A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

Majeed A. Yousif¹, Bewar A. Mahmood², Fadhil H. Easif¹

¹Department of Mathematics, University of Zakho, Zakho, Iraq.
²Department of Mathematics, University of Duhok, Duhok, Iraq.

DOI: 10.4236/ajcm.2015.53024                

Abstract

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

Keywords

Homotopy Perturbation Method, Modified Camassa-Holm Equation, Modified Degasperis-Procesi Equation

1. Introduction

Many varieties of physical, chemical, and biological phenomena can be expressed in terms of nonlinear partial differential equations. In most cases, it is difficult to obtain the exact solution for these equations. Therefore analytical methods have been used to find approximate solutions. In recent years, many analytical methods such as the Adomian decomposition method [1] [2] , the homotopy analysis method [3] [4] , the variational iteration method [5] [6] , the homotopy perturbation method [7] - [10] , and variational homotopy perturbation method [11] [12] have been utilized to solve linear and nonlinear equations.

In this paper, we will use variational homotopy perturbation method to study the Modified Camassa-Holm and Degasperis-Procesi equations and obtain their analytical solutions.

2. Mathematical Models

Wazwaz [13] studies a family of important physically equations which is called modified -equation. It has the following expression:

(1)

where is a positive integer. As is known, when, Equation (1) reduces to modified Camassa- Holm (mCH) equation and modified Degasperis-Procesi (mDP) equation, respectively.

The mCH equation with exact solution [13] :

(2)

(3)

The mDP equation with exact solution [13] :

(4)

(5)

3. Analytical Methods

3.1. Variational Iteration Method (VIM)

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

(6)

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:

(7)

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

(8)

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

3.2. Homotopy Perturbation Method (HPM)

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

(9)

with the boundary condition

(10)

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:

(11)

Homotopy perturbation structure is shown as follows:

(12)

where

(13)

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:

(14)

and the best approximation for solution is:

(15)

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] .

3.3. Variational Homotopy Perturbation Method (VHPM)

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:

(16)

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.

4. Application of VHPM

In this section, we apply the variational homotopy perturbation method to solve mCH and mDP equations.

4.1. Application of VHPM to Modified Camassa-Holm Equation

Consider the mCH equation

(17)

(18)

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

(19)

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:

(20)

Applying the variational homotopy perturbation method, we have:

(21)

Substituting initial condition (18)

(22)

Comparing the coefficient of like powers of, we have

(23)

(24)

(25)

Then

(26)

(27)

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:

(28)

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] .

4.2. Application of VHPM to Modified Degasperis-Procesi

Consider the mDP equation

(29)

(30)

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

(31)

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:

(32)

Applying the variational homotopy perturbation method, we have:

(33)

Substituting initial condition (30)

(34)

Comparing the coefficient of like powers of p, we have

(35)

(36)

(37)

Then

(38)

(39)

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:

(40)

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] .

5. Figures

In this section, we show the accurance of VHPM to finding analytical solution of Modified Camassa-Holm and Degasperis-Procesi equations. Also, we compare between exact and analytical solution (see Figures 1-3).

6. Conclusion

In this paper, we apply variational homotopy perturbation method to obtain the analytical solutions of Modified Camassa-Holm and Degasperis-Procesi equations. The solutions obtained by present method is compared with the exact solution. Also, it was shown that the approximation solution by VHPM had a good agreement with HPM and VIM. We observed that the method is effective for given examples and it can be applied to many other nonlinear equations.

(a) (b)

Figure 1. The surface of mCH equation at different times. (a) Exact solution; (b) Approximate solution.

(a) (b)

Figure 2. The surface of mDP equation at different times. (a) Exact solution; (b) Approximate solution.

(a) (b)

Figure 3. The curve of. (a) Exact and VHPM of mCH equation; (b) Exact and VHPM of mDP equation.

Acknowledgements

We thank the editor and the referee for their comments. Many thanks to University of Zakho for supporting this work.

Conflicts of Interest

The authors declare no conflicts of interest.

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