Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 503137, 13 pages

http://dx.doi.org/10.1155/2013/503137

Research Article

Optimal Homotopy Asymptotic Method to Nonlinear Damped Generalized Regularized Long-Wave Equation

R. Nawaz,1,2 S. Islam,¹ I. A. Shah,² M. Idrees,¹ and H. Ullah¹

¹Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan

²Faculty of Numerical and Physical Sciences, Islamia College Peshawar (Charted University Peshawar), Khyber Pakhtoonkhwa, Pakistan

Received 4 April 2013; Revised 23 July 2013; Accepted 31 July 2013

Academic Editor: Ashraf M. Zenkour

Copyright © 2013 R. Nawaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Long-Wave (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 long-wave, which takes the following form:

Here , is an integer, is known function, and is the amplitude of the long-wave 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 Long-Wave (RLW), and Generalized Regularized Long-Wave (GRLW) equations are special cases of the DGRLW equation [2–4].

The EW equation corresponds to , , whereas the RLW or Benjamin-Bona-Mahony 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 Ritz-Galerkin Method for solving the DGRLW equation [6]. Achouri et al. worked on an article called “A fully Galerkin method for the damped generalized regularized long-wave (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 non-homogenous (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 .

Table 1: (a) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at and various values of t. (b) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at and various values of t. (c) Comparison of absolute errors of 3rd order OHAM solution and 3rd order VIM solution for Example 1 at and various values of t.

(a)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.313486
0.313486

0.04
0.32628
0.32627

0.06
0.339596
0.339571

0.08
0.353455
0.353416

0.10
0.367879
0.367829

(b)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.256661
0.256661

0.04
0.267135
0.267135

0.06
0.278037
0.278037

0.08
0.289384
0.289387

0.10
0.301194
0.301199

(c)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.210136
0.210136

0.04
0.218712
0.218712

0.06
0.227638
0.227635

0.08
0.236928
0.23692

0.10
0.246597
0.246577

Figure 1: Plot of of 3rd order approximate solution for the homogeneous DGRLW equation (2).

Figure 2: Variation profile of exact solution for the homogeneous DGRLW equation (2).

Figure 3: Approximate solution plot for .

Figure 4: Exact solution plot for .

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 .

Table 2: (a) Absolute error of the solution of Example 2 by optimal homotopy asymptotic method (OHAM) at , and and various values of . (b) Absolute error of the solution of Example 2 by optimal homotopy asymptotic method (OHAM) at various values of and .

(a)  

0.01

0.02

0.03

0.04

0.05

(b)  

15

16

17

18

19

20

21

22

23

24

25

Figure 5: Surface 2nd order approximate solution for (4.12).

Figure 6: Exact solution for (4.12).

Figure 7: Approximate solution plot for .

Figure 8: Exact solution plot for .

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:

The first order optimum solution using OHAM is as follows:

The first order OHAM solution yields very encouraging results after comparing with 2nd order VIM solution [5]. Tables 3(a–d), and Figures 9, 10, 11, and 12 show the effectiveness of OHAM for , , and .

Table 3: (a) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at and various values of t. (b) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at and various values of t. (c) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at and various values of t. (d) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 3 at and various values of t.

(a)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.687297
0.687094

0.04
0.682171
0.68177

0.06
0.677041
0.676445

0.08
0.671911
0.67112

0.10
0.666779
0.665796

(b)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.999956
0.997993

0.04
0.999822
0.995986

0.06
0.9996
0.993979

0.08
0.999289
0.991972

0.10
0.99889
0.989965

(c)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.697537
0.697529

0.04
0.702649
0.702639

0.06
0.707754
0.707749

0.08
0.712851
0.712859

0.10
0.71794
0.717969

(d)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.283601
0.283512

0.04
0.286816
0.286609

0.06
0.29006
0.289706

0.08
0.293333
0.292803

0.10
0.296636
0.295901

Figure 9: 1st order approximate solution for (4.12).

Figure 10: Exact solution plot for (4.12).

Figure 11: Approximate solution plot for .

Figure 12: Exact solution plot for .

Example 4. Let us consider the inhomogeneous DGRLW equation:

where and .

The initial condition is

and exact solution given by

According to OHAM scheme presented in Section 2.

Zeroth Order Problem. Consider the following:

Its solution is

First Order Problem. Consider the following:

Its solution is as follows:

Second Order Problem. Consider the following:

Its approximate solution is under

The second order approximate solution is given by the following equation,

Using the method of Least Squares the optimum values of and are computed which are as follows:

The 2nd order OHAM solution yields very encouraging results after comparing with 2nd order VIM solution [5]. Tables 4(a–d), and Figures 13, 14, 15, and 16 show the effectiveness of OHAM for , , , and .

Table 4: (a) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at and various values of t. (b) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at and various values of t. (c) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at and various values of t. (d) Comparison of absolute errors of 1st order OHAM solution and 1st order VIM solution for Example 4 at and various values of t.

(a)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.194735
0.194388

0.04
0.190879
0.189563

0.06
0.1871
0.184295

0.08
0.183395
0.178685

0.10
0.179763
0.172828

(b)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.381707
0.381449

0.04
0.374149
0.373216

0.06
0.36674
0.364862

0.08
0.359478
0.356523

0.10
0.35236
0.348331

(c)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.553462
0.55333

0.04
0.542503
0.542094

0.06
0.53176
0.531101

0.08
0.521231
0.520511

0.10
0.51091
0.510477

(d)  

Exact solution
OHAM solution
Absolute error VIM
Absolute error OHAM

0.02
0.703151
0.703174

0.04
0.689228
0.689438

0.06
0.675581
0.676318

0.08
0.662203
0.663974

0.10
0.649091
0.652558

Figure 13: 2nd order approximate solution for (4.12).

Figure 14: The surface shows Exact solution for (4.12).

Figure 15: Approximate solution plot for for (4.12).

Figure 16: Exact solution plot for for (4.12).

3. Conclusion

In this paper, the OHAM has been successfully implemented for the approximate solution of solutions of the Nonlinear Damped Generalized Regularized Long-Wave (DGRLW) equations. The results obtained by OHAM are very consistent in comparison with VIM.

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