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Applied Mathematics and Computation 216 (2010) 3592–3605
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate /amc
Numerical solutions of partial differential equations by discretehomotopy analysis method
Hongqing Zhu a,*, Huazhong Shu b, Meiyu Ding a
a Department of Electronics and Communications Engineering, East China University of Science and Technology, Shanghai 200237, Chinab School of Computer Science and Engineering, Southeast University, Nanjing 210096, China
a r t i c l e i n f o a b s t r a c t
Keywords:Discrete homotopy analysis methodCrank–NicolsonBurgers’ equationsFinite difference schemeDiffusion equationConvergence region
0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.05.005
* Corresponding author. Address: Department of EMei Long Road, Shanghai 200237, China.
E-mail address: [email protected] (H. Zhu).
This paper introduces a discrete homotopy analysis method (DHAM) to obtain approximatesolutions of linear or nonlinear partial differential equations (PDEs). The DHAM can takethe many advantages of the continuous homotopy analysis method. The proposed DHAMalso contains the auxiliary parameter �h, which provides a simple way to adjust and controlthe convergence region of solution series. The convergence of the DHAM is proved undersome reasonable hypotheses, which provide the theoretical basis of the DHAM for solvingnonlinear problems. Several examples, including a simple diffusion equation and two-dimensional Burgers’ equations, are given to investigate the features of the DHAM. Thenumerical results obtained by this method have been compared with the exact solutions.It is shown that they are in good agreement with each other.
� 2010 Elsevier Inc. All rights reserved.
1. Introduction
The numerical solutions of the linear or nonlinear PDEs play a crucial role in physics and in the engineering fields. Variousmethods for obtaining numerical solutions have been taken a great attention of researches in these fields, such as perturba-tion method [1], artificial small parameter method [2], the d-expansion method [3], the Adomian decomposition method(ADM) [4], and the homotopy perturbation method (HPM) [5,6], etc. In 1992, Liao [7] first introduced the homotopy to pro-pose an analytic technique for strongly nonlinear problems, namely the homotopy analysis method (HAM). Thereafter, theHAM has been improved step by step [8–14] and has been widely applied to solving the nonlinear problems [15–20]. Themain advantage of the HAM is that it always provides one with a simple way to adjust and control the convergence radiusof solution series [15,16]. Otherwise, it is equally valid not only for strongly nonlinear systems but also for weakly nonlinearcases [8,15,16,21,22].
Although the HAM has been widely applied to solve nonlinear problems for many years, to the best of our knowledgeuntil now, the HAM method was regarded only for the continuous equation. There is no discrete version HAM ever usedto solve linear or nonlinear PDEs. Recently, the discrete ADM method was first used to obtain the numerical solutions of dis-crete nonlinear Schrödinger equation [23]. Taking a cue from their work, this study has proposed the DHAM, and found theDHAM has similar advantages of continuous’ HAM. For example, by means of introducing an auxiliary parameter �h, one canadjust and control the convergence region of series solution, etc. In addition, this study has proved DHAM’s convergenceunder suitable and reasonable hypotheses to provide the theoretical basis of the DHAM.
. All rights reserved.
lectronics and Communications Engineering, East China University of Science and Technology, No. 130,
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3593
To illustrate the efficiency and reliability of the proposed method, the DHAM has been used to obtain the numerical solu-tions of diffusion equation and two-dimensional Burgers’ equations. For the former, the continuous diffusion equation is dis-cretized by means of the Crank–Nicolson finite difference method [24]. For diffusion equation, it can be shown the Crank–Nicolson method is unconditionally stable [25]. And two-dimensional Burgers’ equations were discretized in fully implicitfinite difference form introduced by Bahadir [26]. The convergence and stability of this finite difference scheme were dis-cussed in [26]. With the help of symbolic computation software Maple 13, the accuracy of the proposed numerical schemeis examined by comparison with other numerical solutions and exact solutions. The numerical comparison reveals that thenew technique is a promising tool for solving PDEs in science and engineering.
2. Time and space discretizations
The finite difference method is a standard numerical method for the solution of PDEs. It requires discretization with somekinds of mesh. The purpose of this study is to find the approximate solutions of continuous PDEs using discrete version HAM.Thus, it is necessary to discretized them using a certain finite difference method before using DHAM.
2.1. Crank–Nicolson finite difference scheme for diffusion equation
Consider the solution of a simple one-dimensional diffusion equation
@u@t¼ @
2u@x2 ; 0 < x < 1; t > 0; ð2:1Þ
subject to the initial condition
uðx;0Þ ¼ sinpx; 0 6 x 6 1;
and the Dirichlet boundary conditions
uð0; tÞ ¼ 0; uð1; tÞ ¼ 0; t P 0:
The discrete approximation of u(x, t) at the uniform mesh (ih,ns) are denoted as uni , where h and s are the spatial and tem-
poral mesh sizes, respectively. The finite internal, [0,1], is then divided into the mesh intervals by the point xi = ih(i = 0,1, . . . ,N). Then, the Crank–Nicolson finite difference discretization to diffusion equation (2.1) can be given by [24,25]
unþ1i � un
i
s¼ 1
2h2 unþ1iþ1 � 2unþ1
i þ unþ1i�1
� �þ un
iþ1 � 2uni þ un
i�1
� �� �: ð2:2Þ
2.2. An implicit finite difference form for Burgers’ equations
In this study, the following system of two-dimensional Burgers’ equations [26,27] will be considered to illustrate theDHAM, so we address them. One can refer to Ref. [26,27] for more details.
ut þ uux þ vuy ¼1Rðuxx þ uyyÞ;
v t þ uvx þ vvy ¼1Rðvxx þ vyyÞ;
ð2:3Þ
subject to initial conditions:
uðx; y;0Þ ¼ f ðx; yÞ; ðx; yÞ 2 D;
vðx; y;0Þ ¼ gðx; yÞ; ðx; yÞ 2 D;
and boundary conditions:
uðx; y; tÞ ¼ f1ðx; y; tÞ; x; y 2 @D; t > 0;vðx; y; tÞ ¼ g1ðx; y; tÞ; x; y 2 @D; t > 0;
where D = {(x,y) ja 6 x 6 b,a 6 y 6 b} and @D is its boundary, u(x,y, t) and v(x,y, t) are the velocity components to be deter-mined, f, g, f1, and g1 are known functions, and R is the Reynolds number.
To solve system (2.3) with initial conditions, Bahadir proposed a full implicit finite difference scheme as follows [26]:
1s
unþ1i;j � un
i;j
� �þ 1
2hxunþ1
iþ1;j � unþ1i�1;j
� �unþ1
i;j þ1
2hyunþ1
i;jþ1 � unþ1i;j�1
� �vnþ1
i;j
¼ 1
Rh2x
unþ1iþ1;j � 2unþ1
i;j þ unþ1i�1;j
� �þ 1
Rh2y
unþ1i;jþ1 � 2unþ1
i;j þ unþ1i;j�1
� �; ð2:4aÞ
3594 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
and
1s
vnþ1i;j � vn
i;j
� �þ 1
2hxvnþ1
iþ1;j � vnþ1i�1;j
� �unþ1
i;j þ1
2hyvnþ1
i;jþ1 � vnþ1i;j�1
� �vnþ1
i;j
¼ 1
Rh2x
vnþ1iþ1;j � 2vnþ1
i;j þ vnþ1i�1;j
� �þ 1
Rh2y
vnþ1i;jþ1 � 2vnþ1
i;j þ vnþ1i;j�1
� �: ð2:4bÞ
In the above definition, the space domain [0,Nx] � [0,Ny] is divided into an Nx � Ny mesh with the spatial step size hx = 1/Nx inx direction and hy = 1/Ny in y direction, respectively. The time step size s represents the increment in time. The discreteapproximation of u(x,y, t) and v(x,y, t) at the uniform mesh (ihx, jhy,ns) are denoted as un
i;j and vni;j, respectively.
3. Discrete homotopy analysis method
3.1. Fundamentals of discrete homotopy analysis method
To illustrate the basic ideas of this method, let us consider the following difference equation
N uni
� �¼ 0; ð3:1Þ
where N is a linear or nonlinear operator, i and n denote independent variable. For simplicity, we ignore all boundary orinitial conditions, which can be treated in the similar way. Similarly to continuous HAM by Liao [8], the DHAM also con-structs the zeroth-order deformation equation as follows:
ð1� qÞL uni ðqÞ � un
i;0
h i¼ q�hHn
i N uni ðqÞ
� �; ð3:2Þ
where L is an auxiliary linear operator, uni;0 is an initial guess of un
i ; �h is an auxiliary parameter, q 2 [0,1] is an embeddingparameter, Hn
i – 0 is an auxiliary function. uni ðqÞ is an unknown function about i,n,q. It should be emphasized that one
has great freedom to choose the auxiliary parameter in DHAM. Obviously, when q = 0 and q = 1, it holds uni ð0Þ ¼ un
i;0 ¼ u0i
and uni ð1Þ ¼ un
i , respectively. Thus, as q increase from 0 to 1, uni ðqÞ deforms form the initial guess un
i;0 to the solution uni .
Expanding uni ðqÞ in Taylor series with respect to the embedding parameter q, one has
uni ðqÞ ¼ un
i;0 þX1m¼1
uni;mqm; ð3:3Þ
where
uni;m ¼
1m!
@muni ðqÞ
@qm
����q¼0: ð3:4Þ
Similarly continuous HAM by Liao [8], if the auxiliary linear operator, the initial guess, the auxiliary parameter �h, and theauxiliary function Hn
i – 0 are so properly chosen that the above series converges at q = 1, we have
uni ¼ un
i;0 þX1m¼1
uni;m: ð3:5Þ
Which must be one of solutions of original nonlinear equations. It will be discussed in Section 3.2 in detail. According to Eq.(3.4), the governing equation can be deduced from the zeroth-order deformation equation (3.2). Define the vector
uni;m ¼ un
i;0;uni;1; . . . ;un
i;m
n o:
Differentiating the zeroth-order deformation equation (3.2) m-times with respect to q and then dividing them by m! and fi-nally setting q = 0, one obtain the following mth-order deformation problem.
L uni;m � vmun
i;m�1
h i¼ �hHn
iRm uni;m�1
h i; ð3:6Þ
where
Rm uni;m�1
h i¼ 1ðm� 1Þ!
@m�1N uni ðqÞ
� �@qm�1
�����q¼0
; ð3:7Þ
and
vm ¼0 m 6 1;1 m > 1:
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3595
3.2. Convergence theorem
It should be emphasized that it is very important to ensure that the Eq. (3.3) converges at q = 1. Otherwise, the Eq. (3.5)has no meanings. It is present as Theorem 3.1.
Theorem 3.1 (Convergence theorem). As long as the series (3.5) is convergent, where uni;m is governed by the high deformation
equation (3.6) , it must be the solution of original equation (3.1).
Proof. Let
sni ¼ un
i;0 þXþ1m¼1
uni;m; ð3:8Þ
denotes the convergent series. Using the mth-order deformation equation (3.6), we have
Xþ1m¼1
�hHniRm un
i;m�1
h i¼Xþ1m¼1
L uni;m � vmun
i;m�1
h i¼ L
Xþ1m¼1
uni;m �
Xþ1m¼1
vmuni;m�1
" #¼ L
Xþ1m¼1
uni;m �
Xþ1k¼0
vkþ1uni;k
" #
¼ LXþ1m¼1
uni;m �
Xþ1k¼1
uni;k
" #¼ 0: ð3:9Þ
Since �h – 0; Hni – 0, then
Xþ1m¼1
Rm uni;m�1
h i¼ 0: ð3:10Þ
In addition, according to the definition (3.7), one can obtain
Xþ1m¼1
Rm uni;m�1
h i¼Xþ1m¼1
1ðm� 1Þ!
@m�1N uni ðqÞ
� �@qm�1
�����q¼0
¼Xþ1m¼0
1m!
@mN uni ðqÞ
� �@qm
����q¼0¼ 0: ð3:11Þ
Generally, uni ðqÞ don’t satisfy the original difference equation. Let
eni ðqÞ ¼ N un
i ðqÞ� �
; ð3:12Þ
denote the residual error of the original difference equation. Clearly,
eni ðqÞ ¼ 0; ð3:13Þ
corresponds to the exact solution of the original equation. According to the above definition, we expand eni ðqÞ in Maclaurin
series with respect to the embedding parameter q, thus
eni ðqÞ ¼
Xþ1m¼0
1m!
@meni ðqÞ
@qm
����q¼0
!qm ¼
Xþ1m¼0
1m!
@mN uni ðqÞ
� �@qm
����q¼0
!qm: ð3:14Þ
When q = 1, the above expression gives
eni ð1Þ ¼
Xþ1m¼0
1m!
@mN uni ðqÞ
� �@qm
����q¼0¼ 0: ð3:15Þ
According to (3.3) and (3.12)–(3.15), one gets the exact solution of the original equation when q = 1, that is
sni ¼ un
i ð1Þ ¼ uni;0 þ
Xþ1m¼1
uni;m: ð3:16Þ
Thus, as long as the series
uni;0 þ
Xþ1m¼1
uni;m;
is convergent, it must be the solution of the original equation. This ends of the proof. h
3596 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
4. Numerical implementations
Example 1 (DHAM applying to diffusion equation). In order to assess the advantage and accuracy of DHAM, this studyconsiders a simple one-dimensional diffusion equation (2.1). It is discretized by the Crank–Nicolson finite difference methodusing Eq. (2.2). For diffusion equation, it can be shown unconditionally stable [24,25]. Thus, one can rewrite Eq. (2.2) in theoperator form
Dþs uni ¼
12
D2hun
i þ D2hunþ1
i
� �; ð4:1Þ
with the initial conditions:
u0i ¼ fi; i 2 Z:
Here, Dþs uni and D2
huni are standard forward difference and standard second order differences, respectively.
Dþs uni ¼
unþ1i � un
i
s; D2
huni ¼
uniþ1 � 2un
i þ uni�1
h2 : ð4:2Þ
It is straightforward to choose the auxiliary linear operators
L uni ðqÞ
� �¼ Dþs u
ni ðqÞ ¼
unþ1i ðqÞ �un
i ðqÞs
; ð4:3Þ
with the property
L½c� ¼ 0;
where c is constant coefficients. In this example, we define the following linear operators.
N uni ðqÞ
� �¼ Dþs u
ni ðqÞ �
12
D2hu
nþ1i ðqÞ þ D2
huni ðqÞ
� �: ð4:4Þ
Thus, one can construct the zeroth-order deformation equations (3.2) with above definition,When q = 0 in Eq. (3.2) it is clear that
uni ð0Þ ¼ un
i;0:
When q = 1 in Eq. (3.2) it is clear that
uni ð1Þ ¼ un
i :
So, as the embedding parameter q increases from 0 to 1, uni ðqÞ vary form initial guesses un
i;0 to solution series uni . Then, one
can obtain the high-order deformation equation (3.6) for m P 1 with
Rm uni;m�1
h i¼ 1ðm� 1Þ!
@m�1N uni ðqÞ
� �@qm�1
�����q¼0
¼ Dþs uni;m�1 �
12
D2hunþ1
i;m�1 þ D2hun
i;m�1
� �: ð4:5Þ
In this problem, the linear operator is determined as follows:
Lð�Þ ¼ Dþs ð�Þ; ð4:6Þ
and the inverse operator L�1ð�Þ of this system can be defined:
L�1wn ¼ Dþs� ��1
wn ¼ sXn�1
m¼0
wm; n 2 N0: ð4:7Þ
Thus,
Dþs� ��1
Dþs uni ¼ un
i � u0i : ð4:8Þ
For simplicity, we select Hni ¼ 1 in this problem. So, the approximations of un
i are only dependent on auxiliary parameter �h.Applying the inverse operator Dþs
� ��1 on both sides of the system of Eq. (3.6), one can solve this problem by the DHAM withN = 10 and different �h; kðk ¼ s=h2Þ. With the help of symbol software Maple 13, one can obtain the following components:
Table 1The comequatio
k
0.250.51.02.04.0
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3597
uni;0 ¼ u0
i ¼ sinðpihÞ;un
i;1 ¼ �2�hkn sinðpihÞðcosðphÞ � 1Þ;un
i;2 ¼ 2�hkn sinðpihÞ �hkn cosðphÞ � 2�hkn� �h� 1ð Þ cosðphÞ þ 1þ �hþ �hknð Þ;
uni;3 ¼ �
23
�hkn sinðpihÞ ð2n2 þ 1Þ�h2k2 cos3ðphÞ � 6�hn2kþ 3�hkþ 6�hnþ 6n� �
�hk cos2ðphÞ�
þ 6�h2n2k2 þ 3�h2k2 þ 12�h2nkþ 12�hnkþ 3�h2 þ 6�hþ 3� �
cosðphÞ
� 2�h2n2k2 þ �h2k2 þ 6�h2nkþ 6�hnkþ 3�h2 þ 6�hþ 3� ��
;
� � �
So, the approximate solutions corresponding to the above are given by
uni ¼ un
i;0 þ uni;1 þ un
i;2 þ uni;3 þ � � � ð4:9Þ
The exact solutions of this problem are given by Lu and Guang [28]
uðx; tÞ ¼ e�p2t sin px: ð4:10Þ
parison of the results of the DHAM, Lu and Guang [28] and the exact solutions considering different ⁄ and k at u(0.4,0.4), hx = hy = 0.1 for the diffusionn.
Lu and Guang [28] DHAM (⁄ = �0.3) DHAM (⁄ = �0.5) DHAM (⁄ = �0.8) Exact solution
0.0189519 0.0189576 0.0189473 0.0189516 0.01835190.0189407 0.0189467 0.0189406 0.0189409 0.01835190.0188963 0.0188988 0.0188963 0.0188966 0.01835190.0187186 0.0187235 0.0187152 0.0187100 0.01835190.0180093 0.0180156 0.0180071 0.0180088 0.0183519
Fig. 1. ⁄-curve for the 30th-order approximation at k = 2, N = 10, s = 0.02, hx = hy = 0.1, and u(0.4,0.4).
3598 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
The 30th-order approximations of u(x, t) fork = 2, hx = hy = 0.1, s = 0.02 at u(0.4,0.4) are compared with exact solutions andresults are listed in Table 1. The results obtained by DHAM could also be compared with the solutions obtained by Luand Guang [28]. In their methods, the Newton’s method was applied for finding the approximate solutions. Clearly, boththe proposed DHAM and Liu’ method give much better approximations for the exact solutions. Nevertheless, the proposedDHAM has great freedom to adjust and control the convergence region of solution series by an auxiliary parameter ⁄. InFig. 1, the ⁄-curve are plotted for 30th-order approximation fork = 2, s = 0.02 at u(0.4,0.4). It is obvious form Fig. 1 thatthe range for the admissible values of ⁄ is �1.2 6 ⁄ 6 �0.15. We still have freedom to choose the auxiliary parameter ⁄according to ⁄-curve. Thus, the arbitrary point of this interval, i.e. �0.5, is an appropriate selection for ⁄ in which the numer-ical solution converges. Fig. 2 shows the numerical solutions for different time at k = 2, N = 10, s = 0.02, ⁄ = �0.5 which ex-hibit the correct physics behaviors of the problem. The numerical solutions of the considered problem at different time t aregiven in Fig. 3(a). Fig. 3(b) is the comparison of the approximate solutions (4.9) and the exact solutions (4.10).
Example 2 (DHAM applying to two-dimensional Burgers’ equations). In second example, we consider the two-dimensionalBurgers’ equations (2.3), with exact solutions are
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4t = 0.1t = 0.2t = 0.3t = 0.4t = 0.5
Fig. 2. The numerical solutions for the 30th-order approximation by the DHAM at different times for k = 2, N = 10, ⁄ = �0.5, hx = hy = 0.1, s = 0.02.
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
xt
0
0.2
0.4
0.6
0.8
0
0.1
0.2
0.3
0.4
0
0.5
1
1.5
2
x 10−3
xt
a b
Fig. 3. Comparison between numerical solutions and the exact solutions for k = 2, ⁄ = �0.5, N = 10, hx = hy = 0.1, s = 0.02. (a) Approximate solutions, (b)Error = jExact � Approximatej.
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3599
uðx; y; tÞ ¼ 34� 1
4 1þ eðRð�t�4xþ4yÞÞ=32ð Þ ;
vðx; y; tÞ ¼ 34þ 1
4 1þ eðRð�t�4xþ4yÞÞ=32ð Þ :ð4:11Þ
These exact solutions can be obtained using a Hopf–Cole transformation given in [29]. The system of Burgers’ equation (2.3)in the operator form can be written as follows
Dþs uni;j þ Dhx unþ1
i;j
� �unþ1
i;j þ Dhy unþ1i;j
� �vnþ1
i;j ¼1R
D2hx
unþ1i;j þ D2
hyunþ1
i;j
� �;
Dþs vni;j þ Dhx vnþ1
i;j
� �unþ1
i;j þ Dhy vnþ1i;j
� �vnþ1
i;j ¼1R
D2hx
vnþ1i;j þ D2
hyvnþ1
i;j
� �;
ð4:12Þ
The solutions of which are to be obtained subject to the initial conditions:
u0i;j ¼ fi;j; v0
i;j ¼ gi;j; i; j 2 Z:
The numerical approximation of u(x,y, t) and v(x,y, t) are denoted as uni;j � uðihx; jhy;nsÞ and vn
i;j � vðihx; jhy;nsÞ, respectively.The standard forward difference is:
Dþs uni;j ¼ unþ1
i;j � uni;j
� �.s; Dþs vn
i;j ¼ vnþ1i;j � vn
i;j
� �.s; ð4:13Þ
and Dhx unþ1i;j ; Dhy unþ1
i;j ; Dhx vnþ1i;j , and Dhy vnþ1
i;j denote the central differences which are defined as follows:
Dhx unþ1i;j ¼
unþ1iþ1;j � unþ1
i�1;j
2hx; Dhy unþ1
i;j ¼unþ1
i;jþ1 � unþ1i;j�1
2hy; Dhx vnþ1
i;j ¼vnþ1
iþ1;j � vnþ1i�1;j
2hx; Dhy vnþ1
i;j ¼vnþ1
i;jþ1 � vnþ1i;j�1
2hy: ð4:14Þ
The standard second order differences D2hx
unþ1i;j ; D2
hyunþ1
i;j ; D2hx
vnþ1i;j , and D2
hyvnþ1
i;j are written:
D2hx
unþ1i;j ¼ unþ1
iþ1;j � 2unþ1i;j þ unþ1
i�1;j
� �.h2
x ; D2hy
unþ1i;j ¼ unþ1
i;jþ1 � 2unþ1i;j þ unþ1
i;j�1
� �.h2
y ;
D2hx
vnþ1i;j ¼ vnþ1
iþ1;j � 2vnþ1i;j þ vnþ1
i�1;j
� �.h2
x ; D2hy
vnþ1i;j ¼ vnþ1
i;jþ1 � 2vnþ1i;j þ vnþ1
i;j�1
� �.h2
y :ð4:15Þ
According to Eq. (4.7), the inverse operators of this problem are determined as follows:
Dþs� ��1
Dþs uni;j ¼ un
i;j � u0i;j; Dþs
� ��1Dþs vn
i;j ¼ vni;j � v0
i;j: ð4:16Þ
Using Eq. (4.12), it is straightforward to define the auxiliary linear operators:
Lu uni;jðqÞ
h i¼ Dþs u
ni;jðqÞ ¼
unþ1i;j ðqÞ �un
i;jðqÞs
;
Lv wni;jðqÞ
h i¼ Dþs wn
i;jðqÞ ¼wnþ1
i;j ðqÞ � wni;jðqÞ
s ;
ð4:17Þ
with the property
Lu½c1� ¼ 0; Lv ½c2� ¼ 0;
where c1 and c2 are constant coefficients, uni;jðqÞ and wn
i;jðqÞ are real functions about i, j,n,q. The nonlinear operators are de-fined according to Eq. (4.12) as follows:
N u uni;jðqÞ;w
ni;jðqÞ
h i¼ Dþs u
ni;jðqÞ þ Dhxu
nþ1i;j ðqÞ
� �unþ1
i;j ðqÞ þ Dhyunþ1i;j ðqÞ
� �wnþ1
i;j ðqÞ
� 1R
D2hxunþ1
i;j ðqÞ þ D2hyunþ1
i;j ðqÞ� �
; ð4:18aÞ
and
N v uni;jðqÞ;w
ni;jðqÞ
h i¼ Dþs wn
i;jðqÞ þ Dhy wnþ1i;j ðqÞ
� �unþ1
i;j ðqÞ þ Dhy wnþ1i;j ðqÞ
� �wnþ1
i;j ðqÞ �1R
D2hx
wnþ1i;j ðqÞ þ D2
hywnþ1
i;j ðqÞ� �
: ð4:18bÞ
The zeroth-order deformation equations of the DHAM are constructed as follows:
ð1� qÞLu uni;jðqÞ � un
i;j;0
h i¼ q�huH1n
i;jN u uni;jðqÞ;w
ni;jðqÞ
h i;
ð1� qÞLv wni;jðqÞ � vn
i;j;0
h i¼ q�hvH2n
i;jN v uni;jðqÞ;w
ni;jðqÞ
h i:
ð4:19Þ
3600 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
Here, ⁄u and ⁄v are nonzero auxiliary parameters, H1ni;j and H2n
i;j denote nonzero auxiliary functions. Obviously, when q = 0and q = 1 in Eq. (4.19), it holds
uni;jð0Þ ¼ un
i;j;0 ¼ u0i;j ¼ fi;j; wn
i;jð0Þ ¼ vni;j;0 ¼ v0
i;j ¼ gi;j; ð4:20aÞ
and
uni;jð1Þ ¼ un
i;j; wni;jð1Þ ¼ vn
i;j; ð4:20bÞ
respectively. As the embedding parameter p increases from 0 to 1, uni;jðqÞ and wn
i;jðqÞ varies form the initial guess uni;j;0 and vn
i;j;0
to the solutions uni;j and vn
i;j, respectively. Then using Taylor’s theorem and (4.20a), one has the power series of q in the fol-lowing forms
uni;jðqÞ ¼ un
i;j;0 þX1m¼1
uni;j;mqm;
wni;jðqÞ ¼ vn
i;j;0 þX1m¼1
vni;j;mqm:
ð4:21Þ
Here,
uni;j;m ¼
1m!
@muni;jðqÞ
@qm
�����q¼0
;
vni;j;m ¼
1m!
@mwni;jðqÞ
@qm
�����q¼0
:
ð4:22Þ
According to convergence theorem in sub Section 3.2, if series (4.21) is convergence at q = 1, one has
uni;j ¼ un
i;j;0 þX1m¼1
uni;j;m;
vni;j ¼ vn
i;j;0 þX1m¼1
vni;j;m:
ð4:23Þ
The components of series (4.23) in the vector forms is
uni;j;k ¼ un
i;j;0;uni;j;1; . . . ;un
i;j;k
n o;
vni;j;k ¼ vn
i;j;0;vni;j;1; . . . ; vn
i;j;k
n o:
ð4:24Þ
Similarly, the high-order deformation equations are obtained by differentiating the zero-order deformation, (4.19), m timeswith respect to q, then dividing by m!, and finally setting q = 0.
Lu uni;j;m � vmun
i;j;m�1
h i¼ �huH1n
i;jRum un
i;j;m�1;vni;j;m�1
h i;
Lv vni;j;m � vmvn
i;j;m�1
h i¼ �hvH2n
i;jRvm un
i;j;m�1;vni;j;m�1
h i;
ð4:25Þ
where
Rum un
i;j;m�1;vni;j;m�1
h i¼ 1ðm� 1Þ!
@m�1N u uni;jðqÞ;w
ni;jðqÞ
h i@qm�1
������q¼0
¼ Dþs uni;j;m�1 þ
Xm�1
l¼0
unþ1i;j;l Dhx unþ1
i;j;m�1�l þXm�1
l¼0
vnþ1i;j;l Dhy unþ1
i;j;m�1�l �1R
D2hx
unþ1i;j;m�1 þ D2
hyunþ1
i;j;m�1
� �; ð4:26aÞ
Rvm un
i;j;m�1;vni;j;m�1
h i¼ 1ðm� 1Þ!
@m�1N v uni;jðqÞ;w
ni;jðqÞ
h i@qm�1
������q¼0
¼ Dþs vni;j;m�1 þ
Xm�1
l¼0
unþ1i;j;l Dhx vnþ1
i;j;m�1�l þXm�1
l¼0
vnþ1i;j;l Dhy vnþ1
i;j;m�1�l �1R
D2hx
vnþ1i;j;m�1 þ D2
hyvnþ1
i;j;m�1
� �: ð4:26bÞ
In this study, we select H1ni;j ¼ H2n
i;j ¼ 1 and ⁄u = ⁄v = ⁄ for simplicity.
uni;j;m ¼ vmun
i;j;m�1 þ �hL�1u Ru
m uni;j;m�1;vn
i;j;m�1
h i;
vni;j;m ¼ vmvn
i;j;m�1 þ �hL�1v Rv
m uni;j;m�1;v
ni;j;m�1
h i:
ð4:27Þ
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3601
Clearly, the uni;j;m and vn
i;j;m ðm P 1Þ components of this example can been obtained using the symbol software Maple. Theyare listed as follows:
Table 2Compar
Mesh
(0.1,(0.3,(0.9,(0.7,(0.9,(0.8,(0.2,(0.4,(0.2,(0.2,(0.6,(0.1,(0.9,
Table 3Compar
Mesh
(0.1,(0.3,(0.9,(0.7,(0.9,(0.8,(0.2,(0.4,(0.2,(0.2,(0.6,(0.1,(0.9,
uni;j;0 ¼ fi;j ¼
34� 1
4ð1þ e0:5iþ0:5jÞ ;
vni;j;0 ¼ gi;j ¼
34þ 1
4ð1þ e0:5iþ0:5jÞ ;
uni;j;1 ¼ �
1:25� 10�4n�hðe�0:5iþ0:5þ0:5j þ 3e�0:5i�0:5þ0:5j � 4e�0:5iþ0:5j � 2e�iþ0:5þj þ 4e�iþj � 2e�i�0:5þjÞð1þ e�0:5i�0:5þ0:5jÞð1þ e�0:5iþ0:5þ0:5jÞð1þ e�0:5iþ0:5jÞ ;
vni;j;1 ¼
1:25� 10�4n�hðe�0:5iþ0:5þ0:5j þ 3e�0:5i�0:5þ0:5j � 4e�0:5iþ0:5j � 2e�iþ0:5þj þ 4e�iþj � 2e�i�0:5þjÞð1þ e�0:5i�0:5þ0:5jÞð1þ e�0:5iþ0:5þ0:5jÞð1þ e�0:5iþ0:5jÞ ;
uni;j;2 ¼ 3:9� 1047nð1þ nÞ�h2 0:57e�0:5iþ0:5j þ 2:8e�iþj þ 2:9e�1:5iþ1:5j � 1:7e�2iþ2j � 3:9e�2:5iþ2:5j � 2:2e�3iþ3j � 0:52e�3:5iþ3:5j
� �= 1010 þ 6:1� 109e�0:5iþ0:5j� �
109 þ 1:6� 109e�0:5iþ0:5j� �2
ð1þ e�0:5iþ0:5jÞ2 4� 109 þ 2:4� 109e�0:5iþ0:5j
� �
� 2:5� 109 þ 9:2� 108e�0:5iþ0:5j� �
2:5� 108 þ 6:8� 108e�0:5iþ0:5j� ��
þ 1:25� 10�4n�hð1þ �hÞ
��
4e�0:5iþ0:5j þ 2e�iþ0:5þj � 4e�iþj þ 2e�i�0:5þj � e�0:5iþ0:5þ0:5j � 3e�0:5i�0:5þ0:5j�=�ð1þ e�0:5i�0:5þ0:5jÞð1þ e�0:5iþ0:5þ0:5jÞð1þ e�0:5iþ0:5jÞ
�;
ison of the numerical results by DHAM with exact solutions u(x,y, t) at R = 80, s = 10�4, hx = hy = 0.05, ⁄ = �1, and different t.
point t = 0.01 t = 0.1 t = 0.5
Numerical Exact Numerical Exact Numerical Exact
0.1) 0.623468 0.623438 0.609682 0.609456 0.556014 0.5556750.1) 0.529131 0.529151 0.523659 0.523837 0.509224 0.5093320.2) 0.500222 0.500222 0.500175 0.500177 0.500037 0.5000650.3) 0.504381 0.504387 0.503465 0.503516 0.500865 0.5013050.4) 0.501630 0.501632 0.501285 0.501305 0.500291 0.5004820.5) 0.511564 0.511577 0.509219 0.509332 0.502774 0.5035160.6) 0.745390 0.745392 0.744235 0.744256 0.735268 0.7349780.7) 0.737855 0.737858 0.734954 0.734978 0.713696 0.7129880.7) 0.748284 0.748285 0.747846 0.747856 0.744354 0.7442560.8) 0.749366 0.749366 0.749203 0.749207 0.747890 0.7478560.8) 0.719538 0.719537 0.713035 0.712988 0.670091 0.6697950.9) 0.749914 0.749914 0.749892 0.749892 0.749712 0.7497080.9) 0.623468 0.623438 0.609682 0.609456 0.556014 0.555675
ison of the numerical results by DHAM with exact solutions v(x,y, t) at R = 80, s = 10�4, hx = hy = 0.05, ⁄ = �1, and different t.
point t = 0.01 t = 0.1 t = 0.5
Numerical Exact Numerical Exact Numerical Exact
0.1) 0.876532 0.876562 0.890318 0.890544 0.943986 0.9443250.1) 0.970869 0.970849 0.976341 0.976163 0.990776 0.9906680.2) 0.999778 0.999778 0.999825 0.999823 0.999963 0.9999350.3) 0.995619 0.995612 0.996535 0.996484 0.999135 0.9986950.4) 0.998370 0.998368 0.998715 0.998695 0.999708 0.9995180.5) 0.988436 0.988423 0.990781 0.990668 0.997225 0.9964840.6) 0.754610 0.754608 0.755765 0.755744 0.764732 0.7650220.7) 0.762145 0.762142 0.765046 0.765022 0.786304 0.7870120.7) 0.751716 0.751715 0.752154 0.752144 0.755646 0.7557440.8) 0.750634 0.750634 0.750797 0.750793 0.752110 0.7521440.8) 0.780462 0.780463 0.786965 0.787012 0.829908 0.8302050.9) 0.750086 0.750086 0.750108 0.750108 0.750288 0.7502920.9) 0.876532 0.876562 0.890318 0.890544 0.943986 0.944325
Fig. 4.u(x,y,0
Fig. 5.v(0.5,0
3602 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
vni;j;2 ¼ �3:9� 1047nð1þ nÞ�h2 0:57e�0:5iþ0:5j þ 2:8e�iþj þ 2:9e�1:5iþ1:5j � 1:7e�2iþ2j � 3:9e�2:5iþ2:5j � 2:2e�3iþ3j
��0:52e�3:5iþ3:5j
�= 1010 þ 6:1� 109e�0:5iþ0:5j� �
109 þ 1:6� 109e�0:5iþ0:5j� �2
ð1þ e�0:5iþ0:5jÞ2 4� 109 þ 2:4� 109e�0:5iþ0:5j� �
� 2:5� 109 þ 9:2� 108e�0:5iþ0:5j� �
2:5� 108 þ 6:8� 108e�0:5iþ0:5j� ��
þ 1:25� 10�4n�hð1þ �hÞ
��
e�0:5iþ0:5þ0:5j þ 3e�0:5i�0:5þ0:5j � 4e�0:5iþ0:5j � 2e�iþ0:5þj þ 4e�iþj � 2e�i�0:5þj�=�ð1þ e�0:5i�0:5þ0:5jÞ
� ð1þ e�0:5iþ0:5þ0:5jÞð1þ e�0:5iþ0:5jÞ�:
In this discussed problem, the numerical solutions are obtained using the following expressions.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0.5
0.55
0.6
0.65
0.7
xy
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0.8
0.85
0.9
0.95
1
xy
a b
The numerical solutions for the 3rd-order approximation by DHAM at N = 20, R = 80, s = 10�4, hx = hy = 0.05, ⁄ = �1, (a) Approximation solutions of.3), (b) Approximation solutions of v(x,y,0.3).
a b⁄-curve for the 3rd-order approximation for N = 20, R = 80, s = 10�4, hx = hy = 0.05, (a) ⁄-curve in the case u(0.5,0.5, 0.3), (b) ⁄-curve in the case.5,0.3).
Table 4Comparu(x,y, t)
Mesh
(0.1,(0.2,(0.3,(0.2,(0.3,(0.4,(0.1,(0.3,(0.4,(0.2,(0.3,(0.5,(0.5,
Table 5Comparu(x,y, t)
Mesh
(0.1,(0.2,(0.3,(0.2,(0.3,(0.4,(0.1,(0.3,(0.4,(0.2,(0.3,(0.5,(0.5,
Fig. 6.u(x,y,0.
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3603
uni;j � un
i;j;0 þ uni;j;1 þ un
i;j;2;
vni;j � vn
i;j;0 þ vni;j;1 þ vn
i;j;2:ð4:28Þ
ison of numerical solutions by DHAM with the exact solutions for u and v at t = 0.1, N = 20, s = 10�3, hx = hy = 0.025, ⁄ = �1, Error1 = jnumerical � exactj, Error2 = jnumerical � exact v(x,y, t)j.
point Numerical Error1 Numerical Error2
0.1) 0.18371 3.30628E�5 �0.02042 1.06045E�50.1) 0.26535 4.42933E�5 0.08164 1.12272E�50.1) 0.34699 5.55240E�5 0.18371 3.30591E�50.2) 0.36741 6.61254E�5 �0.04084 2.12090E�50.2) 0.44906 7.73562E�5 0.06122 6.22730E�70.2) 0.53070 8.85863E�5 0.16329 2.24544E�50.3) 0.38783 7.67274E�5 �0.26538 7.54773E�50.3) 0.55112 9.91885E�5 �0.06126 3.18136E�50.3) 0.63276 1.10419E�4 0.04081 9.98180E�60.4) 0.57154 1.09790E�4 �0.28580 8.60816E�50.4) 0.65318 1.21021E�4 �0.18374 6.42496E�50.4) 0.81647 1.43481E�4 0.02039 2.05863E�50.5) 0.91853 1.65314E�4 �0.10209 5.30227E�5
ison of numerical solutions by DHAM with the exact solutions for u and v at t = 0.4, N = 20, s = 10�3, hx = hy = 0.025, ⁄ = �1, Error1 = jnumerical � exactj, Error2 = jnumerical � exact v(x,y, t)j.
point Numerical Error1 Numerical Error2
0.1) 0.17678 3.10064E�4 �0.11765 6.67950E�60.1) 0.20634 4.61772E�4 0.02956 1.51679E�40.1) 0.23591 6.13480E�4 0.17678 3.10038E�40.2) 0.35356 6.20129E�4 �0.23531 1.33592E�50.2) 0.38312 7.71836E�4 �0.08809 1.44999E�40.2) 0.41269 9.23544E�4 0.05913 3.03358E�40.3) 0.47121 6.26778E�4 �0.64740 3.36756E�40.3) 0.53034 9.30193E�4 �0.35296 2.00384E�50.3) 0.55990 1.08190E�3 �0.20574 1.38320E�40.4) 0.64800 9.36843E�4 �0.76505 3.43435E�40.4) 0.67756 1.08855E�3 �0.61783 1.85077E�40.4) 0.73669 1.39196E�3 �0.32340 1.31641E�40.5) 0.88390 1.55032E�3 �0.58827 3.33977E�5
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
xy
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
−1−0.8−0.6−0.4−0.2
00.20.40.6
xy
a b
The numerical solutions for the 10th-order approximation by DHAM at N = 20, s = 10�3, hx = hy = 0.025, ⁄ = �1, (a) Approximation solutions of3), (b) Approximation solutions of v(x,y,0.3).
3604 H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605
The example compares the numerical solutions for the 3rd-order approximation with the exact solutions for R = 80, s = 10�4,hx = hy = 0.05, ⁄ = �1 at different t list in Tables 2 and 3. It can be found the numerical solutions are virtually agreed with theexact solutions. The numerical solutions exhibit the physical characteristic for R = 80, s = 10�4, hx = hy = 0.05, ⁄ = �1 areillustrated in Fig. 4. In Fig. 5, the ⁄-curves are plotted for the 3rd-order approximation at R = 80, s = 10�4, hx = hy = 0.05. Itis obvious from this Fig. 5 that the approximation solutions un
i;j and vni;j convergence the exact solutions when ⁄ is
�1.5 6 ⁄ 6 �0.5. Similarly to continuous HAM suggested by Liao [8], by means of the ⁄-curve of DHAM, it is straightforwardto choose an appropriate range for ⁄ which ensures the convergence of the solution series.
Example 3. DHAM applying to two-dimensional Burgers’ equations with the initial conditions u(x,y,0) = x + y andv(x,y,0) = x � y.
In last example, we consider the two-dimensional Burgers’ equations, with thus initial conditions, and the exact solutionsare found in [27]:
Fig. 7.v(0.3,0
uðx; y; tÞ ¼ xþ y� 2xt1� 2t2 ;
vðx; y; tÞ ¼ x� y� 2yt1� 2t2 :
ð4:29Þ
The computational domain has been taken as D ¼ fðx; yÞ : 0 6 x 6 0:5;0 6 y 6 0:5g. By using initial conditions and byfollowing the same procedure as in the Example 2, one obtains the following approximations.
uni;j;0 ¼ ihx þ jhy;
vni;j;0 ¼ ihx � jhy;
uni;j;1 ¼ 2�hnsihx;
vni;j;1 ¼ 2�hnsjhy;
uni;j;2 ¼ 2�h2nsðihx þ ð1þ nÞsðihx þ jhyÞÞ þ 2�hnsihx;
vni;j;2 ¼ 2�h2nsðjhy þ ð1þ nÞsðihx � jhyÞÞ þ 2�hnsjhy;
uni;j;3 ¼ 4�h2ns2ð1þ �hÞð1þ nÞðihx þ jhyÞ þ ð4nþ 6Þð1þ nÞ�h3ns3ihx þ 2�hnsihxð�hþ 1Þ2;
vni;j;3 ¼ 4�h2ns2ð1þ �hÞð1þ nÞðihx � jhyÞ þ ð4nþ 6Þð1þ nÞ�h3ns3jhy þ 2�hnsjhyð�hþ 1Þ2:
Finally, the approximated solutions are given by
a b⁄-curve for the 10th-order approximation for N = 20, s = 10�3, hx = hy = 0.025, (a) ⁄-curve in the case u(0.3,0.4,0.4), (b) ⁄-curve in the case
.4,0.4).
H. Zhu et al. / Applied Mathematics and Computation 216 (2010) 3592–3605 3605
uni;j � un
i;j;0 þ uni;j;1 þ un
i;j;2 þ uni;j;3 þ � � � þ un
i;j;9;
vni;j � vn
i;j;0 þ vni;j;1 þ vn
i;j;2 þ vni;j;3 þ � � � þ vn
i;j;9:ð4:30Þ
Tables 4 and 5 list the 10th-order approximate solutions by the DHAM and the absolute errors between the exact solutionsand the numerical solutions for u and v at N = 20, s = 10�3, hx = hy = 0.025, ⁄ = �1. Fig. 6 shows the 10th-order numerical solu-tions using the proposed DHAM with a high degree of accuracy. Fig. 7 investigates the convergence of the numerical solu-tions. From the Fig. 7, one could find that if auxiliary parameter ⁄ is about in region [�1.1,�0.5] in the case u(0.3,0.4,0.4), andin region [�1.15,�0.7] in the case v(0.3,0.4,0.4), the numerical solution series are convergent. This example also shows thatthe DHAM has an accuracy convergence in large region.
5. Conclusions
In this work, we proposed a discrete version HAM to obtain the numerical solutions of a simple diffusion equation andtwo-dimensional Burgers’ equations. The results of these examples show that the DHAM is effective and reliable. It showsthat DHAM is a promising tool for linear or nonlinear PDEs.
The main contributions of this study are: Firstly, that we have proposed the discrete HAM for solving linear or nonlinearPDEs. Similarly to continuous HAM, the DHAM also provides a simple and easy way to control and adjust the convergenceregion for nonlinear problems. Secondly, the present study proved the convergence property of the proposed DHAM. Thirdly,the basic ideas of this approach can be widely employed to solve other linear or nonlinear PDEs. Fourthly, many modifiedHAM can be easily extended to themselves discrete versions using the proposed method. The further work will focus onthe numerical solution of nonlinear difference equations with uniform mesh.
Acknowledgements
This work has been supported by National Natural Science Foundation of China under Grant No. 60975004. The authorsthank the anonymous referees for their helpful comments and suggestions.
References
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