Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method

The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results.


Introduction
A significant part of the natural technological processes and phenomena are usually modelled by means of partial differential equations. Thus it is very important to find solutions of these equations. However, as in many cases the computation of exact solutions is not possible; numerical or approximate solutions must be computed.
In the present paper we present a new approximation method named optimal homotopy perturbation method (OHPM). As the name suggests, the method is based on the homotopy perturbation method [1,2] and its main feature is an accelerated convergence compared to the regular homotopy perturbation method.
The applications presented show that the approximate solutions obtained by using OHPM requires less iterations in comparison with other iterative methods for approximate solutions of partial differential equations.
Here L is a linear operator, ( , ) is the unknown function, N is a nonlinear operator, ( , ) is a known, given function, and is a boundary operator.
Definition 1. We call an HP-sequence of the problem (1) a sequence of functions { ( , )} ∈N of the form ( , ) = ∑ =0 , where ∈ N, ∈ R. A function of the sequence is called an HP-function of the problem (1).

Definition 2.
We call an -approximate HP-solution of the problem (1) on the real domain Ω an HP-functioñwhich satisfies the following condition: together with the boundary conditions from (1).

Definition 3.
We call a weak -approximate HP-solution of the problem (1) on the real domain Ω an HP-functions atisfying the relation ∫ Ω 2 ( , ,̃) ≤ , together with the boundary conditions from (1).
We will find a weak -approximate HP-solution of the typẽ= ∑ =0 where ≥ 0 and the constants are calculated using the following steps.
Proof. Based on the way the HP-function ( , ) is computed, the following inequality holds: It follows that We obtain From this limit we obtain that ∀ > 0, ∃ 0 ∈ N such that ∀ ∈ N, > 0 it follows that ( , ) is a weakapproximate HP-solution of the problem (1).
The Scientific World Journal 3 Remark 5. Any -approximate HP-solution of the problem (1) is also a weak approximate HP-solution, but the opposite is not always true. It follows that the set of weak approximate HP-solutions of the problem (1) also contains the approximate HP-solutions of the problem.
Taking into account the above remark, in order to findapproximate HP-solutions of the problem (1) by the OHPM method we will first determine weak approximate HPsolutions,̃. If | ( , ,̃)| < theñis also an -approximate HP-solution of the problem.

Applications
In this section we apply OHPM to find approximate analytical solutions for the regularized long wave (RLW) equation.
The RLW equation is a nonlinear evolution equation. These kind of equations are frequently used to model a variety of physical phenomena such as ion-acoustic waves in plasma, magnetohydrodynamics waves in plasma, longitudinal dispersive waves in elastic rods, pressure waves in liquid gas bubble mixtures, and rotating flow down a tube.
The RLW equation was introduced in [3] where it was used to describe the behaviour of the undular bore.
For some restricted initial and boundary conditions, exact analytical solutions for the RLW equation were computed (see, e.g., [4]). However, in most cases it is not possible to find such exact analytical solutions and usually numerical methods are used. Among the numerical methods recently employed for RLW-type equations we mention finite difference methods [5][6][7][8], multistep mixed finite element methods [9], the method of lines [10], and meshless finitepoint methods [11].
Taking into account the usefulness of analytical solutions versus numerical ones, various approximation methods were also employed to find approximate analytical solutions for various RLW-type equations, such as the homotopy perturbation method [12], the variational iteration method [12], the homotopy asymptotic method [13,14], and the Riccati expansion method [15].
In the following, for two test problems presented in [12], we compare solutions obtained by using OHPM with previous results obtained by using the homotopy perturbation method and the variational iteration method.

Application 1.
Our first application is the following RLW problem [12]: (16) In [12] approximate solutions of (16) are computed using the homotopy perturbation method (HPM) and the variational iteration method (VIM).
The fifth order solution computed in [12] The fifth order solution computed in [12] by using the homotopy perturbation method is of the form Using OHPM, the following steps are performed.
We will compute a second order approximate solution, by taking into account the terms from 0 , 1 , and 2 and we will compare this solution with the fifth order solutions from [12]. Our second order approximate solution will have the expression OHPM ( , ) = (ii) Imposing the boundary condition OHPM ( , 0) = we obtain 0 = 1.
Replacing this expression of 0 in the expression of OHPM we obtain the following: We introduce OHPM in the remainder R given by (2) and (9) and we compute the functional ( 1 , 2 , 3 , 4 , 5 ) of (10).
We remark that while the expression of the functional is too long to be included here, the computation is simple and straightforward using a dedicated mathematical software (we used the Wolfram Mathematica 9 software).
(iii) We compute the minimum of the functional and, by replacing the corresponding values of the parameters 1 , 2 , 3 , 4 , 5 , we obtain the following second order approximation: OHPM ( , ) = −0.109895 5 + 0.434798 4 − 0.789112 3 + 0.961938 2 − 0.997729 + . Figure 1 presents the comparison of the absolute errors (computed as the absolute values of the differences between the exact solutions and the approximate solutions) corresponding to the fifth order approximation obtained by using HPM (red surface), to the fifth order approximation obtained by using VIM (blue surface) and to the second order approximation obtained by OHPM (green surface). Table 1 presents the same comparison for several values of and .
It is easy to see that, overall, the approximations obtained by using OHPM are much more accurate than the ones previously computed by using HPM and VIM. Moreover, our approximate solutions are not only more accurate but also, at the same time, present a much simpler expression since they are second order approximate solutions while the previous ones are fifth order approximate solutions.

Application 2.
Our second application is the RLW problem (also from [12]): Again in [12] approximate solutions of (16) are computed using the homotopy perturbation method (HPM) and the variational iteration method (VIM).
The exact solution of this problem is ( , ) = − sin( ).
Using OHPM, the following steps are performed.
(i) Choosing the same homotopy (3) as used in [12] we obtain the same solutions: Hence we will compute a second order approximate solution of the following form: (ii) Imposing the boundary condition OHPM ( , 0) = we obtain 0 = 1.
Replacing this expression of 0 in the expression of OHPM we obtain the following: We introduce OHPM in the remainder R given by (2) and (9) and we compute the functional ( 1 , 2 , 3 ) of (10).
The Scientific World Journal    Figure 2 presents the comparison of the absolute errors corresponding to the third order approximation obtained by using HPM (red surface), to the fourth order approximation obtained by using VIM (blue surface), and to the second order approximation obtained by OHPM (green surface). Table 2 presents the same comparison for several values of and .
Again, overall, the approximations obtained by using OHPM are more accurate than the ones previously computed by using HPM and VIM while, at the same time, they present a much simpler expression.

Conclusions
In the present paper the new optimal homotopy perturbation method is introduced as a straightforward and efficient method to compute approximate solutions for nonlinear partial differential equations.
The optimal homotopy perturbation method has an accelerated convergence compared to the regular homotopy perturbation method, fact proved by the included applications. The method is a powerful one since not only were we capable to find more accurate approximations, but also the approximations computed consist of fewer terms than the previous solutions. 6 The Scientific World Journal