To clarify the basic ideas of VIM, we consider the following differential equation

where is a linear operator defined by, is a nonlinear operator and is a known analytic function. According to (VIM), we can write down a correction functional as follows:

where is a general lagrangian multiplier by [14] defined as:

The subscript indicates the nth approximation and is considered as a restricted variation [15] .

To illustrate the basic idea of this method, we consider the following nonlinear differential equation:

with the boundary condition

where is a general differential operator, a boundary operator, a known analytical function and is the boundary of the domain. can be divided into two parts which are and, where is linear and is nonlinear. Equation (9) can therefore be rewritten as follows:

Homotopy perturbation structure is shown as follows:

where

In Equation (12), is an embedding parameter and is the first approximation that satisfies the boundary condition. We can assume that the solution of Equation (12) can be written as a power series in, as following:

and the best approximation for solution is:

It is well known that series (15) is convergent for most of the cases and also the rate of convergence depends on. We assume that Equation (15) has a unique solution [7] .

To illustrate the concept of the variational homotopy perturbation method, we consider the general differential Equation (6). We construct the correction functional (7) and apply the homotopy perturbation method (14) 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 gives solutions of various orders.

Consider the mCH equation

To solve Equation (17), using VIM, we have the correction functional as:

where is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as Equation (8) is, which yields the following iteration formula:

Applying the variational homotopy perturbation method, we have:

Substituting initial condition (18)

Comparing the coefficient of like powers of, we have

Then

For an arbitrary we can use symbolic software programme such as Mathematica to calculate it in the same manner.

If only the two-term approximation of Equation (15) is sufficient, then the approximate solution of Equation (17) will be expressed as:

From its expression one can see that it is also a solitary wave solution.

Remark 1. It should be remarked that the graph drawn here and approximate solution using VHPM is in excellent agreement with HPM [16] and VIM [17] .

Consider the mDP equation

To solve Equation (29), using VIM, we have the correction functional as:

where is considered as a restricted variation. Making the above functional stationary, the Lagrange multi- plier can be determined as Equation (8) is, which yields the following iteration formula:

Applying the variational homotopy perturbation method, we have:

Substituting initial condition (30)

Comparing the coefficient of like powers of p, we have

Then

For an arbitrary we can use symbolic software programme such as Mathematica to calculate it in the same manner. If only the two-term approximation of Equation (15) is sufficient, then the approximate solution of Equation (29) will be expressed as:

From its expression one can see that it is also a solitary wave solution.

Remark 2. It should be remarked that the graph drawn here and approximate solution using VHPM is in excellent agreement with HPM [16] and VIM [17] .