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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 326473, 12 pages
http://dx.doi.org/10.1155/2013/326473
Research Article

The Application of the Homotopy Analysis Method and the Homotopy Perturbation Method to the Davey-Stewartson Equations and Comparison between Them and Exact Solutions

1Mathematics Department, Faculty of Science, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Mathematics Department, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt

Received 20 November 2012; Accepted 1 February 2013

Academic Editor: Hui-Shen Shen

Copyright © 2013 Hassan A. Zedan 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 introduce two powerful methods to solve the Davey-Stewartson equations: one is the homotopy perturbation method (HPM) and the other is the homotopy analysis method (HAM). HAM is a strong and easy to use analytic tool for nonlinear problems. Comparison of the HPM results with the HAM results, and compute the absolute errors between the exact solutions of the DS equations with the HPM solutions and HAM solutions are obtained.

1. Introduction

Nonlinear partial differential equations are useful in describing the various phenomena in disciplines. Apart from a limited number of these problems, most of them do not have a precise analytical solution, so these nonlinear equations should be solved using approximate methods.

The application of the homotopy perturbation method (HPM) [1, 2] in nonlinear problems has been devoted by scientists and engineers, because this method continuously deforms a simple problem which is easy to solve into the under study problem which is difficult to solve. The homotopy perturbation method was first proposed by He [36].

The HPM yields a very rapid convergence of the solution series in most cases. The method does not depend on a small parameter in the equation. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter which is considered as a “small parameter.”

No need to linearization or discretization, large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximations. These approximations converge rapidly to accurate solutions [710]. The goal of He's homotopy perturbation method was to find a technique to unify linear and nonlinear, ordinary or partial differential equations for solving initial and boundary value problems. The HPM was successfully applied to nonlinear oscillators with discontinuities [4] and bifurcation of nonlinear problem [11]. In [6], a comparison of HPM and homotopy analysis method was made on a simple problem.

In 1992, Liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, homotopy analysis method (HAM) [1215]. This method has been successfully applied to solve many types of nonlinear problems by others [1620].

In this paper, we consider the Davey-Stewartson (DS) equations for the function which are given by (see [21])

The case is called the DSI equation, while is the DSII equation. The parameter characterizes the focusing or defocusing case. The Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear shrödinger (NLS) equation, as well as from physical considerations [22]. Indeed, they appear in many applications, for example, in the description of gravity-capillarity surface wave packets in the limit of the shallow water.

Davey and Stewartson first derived their model in the context of water waves, from purely physical considerations. In the context, is the amplitude of a surface wave packet, while is the velocity potential of the mean flow interacting with the surface wave [22].

In [23], solution of DS equations by (HPM) where the amplitude of a surface wave packet separated into real and imaginary parts, that is, . Consequently, the system (1) rewritten in the following form: with the initial condition where , , , , , and are arbitrary constants.

In this paper, we apply homotopy analysis method (HAM) for the above system. We rewrite system (1) in the following form where we take with the initial condition where , , , , , and are arbitrary constants. After that we will apply homotopy perturbation method and homotopy analysis method. When implementing the homotopy perturbation method (HPM) and the homotopy analysis method (HAM), we get the explicit solutions of the DS equations without using any transformation method. Furthermore, we will show that considerably better approximations related to the accuracy level would be obtained. The homotopy perturbation method can be found in [111, 23]. The homotopy analysis method can be found in details in [1222, 2426] and only the main steps will be summarized here.

2. Application of the Homotopy Perturbation Method

To investigate the traveling wave solution of (4), we first construct a homotopy as follows:

And the initial approximations are as follows: where , are functions yet to be determined. Substituting (8) into (6) and arranging the coefficients of powers, we have To obtain the unknowns , , , we must construct and solve the following system which includes nine equations with nine unknowns, considering the initial conditions of , , :

From (8), if the three approximations are sufficient, we will obtain

To calculate the terms of the homotopy series (19) for , , and , we substitute the initial conditions (5) into the system (9), and using Mathematica software, from (13), we obtain From (10), we obtain From (16), we obtain

In this manner, the other components , , , , , and can be obtained from (14), (11), (17), (15), (12), and (18), respectively, and substituting these components into (19) to obtain , , and .

3. Application of the Homotopy Analysis Method

In order to apply the homotopy analysis method for (2), we choose the linear operator with the property , , where are integral constants to be determined by initial conditions.

Furthermore, (2) suggests to define the nonlinear operators we construct the zero-order deformation equations

When

When

Therefore, as the embedding parameter increases from to , varies from initial guesses to the solutions and , for , respectively.

Expanding in Taylor series with respect to for , one has where

If the auxiliary linear operator, the initial guesses, and the auxiliary parameters are so properly chosen, the above series converge at , and which must be one of solutions of the original nonlinear equations as proved by Liao [13]. Define the vectors

We have the th-order deformation equations where where , , and are functions of , and , and Now, the solutions of the th-order deformation (31) for become

For simplicity, we suppose .

We consider the solutions of (2) with the initial conditions (25). We now obtain at

Obviously, for , the obtained solutions are the same homotopy perturbation method in [2]; we continue to evaluate two terms of HAM.

Now for (4), we choose the linear operator with the property , , where are integral constant to be determined by initial conditions.

Furthermore, (4) suggests to define the nonlinear operators We construct the zero-order deformation equations

When ,

When , Therefore, as the embedding parameter increases from to , varies from initial guesses to the solutions , , and , for , respectively.

Expanding in Taylor series with respect to for , one has where

If the auxiliary linear operator, the initial guesses, and the auxiliary parameters are so properly chosen, the series (40) converge at , has which must be one of solutions of the original nonlinear equation as proved by Liao [13]. Define the vectors We have the th-order deformation equations where where , , and are functions of , , and , and Now, the solutions of the th-order deformation (44) for     1 become

For simplicity, we suppose .

We consider the solutions of (4) with the initial conditions (38) and obtain for Obviously, for , we obtained the same solutions as the one by the homotopy perturbation method in (20)–(22); we continue to evaluate six terms of (47) when .

Using a Taylor series, then the closed form solutions yield as follows [23]: where , and are arbitrary constants.

4. Comparing the HPM Results and the HAM Results with the Exact Solutions

To demonstrate the convergence of the HPM, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. Tables 1 and 2 show the absolute errors between the analytical solutions and the HPM solutions of the DS for the first three approximations with initial conditions (5) for are very small with the present choice of at and , when , , , , and . Tables 3, 4, 5 and 6 help us to compare the HAM results for the first three approximations when with the analytical solution through the absolute errors. Both the analytical solutions, the HPM result, and the HAM result for and are plotted in Figures 1, 2, 3, and 4.

tab1
Table 1: The HPM results for in comparison with the analytical solution with initial conditions (5).
tab2
Table 2: The HPM results for in comparison with the analytical solution with initial conditions (5).
tab3
Table 3: The HAM results for in comparison with the analytical solution with initial conditions (3).
tab4
Table 4: The HAM results for in comparison with the analytical solution with initial conditions (3).
tab5
Table 5: The HAM results for in comparison with the analytical solution with initial conditions (5).
tab6
Table 6: The HAM results for in comparison with the analytical solution with initial conditions (5).
fig1
Figure 1: Comparison between the exact solution, the HPM solution, and the HAM solution for .
fig2
Figure 2: Comparison between the exact solution, the HPM solution,and the HAM solution for.
fig3
Figure 3: The results obtained by HPM and HAM for , at in comparison with the exact solutions.
fig4
Figure 4: The results obtained by HPM and HAM for , at in comparison with the exact solutions.

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