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

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.

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Hassan A. Zedan 1,2 and W. Barakati 1 and Nada Hamad 1

Recommended by Hui-Shen Shen

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

Received 20 November 2012; Accepted 1 February 2013

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.

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 [3-6].

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 p∈[0;1] 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 [7-10]. 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) [12-15]. This method has been successfully applied to solve many types of nonlinear problems by others [16-20].

In this paper, we consider the Davey-Stewartson (DS) equations for the function q=q(x,y,t) which are given by (see [21]) [figure omitted; refer to PDF]

The case σ=1 is called the DSI equation, while σ=i is the DSII equation. The parameter λ 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, q(x,y,t) is the amplitude of a surface wave packet, while [varphi](x,y) 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 q separated into real and imaginary parts, that is, q=u(x,y,t)+iv(x,y,t) . Consequently, the system (1) rewritten in the following form: [figure omitted; refer to PDF] with the initial condition [figure omitted; refer to PDF] where c=_k2 +^σ2_k1 , r=(-2_k3 +^k12σ2 +^k22 )/λ , s=(2_k3 +^k12σ2 +^k22 )/^σ2 , f=2σ-λ/(1-^σ2 ),_k1 , _k2 , and _k3 are arbitrary constants.

In this paper, we apply homotopy analysis method (HAM) for the above system. We rewrite system (1) in the following form [figure omitted; refer to PDF] where we take q=ρ(x,y,t)·^{e-miθ(x,y,t)} with the initial condition [figure omitted; refer to PDF] where c=_k2 +^σ2_k1 , r=(-2_k3 +^k12σ2 +^k22 )/λ , s=(2_k3 +^k12σ2 +^k22 )/^σ2 , f=2σ-λ/(1-^σ2 ),_k1 , _k2 , and _k3 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 [1-11, 23]. The homotopy analysis method can be found in details in [12-22, 24-26] 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: [figure omitted; refer to PDF]

And the initial approximations are as follows: [figure omitted; refer to PDF] where _v(i,j) , i=1,2,3,j=0,1,2,3,... are functions yet to be determined. Substituting (8) into (6) and arranging the coefficients of p powers, we have [figure omitted; refer to PDF] To obtain the unknowns _vi,j (x,y,t) , i , j=1,2,3 , we must construct and solve the following system which includes nine equations with nine unknowns, considering the initial conditions of _vi,j (x,0,t)=0 , i , j=1,2,3 : [figure omitted; refer to PDF]

From (8), if the three approximations are sufficient, we will obtain [figure omitted; refer to PDF]

To calculate the terms of the homotopy series (19) for ρ(x,y,t) , θ(x,y,t) , and [varphi](x,y,t) , we substitute the initial conditions (5) into the system (9), and using Mathematica software, from (13), we obtain [figure omitted; refer to PDF] From (10), we obtain [figure omitted; refer to PDF] From (16), we obtain [figure omitted; refer to PDF]

In this manner, the other components _v1,2 (x,y,t) , _v2,2 (x,y,t) , _v3,2 (x,y,t) , _v1,3 (x,y,t) , _v2,3 (x,y,t) , and _v3,3 (x,y,t) can be obtained from (14), (11), (17), (15), (12), and (18), respectively, and substituting these components into (19) to obtain ρ(x,y,t) , θ(x,y,t) , and [varphi](x,y,t) .

3. Application of the Homotopy Analysis Method

In order to apply the homotopy analysis method for (2), we choose the linear operator L[_{[straight phi]i} (x,y,t;p)]=^∂2_{[straight phi]i} /∂^y2 with the property L[_c1,i +_c2,i y]=0 , i=1,2,3 , where _c1,i ,_c2,i are integral constants to be determined by initial conditions.

Furthermore, (2) suggests to define the nonlinear operators [figure omitted; refer to PDF] we construct the zero -order deformation equations [figure omitted; refer to PDF]

When p=0 [figure omitted; refer to PDF]

When p=1 [figure omitted; refer to PDF]

Therefore, as the embedding parameter p increases from 0 to 1 , _{[straight phi]i} (x,y,t;p) varies from initial guesses _zi,0 (x,y,t) to the solutions u(x,y,t),v(x,y,t) and [varphi](x,y,t) , for i=1,2,3 , respectively.

Expanding _{[straight phi]i} (x,y,t;p) in Taylor series with respect to p for i=1,2,3 , one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

If the auxiliary linear operator, the initial guesses, and the auxiliary parameters _{[Planck's over 2pi]i} are so properly chosen, the above series converge at p=1 , and [figure omitted; refer to PDF] which must be one of solutions of the original nonlinear equations as proved by Liao [13]. Define the vectors [figure omitted; refer to PDF]

We have the m th-order deformation equations [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where _z1 , _z2 , and _z3 are functions of x,y , and t , and [figure omitted; refer to PDF] Now, the solutions of the m th-order deformation (31) for m...5;1 become [figure omitted; refer to PDF]

For simplicity, we suppose _{[Planck's over 2pi]1} =_{[Planck's over 2pi]2} =_{[Planck's over 2pi]3} =[Planck's over 2pi] .

We consider the solutions of (2) with the initial conditions (25). We now obtain at m=1 [figure omitted; refer to PDF]

Obviously, for [Planck's over 2pi]=-1 , 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 L[_{[straight phi]i} (x,y,t;p)]=^∂2_{[straight phi]i} /∂^y2 with the property [_c1,i +_c2,i y]=0 , i=1,2,3 , where _c1,i ,_c2,i are integral constant to be determined by initial conditions.

Furthermore, (4) suggests to define the nonlinear operators [figure omitted; refer to PDF] We construct the zero -order deformation equations [figure omitted; refer to PDF]

When p=0 , [figure omitted; refer to PDF]

When p=1 , [figure omitted; refer to PDF] Therefore, as the embedding parameter p increases from 0 to 1 , _{[straight phi]i} (x,y,t;p) varies from initial guesses _zi,0 (x,y,t) to the solutions ρ(x,y,t) , θ(x,y,t) , and [varphi](x,y,t) , for i=1,2,3 , respectively.

Expanding _{[straight phi]i} (x,y,t;p) in Taylor series with respect to p for i=1,2,3 , one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

If the auxiliary linear operator, the initial guesses, and the auxiliary parameters _{[Planck's over 2pi]i} are so properly chosen, the series (40) converge at p=1 , has [figure omitted; refer to PDF] which must be one of solutions of the original nonlinear equation as proved by Liao [13]. Define the vectors [figure omitted; refer to PDF] We have the m th-order deformation equations [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where _z1 , _z2 , and _z3 are functions of x , y , and t , and [figure omitted; refer to PDF] Now, the solutions of the m th-order deformation (44) for m ...5; 1 become [figure omitted; refer to PDF]

For simplicity, we suppose _{[Planck's over 2pi]1} =_{[Planck's over 2pi]2} =_{[Planck's over 2pi]3} =[Planck's over 2pi] .

We consider the solutions of (4) with the initial conditions (38) and obtain for m=1 [figure omitted; refer to PDF] Obviously, for [Planck's over 2pi]=-1 , 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 m=2,3 .

Using a Taylor series, then the closed form solutions yield as follows [23]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] _k1 ,_k2 , and _k3 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 q(x,y,t),[varphi](x,y,t) are very small with the present choice of t at x=50 and y=0.01 , when _k1 =0.1 , _k2 =0.03 , _k3 =-0.3,σ=I , λ=1 , and m=0.1 . Tables 3, 4, 5 and 6 help us to compare the HAM results for the first three approximations when [Planck's over 2pi]=-1 with the analytical solution through the absolute errors. Both the analytical solutions, the HPM result, and the HAM result for q(x,y,t) and [varphi](x,y,t) are plotted in Figures 1, 2, 3, and 4.

Table 1: The HPM results for q (x,y,t) in comparison with the analytical solution with initial conditions (5).

t	| q e - q |
- 0.6	9.861869456 × 1 0 - 18
- 0.4	1.073360934 × 1 0 - 17
- 0.2	1.161066111 × 1 0 - 17
0	1.248899278 × 1 0 - 17
0.2	1.336466228 × 1 0 - 17
0.4	1.423383001 × 1 0 - 17
0.6	1.509277063 × 1 0 - 17

Table 2: The HPM results for [varphi] (x,y,t) in comparison with the analytical solution with initial conditions (5).

t	| [varphi] e - [varphi] |
- 0.6	3.3574 × 1 0 - 15
- 0.4	1.9466 × 1 0 - 15
- 0.2	1.52954 × 1 0 - 15
0	1.81125 × 1 0 - 15
0.2	2.77765 × 1 0 - 15
0.4	5.04003 × 1 0 - 15
0.6	1.07655 × 1 0 - 15

Table 3: The HAM results for q (x,y,t) in comparison with the analytical solution with initial conditions (3).

t	| q e - q |
- 0.6	1.415900000 × 1 0 - 19
- 0.4	1.400513800 × 1 0 - 19
- 0.2	1.385294700 × 1 0 - 19
0	1.370240800 × 1 0 - 19
0.2	1.355350700 × 1 0 - 19
0.4	1.340622400 × 1 0 - 19
0.6	1.326054200 × 1 0 - 19

Table 4: The HAM results for [varphi] (x,y,t) in comparison with the analytical solution with initial conditions (3).

t	| [varphi] e - [varphi] |
- 0.6	0
- 0.4	0
- 0.2	4 ×10-10
0	0
0.2	0
0.4	2 ×10-10
0.6	0

Table 5: The HAM results for q (x,y,t) in comparison with the analytical solution with initial conditions (5).

t	| q e - q |
- 0.6	9.861809932 × 1 0 - 18
- 0.4	1.073356691 × 1 0 - 17
- 0.2	1.161063556 × 1 0 - 17
0	1.248898384 × 1 0 - 17
0.2	1.336466963 × 1 0 - 17
0.4	1.423385326 × 1 0 - 17
0.6	1.509280934 × 1 0 - 17

Table 6: The HAM results for [varphi] (x,y,t) in comparison with the analytical solution with initial conditions (5).

t	| [varphi] e - [varphi] |
- 0.6	0
- 0.4	0
- 0.2	0
0	0
0.2	0
0.4	0
0.6	5 × 1 0 - 10

Comparison between the exact solution, the HPM solution, and the HAM solution for q(x,y,t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

Comparison between the exact solution, the HPM solution, and the HAM solution for [varphi](x,y,t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

The results obtained by HPM and HAM for q(x,y,t) , at t=0.2 in comparison with the exact solutions.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The results obtained by HPM and HAM for [varphi](x,y,t) , at t=0.2 in comparison with the exact solutions.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

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