numerical solutions of partial differential equations by discrete

Applied Mathematics and Computation 216 (2010) 3592–3605

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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.

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lectronics and Communications Engineering, East China University of Science and Technology, No. 130,

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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:


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


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


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).


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.


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


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


h i;

ð4:25Þ

where

Rum un


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


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


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


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.


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).


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).


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.

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