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International Journal of Differential Equations
Volume 2014 (2014), Article ID 106934, 11 pages
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.
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.
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  used finite difference method, Küçükarslan  used HPM, Bai and Zhang  used quadratic B-Spline finite element method, Fan and Hon  used extended tanh method, Kaya and El-Sayed  used adomian decomposition method (ADM), Doosthoseini and Shahmohamadi  used VIM, Alomari et al.  used homotopy analysis method (HAM), Qing et al.  used element free Galerkin method (EFG), and Golbabai and Safdari-Vaighani  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 [10–14] 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 [15–17] 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 , 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
The solution obtained by variational iteration method  is as follows:
The solution obtained by homotopy perturbation method  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 19–21 that the extended OHAM solution converges rapidly with the increase in the order of approximation and produced better results. From Figures 22–24 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.
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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