξ3dξ,un+1(x,t)=un(x,t)+0tλ2un(x,τ)τ+2pu˜n(x,τ)x-3u˜n(x,τ)x2τ+3u˜n.u˜n(x,τ)x-2u˜n(x,τ)x.2u˜n(x,τ)x2-u˜n.3u˜n(x,τ)x3dτ,where u˜n is a restricted variation. The stationary conditions for this problem takes the form:2pλ1(ξ)=0,1+2pλ1(ξ)|ξ=x=0λ2(τ)=0,1+λ2(τ)|τ=t=0.The Lagrange multipliers can be easily identified:λ1=-12p,λ2=-1.The iteration formulae in x- and t-directions can be written as:un+1(x,t)=un(x,t)-12p0xun(ξ,t)t+2pun(ξ,t)ξ-3un(ξ,t)ξ2t+3un.un(ξ,t)ξ-2un(ξ,t)ξ.2un(ξ,t)ξ2-un.3un(ξ,t)ξ3dξ,un+1(x,t)=un(x,t)-0tun(x,τ)τ+2pun(x,τ)x-3un(x,τ)x2τ+3un.un(x,τ)x-2un(x,τ)x.2un(x,τ)x2-un.3un(x,τ)x3dτ.Now we start with the arbitrary initial approximation: u0  = A ex + B, where A and B are constants or functions to be determined, by the iteration formulae in x-direction, we have:u1(x,t)=Aex+B-12p0x[0+2p.Aeξ-0+3((Aeξ+B).Aeξ-2Aeξ.Aeξ)-(Aeξ+B).Aeξ]dξ=Aex+B-12p0x2pAeξ+2ABeξ]dξ=A+B+1pAB(1-ex).Imposing the boundary conditions at x = 0 and x = 1, we have:-p+a1edt=A+B,-p+a1e1+dt=A+B-1pAB(e-1).Solving Eqs. (50), (51) for A and B we have:A=a1edt,B=-p,orA=-p,B=a1edt.Substituting Eq. (52a) into Eq. (49) we have the following solution:u1(x,t)=-p+a1ex+dt,which is an exact solution to Camassa–Holm equation.

Remark

Camassa–Holm equation has other exact solutions which can be obtained if we change the boundary conditions.

Section snippets

Conclusion

In this paper we have studied some problems, which does not require small parameter in an equation as the perturbation techniques do. The Obtained results show that:

  • 1.

    A correction functional can be easily constructed by a general Lagrange multiplier, and the multiplier can be optimally identified by variational theory.

  • 2.

    The initial approximation can be freely chosen with unknown constants or functions, which can be determined via various methods.

  • 3.

    The approximations obtained by this method are valid

References (11)

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