J. Nig. Soc. Phys. Sci. 2 (2020) 106–114

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

Comparison of the Solution of the Van der Pol Equation Using
the Modified Adomian Decomposition Method and Truncated

Taylor Series Method

J. N. Ndama,∗, O. Adedireb

aDepartment of Mathematics, University of Jos, Nigeria
bFederal College of Forestry, Jos, Nigeria

Abstract

In this paper, we compare the solution of the van der Pol equation obtained by using truncated Taylor series method and modified Adomian
decomposition method with the solution obtained by the Poincare-Lindstedt (P-L) method. The approximating 4-component modified Adomian
decomposition method behaves more like approximate P-L analytic method than tenth-order Taylor series. Also, with addition of just one term,
the approximating 5-component modified Adomian decomposition method produces more convergent solution to that of P-L analytic method
than the twenty second-order Taylor series approximation as the independent variable t representing time progressively increases. A general
comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from Taylor series are greater than
the ones generated from modified Adomian decomposition method. It was further revealed that very few components of the modified Adomian
decomposition could yield a series of about 3 times the order of the one obtained by using the Taylor series method. Hence, we recommend
inclusion of the modified Adomian Decomposition Method in modern mathematical tools.

Keywords: van der Pol oscillator, modified Adomian decomposition method, Taylor series method

Article History :
Received: 01 January 2020
Received in revised form: 04 March 2020
Accepted for publication: 09 March 2020
Published: 14 May 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: T. Latunde

1. Introduction

The van der Pol equation, otherwise called the van der Pol
oscillator is a model which describes the behaviour of electri-
cal circuits [1-19]. It was formulated by an electrical engineer
and physicist, Balthasar van der Pol, and no exact solution has
been obtained for the second order differential equation since
then [12]. However, asymptotic and numerical solutions have
been obtained. One important method of obtaining approximate
analytic solutions to differential equations is the Taylor series

∗Corresponding author tel. no: 07031950138
Email address: ndamj@unijos.edu.ng (J. N. Ndam)

method. However, arguments have been advanced against it
because the method has the problem of producing tedious com-
putational work in determining the coefficient an which can be
observed in deriving recurrence relation, particularly when the
product of two or more infinite series are involved [17].

Recently, another form of series solution procedure for solv-
ing both linear and nonlinear differential equations has been
developed. This procedure, which generates series solutions
to differential equations that converge very rapidly, is called
Adomian decomposition method, named after the author [18].
The Adomian decomposition method has been used to obtain
approximate analytic solutions to a wide range of differential
equations, including linear and nonlinear equations, and has

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 107

been shown to converge more rapidly with proofs of conver-
gence than other forms of series solutions ([1, 4, 13, 17, 19]).
There have been various modifications to this procedure which
have significantly improved the convergence of the method ([1-
3, 5, 8, 9, 14, 18]). Wazwaz [17] compared the solutions of a
linear and nonlinear differential equations using the Taylor se-
ries and the Adomian decomposition methods and concluded
that the solutions obtained by the decomposition method con-
verged much more rapidly than the Taylor series procedure.
However, consideration of Wazwaz [17] of the two series meth-
ods was in the sense of ease of getting Adomian decomposition
series compared with that of Taylor series but did not consider
error analysis of the series methods as his focus was on infinite
series solution.

In this work, we attempt approximate solutions of the van
der Pol equation using few orders of Taylor series and few com-
ponents of the modified Adomian decomposition procedures
with emphasis on finite series solution. The aim is to com-
pare the solution of the Van der Pol equation using the Modified
Adomian Decomposition Method (MADM) and the Truncated
Taylor Series Method (TTSM) with approximate Poincare-Lindstedt
(P-L) technique of pertubation method. The specific objec-
tives are to investigate the behaviour of the errors generated
from both methods for parameter values of van der Pol equa-
tion and to graphically display them. These objectives of the
study will be achieved by comparing the solution of the van der
Pol equation from finite orders of TTSM and finite components
of MADM with that of approximate P-L analytic method. The
choice of P-L analytic method for comparison in this study is
based on its suitability and accuracy for the solution of nonlin-
ear differential equations ([6], [11], [16]).

The remaining parts of this paper are organised as follows:
the second section considers the Taylor series solution, while
section 3 is dedicated to the modified Adomian decomposition
procedure. Results and discussion comes up in section 4. Fi-
nally, conclusion will be the subject of section 5.

2. Taylor series method

The van der Pol oscillator is given by

u′′(t) + µ(u2(t) − 1)u′(t) + u(t) = 0 (1)

with the initial conditions

u(0) = 1, u′(0) = 0,

where µ � 1 or µ � 1. The Taylor series solution about t = 0
is obtained as

u(t) = u(0)+tu′(0)+
t2

2!
u′′(0)+

t3

3!
u′′′(0)+· · ·+

tn

n!
u(n)(0)(2)

Equation (1) can be written as

u′′(t) = µ(1 − u2(t))u′(t) − u(t)

u′′′(t) = −2µu(t)u′2(t) + µ(1 − u2(t))u′′(t) − u′(t)

uiv(t) = −2µu′3 − 6µu(t)u′(t)u′′(t) − 8µu(t)u′(t)u′′′(t)
+µ(1 − u(t)2)uiv(t) − u′′′(t)

...

Computing the derivatives as far as possible and evaluating at
t = 0, we obtain

u(0) = 1

u′(0) = 0

u(2)(0) = −1

u(3)(0) = 0

u(4)(0) = 1

u(5)(0) = −6µ

u(6)(0) = −1

u(7)(0) = 66µ

u(8)(0) = −252µ2 + 1

u(9)(0) = −612µ

u(10)(0) = 11052µ2 − 1

u(11)(0) = 5532µ− 31752µ3

u(12)(0) = −341316µ2 + 1

u(13)(0) = −49818µ + 3714552µ3

u(14)(0) = 6259644µ2 − 8845200µ4 − 1

u(15)(0) = 448398µ− 274369248µ3

u(16)(0) = 2211091344µ4 − 241794792µ2 + 1

u(17)(0) = −4604457312µ5 + 15983461728µ3 − 4035384µ

...

Using the above values, tenth-order Taylor series yields the fol-
lowing approximating polynomial:

u(t) ≈ 1 −
t2

2!
+

t4

4!
− 6µ

t5

5!
+

t6

6!
+ 66µ

t7

7!

−(252µ2 + 1)
t8

8!
− 612µ

t9

9!
+ (11052µ2 − 1)

t10

10!
(3)

Analogously, twenty second-order Taylor series yields the fol-
lowing approximating polynomial:

u(t) ≈ 1 −
t2

2!
+

t4

4!
− 6µ

t5

5!
+

t6

6!
+ 66µ

t7

7!

−(252µ2 + 1)
t8

8!
− 612µ

t9

9!
+ (11052µ2 − 1)

t10

10!

+(5532µ− 31752µ3)
t11

11!
− (341316µ2 − 1)

t12

12!

−(49818µ− 3714552µ3)
t13

13!
+

(
6259644µ2 − 8845200µ4

−1)
t14

14!
+ (448398µ− 274369248µ3)

t15

15!

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 108

+(2211091344µ4 − 241794792µ2 + 1)
t16

16!

+(−4604457312µ5 + 15983461728µ3 − 4035384µ)
t17

17!

+(−325537770960µ4 + −325537770960µ2 − 1)
t18

18!
+(2136491544096µ5 − 924215233680µ3

+36320664µ)
t19

19!
+ (−4018487578560µ6

+37614999137616µ4 − 154525796280µ2 + 1)
t20

20!
+(−557011268077248µ5 + 48491799847920µ3

−326886030µ)
t21

21!
+ (3140584492618944µ6

−3803681636636400µ4 + 3873523538520µ2 − 1)
t22

22!
(4)

3. Adomian decomposition method

In this section, we follow the approach used by El-Kalla [4]
in which he considered kth order nonlinear ordinary differential
equation whose nonlinear term has Adomian polynomial repre-
sentation. This approach is similar to that of Wazwaz [17]. The
Adomian decomposition procedure seeks a solution of the form

u(t) =
∞∑

n=0

un(t) (5)

Equation (1) can be expressed in operator form as

Ltu = µ(1 − u
2)u′ − u (6)

where Lt =
d2
dt2 , and hence the solution of (1) can be expressed

as

u(t) = u(0) + L−1t
{
µ(u′ − u2u′) − u

}
(7)

where L−1t =
∫ ∫

(.)dtdt is the two-fold integration operator.
Initial condition of eq. (1) given by u(0) = 1 is denoted by
u0 = 1 so that from eq. (5), the components that make up the
series becomes

u0 = 1 (8)

un+1(t) =
∫ t

0

∫ t
0

{
µ((un(s))

′
− An(s)) − un(s)

}
d sd s (9)

where

An =
1
n!

dn

dλn

N
 ∞∑

i=0

uiλ
i



λ=0

= u2n(un)
′

n = 0, 1, 2, · · · and λ is a parameter, are called the Adomian
polynomials. However, we shall use the modified procedure for
generating the Adomian polynomials as discussed below:

Suppose the nonlinear term in (1) is denoted by N(u), then
we decompose it as

A0 = N(u0)

A1 = N(u0 + u1) − A0

A2 = N(u0 + u1 + u2) − (A0 + A1)

A3 = N(u0 + u1 + u2 + u3) − (A0 + A1 + A2)

...

Thus we obtain the relation for generating the Adomian poly-
nomials as

An = N

 n∑
i=0

ui

 − n−1∑
i=0

Ai (10)

From the initial value problem (1), N(u) = u2u′ and hence by
using (10), the Adomian polynomials are obtained as

A0 = u
2
0u
′
0

A1 = (u0 + u1)
2(u′0 + u

′
1) − u

2
0u
′
0

= u20u
′
1 + 2u0u1u

′
0 + 2u0u1u

′
1 + u

2
1u
′
0 + u

2
1u
′
1

A2 = (u0 + u1 + u2)
2(u′0 + u

′
1 + u

′
2) − (A0 + A1)

= u20u
′
2 + 2u0u1u

′
2 + 2u0u2u

′
0 + 2u0u2u

′
1

+2u0u2u
′
2 + 2u1u2u

′
0 + 2u1u2u

′
1 + 2u1u2u

′
2

+u22u
′
1 + u

2
2u
′
2 + u

2
1u
′
2 + u

2
2u
′
0

Accordingly, we obtain

u0 = 1

u1 = −
t2

2

u2 =
µ

168
t7 −

µ

20
t5 +

t4

4!

u3 = −
µ4

312947712
t22+

µ4

11289600
t20−

µ4

1299600
t18−

µ3

12870144
t19

+
µ3

399840
t17−

µ

240

(
5µ

32256
−

µ3

1600

)
t16−

617µ3

21168000
t15+

3µ2

125440
t14

−
µ

156

(
−

31µ2

1400
+

1
3456

)
t13−

41µ2

120960
t12−

µ

110

(
−

5
576

+
µ2

40

)
t11

+
19µ2

8400
t10 −

13µ
9072

t9 −
µ2

160
t8 +

µ

140
t7 −

t6

6!
...

The series solution for 4-components Adomian decomposition
then becomes

u(t) = u0 + u1 + u2 + u3 (11)

which gives

u(t) = 1 −
t2

2
+

µ

168
t7 −

µ

20
t5 +

t4

4!

−
µ4

312947712
t22 +

µ4

11289600
t20

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 109

−
µ4

1299600
t18 −

µ3

12870144
t19 +

µ3

399840
t17

−
µ

240

(
5µ

32256
−

µ3

1600

)
t16

−
617µ3

21168000
t15 +

3µ2

125440
t14

−
µ

156

(
−

31µ2

1400
+

1
3456

)
t13

−
41µ2

120960
t12 −

µ

110

(
−

5
576

+
µ2

40

)
t11 +

19µ2

8400
t10

−
13µ
9072

t9 −
µ2

160
t8 +

µ

140
t7 −

t6

6!
(12)

and the series solution for 5-component Adomian decomposi-
tion then gives

u(t) = u0 + u1 + u2 + u3 + u4 (13)

We simplify (13) and choose µ to be arbitrarily small because
approximate P-L method to be used for comparison can work
effectively on differential equations of the form

u′′ + w20u = µF(t, u, u
′), 0 < µ << 1

whose leading order is oscillatory with frequency w0 [10]. So
eq.(13) using µ = 2 x 10−2 gives

u(t) = 1 − 1/2 t2 + 0.0002619047619 t7

−0.001000000000 t5 −
t6

720
+ 1/24 t4

+1.329776164 × 10−50 t67 − 1.139867761 × 10−48 t65

+5.077535944 × 10−47 t64 + 4.357672234 × 10−47 t63

−4.592882729 × 10−45 t62 + 8.593423967 × 10−44 t61

−8.269818866 × 10−42 t59 − 1.845062816 × 10−13 t23

+1.866090683 × 10−43 t60 + 8.309132396 × 10−41 t58

+3.568516858 × 10−40 t57 − 8.714821000 × 10−39 t56

+4.811105327 × 10−38 t55 + 4.002845402 × 10−37 t54

−5.886109478 × 10−36 t53 + 1.424302351 × 10−35 t52

+2.902430667 × 10−34 t51 − 2.595201219 × 10−33 t50

−1.043753849 × 10−33 t49 + 1.407479522 × 10−31 t48

−7.032084376 × 10−31 t47 − 3.022579830 × 10−30 t46

+4.519343938 × 10−29 t45 − 8.093847241 × 10−29 t44

−1.418636597 × 10−27 t43 + 8.881156719 × 10−27 t42

+1.617446587 × 10−26 t41 − 3.583414553 × 10−25 t40

+7.270414170 × 10−25 t39 + 8.689959175 × 10−24 t38

−4.931206414 × 10−23 t37 − 1.020281668 × 10−22 t36

+1.652954579 × 10−21 t35 − 1.931801068 × 10−21 t34

−3.707212443 × 10−20 t33 + 1.482428707 × 10−19 t32

+5.109844495 × 10−19 t31 − 4.724699345 × 10−18 t30

+2.377613320 × 10−19 t29 + 1.016506228 × 10−16 t28

−2.556365546 × 10−16 t27 − 1.543943729 × 10−15 t26

+8.696083800 × 10−15 t25 + 1.257108256 × 10−14 t24

+1.611259824 × 10−13 t22 − 9.420685884 × 10−12 t20

+0.00002230158730 t8 + 2.799612144 × 10−12 t21

−3.001809900 × 10−11 t19 + 0.0000000002238474420 t18

+0.0000000001499617141 t17 − 0.000000003494172630 t16

+0.000000003368584091 t15 + 0.00000003773049937 t14

−0.0000001349519000 t13 − 0.0000002741969964 t12

+0.000002674211961 t11 + 0.000001218253968 t10

−0.00003373015873 t9 (14)

4. Results and Discussion

Here, comparison of results from the approximating poly-
nomials (3), (4), (12) and (14) with approximate P-L method
for the solution of (1) and its initial conditions are shown in
Tables 1, 2, 3 and 4. The choice of approximate P-L method is
because exact solution has not been obtained for the Van der Pol
equation. Details of the approximate method used here can be
obtained from ([10], [15]) and references contained therein. It
should be noted that the initial conditions used in this research
suggest that the periodic behaviour exhibits the amplitude 1 as
against the amplitude 2 used in [15]. Absolute errors (ABS Er-
rors) are also obtained and are tabulated as follows:

Figure 1: Graphical solutions using P-L method and approximating polynomi-
als (3) and (12) of TTSM and 4-component MADM.

While the graphs of solutions of equation (1) and its initial
conditions using approximate P-L analytic method and approx-
imating polynomials (3), (4), (12) and (14) are shown in Figures

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 110

Table 1: Solutions and errors from (3) and (12) of TTSM series and 4-component MADM

t Poincare-Lindstedt TTSM MADM ABS Error TTSM ABS Error MADM
0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000
0.3 0.955854836508 0.955334115584 0.955334114058 0.000520720924 0.000520722450
0.6 0.828944246584 0.825264812016 0.825264451699 0.003679434568 0.003679794885
0.9 0.631240082296 0.621131026775 0.621122751091 0.010109055521 0.010117331205
1.2 0.378578337374 0.360630621478 0.360560141843 0.017947715896 0.018018195531
1.5 0.0906245246636 0.066327550293 0.065998964725 0.024296974371 0.024625559939
1.8 -0.20868615269 -0.23659702897 -0.23752492072 0.027910876280 0.028838768030
2.1 -0.491935529476 -0.52423546969 -0.52542008925 0.032299940214 0.033484559774
2.4 -0.731188366817 -0.78110829991 -0.77773900461 0.049919933093 0.046550637793
2.7 -0.902481373797 -1.01309133371 -0.98448885633 0.110609959913 0.082007482533
3.0 -0.989925613354 -1.26963992857 -1.15176986217 0.279714315216 0.161844248816

Table 2: Solutions and errors from (4) and (12) of TTSM and 4-component MADM

t Poincare-Lindstedt TTSM MADM ABS Error TTSM ABS Error MADM
0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000
0.3 0.955854836508 0.955334115588 0.955334114058 0.000520720920 0.000520722450
0.6 0.828944246584 0.825264821264 0.825264451699 0.003679425320 0.003679794885
0.9 0.631240082296 0.621131785279 0.621122751091 0.010108297017 0.010117331205
1.2 0.378578337374 0.360647539325 0.360560141843 0.017930798049 0.018018195531
1.5 0.0906245246636 0.066512092157 0.065998964725 0.024112432507 0.024625559939
1.8 -0.20868615269 -0.23531787096 -0.23752492072 0.026631718270 0.028838768030
2.1 -0.491935529476 -0.51774914669 -0.52542008925 0.025813617214 0.033484559774
2.4 -0.731188366817 -0.75481924354 -0.77773900461 0.023630876723 0.046550637793
2.7 -0.902481373797 -0.9216069535 -0.98448885633 0.019125579703 0.082007482533
3.0 -0.989925613354 -0.97208752941 -1.15176986217 0.017838083944 0.161844248816

Table 3: Solutions and errors from (3) and (14) of TTSM and 5-component MADM

t Poincare-Lindstedt TTSM MADM ABS Error TTSM ABS Error MADM
0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000
0.3 0.955854836508 0.955334115584 0.95533411559 0.000520720924 0.000520720918
0.6 0.828944246584 0.825264812016 0.825264822641 0.003679434568 0.003679423943
0.9 0.631240082296 0.621131026775 0.621131859045 0.010109055521 0.010108223251
1.2 0.378578337374 0.360630621478 0.360648788531 0.017947715896 0.017929548843
1.5 0.0906245246636 0.066327550293 0.066523502482 0.024296974371 0.024101022182
1.8 -0.20868615269 -0.23659702897 -0.23524709918 0.027910876280 0.026560946490
2.1 -0.491935529476 -0.52423546969 -0.51742153967 0.032299940214 0.025486010194
2.4 -0.731188366817 -0.78110829991 -0.75375297048 0.049919933093 0.022564603663
2.7 -0.902481373797 -1.01309133371 -0.92099274209 0.110609959913 0.018511368293
3.0 -0.989925613354 -1.26963992857 -0.99998289494 0.279714315216 0.010057281586

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 111

Table 4: Solutions and errors from (4) and (14) of TTSM and 5-component MADM

t Poincare-Lindstedt TTSM MADM ABS Error TTSM ABS Error MADM
0 1.000000000000 1.000000000000 1.000000000000 0.000000000000 0.000000000000
0.3 0.955854836508 0.955334115588 0.95533411559 0.000520720920 0.000520720918
0.6 0.828944246584 0.825264821264 0.825264822641 0.003679425320 0.003679423943
0.9 0.631240082296 0.621131785279 0.621131859045 0.010108297017 0.010108223251
1.2 0.378578337374 0.360647539325 0.360648788531 0.017930798049 0.017929548843
1.5 0.0906245246636 0.066512092157 0.066523502482 0.024112432507 0.024101022182
1.8 -0.20868615269 -0.23531787096 -0.23524709918 0.026631718270 0.026560946490
2.1 -0.491935529476 -0.51774914669 -0.51742153967 0.025813617214 0.025486010194
2.4 -0.731188366817 -0.75481924354 -0.75375297048 0.023630876723 0.022564603663
2.7 -0.902481373797 -0.9216069535 -0.92099274209 0.019125579703 0.018511368293
3.0 -0.989925613354 -0.97208752941 -0.99998289494 0.017838083944 0.010057281586

1, 2, 3 and 4 for µ = 2 x 10−2, the graphs of absolute errors are
also indicated in Figures 5, 6, 7 and 8.

Figure 2: Graphical solutions using P-L method and approximating polynomi-
als (4) and (12) of TTSM and 4-component MADM.

Figure 3: Graphical solutions using P-L method and approximating polynomi-
als (3) and (14) of TTSM and 5-component MADM.

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J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 112

Figure 4: Graphical solutions using P-L method and approximating polynomi-
als (4) and (14) of TTSM and 5-component MADM.

Figure 5: Plot of absolute errors from approximating polynomials (3) and (12)
of TTSM and 4-component MADM.

Figure 6: Plot of absolute errors from approximating polynomials (4) and (12)
of TTSM and 4-component MADM.

Figure 7: Plot of absolute errors from approximating polynomials (3) and (14)
of TTSM and 5-component MADM.

112



J. N. Ndam & O. Adedire / J. Nig. Soc. Phys. Sci. 2 (2020) 106–114 113

Figure 8: Plot of absolute errors from approximating polynomials (4) and (14)
of TTSM and 5-component MADM.

Equation (12) of approximating 4-component MADM be-
have more like approximate P-L method than approximating
polynomial (3) of TTSM as shown in Figure 1 and Table 1. On
the other hand, many more terms are added to approximating
TTSM (3) to obtain TTSM (4). Solutions from approximating
polynomial (4) of TTSM with twenty-two order behave more
closely as that of approximate P-L method than equation (12)
of approximating 4-component MADM series and are indicated
in Figure 2 and Table 2. However, results from 5-component
MADM (14) shown in Tables 3 and 4 are similar to the results
obtained from approximate P-L method than those got from
TTSM of equations (3) and (4) as shown in Figures 3 and 4.
The absolute errors generated by the approximating polynomial
(3) are greater than the ones generated by (12) as shown in Fig-
ure 5. Absolute errors obtained from TTSM (4) are generally
lower than the ones generated from (12) of MADM due to many
numbers of terms added to TTSM (3) as shown in Figure 6. Fi-
nally, Figures 7 and 8 indicate that (14) shows smaller absolute
errors compared with (3) and (4). At t=3 from Table 1, equation
(3) gives u(3) ≈ −1.27 and (12) gives u(3) ≈ −1.15. Observe
that with several terms added to (3) to obtain (4), the solution
obtained in Table 4 at t = 3 for (4) is not as close to that of
approximate P-L method as (14) which is obtained from (12)
by addition of just one component. The theoretical implication
of this is that few components from MADM - obtained with
ease - can produce solution that is more convergent to that of P-
L method than for tediuos computational work of many terms
added to TTSM.

5. Conclusion

In this paper, approximate series solutions of the van der Pol
equation have been obtained using the truncated Taylor series
method and the modified Adomian decomposition method. The
two series solutions for Van der Pol equation obtained as ap-
proximating polynomials (3), (4), (12) and (14) agree. From the
approximating polynomials of the two series solutions, MADM
produces more convergent results to the P-L methods than the
TTSM, as only very few components of the MADM yielded a
series much longer than order twenty of Taylor series. Hence,
we recommend inclusion of the MADM in modern mathemati-
cal tools.

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