Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 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
Hassan A. Zedan,1,2 W. Barakati,1 and Nada Hamad1
1
2
Mathematics Department, Faculty of Science, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Mathematics Department, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt
Correspondence should be addressed to Hassan A. Zedan; hassanzedan2003@yahoo.com
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.
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 [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])
1
󵄨 󵄨2
𝑖𝑞𝑡 + 𝜎2 (𝑞𝑥𝑥 + 𝜎2 𝑞𝑦𝑦 ) + 𝜆󵄨󵄨󵄨𝑞󵄨󵄨󵄨 𝑞 − 𝜙𝑥 𝑞 = 0,
2
󵄨 󵄨2
𝜙𝑥𝑥 − 𝜎2 𝜙𝑦𝑦 − 2𝜆(󵄨󵄨󵄨𝑞󵄨󵄨󵄨 )𝑥 = 0, 𝜆 = ±1, 𝜎2 = ±1.
(1)
The case 𝜎 = 1 is called the DSI equation, while 𝜎 =
𝑖 is the DSII equation. The parameter 𝜆 characterizes the
2
Journal of Applied Mathematics
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 𝜙(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:
1
𝑢𝑡 = − 𝜎2 (V𝑥𝑥 + 𝜎2 V𝑦𝑦 ) − 𝜆 (𝑢2 + V2 ) V + 𝜙𝑥 V,
2
1
V𝑡 = 𝜎2 (𝑢𝑥𝑥 + 𝜎2 𝑢𝑦𝑦 ) + 𝜆 (𝑢2 + V2 ) 𝑢 − 𝜙𝑥 𝑢,
2
𝜙𝑥𝑥 − 𝜎2 𝜙𝑦𝑦 − 2𝜆(𝑢2 + V2 )𝑥 = 0,
(2)
𝜆 = ±1, 𝜎2 = ±1,
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:
2
𝜕2 V1,0
𝜕2 𝑉1 𝜕 V1,0
−
+
𝑝
𝜕𝑦2
𝜕𝑦2
𝜕𝑦2
+ 𝑝 [2𝑚𝑉1
with the initial condition
𝑢 (𝑥, 0, 𝑡) = 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] cos [𝑘1 𝑥 + 𝑘3 𝑡] ,
V (𝑥, 0, 𝑡) = 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] sin [𝑘1 𝑥 + 𝑘3 𝑡] ,
− 𝑚2 𝑉1 (
(3)
𝜙 (𝑥, 0, 𝑡) = 𝑓 tanh [𝑠 (𝑥 − 𝑐𝑡)] ,
where 𝑐 = 𝑘2 + 𝜎2 𝑘1 , 𝑟 = √(−2𝑘3 + 𝑘12 𝜎2 + 𝑘22 )/𝜆, 𝑠 =
√(2𝑘3 + 𝑘12 𝜎2 + 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
1
1
𝜌𝑡 = 𝜎2 𝑚 (𝜃𝑥 𝜌𝑥 + 𝜌𝜃𝑥𝑥 ) + 𝑚 (𝜃𝑦 𝜌𝑦 + 𝜌𝜃𝑦𝑦 ) ,
2
2
𝜃𝑡 =
−1
𝑚
1
(𝜎2 𝜌𝑥𝑥 + 𝜌𝑦𝑦 ) + (𝜎2 𝜃𝑥2 + 𝜃𝑦2 ) − (𝜆𝜌2 + 𝜙𝑥 ) ,
2𝑚𝜌
2
𝑚
𝜙𝑥𝑥 − 𝜎2 𝜙𝑦𝑦 − 4𝜆𝜌𝜌𝑥 = 0,
(4)
where we take 𝑞 = 𝜌(𝑥, 𝑦, 𝑡) ⋅ 𝑒−𝑚𝑖𝜃(𝑥,𝑦,𝑡) with the initial
condition
𝜌 (𝑥, 0, 𝑡) = 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] cos [𝑘1 𝑥 + 𝑘3 𝑡] ,
𝜃 (𝑥, 0, 𝑡) = 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] sin [𝑘1 𝑥 + 𝑘3 𝑡] ,
𝜕𝑉2
𝜕𝑉 2 𝜕2 𝑉1
)
+ 𝜎2 (−𝑚2 𝑉1 ( 2 ) +
𝜕𝑡
𝜕𝑥
𝜕𝑥2
(5)
𝜙 (𝑥, 0, 𝑡) = 𝑓 tanh [𝑠 (𝑥 − 𝑐𝑡)] ,
where 𝑐 = 𝑘2 + 𝜎2 𝑘1 , 𝑟 = √(−2𝑘3 + 𝑘12 𝜎2 + 𝑘22 )/𝜆, 𝑠 =
√(2𝑘3 + 𝑘12 𝜎2 + 𝑘22 )/𝜎2 , 𝑓 = 2𝜎√−𝜆/(1 − 𝜎2 ), 𝑘1 , 𝑘2 , and k3
𝜕𝑉
𝜕𝑉2 2
) + 2𝜆𝑉13 + 2𝑉1 3 ] = 0,
𝜕𝑦
𝜕𝑥
2
𝜕2 V2,0
𝜕2 𝑉2 𝜕 V2,0
−
+
𝑝
𝜕𝑦2
𝜕𝑦2
𝜕𝑦2
+ 𝑝[
−2 𝜕𝑉1 2𝜎2 𝜕𝑉2 𝜕𝑉1 1 𝜕2 𝑉2
(
)
+
+ 𝑉
𝑚𝑉1 𝜕𝑡
𝑉1
𝜕𝑥 𝜕𝑥 2 1 𝜕𝑥2
+
2 𝜕𝑉2 𝜕𝑉1
] = 0,
𝑉1 𝜕𝑦 𝜕𝑦
2
𝜕2 V3,0
𝜕2 𝑉3 𝜕 V3,0
1 𝜕2 𝑉3 4𝜆 𝜕𝑉1
−
+
𝑝
+ 𝑝 [− 2
+ 2 𝑉1
] = 0.
2
2
2
𝜕𝑦
𝜕𝑦
𝜕𝑦
𝜎 𝜕𝑥2
𝜎
𝜕𝑥
(6)
And the initial approximations are as follows:
V1,0 (𝑥, 𝑦, 𝑡) = 𝜌0 (𝑥, 𝑦, 𝑡) = 𝜌 (𝑥, 0, 𝑡) ,
V2,0 (𝑥, 𝑦, 𝑡) = 𝜃0 (𝑥, 𝑦, 𝑡) = 𝜃 (𝑥, 0, 𝑡) ,
(7)
V3,0 (𝑥, 𝑦, 𝑡) = 𝜙0 (𝑥, 𝑦, 𝑡) = 𝜙 (𝑥, 0, 𝑡) ,
𝑉1 = V1,0 + 𝑝V1,1 + 𝑝2 V1,2 + 𝑝3 V1,3 + ⋅ ⋅ ⋅ ,
𝑉2 = V2,0 + 𝑝V2,1 + 𝑝2 V2,2 + 𝑝3 V2,3 + ⋅ ⋅ ⋅ ,
𝑉3 = V3,0 + 𝑝V3,1 + 𝑝2 V3,2 + 𝑝3 V3,3 + ⋅ ⋅ ⋅ ,
(8)
Journal of Applied Mathematics
3
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−10
−0.4
−10
−4
−0.2
0
0
𝑥
0.2
−2
0
𝑡
0
𝑥
0.4
The exact solution of ∣𝑞(𝑥, 𝑦, 𝑡)∣
2
𝑡
HAM solution for ∣𝑞∣ = ∣𝑢 + 𝑖𝑣∣
(a)
(b)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−10
−0.4
−10
−0.4
−0
−0.2
−0.2
0
0
𝑥
0.2
0
𝑡
0
𝑥
0.2
HPM solution for ∣𝑞∣ = ∣𝜌𝑒−0.1𝑖𝜃 ∣
(c)
𝑡
00.44
0.4
HAM solution for ∣𝑞∣ = ∣𝜌𝑒−0.1𝑖𝜃 ∣
(d)
Figure 1: Comparison between the exact solution, the HPM solution, and the HAM solution for 𝑞(𝑥, 𝑦, 𝑡).
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
0 = (−2V1,0𝑡 + 𝑚V1,0 V2,1𝑦𝑦 + 2𝜎2 𝑚V2,0𝑥 V1,0𝑥
+ 𝜎2 𝑚V2,0𝑥𝑥 V1,0 + 𝑚V2,0𝑦𝑦 V1,0 + 2𝑚V2,0𝑦 V1,0𝑦 ) 𝑝
+ (𝜎2 𝑚V2,0𝑥𝑥 V1,1 + 2𝜎2 𝑚V2,1𝑥 V1,0𝑥 + 𝜎2 𝑚V2,1𝑥𝑥 V1,0
+2𝑚V2,0𝑦 V1,1𝑦 + 2𝑚V2,1𝑦 V1,0𝑦 ) 𝑝2
+ (2𝑚V2,2𝑦 V1,0𝑦 + 2𝜎2 𝑚V2,0𝑥 V1,2𝑥 + 2𝜎2 𝑚V2,1𝑥 V1,1𝑥
+ 𝑚V2,1𝑦𝑦 V1,2 − 2V1,2𝑡 + 𝑚V2,0𝑦𝑦 V1,2
+ 𝜎2 𝑚V2,0𝑥𝑥 V1,2 + 𝜎2 𝑚V2,1𝑥𝑥 V1,1
+ 𝑚V2,3𝑦𝑦 V1,0 + V2,2𝑦𝑦 𝑚V1,1
+ 𝑚V2,1𝑦𝑦 V1,1 − 2V1,1𝑡 + 𝑚V2,0𝑦𝑦 V1,1
+ 2𝑚V2,0𝑦 V1,2𝑦 + 2𝜎2 𝑚V2,2𝑥 V1,0𝑥
+ 2𝜎2 𝑚V2,0𝑥 V1,1𝑥 + 𝑚V2,2𝑦𝑦 V1,0
+ 2𝑚V2,1𝑦 V1,1𝑦 + 𝜎2 𝑚V2,2𝑥𝑥 V1,0 ) 𝑝3 + ⋅ ⋅ ⋅ ,
4
Journal of Applied Mathematics
1
1
0.5
0.5
0
0
−0.5
−0.5
−10
−0.4
−10
−4
−2
−0.2
𝑥
0
0
0.4
0.2
𝑥
𝑡
0
0
2
𝑡
HAM solution for 𝜙 when 𝑞 = 𝑢 + 𝑖𝑣
The exact solution for 𝜙(𝑥, 𝑦, 𝑡)
(a)
(b)
1
1
0.5
0.5
0
0
−0.5
−0.5
−10
−0.4
−10
−0.4
−0.2
𝑥
0
0
0.4
0.2
𝑡
HPM solution for 𝜙 when 𝑞 = 𝜌𝑒−0.1𝑖𝜃
(c)
−0.2
𝑥
0
0
0.2
𝑡
0.4
HAM solution for 𝜙 when 𝑞 = 𝜌𝑒−0.1𝑖𝜃
(d)
Figure 2: Comparison between the exact solution, the HPM solution, and the HAM solution for 𝜙(𝑥, 𝑦, 𝑡).
0 = (2𝑚V1,0 V2,0𝑡 + 2V1,0 V3,0𝑥 + 𝜎2 V1,0𝑥𝑥
2
+ (6𝜆V1,0
V1,2 + V1,3𝑦𝑦 − 2𝑚2 𝜎2 V1,1 V2,0𝑥 V2,1𝑥
3
2
+ 2𝜆V1,0
− 𝑚2 𝜎2 V1,0 V2,0𝑥
+ 2𝑚V1,0 V2,2𝑡 + 2V1,1 V3,1𝑥
2
− 𝑚2 V1,0 V2,0𝑦
+ V1,0𝑦𝑦 + V1,1𝑦𝑦 ) 𝑝
2
+ 2𝑚V1,2 V2,0𝑡 − 𝑚2 V1,0 V2,1𝑦
+ (2V1,1 V3,0𝑥 + V1,2𝑦𝑦 + 2𝑚V1,0 V2,1𝑡
+ 2V1,2 V3,0𝑥 + 𝜎2 V1,2𝑥𝑥
2
+ 𝜎2 V1,1𝑥𝑥 + 6𝜆V1,0
V1,1
2
− 𝑚2 𝜎2 V1,0 V2,1𝑥
+ 2𝑚V1,1 V2,1𝑡
− 2𝑚2 V1,0 V2,0𝑦 V2,1𝑦 − 2𝑚2 𝜎2 V1,0 V2,0𝑥 V2,1𝑥
2
− 𝑚2 𝜎2 V1,2 V2,0𝑥
− 2𝑚2 V1,1 V2,0𝑦 V2,1𝑦
+ 2V1,0 V3,1𝑥 + 2𝑚V1,1 V2,0𝑡
2
+ 6𝜆V1,0 V1,1
+ 2V1,0 V3,2𝑥
2
2
− 𝑚2 𝜎2 V1,1 V2,0𝑥
− 𝑚2 V1,1 V2,0𝑦
) 𝑝2
− 2𝑚2 V1,0 V2,0𝑦 V2,2𝑦 − 2𝑚2 𝜎2 V1,0 V2,0𝑥 V2,2𝑥
Journal of Applied Mathematics
5
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 1 2
𝑥
For ∣𝑞∣ = ∣𝑢 + 𝑖𝑣∣
−6 −5 −4 −3 −2 −1
3
4
5
0
−6 −5 −4 −3 −2 −1
6
HAM
Exact
1
2
3
4
5
6
𝑥
For ∣𝑞∣ = ∣𝜌𝑒−0.1𝑖𝜃 ∣
HPM
Exact
(a)
(b)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−2𝜋
−
3𝜋
2
−𝜋
−
𝜋
2
0
𝜋
2
𝜋
3𝜋
2
2𝜋
𝑥
For ∣𝑞∣ = ∣𝜌𝑒−0.1𝑖𝜃 ∣
(c)
Figure 3: The results obtained by HPM and HAM for 𝑞(𝑥, 𝑦, 𝑡), at 𝑡 = 0.2 in comparison with the exact solutions.
2
−𝑚2 V1,2 V2,0𝑦
) 𝑝3 + ⋅ ⋅ ⋅ ,
nine equations with nine unknowns, considering the initial
conditions of V𝑖,𝑗 (𝑥, 0, 𝑡) = 0, 𝑖, 𝑗 = 1, 2, 3:
0 = (𝜎2 V3,1𝑦𝑦 + 𝜎2 V3,0𝑦𝑦 − V3,0𝑥𝑥 + 4𝜆V1,0 V1,0𝑥 ) 𝑝
2
− 2V1,0𝑡 + 𝑚V1,0 V2,1𝑦𝑦 + 2𝜎2 𝑚V2,0𝑥 V1,0𝑥
2
+ (𝜎 V3,2𝑦𝑦 − V3,1𝑥𝑥 + 4𝜆V1,0 V1,1𝑥 + 4𝜆V1,1 V1,0𝑥 ) 𝑝
+ 𝜎2 𝑚V2,0𝑥𝑥 V1,0 + 𝑚V2,0𝑦𝑦 V1,0 + 2𝑚V2,0𝑦 V1,0𝑦 = 0,
+ (𝜎2 V3,3𝑦𝑦 − V3,2𝑥𝑥 + 4𝜆V1,0 V1,2𝑥 + 4𝜆V1,1 V1,1𝑥
(10)
𝜎2 𝑚V2,0𝑥𝑥 V1,1 + 2𝜎2 𝑚V2,1𝑥 V1,0𝑥 + 𝜎2 𝑚V2,1𝑥𝑥 V1,0
+ 4𝜆V1,2 V1,0𝑥 ) 𝑝3 + ⋅ ⋅ ⋅ .
(9)
To obtain the unknowns V𝑖,𝑗 (𝑥, 𝑦, 𝑡), 𝑖, 𝑗 = 1, 2, 3, we must
construct and solve the following system which includes
+ 𝑚V2,1𝑦𝑦 V1,1 − 2V1,1𝑡 + 𝑚V2,0𝑦𝑦 V1,1
+ 2𝜎2 𝑚V2,0𝑥 V1,1𝑥 + 𝑚V2,2𝑦𝑦 V1,0 + 2𝑚V2,0𝑦 V1,1𝑦
+ 2𝑚V2,1𝑦 V1,0𝑦 = 0,
(11)
6
Journal of Applied Mathematics
1
1
0.5
0.5
0
−6 −5 −4 −3 −2 −1
1
2
3
4
5
6
0
−6 −5 −4 −3 −2 −1
−0.5
1
2
3
4
5
6
−0.5
−1
−1
When 𝑞 = 𝜌𝑒−0.1𝑖𝜃
When 𝑞 = 𝑢 + 𝑖𝑣
HAM
Exact
HPM
Exact
(a)
(b)
1
0.5
0
−6 −5 −4 −3 −2 −1
1
2
3
4
5
6
−0.5
−1
When 𝑞 = 𝜌𝑒−0.1𝑖𝜃
HAM
Exact
(c)
Figure 4: The results obtained by HPM and HAM for 𝜙(𝑥, 𝑦, 𝑡), at 𝑡 = 0.2 in comparison with the exact solutions.
2𝑚V2,2𝑦 V1,0𝑦 + 2𝜎2 𝑚V2,0𝑥 V1,2𝑥 + 2𝜎2 𝑚V2,1𝑥 V1,1𝑥
2V1,1 V3,0𝑥 + V1,2𝑦𝑦 + 2𝑚V1,0 V2,1𝑡
2
+ 𝜎2 V1,1𝑥𝑥 + 6𝜆V1,0
V1,1 − 2𝑚2 V1,0 V2,0𝑦 V2,1𝑦
+ 𝑚V2,1𝑦𝑦 V1,2 − 2V1,2𝑡 + 𝑚V2,0𝑦𝑦 V1,2 + 𝜎2 𝑚V2,0𝑥𝑥 V1,2
+ 𝜎2 𝑚V2,1𝑥𝑥 V1,1 + 𝑚V2,3𝑦𝑦 V1,0 + V2,2𝑦𝑦 𝑚V1,1
(12)
2
2
− 𝑚2 𝜎2 V1,1 V2,0𝑥
− 𝑚2 V1,1 V2,0𝑦
= 0,
+ 2𝑚V2,0𝑦 V1,2𝑦 + 2𝜎2 𝑚V2,2𝑥 V1,0𝑥 + 2𝑚V2,1𝑦 V1,1𝑦
2
V1,2 + V1,3𝑦𝑦 − 2𝑚2 𝜎2 V1,1 V2,0𝑥 V2,1𝑥
6𝜆V1,0
+ 𝜎2 𝑚V2,2𝑥𝑥 V1,0 = 0,
3
2𝑚V1,0 V2,0𝑡 + 2V1,0 V3,0𝑥 + 𝜎2 V1,0𝑥𝑥 + 2𝜆V1,0
2
2
− 𝑚2 𝜎2 V1,0 V2,0𝑥
− 𝑚2 V1,0 V2,0𝑦
+ V1,0𝑦𝑦 + V1,1𝑦𝑦 = 0,
− 2𝑚2 𝜎2 V1,0 V2,0𝑥 V2,1𝑥 + 2V1,0 V3,1𝑥 + 2𝑚V1,1 V2,0𝑡
+ 2𝑚V1,0 V2,2𝑡 + 2V1,1 V3,1𝑥 + 2𝑚V1,2 V2,0𝑡
(13)
2
− 𝑚2 V1,0 V2,1𝑦
+ 2V1,2 V3,0𝑥 + 𝜎2 V1,2𝑥𝑥
(14)
Journal of Applied Mathematics
7
2
2
+ 2𝑚V1,1 V2,1𝑡 − 𝑚2 𝜎2 V1,2 V2,0𝑥
− 𝑚2 𝜎2 V1,0 V2,1𝑥
+ 𝑠2 𝜎2 cos [𝑘3 𝑡 + 𝑘1 𝑥]
2
− 2𝑚2 V1,1 V2,0𝑦 V2,1𝑦 + 6𝜆V1,0 V1,1
+ 2V1,0 V3,2𝑥
× (−1 + 𝑚2 𝑟2 sech2 [𝑠 (−𝑐𝑡 + 𝑥)] (sin2 [𝑘3 𝑡 + 𝑘1 𝑥]))
− 2𝑚2 V1,0 V2,0𝑦 V2,2𝑦 − 2𝑚2 𝜎2 V1,0 V2,0𝑥 V2,2𝑥
×tanh2 [𝑠 (−𝑐𝑡 + 𝑥)]) .
(20)
2
− 𝑚2 V1,2 V2,0𝑦
= 0,
(15)
From (10), we obtain
𝜎2 V3,1𝑦𝑦 + 𝜎2 V3,0𝑦𝑦 − V3,0𝑥𝑥 + 4𝜆V1,0 V1,0𝑥 = 0,
(16)
V2,1
𝜎2 V3,2𝑦𝑦 − V3,1𝑥𝑥 + 4𝜆V1,0 V1,1𝑥 + 4𝜆V1,1 V1,0𝑥 = 0,
(17)
=
𝜎2 V3,3𝑦𝑦 − V3,2𝑥𝑥 + 4𝜆V1,0 V1,2𝑥 + 4𝜆V1,1 V1,1𝑥
+ 4𝜆V1,2 V1,0𝑥 = 0,
1 2
𝑦 (𝑚𝑟𝑠2 𝜎2 sech3 [𝑠 (−𝑐𝑡 + 𝑥)] sin [𝑘3 𝑡 + 𝑘1 𝑥]
2𝑚
− 2𝑘3 tan [𝑘3 𝑡 + 𝑘1 𝑥] + 2𝑐𝑠 tanh [𝑠 (−𝑐𝑡 + 𝑥)]
(18)
+ 𝑚𝑟𝜎2 sech [𝑠 (−𝑐𝑡 + 𝑥)]
× (2𝑘12 sin [𝑘3 𝑡 + 𝑘1 𝑥] + 𝑠 tanh [𝑠 (−𝑐𝑡 + 𝑥)]
From (8), if the three approximations are sufficient, we
will obtain
× (𝑘1 (1 + 3 cos [2 (𝑘3 𝑡 + 𝑘1 𝑥)])
3
× sec [𝑘3 𝑡 + 𝑘1 𝑥] − 3𝑠 sin [𝑘3 𝑡 + 𝑘1 𝑥]
𝜌 (𝑥, 𝑦, 𝑡) = lim 𝑉1 (𝑥, 𝑦, 𝑡) = ∑ V1,𝑘 (𝑥, 𝑦, 𝑡) ,
𝑝→1
𝑘=0
× tanh [𝑠 (−𝑐𝑡 + 𝑥)]))) .
3
𝜃 (𝑥, 𝑦, 𝑡) = lim 𝑉2 (𝑥, 𝑦, 𝑡) = ∑ V2,𝑘 (𝑥, 𝑦, 𝑡) ,
𝑝→1
(19)
𝑘=0
From (16), we obtain
3
V3,1
𝑘=0
=
𝜙 (𝑥, 𝑦, 𝑡) = lim 𝑉3 (𝑥, 𝑦, 𝑡) = ∑ V3,𝑘 (𝑥, 𝑦, 𝑡) .
𝑝→1
(21)
To calculate the terms of the homotopy series (19) for
𝜌(𝑥, 𝑦, 𝑡), 𝜃(𝑥, 𝑦, 𝑡), and 𝜙(𝑥, 𝑦, 𝑡), we substitute the initial
conditions (5) into the system (9), and using Mathematica
software, from (13), we obtain
V1,1
1
= 𝑟𝑦2 sech [𝑠 (−𝑐𝑡 + 𝑥)]
2
× (cos [𝑘3 𝑡 + 𝑘1 𝑥]
× (𝑘12 𝜎2 − 2𝑘3 𝑚𝑟 cos [𝑘3 𝑡 + 𝑘1 𝑥]
× sech [𝑠 (−𝑐𝑡 + 𝑥)]
+ (𝑠 (−2𝑓 + 𝑠𝜎2 )
+ 𝑟2 (−2𝜆 + 𝑘12 𝑚2 𝜎2 )
× cos2 [𝑘3 𝑡 + 𝑘1 𝑥])
× (sech2 [𝑠 (−𝑐𝑡 + 𝑥)]))
− 𝑠 (2𝑘1 𝜎2 sin [𝑘3 𝑡 + 𝑘1 𝑥] + 𝑚𝑟 sech [𝑠 (−𝑐𝑡 + 𝑥)]
× (𝑐 + 𝑘1 𝑚𝑟𝜎2 cos [𝑘3 𝑡 + 𝑘1 𝑥] sech [𝑠 (−𝑐𝑡 + 𝑥)])
× sin [2 (𝑘3 𝑡 + 𝑘1 𝑥)]) tanh [𝑠 (−𝑐𝑡 + 𝑥)]
1
(𝑦2 sech2 [𝑠 (−𝑐𝑡 + 𝑥)] (𝑘1 𝑟2 𝜆 sin [2 (𝑘3 𝑡 + 𝑘1 𝑥)]
𝜎2
+ 𝑠 (−𝑓𝑠 + 2𝑟2 𝜆 cos2 [𝑘3 𝑡 + 𝑘1 𝑥]) tanh [𝑠 (−𝑐𝑡 + 𝑥)]) .
(22)
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 𝜙(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[𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝)] = 𝜕2 𝜑𝑖 /𝜕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
𝑁1 = 𝜎4
𝜕𝜑
𝜕2 𝜑2
𝜕2 𝜑
𝜕𝜑
+ 𝜎2 22 + 2 1 − 2𝜑2 3 + 2𝜆 (𝜑12 + 𝜑22) 𝜑2 ,
2
𝜕𝑦
𝜕𝑥
𝜕𝑡
𝜕𝑥
𝑁2 = 𝜎4
2
𝜕𝜑
𝜕2 𝜑1
𝜕𝜑
2 𝜕 𝜑1
+
𝜎
− 2 2 − 2𝜑1 3 + 2𝜆 (𝜑12 + 𝜑22) 𝜑1 ,
2
2
𝜕𝑦
𝜕𝑥
𝜕𝑡
𝜕𝑥
𝑁3 =
2
2
𝜕2 𝜑3
1 𝜕2 𝜑3 2𝜆 𝜕 (𝜑1 + 𝜑2 )
−
+
𝜕𝑦2
𝜎2 𝜕𝑥2
𝜎2
𝜕𝑥
(23)
8
Journal of Applied Mathematics
We have the 𝑚th-order deformation equations
we construct the zero-order deformation equations
(1 − 𝑝) 𝐿 [𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) − 𝑧𝑖,0 (𝑥, 𝑦, 𝑡)]
= 𝑝ℏ𝑖 𝑁𝑖 [𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝)] ,
𝑖 = 1, 2, 3.
(24)
𝐿 [𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) − 𝜒𝑚 𝑧𝑖,𝑚−1 (𝑥, 𝑦, 𝑡)]
= ℏ𝑖 𝑅𝑖,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 ) ,
𝑖 = 1, 2, 3,
(31)
When 𝑝 = 0
where
𝜑1 (𝑥, 𝑦, 𝑡; 0) = 𝑧1,0 (𝑥, 𝑦, 𝑡) = 𝑢0 (𝑥, 𝑦, 𝑡)
𝑅1,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
= 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] cos [𝑘1 𝑥 + 𝑘3 𝑡] ,
𝜑2 (𝑥, 𝑦, 𝑡; 0) = 𝑧2,0 (𝑥, 𝑦, 𝑡) = V0 (𝑥, 𝑦, 𝑡)
= 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] sin [𝑘1 𝑥 + 𝑘3 𝑡] ,
(25)
= 𝜎4
2
𝜕2 𝑧2,𝑚−1
𝜕𝑧1,𝑚−1
2 𝜕 𝑧2,𝑚−1
+
𝜎
+2
2
2
𝜕𝑦
𝜕𝑥
𝜕𝑡
𝑚−1
− 2 ∑ 𝑧2,𝑛
𝜑3 (𝑥, 𝑦, 𝑡; 0) = 𝑧3,0 (𝑥, 𝑦, 𝑡) = 𝜙0 (𝑥, 𝑦, 𝑡)
𝑛=0
= 𝑓 tanh [𝑠 (𝑥 − 𝑐𝑡)] .
𝜕𝑧3,𝑚−1−𝑛
𝜕𝑥
𝑚−1 𝑛
+ 2𝜆 ∑ ∑ (𝑧1,𝑘 𝑧1,𝑛−𝑘 + 𝑧2,𝑘 𝑧2,𝑛−𝑘 ) 𝑧2,𝑚−1−𝑛 ,
When 𝑝 = 1
𝑛=0 𝑘=0
𝜑1 (𝑥, 𝑦, 𝑡; 1) = 𝑢 (𝑥, 𝑦, 𝑡) ,
𝜑2 (𝑥, 𝑦, 𝑡; 1) = V (𝑥, 𝑦, 𝑡) ,
𝑅2,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
(26)
𝜑3 (𝑥, 𝑦, 𝑡; 1) = 𝜙 (𝑥, 𝑦, 𝑡) .
= 𝜎4
𝑚−1
Therefore, as the embedding parameter 𝑝 increases from
0 to 1, 𝜑𝑖 (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 𝜙(𝑥, 𝑦, 𝑡), for 𝑖 = 1, 2, 3,
respectively.
Expanding 𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) in Taylor series with respect to 𝑝
for 𝑖 = 1, 2, 3, one has
𝑛=0
𝑚−1 𝑛
𝑛=0 𝑘=0
=
where
󵄨
𝑚
1 𝜕 𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) 󵄨󵄨󵄨
󵄨󵄨 .
󵄨󵄨
𝑚!
𝜕𝑝𝑚
󵄨𝑝=0
(28)
If the auxiliary linear operator, the initial guesses, and
the auxiliary parameters ℏ𝑖 are so properly chosen, the above
series converge at 𝑝 = 1, and
+∞
𝑢 (𝑥, 𝑦, 𝑡) = 𝑧1,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧1,𝑚 (𝑥, 𝑦, 𝑡) ,
𝑚=1
+∞
V (𝑥, 𝑦, 𝑡) = 𝑧2,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧2,𝑚 (𝑥, 𝑦, 𝑡) ,
𝜕𝑧3,𝑚−1−𝑛
𝜕𝑥
+ 2𝜆 ∑ ∑ (𝑧1,𝑘 𝑧1,𝑛−𝑘 + 𝑧2,𝑘 𝑧2,𝑛−𝑘 ) 𝑧1,𝑚−1−𝑛 ,
(27)
𝑚=1
𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) =
− 2 ∑ 𝑧1,𝑛
(32)
𝑅3,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
+∞
𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) = 𝑧𝑖,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) 𝑝𝑚 ,
2
𝜕2 𝑧1,𝑚−1
𝜕𝑧2,𝑚−1
2 𝜕 𝑧1,𝑚−1
+
𝜎
−2
𝜕𝑦2
𝜕𝑥2
𝜕𝑡
(29)
𝑚=1
2
𝜕2 𝑧3,𝑚−1
1 𝜕 𝑧3,𝑚−1
−
𝜕𝑦2
𝜎2 𝜕𝑥2
+
2𝜆 𝜕 𝑚−1
+ 𝑧2,𝑛 𝑧2,𝑚−1−𝑛 ) ,
∑ (𝑧 𝑧
𝜎2 𝜕𝑥 𝑛=0 1,𝑛 1,𝑚−1−𝑛
where 𝑧1 , 𝑧2 , and 𝑧3 are functions of 𝑥, 𝑦, and 𝑡, and
0,
𝜒𝑚 = {
1,
𝑚≤1
𝑚 > 1.
(33)
Now, the solutions of the 𝑚th-order deformation (31) for 𝑚 ≥
1 become
𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) = 𝜒𝑚 𝑧𝑖,𝑚−1 (𝑥, 𝑦, 𝑡)
+∞
𝜙 (𝑥, 𝑦, 𝑡) = 𝑧3,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧3,𝑚 (𝑥, 𝑦, 𝑡) ,
𝑦
+ ℏ𝑖 ∬ [𝑅𝑖,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )] 𝑑𝑦 𝑑𝑦
𝑚=1
which must be one of solutions of the original nonlinear
equations as proved by Liao [13]. Define the vectors
𝑧⃗𝑖,𝑛 = {𝑧𝑖,0 (𝑥, 𝑦, 𝑡) , 𝑧𝑖,1 (𝑥, 𝑦, 𝑡) , . . . , 𝑧𝑖,𝑛 (𝑥, 𝑦, 𝑡)} ;
𝑖 = 1, 2, 3.
0
+ 𝑐1,𝑖 + 𝑐2,𝑖 𝑦.
(34)
(30)
For simplicity, we suppose ℏ1 = ℏ2 = ℏ3 = ℏ.
Journal of Applied Mathematics
9
We consider the solutions of (2) with the initial conditions (25). We now obtain at 𝑚 = 1
We construct the zero-order deformation equations
(1 − 𝑝) 𝐿 [𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) − 𝑧𝑖,0 (𝑥, 𝑦, 𝑡)]
1
𝑧1,1 (𝑥, 𝑦, 𝑡) = ℏ𝑟𝑦2 sech [𝑠 (𝑥 − 𝑐𝑡)]
2
= 𝑝ℏ𝑖 𝑁𝑖 [𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝)] ,
× {− [2𝑘3 + 𝑘12 𝜎2 + (2𝑓𝑠 − 2𝑟2 𝜆 + 𝑠2 𝜎2 )
𝜑1 (𝑥, 𝑦, 𝑡; 0) = 𝑧1,0 (𝑥, 𝑦, 𝑡) = 𝜌0 (𝑥, 𝑦, 𝑡)
= 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] cos [𝑘1 𝑥 + 𝑘3 𝑡] ,
− 2𝑠 (𝑐 − 𝑘1 𝜎2 ) sin [𝑘1 𝑥 + 𝑘3 𝑡]
𝜑2 (𝑥, 𝑦, 𝑡; 0) = 𝑧2,0 (𝑥, 𝑦, 𝑡) = 𝜃0 (𝑥, 𝑦, 𝑡)
× tanh [𝑠 (𝑥 − 𝑐𝑡)] + 𝑠2 𝜎2
= 𝑟 sech [𝑠 (𝑥 − 𝑐𝑡)] sin [𝑘1 𝑥 + 𝑘3 𝑡] ,
2
× cos [𝑘1 𝑥 + 𝑘3 𝑡] tanh [𝑠 (𝑥 − 𝑐𝑡)]} ,
= 𝑓 tanh [𝑠 (𝑥 − 𝑐𝑡)] .
× {− [2𝑘3 + 𝑘12 𝜎2 + (2𝑓𝑠 − 2𝑟2 𝜆 + 𝑠2 𝜎2 )
When 𝑝 = 1,
× sech2 [𝑠 (𝑥 − 𝑐𝑡)]] sin [𝑘1 𝑥 + 𝑘3 𝑡]
𝜑1 (𝑥, 𝑦, 𝑡; 1) = 𝜌 (𝑥, 𝑦, 𝑡) ,
+ 2𝑠 (𝑐 − 𝑘1 𝜎2 ) cos [𝑘1 𝑥 + 𝑘3 𝑡]
𝜑2 (𝑥, 𝑦, 𝑡; 1) = 𝜃 (𝑥, 𝑦, 𝑡) ,
× tanh [𝑠 (𝑥 − 𝑐𝑡)]
𝜑3 (𝑥, 𝑦, 𝑡; 1) = 𝜙 (𝑥, 𝑦, 𝑡) .
× tanh2 [𝑠 (𝑥 − 𝑐𝑡)]} ,
1
(ℏ𝑠𝑦2 (𝑓𝑠 − 2𝑟2 𝜆) sech2
𝜎2
+∞
× [𝑠 (𝑥 − 𝑐𝑡)] tanh [𝑠 (𝑥 − 𝑐𝑡)]) .
𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) = 𝑧𝑖,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) 𝑝𝑚 ,
Obviously, for ℏ = −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[𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝)] = 𝜕2 𝜑𝑖 /𝜕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
2
𝜕𝜑 𝜕𝜑
𝜕 𝜑2
𝜕𝜑 𝜕𝜑
+ 2 2 1 + 2𝜎2 2 1
2
𝜕𝑦
𝜕𝑦 𝜕𝑦
𝜕𝑥 𝜕𝑥
+ 𝜎2
𝑁3 =
𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) =
󵄨
𝑚
1 𝜕 𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) 󵄨󵄨󵄨
󵄨󵄨 .
󵄨󵄨
𝑚!
𝜕𝑝𝑚
󵄨𝑝=0
(41)
If the auxiliary linear operator, the initial guesses, and the
auxiliary parameters ℏ𝑖 are so properly chosen, the series (40)
converge at 𝑝 = 1, has
+∞
𝜌 (𝑥, 𝑦, 𝑡) = 𝑧1,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧1,𝑚 (𝑥, 𝑦, 𝑡) ,
+∞
𝜃 (𝑥, 𝑦, 𝑡) = 𝑧2,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧2,𝑚 (𝑥, 𝑦, 𝑡) ,
𝜕 𝜑2 2 𝜕𝜑1
−
,
𝜕𝑥2
𝑚 𝜕𝑡
𝜕2 𝜑3
1 𝜕2 𝜑3 4𝜆 𝜕𝜑1
−
+ 2 𝜑1
.
𝜕𝑦2
𝜎2 𝜕𝑥2
𝜎
𝜕𝑥
where
𝑚=1
2
𝜕2 𝜑1
𝜕𝜑
𝜕𝜑
+ 2𝑚𝜑1 2 + 2𝜆𝜑13 + 2𝜑1 3 ,
𝜕𝑥2
𝜕𝑡
𝜕𝑥
(40)
𝑚=1
(35)
2
𝜕2 𝜑1
𝜕𝜑2 2
2
2 𝜕𝜑2
−
𝑚
𝜑
((
+
𝜎
(
)
𝑁2 =
)
)
1
𝜕𝑦2
𝜕𝑦
𝜕𝑥
(39)
Therefore, as the embedding parameter 𝑝 increases from 0
to 1, 𝜑𝑖 (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 𝜙(𝑥, 𝑦, 𝑡), for 𝑖 = 1, 2, 3,
respectively.
Expanding 𝜑𝑖 (𝑥, 𝑦, 𝑡; 𝑝) in Taylor series with respect to 𝑝
for 𝑖 = 1, 2, 3, one has
+ 𝑠2 𝜎2 sin [𝑘1 𝑥 + 𝑘3 𝑡]
+ 𝜎2 𝜑1
(38)
𝜑3 (𝑥, 𝑦, 𝑡; 0) = 𝑧3,0 (𝑥, 𝑦, 𝑡) = 𝜙0 (𝑥, 𝑦, 𝑡)
1
𝑧2,1 (𝑥, 𝑦, 𝑡) = ℏ𝑟𝑦2 sech [𝑠 (𝑥 − 𝑐𝑡)]
2
𝑁1 = 𝜑1
(37)
When 𝑝 = 0,
× sech2 [𝑠 (𝑥 − 𝑐𝑡)]] cos [𝑘1 𝑥 + 𝑘3 𝑡]
𝑧3,1 (𝑥, 𝑦, 𝑡) =
𝑖 = 1, 2, 3.
(42)
𝑚=1
+∞
(36)
𝜙 (𝑥, 𝑦, 𝑡) = 𝑧3,0 (𝑥, 𝑦, 𝑡) + ∑ 𝑧3,𝑚 (𝑥, 𝑦, 𝑡) ,
𝑚=1
which must be one of solutions of the original nonlinear
equation as proved by Liao [13]. Define the vectors
𝑧⃗𝑖,𝑛 = {𝑧𝑖,0 (𝑥, 𝑦, 𝑡) , 𝑧𝑖,1 (𝑥, 𝑦, 𝑡) , . . . , 𝑧𝑖,𝑛 (𝑥, 𝑦, 𝑡)} ;
𝑖 = 1, 2, 3.
(43)
10
Journal of Applied Mathematics
We have the 𝑚th-order deformation equations
𝐿 [𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) − 𝜒𝑚 𝑧𝑖,𝑚−1 (𝑥, 𝑦, 𝑡)]
= ℏ𝑖 𝑅𝑖,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 ) ,
𝑖 = 1, 2, 3,
Table 1: The HPM results for 𝑞 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (5).
(44)
where
𝑅1,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
𝑚−1
= ∑ 𝑧1,𝑛
𝑛=0
𝑚−1
𝜕2 𝑧2,𝑚−1−𝑛
𝜕𝑧2,𝑛 𝜕𝑧1,𝑚−1−𝑛
+
2
∑
2
𝜕𝑦
𝜕𝑦
𝑛=0 𝜕𝑦
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
|𝑞𝑒 − 𝑞|
9.861869456 × 10−18
1.073360934 × 10−17
1.161066111 × 10−17
1.248899278 × 10−17
1.336466228 × 10−17
1.423383001 × 10−17
1.509277063 × 10−17
𝑚−1
𝜕𝑧2,𝑛 𝜕𝑧1,𝑚−1−𝑛 2 𝜕𝑧1,𝑚−1
−
𝜕𝑥
𝑚 𝜕𝑡
𝑛=0 𝜕𝑥
+ 2𝜎2 ∑
𝑚−1
+ 𝜎2 ∑ 𝑧1,𝑛
𝑛=0
Table 2: The HPM results for 𝜙 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (5).
𝜕2 𝑧2,𝑚−1−𝑛
,
𝜕𝑥2
𝑅2,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
=
𝑚−1 𝑛
𝜕2 𝑧1,𝑚−1
𝜕𝑧2,𝑛−𝑘 𝜕𝑧2,𝑚−1−𝑛
2
−
𝑚
∑ ∑ 𝑧1,𝑘
2
𝜕𝑦
𝜕𝑦
𝜕𝑦
𝑛=0 𝑘=0
𝑚−1 𝑛
− 𝑚2 𝜎2 ∑ ∑ 𝑧1,𝑘
𝑛=0 𝑘=0
+ 𝜎2
(45)
𝜕𝑧2,𝑛−𝑘 𝜕𝑧2,𝑚−1−𝑛
𝜕𝑥
𝜕𝑥
𝑚−1 𝑛
𝜕2 𝑧1,𝑚−1
+
2𝜆
∑ ∑ 𝑧1,𝑘 𝑧1,𝑛−𝑘 𝑧1,𝑚−1−𝑛
𝜕𝑥2
𝑛=0 𝑘=0
|𝜙𝑒 − 𝜙|
3.3574 × 10−15
1.9466 × 10−15
1.52954 × 10−15
1.81125 × 10−15
2.77765 × 10−15
5.04003 × 10−15
1.07655 × 10−15
Table 3: The HAM results for 𝑞 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (3).
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
𝑚−1
𝑚−1
𝜕𝑧3,𝑚−1−𝑛
𝜕𝑧2,𝑚−1−𝑛
+ 2 ∑ 𝑧1,𝑛
+ 2𝑚 ∑ 𝑧1,𝑛
,
𝜕𝑥
𝜕𝑡
𝑛=0
𝑛=0
𝑅3,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )
=
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
2
𝜕2 𝑧3,𝑚−1
𝜕𝑧1,𝑚−1−𝑛
1 𝜕 𝑧3,𝑚−1 4𝜆 𝑚−1
−
+ 2 ∑ 𝑧1,𝑛
,
2
2
2
𝜕𝑦
𝜎
𝜕𝑥
𝜎 𝑛=0
𝜕𝑥
|𝑞𝑒 − 𝑞|
1.415900000 × 10−19
1.400513800 × 10−19
1.385294700 × 10−19
1.370240800 × 10−19
1.355350700 × 10−19
1.340622400 × 10−19
1.326054200 × 10−19
where 𝑧1 , 𝑧2 , and 𝑧3 are functions of 𝑥, 𝑦, and 𝑡, and
0,
𝜒𝑚 = {
1,
𝑚≤1
𝑚 > 1.
(46)
Now, the solutions of the 𝑚th-order deformation (44) for
𝑚 ≥ 1 become
𝑧1,1 (𝑥, 𝑦, 𝑡)
1
= ℏ𝑟𝑦2 sech [𝑠 (−𝑐𝑡 + 𝑥)]
2
𝑧𝑖,𝑚 (𝑥, 𝑦, 𝑡) = 𝜒𝑚 𝑧𝑖,𝑚−1 (𝑥, 𝑦, 𝑡)
𝑦
+ ℏ𝑖 ∬ [𝑅𝑖,𝑚 (𝑧⃗1,𝑚−1 , 𝑧⃗2,𝑚−1 , 𝑧⃗3,𝑚−1 )] 𝑑𝑦 𝑑𝑦
0
× (cos [𝑘3 𝑡 + 𝑘1 𝑥]
× (−𝑘12 𝜎2 + 2𝑘3 𝑚𝑟 cos [𝑘3 𝑡 + 𝑘1 𝑥] sech [𝑠 (−𝑐𝑡 + 𝑥)]
+ 𝑐1,𝑖 + 𝑐2,𝑖 𝑦.
(47)
For simplicity, we suppose ℏ1 = ℏ2 = ℏ3 = ℏ.
We consider the solutions of (4) with the initial conditions (38) and obtain for 𝑚 = 1
+ (𝑠 (2𝑓 − 𝑠𝜎2 ) + 𝑟2 (2𝜆 − 𝑘12 𝑚2 𝜎2 )
× cos2 [𝑘3 𝑡 + 𝑘1 𝑥]) sech2 [𝑠 (−𝑐𝑡 + 𝑥)])
Journal of Applied Mathematics
11
+ 𝑠 (2𝑘1 𝜎2 sin [𝑘3 𝑡 + 𝑘1 𝑥] + 𝑚𝑟 sech [𝑠 (−𝑐𝑡 + 𝑥)]
Table 4: The HAM results for 𝜙 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (3).
× (𝑐 + 𝑘1 𝑚𝑟𝜎2 cos [𝑘3 𝑡 + 𝑘1 𝑥] sech [𝑠 (−𝑐𝑡 + 𝑥)])
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
× sin [2 (𝑘3 𝑡 + 𝑘1 𝑥)]) tanh [𝑠 (−𝑐𝑡 + 𝑥)]
+ 𝑠2 𝜎2 cos [𝑘3 𝑡 + 𝑘1 𝑥]
× (1 − 𝑚2 𝑟2 sech2 [𝑠 (−𝑐𝑡 + 𝑥)] sin2 [𝑘3 𝑡 + 𝑘1 𝑥])
×tanh2 [𝑠 (−𝑐𝑡 + 𝑥)]) ,
|𝜙𝑒 − 𝜙|
0
0
4 ×10−10
0
0
2 ×10−10
0
(48)
𝑧2,1 (𝑥, 𝑦, 𝑡)
Table 5: The HAM results for 𝑞 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (5).
1
=
ℏ𝑦2 (2𝑘3 tan [𝑘3 𝑡 + 𝑘1 𝑥] + 𝑚𝑟𝜎2 sech [𝑠 (−𝑐𝑡 + 𝑥)]
2𝑚
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
× ((−3𝑘12 + 3𝑠2 − 4𝑠2 sech2 [𝑠 (−𝑐𝑡 + 𝑥)])
× sin [𝑘3 𝑡 + 𝑘1 𝑥]
+ 𝑘1 𝑠 (1 + 3 cos [2 (𝑘3 𝑡 + 𝑘1 𝑥)])
× sec [𝑘3 𝑡 + 𝑘1 𝑥] tanh [𝑠 (𝑐𝑡 − 𝑥)])
−2𝑐𝑠 tanh [𝑠 (−𝑐𝑡 + 𝑥)]) ,
|𝑞𝑒 − 𝑞|
9.861809932 × 10−18
1.073356691 × 10−17
1.161063556 × 10−17
1.248898384 × 10−17
1.336466963 × 10−17
1.423385326 × 10−17
1.509280934 × 10−17
(49)
Table 6: The HAM results for 𝜙 (𝑥, 𝑦, 𝑡) in comparison with the
analytical solution with initial conditions (5).
𝑧3,1 (𝑥, 𝑦, 𝑡)
=
−1
(ℏ𝑦2 sech2 [𝑠 (−𝑐𝑡 + 𝑥)]
𝜎2
× (𝑘1 𝑟2 𝜆 sin [2 (𝑘3 𝑡 + 𝑘1 𝑥)]
+𝑠 (−𝑓𝑠 + 2𝑟2 𝜆 cos2 [𝑘3 𝑡 + 𝑘1 𝑥]) tanh [𝑠 (−𝑐𝑡 + 𝑥)]) .
(50)
Obviously, for ℏ = −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]:
𝑞 (𝑥, 𝑦, 𝑡) = 𝑟 sech [𝑠 (𝑥 + 𝑦 − 𝑐𝑡)] exp [𝑖 (𝑘1 𝑥 + 𝑘2 𝑦 + 𝑘3 𝑡)] ,
(51)
𝜙 (𝑥, 𝑦, 𝑡) = 𝑓 tanh [𝑠 (𝑥 + 𝑦 − 𝑐𝑡)] ,
(52)
where
𝑟 = √−
𝑐 = 𝑘2 + 𝜎2 𝑘1 ,
𝑠=√
2𝑘3 +
𝜎2 𝑘12
𝜆
+
𝑘22
2𝑘3 + 𝜎2 𝑘12 + 𝑘22
,
𝜆
,
𝑘1 , 𝑘2 , and 𝑘3 are arbitrary constants.
2𝜎√−𝜆
,
𝑓=
1 − 𝜎2
(53)
𝑡
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
|𝜙𝑒 − 𝜙|
0
0
0
0
0
0
5 × 10−10
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), 𝜙(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 ℏ = −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 𝜙(𝑥, 𝑦, 𝑡) are plotted in Figures 1, 2, 3, and 4.
12
Journal of Applied Mathematics
References
[1] J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp.
257–262, 1999.
[2] H. A. Zedan, “Symmetry analysis of an integrable Ito coupled
system,” Computers and Mathematics with Applications, vol. 60,
no. 12, pp. 3088–3097, 2010.
[3] J. H. He, “Asymptotology by homotopy perturbation method,”
Applied Mathematics and Computation, vol. 156, no. 3, pp. 591–
596, 2004.
[4] J. H. He, “The homotopy perturbation method for nonlinear
oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
[5] J. H. He, “Application of homotopy perturbation method to
nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26,
no. 3, pp. 695–700, 2005.
[6] J. H. He, “Comparison of homotopy perturbation method
and homotopy analysis method,” Applied Mathematics and
Computation, vol. 156, no. 2, pp. 527–539, 2004.
[7] M. Rafei and D. D. Ganji, “Explicit solutions of Helmholtz
equation and fifth-order KdV equation using homotopy perturbation method,” International Journal of Nonlinear Sciences and
Numerical Simulation, vol. 7, no. 3, pp. 321–328, 2006.
[8] H. A. Zedan, “Exact solutions for the generalized KdV equation
by using Backlund transformations,” Journal of the Franklin
Institute, vol. 348, no. 8, pp. 1751–1768, 2011.
[9] S. Abbasbandy, “Homotopy perturbation method for quadratic
Riccati differential equation and comparison with Adomian’s
decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 485–490, 2006.
[10] S. Abbasbandy, “Numerical solutions of the integral equations:
homotopy perturbation method and Adomian’s decomposition
method,” Applied Mathematics and Computation, vol. 173, no. 1,
pp. 493–500, 2006.
[11] J. H. He, “Homotopy perturbation method for bifurcation of
nonlinear problems,” International Journal of Nonlinear Sciences
and Numerical Simulation, vol. 6, no. 2, pp. 207–208, 2005.
[12] S. J. Liao, The proposed homotopy analysis technique fot the
solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong
University, 1992.
[13] S. Liao, Beyond Perturbation: Introduction to the Homotopy
Analysis Method, CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.
[14] S. Liao, “On the homotopy analysis method for nonlinear
problems,” Applied Mathematics and Computation, vol. 147, no.
2, pp. 499–513, 2004.
[15] S. J. Liao, “Comparison between the homotopy analysis method
and homotopy perturbation method,” Applied Mathematics and
Computation, vol. 169, no. 2, pp. 1186–1194, 2005.
[16] S. Abbasbandy, “The application of homotopy analysis method
to solve a generalized Hirota-Satsuma coupled KdV equation,”
Physics Letters A, vol. 361, no. 6, pp. 478–483, 2007.
[17] S. Abbasbandy, “The application of homotopy analysis method
to nonlinear equations arising in heat transfer,” Physics Letters
A, vol. 360, no. 1, pp. 109–113, 2006.
[18] H. A. Zedan, “(𝐺󸀠 /𝐺)-expansion method for the generalized
Zakharov equations,” Ricerche di Matematica, vol. 60, no. 2, pp.
203–217, 2011.
[19] M. Sajid and T. Hayat, “Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations,”
[20]
[21]
[22]
[23]
[24]
[25]
[26]
Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp.
2296–2301, 2008.
M. M. Rashidi, G. Domairry, and S. Dinarvand, “Approximate
solutions for the Burger and regularized long wave equations
by means of the homotopy analysis method,” Communications
in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp.
708–717, 2009.
H. A. Zedan and S. M. Al-Tuwairqi, “Painlevé analysis of
generalized zakharov equations,” Pacific Journal of Mathematics,
vol. 247, no. 2, pp. 497–510, 2010.
A. Davey and K. Stewartson, “On three-dimensional packets of
surface waves,” Proceedings of the Royal Society London A, vol.
338, pp. 101–110, 1974.
H. A. Zedan and S. S. Tantawy, “Solution of Davey-stewartson
equations by homotopy perturbation method,” Computational
Mathematics and Mathematical Physics, vol. 49, no. 8, pp. 1382–
1388, 2009.
S. J. Liao, “An approximate solution technique not depending
on small parameters: a special example,” International Journal
of Non-Linear Mechanics, vol. 30, no. 3, pp. 371–380, 1995.
S. J. Liao, “A kind of approximate solution technique which does
not depend upon small parameters—II: an application in fluid
mechanics,” International Journal of Non-Linear Mechanics, vol.
32, no. 5, pp. 815–822, 1997.
S. J. Liao, “An explicit, totally analytic approximate solution for
Blasius’ viscous flow problems,” International Journal of NonLinear Mechanics, vol. 34, no. 4, pp. 759–778, 1999.
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