Research Article  Open Access
Optimal Homotopy Asymptotic Method to Nonlinear Damped Generalized Regularized LongWave Equation
Abstract
A new semianalytical technique optimal homology asymptotic method (OHAM) is introduced for deriving approximate solution of the homogeneous and nonhomogeneous nonlinear Damped Generalized Regularized LongWave (DGRLW) equation. We tested numerical examples designed to confine the features of the proposed scheme. We drew 3D and 2D images of the DGRLW equations and the results are compared with that of variational iteration method (VIM). Results reveal that OHAM is operative and very easy to use.
1. Introduction
Partial differential equations used in modeling different problems in physics, biology, chemical reactions, and engineering sciences problems are frequently too difficult to be solved exactly, and even if an exact solution is possible, the required calculations may be too difficult.
The DGRLW equation is a partial differential equation that describes the amplitude of longwave, which takes the following form: Here , is an integer, is known function, and is the amplitude of the longwave at the position and time . For , (1) features a balance between nonlinear and dispersive effects but also takes into account mechanisms of dissipation. In the physical sense, (1) with the dissipative term is suggested if the good predictive power is preferred; such type of problem arises in the bore propagation as well as for water waves [1].
The Equal Width (EW) Wave, Regularized LongWave (RLW), and Generalized Regularized LongWave (GRLW) equations are special cases of the DGRLW equation [2–4].
The EW equation corresponds to , , whereas the RLW or BenjaminBonaMahony equation corresponds to , , and . On the other hand, the GRLW equation corresponds to , , and .
Different methods have been used for numerical solutions of DGRLW equations. VIM was presented by Demir et al. for numerical solutions for the DGRLW equation [5]. Yousefi et al. used Bernstein RitzGalerkin Method for solving the DGRLW equation [6]. Achouri et al. worked on an article called “A fully Galerkin method for the damped generalized regularized longwave (DGRLW) equation,” namely, in [7]. For the mathematical theory and physical significance of DGRLW equation see [8–16] and references therein.
Marinca and Herişanu proposed semianalytical technique OHAM for deriving approximate solution of nonlinear problems of thin film flow of a fourth grade fluid down a vertical cylinder [17–19]. The method has been used by many researchers for obtaining numerical approximations of linear and nonlinear differential equations [20–24]. The author has successfully applied the OHAM for deriving approximate solution of Equal Width Wave equation, Burger equations, and tenth order boundary value problems [25, 26]. The convergence criterion of proposed method is similar to that of homotopy analysis method (HAM) and homotopy perturbation method (HPM), but this method is more efficient and flexible. To improve the efficiency and accuracy of OHAM, Herişanu and Marinca introduced more generalized and new advances in OHAM which shows that the auxiliary function includes the functions of physical parameters in addition to the convergence control parameter [27, 28].
Here, we investigate the approximate solution of the DGRLW equation with a variable coefficient using OHAM. The whole paper is divided into 3 sections. Section 2 is devoted to the analysis of the proposed method. In Section 3, solution of homogeneous and nonhomogenous (DGRLW) equations is presented by OHAM, and absolute errors are also compared with VIM. The 3D and 2D images of the approximate solution and exact solution are also drawn. In all cases, the proposed method yields very encouraging results.
2. Fundamental Theory of OHAM
Consider the partial differential equation of the following form: where is a linear operator and is nonlinear operator. is boundary operator, is an unknown function, and denote spatial and time variables, respectively, is the problem domain, and is a known function.
Using the basic idea of OHAM, the optimal homotopy is constructed which satisfies the following condition: where is an embedding parameter and is a nonzero auxiliary function for , . In such a case, (3) is called optimal homotopy equation. Clearly, we have when and , then and hold. Thus, as varies from , the solution approaches from to , where is obtained from (3) for : Next, we choose auxiliary function of the following general form:
Here are constants to be determined later.
To get an approximate solution, we expand in Taylor’s series about in the following manner,
Substituting (8) into (3) and equating the coefficient of like powers of , we obtain Zeroth order problem, given by (6); the first and second order problems are given by (9) and (10), respectively, and the general governing equations for are given by (11) as follows: where is the coefficient of in the expansion of about the embedding parameter :
Here for are a set of linear equations with the linear boundary conditions, which can be easily solved.
The convergence of the series in (8) depends upon the auxiliary constants . If it is convergent at , then
Substituting (13) into (1), it results with the following expression for residual:
If , then will be the exact solution.
For computing the optimal values of auxiliary constants, , there are many methods available like Galerkin’s, Ritz, Least Squares, and Collocation method. One can apply the method of Least Squares as under the following: where is the residual,
The constants can also be determined by another method as under: at any time , where . The convergence depends upon constants , which can be optimally identified and minimized by (18).
Example 1. Consider (1) with , , and which in the simplest form is given as The initial condition is and exact solution given by
Zeroth Order Problem. Consider the following: Its solution is given as under
First Order Problem. Consider the following: Its solution is as follows:
Second Order Problem. Consider the following: Its solution is under
Third Order Problem. Consider the following:
Its solution is given as follows: The third order approximate solution is given by the following equation:
The constants , , and are calculated using the Least Squares, we have their optimal values as follows:
The 3rd order OHAM solution yields very encouraging results after comparing with 3rd order VIM solution [5]. Tables 1(a–c), and Figures 1, 2, 3, and 4 show the effectiveness of OHAM for , and .
(a)  
 
(b)  
 
(c)  

Example 2. Consider (1) with , and which in the simplest form is given as
The initial condition is and exact solution given by
Zeroth Order Problem. Consider the following: Its solution is given by
First Order Problem. Consider the following: Its solution is
Second Order Problem
Its approximate solution is obtained in similar manner. The second order approximate solution is given by
Using method of Least Squares, the optimal values of constants are computed and are given as under
Table 2(a) shows the effectiveness of OHAM for , , and , while Table 2(b), and Figures 5, 6, 7, and 8 shows the effectiveness of OHAM for various values of and .
(a)  
 
(b)  

Example 3. Consider (1) with , , and − which in the simplest form is given as The initial condition is and exact solution given by
Zeroth Order Problem. Consider the following: Its solution is as follows:
First Order Problem. Consider the following:
Its solution is
The first order approximate solution is given by The constants is calculated using the Least Squares that we have its optimal values as follows: