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International Journal of Differential Equations

Volume 2014 (2014), Article ID 106934, 11 pages

http://dx.doi.org/10.1155/2014/106934
Research Article

An Extension of the Optimal Homotopy Asymptotic Method to Coupled Schrödinger-KdV Equation

Department of Mathematics, Abdul Wali Khan University Mardan, 23200, Pakistan

Received 26 December 2013; Accepted 13 March 2014; Published 7 May 2014

Academic Editor: Patricia J. Y. Wong

Copyright © 2014 Hakeem Ullah 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

We consider the approximate solution of the coupled Schrödinger-KdV equation by using the extended optimal homotopy asymptotic method (OHAM). We obtained the extended OHAM solution of the problem and compared with the exact, variational iteration method (VIM) and homotopy perturbation method (HPM) solutions. The obtained solution shows that extended OHAM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

1. Introduction

The nonlinear Schrödinger equations are of great interest due to their numerous applications in physical phenomena. The coupled Schrödinger-KdV equations are extensively used to model nonlinear dynamics of one-dimensional Langmuir and ion acoustic waves in the system of coordinates moving at the speed of ion acoustic. This problem remains under consideration from many years and has been investigated by many researchers. Many authors have investigated the nonlinear Schrödinger-KdV equation by various techniques such as the following: Wang [1] used finite difference method, Küçükarslan [2] used HPM, Bai and Zhang [3] used quadratic B-Spline finite element method, Fan and Hon [4] used extended tanh method, Kaya and El-Sayed [5] used adomian decomposition method (ADM), Doosthoseini and Shahmohamadi [6] used VIM, Alomari et al. [7] used homotopy analysis method (HAM), Qing et al. [8] used element free Galerkin method (EFG), and Golbabai and Safdari-Vaighani [9] used meshless method using RBF collocation scheme. The perturbation methods like HPM required a small parameter and are difficult to determine.

Recently, Marinca et al. introduced OHAM [1014] for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters and huge computational work.

The motivation of this paper is to extend the OHAM formulation for a system of three partial differential equations and to apply the extended OHAM formulation to coupled nonlinear Schrödinger-KdV equation. In [1517] OHAM has been proved to be valuable for obtaining an approximate solution of ordinary/partial differential equations (O/PDEs). Before, this system of nonlinear partial differential equations (NPDEs) was not solved by OHAM. We have proved that extended OHAM is useful and reliable for NPDEs, showing its validity and great potential for the solution of transient physical phenomenon in science and engineering.

In the succeeding section, the basic idea of extended OHAM is formulated for the solution of system NPDEs. The effectiveness and efficiency of OHAM are shown in Section 3.

2. Extended Mathematical Formulation of OHAM

Consider a system of three partial differential equations: where , , are differential operators, , , are unknown functions, and denote spatial and temporal independent variables, respectively, is the boundary of , and , , are known analytic functions. , , can be divided into two parts: , , contain the linear parts while , , contain the nonlinear parts of the system of partial differential equations.

According to OHAM, we construct satisfying the following homotopies: where the auxiliary functions , , are nonzero for and , , . Equation (4) is called optimal homotopy equation. Clearly, we have

Obviously, when and we obtain respectively. When varies from to , the solution , , approaches from , , to , , , where , , are obtained from (4) for :

We choose auxiliary functions , , in the form

To get the approximate solutions, we expand , , by Taylor’s series about in the following manner: where . Now substituting (8)-(9) into (4) and equating the coefficient of like powers of , we obtain zeroth order system, given by (7) and the first and second order systems given by (10)-(11), respectively, and the general governing equations for are given by (12):

It has been observed that the convergence of the series (9) depends upon the auxiliary constants , , , , , , , , , . If it is convergent at , one has

Substituting (14) into (1.1), the following expression for residuals results:

If , , then , , and will be the exact solutions of the problem. Generally it does not happen, especially in nonlinear problems.

For the computation of auxiliary constants, , , , , there are different methods like Galerkin’s method, Ritz method, least squares method, and collocation method. One can apply the method of least squares as follows:

The th order approximate solution can be obtained by these constants so-obtained. The more general auxiliary functions , , are useful for convergence, which depends upon constants , , , , , , can be optimally identified by (17), and is useful in error minimization.

3. Application of Extended OHAM to Coupled Schrödinger-KdV Equation

To demonstrate the effectiveness of the extended OHAM for coupled Schrödinger- KdV equation taken from [6], we have with boundary conditions

The exact solution of (19) for and is given by

Applying the extended OHAM technique discussed in Section 2,

We consider Zeroth Order System. Consider with initial conditions

Its solution is First Order System. Consider with

Its solution is Second Order System. Consider with

The solution of second order system is

Adding (26), (29), and (32), we obtain

For the calculation of the constants , , , , and using (33) in (19) and applying the method of least square mentioned in (16)-(17) by taking, we get

The solution obtained by variational iteration method [6] is as follows:

The solution obtained by homotopy perturbation method [2] is as follows:

4. Results and Discussions

The extended mathematical theory presented in Section 2 gives highly accurate solutions for the BVP presented in Section 3. We have used Mathematica 7 for most of our computational work. In Tables 1 and 2, the absolute errors of extended OHAM results corresponding to the exact solution of the problem given in (19) are given for , at , , , , . In Table 3, the extended OHAM results are compared with exact solution for . In Figures 1, 2, 3, 4, 5, and 6, 3D plots of the results obtained by extended OHAM and exact solutions are plotted for , , at fixed value of . While the 3D plots of the residuals at are given in Figures 7, 8, and 9, Figures 10, 11, 12, 13, 14, and 15 show the 2D plots of extended OHAM and exact solutions for , , at . The 2D residual plots are given in Figures 16, 17, and 18 at . In Figures 19, 20, and 21 the 2D plots of ordered solutions are given at for , , . The extended OHAM solutions are compared with VIM and HPM solution in 2D, Figures 22, 23, and 24. Here we observed from Figures 1921 that the extended OHAM solution converges rapidly with the increase in the order of approximation and produced better results. From Figures 2224 it is evident that the extended OHAM results are nearly identical to the exact, VIM, and HPM solutions. Here the results are very consistent with the decreasing time as evident from Tables 1-2.

tab1
Table 1: Absolute error of OHAM solution corresponding to the exact solution.
tab2
Table 2: Absolute error of OHAM solution corresponding to the exact solution.
tab3
Table 3: Comparison of solutions obtained by OHAM to the exact solution.
106934.fig.001
Figure 1: 3D, OHAM solution of at .
106934.fig.002
Figure 2: 3D, exact solution of at .
106934.fig.003
Figure 3: 3D, OHAM solution of at .
106934.fig.004
Figure 4: 3D, exact solution of at .
106934.fig.005
Figure 5: 3D, OHAM solution of at .
106934.fig.006
Figure 6: 3D, exact solution of at .
106934.fig.007
Figure 7: 3D, residual of at .
106934.fig.008
Figure 8: 3D, residual of at .
106934.fig.009
Figure 9: 3D, residual of at .
106934.fig.0010
Figure 10: 2D, OHAM solution of at .
106934.fig.0011
Figure 11: 2D, exact solution of at .
106934.fig.0012
Figure 12: 2D, OHAM solution of at .
106934.fig.0013
Figure 13: 2D, exact solution of at .
106934.fig.0014
Figure 14: 2D, OHAM solution of at .
106934.fig.0015
Figure 15: 2D, exact solution of at .
106934.fig.0016
Figure 16: 2D, residual of at .
106934.fig.0017
Figure 17: 2D, residual of at .
106934.fig.0018
Figure 18: 2D, residual of at .
106934.fig.0019
Figure 19: 2D, comparison of ordered solutions for at .
106934.fig.0020
Figure 20: 2D, comparison of ordered solutions for at .
106934.fig.0021
Figure 21: 2D, comparison of ordered solutions for at .
106934.fig.0022
Figure 22: 2D, comparison of OHAM, VIM, HPM, and exact solutions for at .
106934.fig.0023
Figure 23: 2D, comparison of OHAM, VIM, HPM, and exact solutions for at .
106934.fig.0024
Figure 24: 2D, comparison of OHAM, VIM, HPM, and exact solutions for at .

5. Conclusion

In this paper, we have seen the effectiveness of extended OHAM to coupled Schrödinger-KdV equation. By applying the basic idea of extended OHAM to coupled Schrödinger-KdV equation, we found it simpler in applicability and more convenient to control convergence and that it involved less computational overhead. Therefore, extended OHAM shows its validity and great potential for the solution of nonlinear system of PDEs problems in science and engineering.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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