Modified Adomian Polynomial for Nonlinear Functional with Integer Exponent

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. V (Nov. - Dec. 2015), PP 40-45
www.iosrjournals.org
DOI: 10.9790/5728-11654045 www.iosrjournals.org 40 | Page
Modified Adomian Polynomial for Nonlinear Functional with
Integer Exponent
E. U. Agom1
, F. O. Ogunfiditimi2
1
(Department of Mathematics, University of Calabar, Calabar, Nigeria)
2
(Department of Mathematics, University of Abuja, Abuja, Nigeria)
Email: agomeunan@gmail.com
Abstract :Successful application of Adomian decomposition method (ADM) in solving problems in nonlinear
ordinary and partial differential equations depend strictly on the Adomian polynomial. In this paper, we present
a simple modified known Adomian polynomial for nonlinear polynomial functionals with index as integers. The
simple modified Adomian polynomial was tested for nonlinear functional with index 3 and 4 respectively. The
result shows remarkable exact results as that given by Adomian himself. Also, the modifed simple Adomian
polynomial was further tested on concrete problems and the numerical results were exactly the same as the
exact solution. The large scale computation and evaluation was made possible by Maple software package.
Keywords - Adomian Polynomial, Adomian Decomposition Method.
I. INTRODUCTION
The Adomian Polynomial in ADM has been subject of some studies [1] to [9]. This method generates a
solution in form of a series whose terms are determined by a recursive relationship using the Adomian
Polynomial. Several authors have suggested different algorithms for computing Adomian Polynomial,
prominent among them are [2], [3]. Using the algorithm presented by Adomian himself [1] requires
classification of terms in both the ordinary and the accelerated form, which is very complicated for large n
(order of the derivative).
Algorithm presented by [2] uses Taylor series expansion of the functional which is complicated
especially when the unknown appears at the denominator. Calculation of the nth Adomian Polynomials using
[3] requires computing the nth order derivative which is complicated for large n. That is why most literatures
gives, at most, the first five generated Adomian Polynomial. Despite all the difficulties in applying the used
method in [5], it cannot be applied to functionals with several variables. Here we suggest a new simplified
single line algorithm that can be implemented in any computer algebraic system. To generate the Adomian
Polynomial without resulting to writing codes before implementation
II. THE ADOMIAN POLYNOMIAL IN ADM
Consider the general nonlinear differential equation;
fFu  (1)
F is nonlinear differential operator and u, f are functions of t. Equation (1) in operator form is given as;
fNuRuLu  (2)
where L is an operator representing linear portion of f which is easily invertible, R is a linear operator for the
remainder of the linear portion. N is a nonlinear operator representing the nonlinear term in f. Applying the
inverse operator L-1
on equation (2) we have;
NuLRuLfLLuL
1111 
 (3)
By virtue of L, L-1
would represent integration with any given initial/boundary conditions. Equation (3)
becomes;
NuLRuL)t(g)t(u
11 
 (4)
where g(t) represent the function generated by integrating f and using the initial/boundary conditions.
ADM admit the decomposition into an infinite series with equation (4) given as;










0n
n
1
0n
n
1
0
0n
n
AL)t(RuLu)t(u
where An is the Adomian Polynomial which is given as;
0n
n
n
)u(N
d
d
!n
1
A


 (5)

Modified Adomian Polynomial for Nonlinear Functional with Integer Exponent
The recursive relation is found to be
)t(gu
0
 (6)
n
11
1n
ALRuLu


 (7)
Having determined the components un; n ≥ 0 the solution




0n
n
)t(uu (8)
is in series form. The series may be summed to provide the solution in closed form. Or, for concrete problems
the nth partial sum may be used to give the approximate solution.
We give the simple modification to the Adomian Polynomial of equation (5) as;






0n
n
n
n
n
d
d
!n
1
A (9)
where
 
 


n
0i
i
0j
injij
)u...uu(...
III. IMPLEMENTATION OF THE SIMPLE MODIFIED ADOMIAN POLYNOMIAL
In this section, we present some examples that resulted from the use of the simple modified Adomian
polynomial.
1. For N(u) = 3,u 

and using equation (9), the first ten plus one Adomian Polynomials are given as;
3
00
uA 
1
2
01
uu3A 
2
102
2
02
uu3uu3A 
3
12103
2
03
uuuu6uu3A 
2
2
1
2
203104
2
04
uu3uu3uuu6uu3A 
2
213
2
15204105
2
05
uu3uu3uuu3uuu6uu3A 
3
2321
2
304
2
14205106
2
06
uuuu6uu3uu3uuu6uuu6uu3A 
3
2
2
2
314214305
2
15206107
2
07
uu3uu3uuu6uuu6uu3uuu6uuu6uu3A 
4
2
2431
2
405215306
2
16207108
2
08
uu3uuu6uu3uuu6uuu6uu3uuu6uuu6uu3A 
2
32
uu3
5
2
25315406216307
2
17208109
2
09
uu3uuu6uuu6uuu6uuu6uu3uuu6uuu6uu3A 
3
34321
2
4
uuuu6uu3 
6
2
26316407217308
2
182091010
2
010
uu3uuu6uuu6uuu6uuu6uu3uuu6uuu6uu3A 
4
2
32
2
45325410
2
5
uu3uu3uuu6uuu6uu3 
2. For N(u) = 4,u 

and using equation (9), the first ten plus one Adomian Polynomials are also given as;
4
00
uA 
1
3
01
uu4A 
2
1
2
02
3
02
uu6uu4A 
3
1021
2
03
3
03
uu4uuu12uu4A 

4
12
2
10
2
2
2
031
2
04
3
04
uuuu12uu6uuu12uu4A 
3
12
2
2103
2
1032
2
041
2
05
3
05
uu4uuu12uuu12uuu12uuu12uu4A 
3
203
3
13210
2
3
2
04
2
1042
2
051
2
06
3
06
uu4uu4uuuu24uu6uuu12uuu12uuu12uu4A 
2
2
2
1
uu6
6
2
014
3
11
3
22
2
103
2
2010
2
350
2
17
3
07
uuu12uu4uu4uuu12uuu12uuu12uuu12uu4A 
42103
2
045
2
02
uuuu24uuu12uuu12 
5
3
17
2
014
2
124310521060
2
1
4
28
3
08
uu4uuu12uuu12uuuu24uuuu24uuu12uuu4A 
6
2
025
2
032
2
304
2
20
2
4
2
03
2
21
2
3
2
1
uuu12uuu12uuu12uuu12uu6uuu12uu6 
531054
2
0621063
2
072
2
081
2
09
3
09
uuuu24uuu12uuuu24uuu12uuu12uuu12uu4A 
21
2
352
2
110
2
450
2
270
2
141
2
24320
uuu12uuu12uuu12uuu12uuu12uuu12uuuu24 
3
23
3
16
3
3043
2
1
uu4uu4uu4uuu12 
432191
2
0532064
2
073
2
082
2
010
3
010
uuuu24uuu12uuuu24uuu12uuu12uuu12uu4A 
40
2
380
2
153
2
151
2
262
2
141
2
26310
uuu12uuu12uuu12uuu12uuu12uuu12uuuu24 
3
24
3
31
3
1720
2
460
2
254107210
uu4uu4uu4uuu12uuu12uuuu24uuuu24 
2
3
2
2
2
5
2
0
2
4
2
1
uu6uu6uu6 
IV. APPLICATION OF THE SIMPLE MODIFIED ADOMIAN POLYNOMIAL TO
CONCRETE PROBLEMS
In this section, we apply the modified Adomian polynomials to concrete problems.
Problem 1
Consider
3
uu5
dt
du
 , u(0) = 1 (10)
The exact solution of equation (10) is given as;
1e6
5
u
t10



(11)
And in series form equation (11) is given as

432
t470t100t24t61u (12)
Applying the ADM to equation (10), we have;
)A(L)u5(L)0(u)t(u
n
11 

where An in this case is given as
3
n
u)u(NA 
Applying equation (6), (7) and (9) to equation (10), we obtain;
1u
0

 
t
0
3
001
t6dt)uu5(u
 
t
0
2
1
3
012
t24dt)uu3u5(u
 
t
0
32
102
2
023
t100dt)uu3uu3u5(u
 
t
0
43
12103
2
034
t470dt)uuuu6uu3u5(u

 
t
0
5
2
2
1
2
203104
2
045
t2336dt)uu3uu3uuu6uu3u5(u
 
t
0
62
213
2
15204105
2
056
t
3
35588
dt)uu3uu3uuu3uuu6uu3u5(u
 
t
0
73
2321
2
304
2
14205106
2
067
t
21
1282984
dt)uuuu6uu3uu3uuu6uuu6uu3u5(u
 
t
0 3
2
2
2
314214305
2
15206107
2
078
dt)uu3uu3uuu6uuu6uu3uuu6uuu6uu3u5(u
8
t
21
6681580

Continuing in this order, the sum of the first few terms of un, is given as;
 


765432
0n
n
t
21
1282984
t
3
35588
t2336t470t100t24t61uu . . .
This is obviously the same as the series form of the exact solution given in equations (12). The similarity
between the exact solution, equation (11) and the numerical solution of the first 12th terms is further given in
Fig. 1 and Fig. 2 respectively.
Problem 2
Consider
1)0(u,uu
dt
du 4
 (13)
The exact solution of equation (13) is
1e2
1
u
t3
3



(14)
And in series form equation (14) is given as

432
t
12
701
t
3
49
t5t21u . . . (15)
Applying the Adomian decomposition method to equation (13) we have
)A(L)u(L)0(u)t(u
n
11 

where An in this case is given as;
An = N(u) = u4
Also, applying the recursive relation and the simple modified Adomian polynomial to equation (13), we obtain;
1u
0


 
t
0
4
001
t2dt)uu(u
 
t
0
2
1
3
012
t5dt)uu4u(u
 
t
0
32
1
2
02
3
023
t
3
49
dt)uu6uu4u(u
 
t
0
43
1021
2
03
3
034
t
12
701
dt)uu4uuu12uu4u(u
 
t
0
54
12
2
10
2
2
2
031
2
04
3
045
t
60
13081
dt)uuuu12uu6uuu12uu4u(u
 
t
0
63
12
2
2103
2
1032
2
041
2
05
3
056
t
72
60193
dt)uu4uuu12uuu12uuu12uuu12uu4u(u
 
t
0
3
203
3
13210
2
3
2
04
2
1042
2
051
2
06
3
067
uu4uu4uuuu24uu6uuu12uuu12uuu12uu4u(u
72
2
2
1
t
2520
8231329
dt)uu6 
Continuing in this order, we have;
 


765432
0n
n
t
2520
8231329
t
72
60193
t
60
13081
t
12
701
t
3
49
t5t21uu . . .
The first few terms of the series are obviously the same as equation (15) of the exact solution of Problem 2.
The resemblance of the numerical solution using the simple modified Adomian polynomial of equation (9) and
the exact result is further depicted in Fig. 3 and Fig. 4.
In Fig. 2 and Fig. 4, finite terms of the series, 


11
0n
n
uu , were used in the plot. The remarkable similarities
between the exact and ADM using equation (9) (the simple modified Adomian polynomial) of problems 1and 2
is further shown in Tables 1 and 2.

Table I: Exact versus ADM solution of Problem 1
t Exact solution Solution by ADM,  

11
0n n
uu
-0.14 4.5480323980 x 10-1
4.6293121480 x 10-1
-0.13 4.8432134540 x 10-1
4.8776680970 x 10-1
-0.12 5.1270154680 x 10-1
5.1406304520 x 10-1
-0.11 5.4143250440 x 10-1
5.4192787600 x 10-1
-0.10 5.7131765330 x 10-1
5.7148099190 x 10-1
0.00 1.0000000000 x 10-1
1.0000000000 x 10-1
0.10 2.0345383800 x 100
2.0350805430 x 100
0.11 2.2372482020 x 100
2.2391733070 x 100
0.12 2.4825644110 x 100
2.4888789140 x 100
0.13 2.7861415940 x 100
2.8056466280 x 100
0.14 3.1707781130 x 100
3.2288930710 x 100
Table II: Exact versus ADM solution of Probem 2
t Exact solution Solution by ADM,  

11
0n n
uu
-0.14 7.8784892490 x 10-1
7.8797386830 x 10-1
-0.13 7.9983309430 x 10-1
7.9988581050 x 10-1
-0.12 8.1214566950 x 10-1
8.1216639740 x 10-1
-0.11 8.2483167170 x 10-1
8.2483917380 x 10-1
-0.10 8.3792754430 x 10-1
8.3793000380 x 10-1
0.00 1.0000000000 x 10-1
1.0000000000 x 10-1
0.10 1.2757283730 x 100
1.2757342650 x 100
0.11 1.3168990780 x 100
1.3169189670 x 100
0.12 1.3624449460 x 100
1.3625060770 x 100
0.13 1.4113293205 x 100
1.4137672060 x 100
0.14 1.4706510740 x 100
1.4711161140 x 100
V. CONCLUSION
In this paper, we proposed an efficient simple modification of the standard Adomian Polynomial in
the popular Adomian decomposition method for solving nonlinear functional whose nonlinear term is of the
form N(u) = un
. The study showed that the modified Adomian polynomial is simple and is efficient, and also
effective in any computer algebra system to get as many term of the Adomian polynomials as required without
difficulties. The outcome from the modifications is the same as those presented by Adomian himself. And when
applied to concrete problems the results were remarkable.
REFERENCES
[1]. G. Adomian, A Review of the Decomposition Method in Applied Mathematics, Journal of Mathematical Analysis and
Applications, 135, 1988, 501 - 544.
[2]. J. Biazar, E. Babolian, A. Nouri, S. Islam, An Alternative Algorithm for Computing Adomian Polynomials in Special cases, Applied
Mathematics and Computation 138, 2003, 523 - 529.
[3]. A. M. Wazwaz, A new Algorithm for Calculating Adomian Polynomials for Nonlinear Operator, Applied Mathematics and
Computation, 111, 2000, 53 - 69.
[4]. G. Adomian and R. Rach, Polynomials Nonlinearity in Differential Equations, Journal of Mathematical Analysis and Application,
109(1), 1985
[5]. G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Springer New York, 1993
[6]. K. Abaoui and Y. Cherruault, Convergence of Adomian's Method Applied to Differential Equations, Computer and Mathematics
Applications, 28(5), 1994, 103 - 109.
[7]. E. U. Agom and F. O. Ogunfiditimi, Adomian Decomposotion Method for Bernoulli Differential Equations, International Journal
of Science and Research, 4(12), 2015, 1581 - 1584.
[8]. R. Rach, G. Adomian and R. E. Meyers, A Modified Decomposition, Computers and Mathematics with Applications, 23(1), 1992,
12 - 23.
[9]. M. Almazmumy, F. A. Hendi, H. O. Bakodah and H. Alzumi, Recent Modifications of Adomian Decompositon Method for Initial
Value Problems in Ordinary Differential Equations, American Journal of Computational Mathematics, 2, 2012, 228 - 234.

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