Variational Homotopy Perturbation Method for Solving Benjamin-Bona-Mahony Equation

In this article, the application of variational homotopy perturbation method is applied to solve Benjamin-Bona-Mahony equation. Then, we obtain the numerical solution of BBM equation using the initial condition. Comparison with Adomian’s decomposition method, homotopy perturbation method, and with the exact solution shows that VHPM is more effective and accurate than ADM and HPM, and is reliable and manageable for this type of equation.


Introduction
Most scientific problems arise in real-world physical problems such as plasma physics, fluid mechanic, solid state physics and in many branches of chemistry [1].The Benjamin-Bona-Mahony (BBM) equation is inherently of nonlinearity.We know that except a limited number of these problems, most of them do not have analytical solution.Therefore, these nonlinear equations should be solved using numerical methods: variational iteration method (VIM) [2] and homotopy-perturbation method (HPM) [3].These methods are the most effective and convenient ones for both weakly and strongly nonlinear equations.In this article, VHPM is used to solve nonlinear Benjamin-Bona-Mahony [4] and [5].The Benjamin-Bona-Mahony equation (BBM equation) is the partial differential equation 0 This equation was introduced in (Benjamin T. B., Bona J. L. & Mahony J. J. 1972) [5] as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in 1 + 1 dimensions.
The main goal of this paper is to find the approximate solution of the BBM by the variational homotopy perturbation method that has already been successfully applied to several nonlinear problems.

Mathematical Model
The generalized BBM (Benjamin-Bona-Mahony) equation has a higher order nonlinearity of the form 0, 1, where a is constant.
The case 1 n = corresponds to the BBM equation 0, which was first proposed in 1972 by Benjamin et al. [5].This equation is an alternative to the Korteweg-de Vries (KdV) equation, and describes the unidirectional propagation of small-amplitude long waves on the surface of water in channel.The BBM equation is not only convenient for shallow water waves but also for hydro magnetic waves, acoustic waves, and therefore it has more advantages compared with the KdV equation.When 2 n = , Equation (2.1) is called the modified BBM equation:

Variational Iteration Method
To clarify the basic ideas of VIM, we consider the following differential equation ( ) where L is a linear operator is defined by , m ∈  , N is a nonlinear operator and ( ) , g x t is a known analytic function.According to VIM, we can write down a correction functional as follows: where λ is a general lagrangian multiplier [6] defined as: The subscript n indicates the nth approximation and n u  is considered as a restricted variation [7].

Homotopy Perturbation Method
To illustrate the basic ideas of this method, we consider the following nonlinear differential equation With the following boundary conditions where A is a general differential operator, B a boundary operator,

( )
f r is a known analytical function and τ is the boundary of the domain δ .The operator A can be decomposed into two operators L and N, where L is a linear, and N a nonlinear operator.Equation (3.2.1) can be written as follows: By using the homotopy technique, we construct a homotopy: Which are satisfies: where is an embedding parameter, 0 u is an initial approximation for the solution of Equation (3.2.1), which satisfies the boundary conditions.Obviously, from Equations (3.2.5) and (3.2.6) we have: The changing process of p forms zero to unity is just that of ( ) In topology, this is called homotopy.According to the (HPM), we can first use the embedding parameter as a small parameter, And assume that the solution of Equations (3.2.5) and (3.2.6) can be written as a power series in p: The combination of the perturbation method and the homotopy method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods.On the other hand, this technique can have full advantage of the traditional perturbation techniques.
The series (3.2.10) is convergent for most cases.Some criteria are suggested for convergence of the series (3.2.10) [8].

Variational Homotopy Perturbation Method
To illustrate the concept of the variational homotopy perturbation method, we consider the general differential Equation (3.1).We construct the correction functional (3.2) and apply the homotopy perturbation method (3.2.9) to obtain: As we see, the procedure is formulated by the coupling of variational iteration method and homotopy perturbation method.A comparison of like powers of p gives solutions of various orders [9].
The correct functional is given as where n u  is considered as a restricted variation.Making the above functional stationary, the Lagrange multip- lier can be determined as Equation (3.3) is ( ) , 1 x t λ = − , which yields the following iteration formula: Applying the variational homotopy perturbation method, we have: ( ) ( ) Comparing the coefficient of like powers of p, we have  The other components of the VHPM can be determined in a similar way.Finally, the approximate solution of Equation (4.1) in a series form is

Figures and Tables
In this section, the results obtained by VHPM are tabulated in the following tables, followed by their figures (Figures 1-12) and comparisons.

Conclusions
In this paper, we have successfully used variational homotopy perturbation method for solving the Benjamin-Bona-Mahony equation, it is apparently seen that VHPM is very powerful and efficient technique in finding

Table 2 .
Absolute error of VHPM with exact solution.

Table 4 .
Absolute error between VHPM and exact solution.