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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.

Most scientific problems arise in real-world physical problems such as plasma physics, fluid mechanic, solid state physics and in many branches of chemistry [

This equation was introduced in (Benjamin T. B., Bona J. L. & Mahony J. J. 1972) [

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.

The generalized BBM (Benjamin-Bona-Mahony) equation has a higher order nonlinearity of the form

where a is constant.

The case

which was first proposed in 1972 by Benjamin et al. [

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

where L is a linear operator is defined by

analytic function. According to VIM, we can write down a correction functional as follows:

where

The subscript n indicates the nth approximation and

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,

Equation (3.2.1) can be written as follows:

By using the homotopy technique, we construct a homotopy:

Which are satisfies:

or

where

The changing process of p forms zero to unity is just that of

And assume that the solution of Equations (3.2.5) and (3.2.6) can be written as a power series in p:

Setting

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) [

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 [

Example:

Consider the nonlinear Benjamin-Bona-Mahony equation where

And initial condition

And exact solution

The correct functional is given as

where

Applying the variational homotopy perturbation method, we have:

Comparing the coefficient of like powers of p, we have

Then

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

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

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

x® | 0.03 | 0.04 | 0.05 | |||
---|---|---|---|---|---|---|

t | ADM & HPM | VHPM | ADM & HPM | VHPM | ADM & HPM | VHPM |

0.01 | 2.26646e−004 | 1.1543e−004 | 2.77073e−004 | 1.4926e−004 | 3.27453e−004 | 1.8307e−004 |

0.02 | 6.03525e−004 | 2.5862e−004 | 7.04304e−004 | 3.2626e−004 | 8.04969e−004 | 3.9387e−004 |

0.03 | 1.13061e−003 | 4.2956e−004 | 1.28165e−003 | 5.3101e−004 | 1.43250e−003 | 6.3239e−004 |

0.04 | 1.80786e−003 | 6.2827e−004 | 2.00908e−003 | 7.6350e−004 | 2.20999e−003 | 8.9864e−004 |

0.05 | 2.63524e−003 | 8.5474e−004 | 2.88653e−003 | 1.0237e−003 | 3.13739e−003 | 1.1926e−003 |

t = 0.1 | t = 0.3 | t = 0.5 | |
---|---|---|---|

x = 50 | 5.631561237089383e−011 | 5.825992451205052e−011 | 6.075975440786746e−011 |

x = 30 | 1.240433164768256e−006 | 1.283259519582440e−006 | 1.338321997161011e−006 |

x = 10 | 2.696255591595309e−002 | 2.791563560904538e−002 | 2.915192190432746e−002 |

x = −10 | 2.629268246421708e−002 | 2.590601471553702e−002 | 2.580255186686319e−002 |

x = −30 | 1.209842932718460e−006 | 1.191488823433051e−006 | 1.185370836912030e−006 |

x = −50 | 5.492681798439742e−011 | 5.409354135256130e−011 | 5.381578247538546e−011 |

t = 0 | t = 0.5 | t = 1 | |
---|---|---|---|

x = −5 | 5.551115123125783e−017 | 7.314427490900022e−002 | 1.560686720003175e−001 |

x = 0 | 0 | 3.522864401908321e−002 | 1.466295596049143e−001 |

x = 5 | 5.551115123125783e−017 | 6.534308332294098e−002 | 1.230071227985541e−001 |

t = 0 | t = 2.5 | t = 5 | |
---|---|---|---|

x = 0 | 0 | 1.097018849581008e+000 | 5.383034896646911e+000 |

x = 25 | 0 | 4.770884185166814e−005 | 3.469615990472011e−004 |

x = 50 | 0 | 1.778067925425482e−010 | 1.293335597251119e−009 |

analytical solutions for wide classes of nonlinear problems. They also do not require large computer memory. This method is reliable and manageable. The results show that:

・ As shown in (

・ As shown in (

・ As shown in (

・ As shown in (

・ In general, whenever a space gets extended the error decreases and closer gets to zero.

The authors thank the University of Zakho for their support.

Fadhil H.Easif,Saad A.Manaa,Bewar A.Mahmood,Majeed A.Yousif, (2015) Variational Homotopy Perturbation Method for Solving Benjamin-Bona-Mahony Equation. Applied Mathematics,06,675-683. doi: 10.4236/am.2015.64062