Applied Mathematics and Computation 171 (2005) 1095–1103

www.elsevier.com/locate/amc

A numerical implementation ofthe decomposition method for the

Lienard equation

Dogan Kaya a, Salah M. El-Sayed b,*

a Department of Mathematics, Firat University, Elazig 23119, Turkeyb Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Abstract

In this study, a kind of explicit exact and numerical solutions to the Lienard equation

is obtained, and the applications of the results to this equation has been compared with

their known theoretical solution. This paper is particularly concerned with the Adomian

decomposition method and the numerical results demonstrate that the new method is

relatively accurate and easily implemented.

� 2005 Elsevier Inc. All rights reserved.

Keywords: The Adomian decomposition method; Numerical solutions; The Lienard equation

1. Introduction

In this work, we consider the Lienard equation

x00 þ f ðxÞx0 þ gðxÞ ¼ eðtÞ; ð1Þ

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.01.104

* Corresponding author. Address: Department of Mathematics, Scientific Departments, Educa-

tion College for Girls, Al-Montazah, Buraydah, Al-Qassim, Kingdom of Saudi Arabia.

E-mail addresses: dkaya@firat.edu.tr (D. Kaya), ms4elsayed@yahoo.com (S.M. El-Sayed).

1096 D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103

which is not only regarded as a generalization of the damped pendulum equa-

tion or a damped spring-mass system (where f(x)x 0 is the damping force, g(x) is

the restoring force, and e(t) is the external force), but also used as nonlinear

models in many physically significant fields when taking different choices for

f(x), g(x) and e(t). For example, the choices f(x) = �(x2 � 1), g(x) = x, and

e(t) = 0 lead Eq. (1) to the Van der Pol equation served as a nonlinear modelof electronic oscillation [1,2]. Therefore, studying Eq. (1) is of physical signif-

icance. In the general case, it is commonly believed that it is very difficult to

find its exact solution by usual ways [3]. Kong studied the following special

case of Eq. (1) [4,5]:

a00ðnÞ þ ‘aðnÞ þ ma3ðnÞ þ na5ðnÞ ¼ 0; ð2Þ

where ‘, m, n are real coefficients. Finding explicit exact and numerical solu-

tions of nonlinear equations efficiently is of major importance and has wide-

spread applications in numerical methods and applied mathematics. In this

study, we will implement the Adomian decomposition method (in short

ADM) [6–10] to find exact solution and approximate solutions to the Lienardequation for a given nonlinearity.

Unlike classical techniques, the decomposition method leads to an analytical

approximation and exact solution of the nonlinear equations easily and ele-

gantly without transforming the equation or linearization of the problem

and with high accuracy, minimal calculation, and avoidance of physically

unrealistic assumptions. As numerical tool the method provide us with numer-

ical solution without discretization of the given equation and therefore it is not

effected by computation round off errors and one is not faced with necessity oflarge computer memory and time—straightforward to write computer codes in

any symbolic programming language—not faced with necessity of large com-

puter memory and time. The method has features in common with many other

methods, but it is distinctly different on close examination, and one should not

be mislead by apparent simplicity into superficial conclusions [7].

In this paper, various initial values of the Lienard equation [4,5] can be han-

dled more easily, quickly, and elegantly by implementing the ADM rather than

the traditional methods for finding analytical as well as numerical solutions.

2. Analysis of the method

In this section we outline the steps to obtain analytic solution of the Lienard

equation (1) is using the ADM. First we write the Lienard equation in the stan-

dard operator form

La ¼ �ð‘aþ ma3 þ na5Þ; ð3Þ

D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103 1097

where the notation L ¼ d2

dn2symbolizes the linear differential operators. The in-

verse operator L�1 exists and it can conveniently be taken as the twofold inte-

gration operator L�1. Thus, applying the inverse operator L�1 to (2) yields

L�1La ¼ �L�1ð‘aþ ma3 þ na5Þ: ð4ÞTherefore, it follows that

aðnÞ ¼ að0Þ þ na0ð0Þ � L�1ð‘aþ ma3 þ na5Þ: ð5ÞNow we decompose the unknown function a(n) a sum of components defined

by the series

aðnÞ ¼X1i¼0

aiðnÞ: ð6Þ

The zeroth component is usually taken to be all terms arise from the initial

conditions, i.e.,

a0 ¼ að0Þ þ na0ð0Þ: ð7ÞThe remaining components ai(n), iP 1, can be completely determined such

that each term is computed by using the previous term. Since a0 is known,

ai ¼ L�1½‘ai�1 þ mAi�1 þ nBi�1�; i P 1; ð8Þwhere f ðaÞ ¼

P1i¼0Aiða0; a1; . . . ; aiÞ. The components Ai are called the Ado-

mian polynomials, these polynomials can be calculated for all forms of nonlin-

earity according to specific algorithms constructed by Adomian [6,9]. For this

specific nonlinearity, we use the general formula for Ai polynomials as

Ai ¼1

i!di

dkifX1k¼0

kkak

! !k¼0

; i P 0: ð9Þ

This formula makes it easy to set computer code to get as many polynomial as

we need in the calculation of the numerical as well as analytical solutions. Thefirst few Adomian polynomials for the nonlinearity f(a) are

A0 ¼ f ð1Þða0Þ; A1 ¼ a1f ð1Þða0Þ; A2 ¼ a2f ð1Þða0Þ þ1

2!a21f

ð2Þða0Þ;

A3 ¼ a3f ð1Þða0Þ þ a2a1f ð2Þða0Þ þ1

3!a31f

ð3Þða0Þ

and so on, the rest of the polynomials can be constructed in a similar manner.

Finally an N-term approximate solution is given by

UN ¼XN�1

i¼0

ai ð10Þ

and the exact solution is a(n) = limN!1UN.

1098 D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103

Moreover, the convergence of the decomposition series has investigated by

several authors. The theoretical treatment of convergence of the decomposition

method has been considered in the literature [11–17]. They obtained some re-

sults about the speed of convergence of this method. In recent work of Ngarh-

asta et al. [18] have proposed a new approach of convergence of the

decomposition series. The authors have given a new condition for obtainingconvergence of the decomposition series to the classical presentation of the

ADM in [16]. Here, we will introduce a brief discussion of the convergence

analysis in Hilbert space H as same manner in [18] of the ADM applied to

the Lienard equation. Since we have

La ¼ �ð‘aþ ma3 þ na5Þand by using the identity arþ1 � brþ1 ¼ ða� bÞ

Prþ1

i¼1ar�iþ1bi�1 and according the

Schwartz inequality, we have

ðarþ1 � brþ1Þ; a� b� �

6 karþ1 � brþ1kka� bk

¼ kða� bÞXrþ1

i¼1

ar�iþ1bi�1kka� bk

6 ðr þ 1ÞMrka� bk2;

where kak, kbk 6M. Also, we can prove the above inequality by using mean

value theorem. Therefore

ðarþ1 � brþ1Þ; a� b� �

6 ðr þ 1ÞMrka� bk2 () �ðarþ1 � brþ1Þ; a� b� �

P ðr þ 1ÞMrka� bk2:

From the above conditions (H1) and (H2) are fulfilled easily, see [18].

Numerical computations of the Lienard equation have often been repeated

in the literature. However, to show the effectiveness of the proposed decompo-

sition method and to give a clear overview of the methodology some examplesof the Lienard equation (1) will be discussed in the following section.

3. Applications of the Lienard equation

In this section we will be concerned with the general initial values of the

Lienard equation (1)

að0Þ ¼ffiffiffiffiffiffiffiffiffi�2‘

m

r; a0ð0Þ ¼ �

ffiffiffiffiffiffi�‘

p‘ffiffiffiffiffiffi

�2‘m

qm; ð11Þ

in which, m and ‘ are arbitrary constants.

D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103 1099

To find the solution of the initial value problem (11) and (12) we apply the

Scheme (9). The Adomian polynomials Ai are computed according to (8) with

f(a) = ap and this gives

A0 ¼ ap0; A1 ¼ pap�10 a1; A2 ¼ ðp � 1Þpap�2

0

1

2!a21 þ pap�1

0 a2;

A3 ¼ ðp � 2Þðp � 1Þpap�30

1

3!a31 þ ðp � 1Þpap�2

0 a1a2 þ pap�10 a3

and so on the other polynomials can be constructed. In the case of our imple-

mentation, the Adomian polynomials Ai and Bi are computed according to

(8) with a3 and a5, respectively. Performing the integration we obtain the

following:

a0 ¼

ffiffiffiffiffiffi�2‘m

qð2þ

ffiffiffiffiffiffi�‘

pnÞ

2; ð12Þ

a1 ¼‘2n2

840ffiffiffiffiffiffi�2‘m

qm3

ð�840m2 þ 3360‘n� 700ffiffiffiffiffiffi�‘

pm2nþ 2800

ffiffiffiffiffiffi�‘

p‘nn

þ 210‘m2n2 � 1400‘2nn2 þ 21ffiffiffiffiffiffi�‘

p‘m2n3 � 420

ffiffiffiffiffiffi�‘

p‘2nn3

þ 70‘3nn4 þ 5ffiffiffiffiffiffi�‘

p‘3nn5Þ; ð13Þ

a2 ¼‘3n4

5; 765; 760ffiffiffiffiffiffi�2‘m

qm5

��2; 402; 400m4 þ 19; 219; 200‘m2n

� 38; 438; 400‘2n2 � 2; 930; 928ffiffiffiffiffiffi�‘

pm4nþ 28; 060; 032

�ffiffiffiffiffiffi�‘

p‘m2nn� 65; 345; 280

ffiffiffiffiffiffi�‘

p‘2n2nþ 1; 489; 488‘m4n2

� 19; 731; 712‘2m2nn2 þ 55; 095; 040‘3n2n2 þ 394; 680

�ffiffiffiffiffiffi�‘

p‘m4n3 � 8; 648; 640

ffiffiffiffiffiffi�‘

p‘2m2nn3 þ 29; 744; 000

�ffiffiffiffiffiffi�‘

p‘3n2n3 � 54; 054‘2m4n4 þ 2; 471; 040‘3m2nn4

� 11; 325; 600‘4n2n4 � 3003ffiffiffiffiffiffi�‘

p‘2m4n5 þ 446; 160

�ffiffiffiffiffiffi�‘

p‘3m2nn5 � 3; 146; 000

ffiffiffiffiffiffi�‘

p‘4n2n5 � 46; 332‘4m2nn6

þ 629; 200‘5n2n6 � 2106ffiffiffiffiffiffi�‘

p‘4m2nn7 þ 85; 800

�ffiffiffiffiffiffi�‘

p‘5n2n7 � 7150‘6n2n8 � 275

ffiffiffiffiffiffi�‘

p‘6n2n9

�; ð14Þ

1100 D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103

a3 ¼‘3

ffiffiffiffiffiffi�2‘m

qð61m2�340‘nÞð�m2þ4‘nÞ2n6

720m6

�ffiffiffiffiffiffi�‘

p‘4ð277m2�2260‘nÞð�m2þ4‘nÞ2n7

1008ffiffiffiffiffiffi�2‘m

qm7

� ‘5ð�m2þ4‘nÞð663m4�11;200‘m2nþ35;600‘2n2Þn8

3360ffiffiffiffiffiffi�2‘m

qm7

þffiffiffiffiffiffi�‘

p‘5ð4911m6�144;804‘m4nþ997;840‘2m2n2�1;988;800‘3n3Þn9

60;480ffiffiffiffiffiffi�2‘m

qm7

� ‘6ð12;483m6�572;472‘m4nþ4;893;360‘2m2n2�11;264;000‘3n3Þn10

604;800ffiffiffiffiffiffi�2‘m

qm7

�ffiffiffiffiffiffi�‘

p‘6ð2865m6�231;272‘m4nþ2;619;152‘2m2n2�7;180;800‘3n3Þn11

887;040ffiffiffiffiffiffi�2‘m

qm7

þ ‘7ð7623m6�1;316;340‘m4nþ21;658;000‘2m2n2�73;656;000‘3n3Þn12

26;611;200ffiffiffiffiffiffi�2‘m

qm7

þffiffiffiffiffiffi�‘

p‘7ð2541m6�1;425;220‘m4nþ38;766;000‘2m2n2�173;272;000‘3n3Þn13

230;630;400ffiffiffiffiffiffi�2‘m

qm7

þ ‘9nð24;661m4�1;359;300‘m2nþ8;676;800‘2n2Þn14

53;813;760ffiffiffiffiffiffi�2‘m

qm7

þffiffiffiffiffiffi�‘

p‘9nð24;661m4�4;212;100‘m2nþ43;384;000‘2n2Þn15

1;614;412;800ffiffiffiffiffiffi�2‘m

qm7

þ ‘11n2ð5349m2�108;460‘nÞn16

32;288;256ffiffiffiffiffiffi�2‘m

qm7

þffiffiffiffiffiffi�‘

p‘11n2ð1783m2�108;460‘nÞn17

365;933;568ffiffiffiffiffiffi�2‘m

qm7

þ 145‘13n3n18

8;805;888ffiffiffiffiffiffi�2‘m

qm7

þ 145ffiffiffiffiffiffi�‘

p‘13n3n19

334;623;744ffiffiffiffiffiffi�2‘m

qm7

;

ð15Þ

in this manner the components of the decomposition series (5) are obtained

as far as we like. This series is exact to the last term, as one can verify, of

the Taylor series of the exact closed form solution [5]

aðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2‘ 1þ tanhð

ffiffiffiffiffiffi�‘

pnÞ

� �m

s: ð16Þ

D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103 1101

In the second example, we will consider the Lienard equation (11) with the

initial conditions

að0Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiK

2þ D

r; a0ð0Þ ¼ 0; ð17Þ

where

K ¼ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3‘2

3m2 � 16n‘

s; D ¼ �1þ

ffiffiffi3

pmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3m2 � 16n‘p :

Again, to find the solution of this equation, we substitute in Scheme (9)

a0 ¼ að0Þ þ na0ð0Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiK

2þ D

r; ð18Þ

a1 ¼ �Z t

0

Z t

0

½‘a0 þ mA0 þ nB0�dtdt; ð19Þ

..

.

ai ¼ �Z t

0

Z t

0

½‘ai�1 þ mAi�1 þ nBi�1�dtdt; i P 2; ð20Þ

where the Adomian polynomials Ai�1 and Bi�1 are constructed by (9) for thefunction f(a) is equal to a3 and a5 as in the first example. Performing the cal-

culations in (8) using Mathematica and substituting into (10) gives the exact

solution [5]

aðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKsech2ð

ffiffiffiffiffiffi�‘

pnÞ

2þ Dsech2ðffiffiffiffiffiffi�‘

pnÞ

s; ð21Þ

where

K ¼ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3‘2

3m2 � 16n‘

sand D ¼ �1þ

ffiffiffi3

pmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3m2 � 16n‘p :

4. Experimental results for the Lienard equation

In this section we verify numerically whether the proposed methodology

lead to accurate solutions, we will evaluate the ADM solutions for Lienard

equation with various initial conditions for the equations solved in the previous

section. The solutions are very rapidly convergent by utilizing the ADM. Thenumerical results we obtained justify the advantage of this methodology.

1102 D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103

In order to verify numerically whether the proposed methodology lead to

accurate solutions, we will evaluate the ADM solutions using the N-term

approximation for some examples of the Lienard equation solved in the previ-

ous section. First we consider Eq. (2) with m = 4, n = �3, ‘ = �1 with initial

conditions (11) and the second example with m = 4, n = 3, ‘ = �1 with initial

conditions (17). The differences between the N-terms (N = 1,4,6,8) solutionand the exact solution (16) are shown in Tables 1 and 2. Tables 1 and 2 show

that we achieved a very good approximation to the actual solution of the equa-

tions by using only few terms of the decomposition series solution derived

above. It is evident that the overall errors can be made smaller by adding

new terms of the decomposition series.

The solutions are very rapidly convergent by utilizing the ADM. The

numerical results we obtained justify the advantage of this methodology. Fur-

thermore, as the decomposition method does not require discretization of thevariables, i.e. time and space, it is not effected by computation round off errors

and necessity of large computer memory and time. Clearly, the series solution

Table 1

The decompositions solution (/K) for the first example (16)

n ja(n) � /1j ja(n) � /4j ja(n) � /6j ja(n) � /8j0.1 8.83121E�07 4.66294E�15 2.22045E�16 2.22045E�16

0.2 1.29319E�05 5.00344E�12 1.11022E�16 1.11022E�16

0.3 5.61401E�05 2.75613E�10 1.06137E�13 0.00000E+00

0.4 1.37208E�04 3.72567E�09 8.52762E�12 3.33067E�15

0.5 2.10312E�04 1.24172E�08 2.38155E�10 4.67848E�13

0.6 1.23680E�04 1.43725E�07 3.11725E�09 2.22292E�11

0.7 4.45513E�04 2.10636E�06 1.96868E�08 4.85821E�10

0.8 2.06490E�03 1.48488E�05 8.92164E�09 5.40902E�09

0.9 5.62644E�03 7.35220E�05 1.00235E�06 2.23907E�08

1.0 1.24297E�02 2.84971E�04 1.06531E�05 2.13460E�07

Table 2

The decompositions solution (/K) for the second example (21)

n ja(n) � /1j ja(n) � /4j ja(n) � /6j ja(n) � /8j0.1 1.18605E�07 4.62874E�12 1.11022E�16 1.11022E�16

0.2 7.45453E�06 4.64951E�09 3.21942E�12 2.33147E�15

0.3 8.24511E�05 2.59862E�07 9.10129E�10 3.35265E�12

0.4 4.45245E�04 4.42415E�06 4.89135E�08 5.69094E�10

0.5 1.61782E�03 3.91301E�05 1.05471E�06 2.99375E�08

0.6 4.56661E�03 2.28269E�04 1.27380E�05 7.49040E�07

0.7 1.08186E�02 9.98255E�04 1.03025E�04 1.12133E�05

0.8 2.25384E�02 3.53544E�03 6.21182E�04 1.15230E�04

0.9 4.25659E�02 1.06513E�02 2.99319E�03 8.88545E�04

1.0 7.44244E�02 2.82762E�02 1.20897E�02 5.46495E�03

D. Kaya, S.M. El-Sayed / Appl. Math. Comput. 171 (2005) 1095–1103 1103

methodology can be applied to various type of linear or nonlinear differential

equations [19–23].

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