Applied Mathematics and Computation 169 (2005) 355–365

www.elsevier.com/locate/amc

Solving the Klein–Gordon equation bymeans of the homotopy analysis method

Qiang Sun

School of Naval Architecture, Ocean and Engineering, Shanghai Jiao Tong University,

Shanghai 200030, China

Abstract

An analytic technique, namely the homotopy analysis method, is applied to solve the

nonlinear travelling waves governed by the Klein–Gordon equation. The phase speed

and the solution, which are dependent on the amplitude a, are given and valid in the

whole region 0 6 a < + 1.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Klein–Gordon equation; Travelling waves; Phase speed; Analytic solution; Homotopy

analysis method

1. Introduction

Consider the travelling waves governed by the Klein–Gordon equation [1–3]

utt � a2uxx þ c2u ¼ bu3; ð1Þwhere a, b, and c are physical constants.

If the nonlinear term bu3 is neglected, the above equation describes the trav-elling harmonic wave [1,2]

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.09.056

E-mail address: kensun@sjtu.edu.cn

356 Q. Sun / Appl. Math. Comput. 169 (2005) 355–365

u ¼ a cosðkx� xtÞ; x2 ¼ a2k2 þ c2;

where k is wave-number, and x is frequency. The phase speed, x/k, is indepen-

dent on the amplitude a. However, when the nonlinear term is considered, the

phase speed is in general a function of the amplitude a.

Many researchers have applied perturbation techniques [1,4,5] to solve the

Klein–Gordon equation. However, nearly all of these perturbation results be-

come invalid for large amplitude a. In this paper, the homotopy analysis

method (HAM) [6–15] is applied to give the phase speed c valid for all possibleamplitude a.

2. Mathematical formulation

2.1. Rule of the solution expression

Obviously, the solution of the travelling waves of Eq. (1) can be expressed by

umðx; tÞ ¼Xþ1

m¼1

am cosmðkx� xtÞ; m P 1: ð2Þ

Setting

n ¼ kx� xt; ð3Þit holds

uðnÞ ¼Xþ1

m¼1

am cosmn; m P 1; ð4Þ

and Eq. (1) becomes

ðc2 � a2Þu00 þ c2

k2u ¼ b

k2u3; ð5Þ

where the prime denotes the differentiation with respect to n. Eq. (4) providesus with the Rule of Solution Expression.

2.2. Initial guess and auxiliary linear operator

According to the Rule of Solution Expression denoted by (4), it is natural to

choose

u0 ¼ a cos n ð6Þas the initial approximation of u(n), where a is the amplitude. Let c0 denotethe initial guess of the phase speed c. Under the Rule of Solution Expression

denoted by (4), it is obvious to choose the auxiliary linear operator

Q. Sun / Appl. Math. Comput. 169 (2005) 355–365 357

L½Uðn; qÞ ¼ o2U

on2þ Uðn; qÞ ð7Þ

with the property

L½C1 sinðnÞ þ C2 cosðnÞ ¼ 0; ð8Þwhere C1 and C2 are coefficients.

2.3. Zero-order deformation equation

The homotopy analysis method is based on such continuous variations

U(n;q) and X(q) that, as the embedding parameter q increases from 0 to 1,U(n;q) and X(q) vary from the initial guess u0(n) and c0 to the exact solution

u(n) and c, respectively.To ensure this, let �h 5 0 denote an auxiliary parameter, H(n) an auxiliary

function, q 2 [0,1] an embedding parameter. From Eq. (1), we define the non-

linear operator

N½Uðn; qÞ;XðqÞ ¼ ½X2ðqÞ � a2 o2U

on2þ c2

k2Uðn; qÞ � b

k2U3ðn; qÞ; ð9Þ

and then construct such a homotopy

H½Uðn; qÞ;XðqÞ ¼ ð1� qÞL½Uðn; qÞ � u0� �hqHðnÞN½Uðn; qÞ;XðqÞ: ð10Þ

Setting

H½Uðn; qÞ;XðqÞ ¼ 0:

We have the zero-order deformation equation

ð1� qÞL½Uðn; qÞ � u0 ¼ �hqHðnÞN½Uðn; qÞ;XðqÞ: ð11ÞWhen q = 0 and 1, we have from Eq. (11) that

Uðn; 0Þ ¼ u0ðnÞ; Uðn; 1Þ ¼ uðnÞ; ð12Þ

Xð0Þ ¼ c0; Xð1Þ ¼ c; ð13Þrespectively. Thus, U(n;q) and X(q) can be expanded in the Maclaurin series

with respect to q in the form

Uðn; qÞ ¼ Uðn; 0Þ þXþ1

m¼1

umðnÞqm; ð14Þ

XðqÞ ¼ Xð0Þ þXþ1

m¼1

cmqm; ð15Þ

358 Q. Sun / Appl. Math. Comput. 169 (2005) 355–365

where

umðnÞ ¼1

m!omUðn; qÞ

oqm

����q¼0

; ð16Þ

cm ¼ 1

m!omXðqÞoqm

����q¼0

: ð17Þ

Note that the zero-order deformation equation (11) contains the auxiliary

parameter ⁄ and the auxiliary function H(n) so that u(n;q) is dependent uponh and H(n). Assuming that both ⁄ and H(n) are so properly chosen that the ser-

ies (16) and (17) are convergent at q = 1, we have from (14) and (15) that

uðnÞ ¼ u0 þXþ1

m¼1

um; ð18Þ

c ¼ c0 þXþ1

m¼1

cm: ð19Þ

a

c

0 10 20 30 40 50 600

10

20

30

40

50

Fig. 1. Analytic approximations by the homotopy analysis method with different �h when k = 1,

a = 1, b = �1, and c = 1. Solid line with circle symbols: 19th-order approximation when �h = �1/10;

dash-dotted line: 24th-order approximation when �h = �1/100; dash-dot-dotted line: 31st-order

approximation when �h = �1/1000; solid line: 40th-order approximation when �h = �1/10,000.

Q. Sun / Appl. Math. Comput. 169 (2005) 355–365 359

2.4. High-order deformation equation

Differentiating the zero-order deformation equation (11) m times with re-

spect to q, then setting q = 0, and finally dividing it by m!, we have the so-called

mth-order deformation equation

L½umðnÞ � vmum�1ðnÞ ¼ �hHðnÞRmðnÞ; ð20Þ

where

RmðnÞ ¼Xm¼1

i¼0

u00m�1�i

Xi

j¼0

cjci�j � a2u00m�1 þc2

k2um�1 �

b

k2Xm�1

i¼0

um�1�i

Xi

j¼0

ujui�j

ð21Þ

and

vm ¼0; m 6 1;

1; m > 1:

�ð22Þ

a

c

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

k=1

k=2

k=5

k=1/2

k=1/5

Fig. 2. Comparison of perturbation results with homotopy analysis approximations when

�h ¼ �1=c20, a = 1, b = �1, and c = 1. Solid line with square symbols: perturbation solutions

O(a12); solid line: 29th-order approximations by the homotopy analysis method.

360 Q. Sun / Appl. Math. Comput. 169 (2005) 355–365

2.5. Rule of coefficient-ergodicity

Under the Rule of Solution Expression denoted by (4), the auxiliary func-

tion H(n) can be chosen as H(n) = 1. Note that we have freedom to choose

the value of the auxiliary parameter ⁄, which provides us with a convenient

way to adjust the convergence region of solution series, as shown in the follow-ing section.

3. Results

3.1. Solution expression

According to the Rule of the Solution denoted by (4), the term Rm(n) can berewritten as

RmðnÞ ¼XWðmÞ

n¼1

bm;n cosðnnÞ; ð23Þ

a

c

0 1 2 3 4 50

5

10

15

20

25

30

k=1

k=2

k=5

k=1/2

k=1/5

Fig. 3. Comparison of perturbation results with homotopy analysis approximations when

�h ¼ �1=c20, a = 2, b = �5, and c = 3. Solid line with square symbols: perturbation solutions

O(a12); solid line: 29th-order approximations by the homotopy analysis method.

Q. Sun / Appl. Math. Comput. 169 (2005) 355–365 361

where the integer W(m) is dependent on m and the nonlinear terms of the ori-

ginal Eq. (5), and the coefficient

bm;n ¼2

p

Z p

0

RmðnÞ cosðnnÞ dn ð24Þ

becomes zero when n > W(m). Due to the definition of L denoted by (7), the

solution of the mth-order deformation equation involves the secular terms

n cosðnÞ if bm,1 5 0. However, this disobeys the Rule of Solution Expression

denoted by (4). Thus, bm,1 must be enforced to zero, i.e.

bm;1 ¼ 0: ð25ÞThis provides us with an algebraic equation for cm�1.

Then, the solution of the mth-order deformation equation (20) is

umðnÞ ¼ vmum�1ðnÞ þ �hXWðmÞ

n¼2

bm;nð1� n2Þ cosðnnÞ þ C1 sinðnÞ þ C2 cosðnÞ;

ð26Þ

a

c

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

k=1

k=2k=5

k=1/2

k=1/5

Fig. 4. Comparison of perturbation results with analytic approximations when �h ¼ �1=c20, a = 3,

b = �2/3, and c = 2. Solid line with square symbols: perturbation solutions O(a12); solid line: 29th-

order approximations by the homotopy analysis method.

362 Q. Sun / Appl. Math. Comput. 169 (2005) 355–365

where C1 and C2 are two integral constants. Using the Rule of Solution Expres-

sion denoted by (4), we have C1 = 0. To ensure that the amplitude of the wave

equals to a, it holds

umðpÞ � umð0Þ ¼ 0; ð27Þwhich determines the value of C2. In this way, we obtain um(n) and cm�1 suc-

cessfully. The Mth-order approximation is given by

uðnÞ ¼XMm¼0

umðnÞ; ð28Þ

c ¼XM�1

m¼0

cm: ð29Þ

Note that the mth-order deformation equations (20) are linear equations,

and cm�1 is only governed by algebraic equations (25). Thus um and cm�1

can be easily solved, especially by means of symbolic software such as Math-

ematica, Maple, MathLab, and so on.

a

c

101 102 103100

101

102

103

k=5

k=2

k=1

k=1/2

k=1/5

Fig. 5. Comparison of numerical results with analytic approximations when �h ¼ �1=c20, a = 1,

b = �1, and c = 1. Solid lines: numerical results; square symbols: first-order approximations by the

homotopy analysis method.

Q. Sun / Appl. Math. Comput. 169 (2005) 355–365 363

When m = 1, we have

c0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2a2 þ 4c2 � 3a2b

p2k

: ð30Þ

If the above expression is expanded in the Maclaurin series of a, its first two

terms are the same as the first two terms of the perturbation method result [1]

c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ c2k�2

q1� 3a2b

8ða2k2 þ c2Þ

� þ � � � : ð31Þ

However, it is obvious that there is great difference between the results (30)

by the homotopy analysis method and the results (31) by the perturbation

method. Only when a is small, the solutions by these two methods are the same,as shown in Figs. 2–4.

3.2. The effect of �h

As pointed by Liao [6], the auxiliary parameter �h can be employed to adjust

the convergence region of homotopy analysis solutions. It is found that the

convergence regions of the series (28) and (29) are enlarged as �h tends to zero

a

c

101 102 103100

101

102

103

k=5

k=2

k=1

k=1/2

k=1/5

Fig. 6. Comparison of numerical results with analytic approximations when �h ¼ �1=c20, a = 2,

b = �5, and c = 3. Solid lines: numerical results; square symbols: first-order approximations by the

homotopy analysis method.

a

c

101 102 103100

101

102

103

k=1

k=2

k=5

k=1/2

k=1/5

Fig. 7. Comparison of numerical results with homotopy analysis approximations when �h ¼ �1=c20,a = 3, b = �2/3, and c = 2. Solid lines: numerical results; square symbols: first-order approxima-

tions by the homotopy analysis method.

364 Q. Sun / Appl. Math. Comput. 169 (2005) 355–365

from below, as shown in Fig. 1. Thus, one can adjust the convergence regionsof the series (28) and (29) simply by choosing a proper value of the auxiliary

parameter �h.Obviously, �h can be a function of a. It is found that, when �h ¼ �1=c20, even

the first-order approximation

c � c0 þ c1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4k2a2 þ 4c2 � 3a2b

p2k

� 3ka4b2

8ð4k2a2 � 3a2b þ 4c2Þ5=2ð32Þ

agrees very well with numerical results in the whole region 0 6 a < + 1, as

shown in Figs. 5–7.

4. Conclusions

In this paper, the homotopy analysis method is applied to obtain the phase

speed c, and the valid solution of the travelling waves governed by the Klein–

Gordon equation.

Different from the perturbation results [1], by homotopy analysis method,even the first-order approximation (32) of the phase speed agrees well with

Q. Sun / Appl. Math. Comput. 169 (2005) 355–365 365

numerical results for all possible wave amplitude 0 6 a < + 1. Therefore,

unlike perturbation solution, solution given by the homotopy analysis method

is valid for 0 6 a < + 1. This indicates that the homotopy analysis method is

valid for travelling waves problems with strong nonlinearity.

References

[1] A.H. Nayfeh, Perturbation Methods, Wiley Classics Library Edition, 2000.

[2] E.Y. Deeba, S.A. Khuri, A decomposition method for solving the nonlinear Klein–Gordon

equation, J. Comput. Phys. 124 (1996) 442–448.

[3] K. Narita, New solution for difference-difference models of the U4 equation and nonlinear

Klein–Gordon equation in 1 + 1, Chaos Solitons Fract. 7 (1996) 365–369.

[4] A. Maccari, Nonresonant interacting waves for the nonlinear Klein–Gordon equation in three-

dimensional space, Physica D 135 (2000) 331–344.

[5] A. Maccari, Solitons trapping for the nonlinear Klein–Gordon equation with an external

excitation, Chaos Solitons Fract. 17 (2003) 145–154.

[6] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman &

Hall/CRC, Boca Raton, 2003.

[7] S.J. Liao, A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate,

J. Fluid Mech. 385 (1999) 101–128.

[8] S.J. Liao, A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow

problems, J. Fluid Mech. 453 (2002) 411–425.

[9] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput.

147 (2) (2004) 499–513.

[10] S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids

over a stretching sheet, J. Fluid Mech. 488 (2003) 189–212.

[11] H. Xu, An explicit analytic solution for free convection about a vertical flat plate embedded in

a porous medium by means of homotopy analysis method, Appl. Math. Comput. 158 (2004)

433–443.

[12] M. Ayub, A. Rasheed, T. Hayat, Exact flow of a third grade fluid past a porous plate using

homotopy analysis method, Int. J. Engng. Sci. 41 (2003).

[13] C. Wang, Shijun, Liao, Jimao Zhu, An explicit solution for the combined heat and mass

transfer by natural convection from a vertical wall in a non-Darcy porous medium, Int. J. Heat

Mass Transfer 46 (2003) 4813–4822.

[14] C. Wang, J.M. Zhu, S.J. Liao, On the explicit analytic solution of Cheng–Chang equation, Int.

J. Heat Mass Transfer 46 (2003) 1855–1860.

[15] C. Wang, S.J. Liao, J.M. Zhu, An explicit analytic solution for non-Darcy natural convection

over horizontal plate with surface mass flux and thermal dispersion effects, Acta Mech. 165

(2003) 139–150.