In this paper, we state the Adomian Decomposition Method (ADM) for third order time-invariant linear homogeneous differential equations. And we applied it to find solutions to the same class of equations. Three test problems were used as concrete examples to validate the reliability of the method, and the result shows remarkable solutions as those that are obtained by any knows analytical method(s).

1. The International Journal Of Engineering And Science (IJES)
|| Volume || 5 || Issue || 6 || Pages || PP -81-85 || 2016 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 81
Numerical Solution of Third Order Time-Invariant Linear
Differential Equations by Adomian Decomposition Method
E. U. Agom, F. O. Ogunfiditimi
Department Of Mathematics, University Of Calabar, Calabar, Nigeria.
Department Of Mathematics, University Of Abuja, Abuja, Nigeria
--------------------------------------------------------ABSTRACT-----------------------------------------------------------
In this paper, we state the Adomian Decomposition Method (ADM) for third order time-invariant linear
homogeneous differential equations. And we applied it to find solutions to the same class of equations. Three test
problems were used as concrete examples to validate the reliability of the method, and the result shows
remarkable solutions as those that are obtained by any knows analytical method(s).
Keywords: Adomian Decomposition Method, Third Order Time-Invariant Differential Equations.
-------------------------------------------------------------------------------------------------------------------------------------
Date of Submission: 17 May 2016 Date of Accepted: 30 June 2016
I. INTRODUCTION
Linear time-invariant differential equations plays important role in Physics because it assumes the law of nature
which hold now and are identical to those for times in the past or future. This class of equation has wide
application in automated theory, digital signal processing, telecommunication engineering, spectroscopy,
seismology, circuit and other technical areas. Specifically in telecommunication, the propagation medium for
wireless communication systems is often modeled with this class of equations. Also, this class of equation has
tremendous application in dynamics.
Many physical systems are either time-invariant or approximately so, and spectral analysis is an efficient tool in
investigating linear time-invariant differential equations. In this paper, we apply the powerful ADM to find
solution to third order linear time-invariant differential equations. The ADM has been a subject of several
studies [1-6]. It seeks to make possible physically realistic solution to complex real life problems without using
modeling and mathematical compromises to achieve results. It has been judged to provide the best and
sometimes the only realistic simulation phenomena.
In compact form, the general third order time-invariant linear differential equation is given as
))t(x),t(x,t(f)t(x   (1)
With initial conditions given as
C)t(xandB)t(x,A)(x  
ADM gives a series solution of x which must be truncated for practical application. In addition, the rate and
region of convergence of the solutions are potentially short or long coming. The series of x can rapidly converge
in small and wide region depending on the problem at hand.
II. THE THEORY OF ADM
By [5] the ADM of equation (1) is given as;
 x (2)
Where  is a differential operator, x and  are functions of t in this case. In operator form, equation (2) is
given as;
 xxx (3)
 ,  and  are given respectively as; highest order differential operator with respect to t of  to be inverted,
the linear remainder operator of  and the nonlinear operator of  which is assumed to be analytic. The
choice of the linear operator is designed to yield an easily invertible operator with resulting trivial integration. In
this article 3
3
dt
d
 . Furthermore, we emphasize that the choice of  and concomitantly it inverse 1
 are
determined by a particular equation to be solved. Hence, the choice is non-unique. For equation (1),


.dtdtdt(.)
1
That is, a three-fold definite integral operator from t0 to t. For a linear form of equation (3),
0x  and we have;

2. Solution Of Third Order Time-Invariant Linear Differential Equations By Adomian Decomposition…
www.theijes.com The IJES Page 82
]x[x
1


(4)
ADM decomposes the solution of equation (1) into a series;




0n
n
xx (5)
Where  incorporates all the initial conditions which is considered as x0. For details of ADM theory see [1-6].
III. ILLUSTRATION
In this section we give examples on how ADM can be used to find solutions to third order linear time-invariant
differential equations.
Problem 1
0xxxx   (6)
With initial conditions given as
1)0(xand0)0(x,1)0(x  
The exact solution of (6) is;
tcosx  (7)
The series form of (7) is ;

108642
t
!10
1
t
!8
1
t
!6
1
t
!4
1
t
!2
1
!0
1
x . . . (8)
Applying equations (2) to (5) on (6), we obtain;
2
0
t
2
1
1x 
54
1
t
120
1
t
24
1
x 
8765
2
t
40320
1
t
2520
1
t
360
1
t
120
1
x 
11109876
3
t
399168
1
t
1209600
1
t
72576
1
t
8064
1
t
1680
1
t
720
1
x 
121110987
4
t
53222400
1
t
39916800
13
t
3628800
13
t
40320
1
t
10080
1
t
5040
1
x 
1413
t
08717829120
1
t
1556755200
1

1312111098
5
t
1779148800
1
t
13685760
1
t
19958400
13
t
259100
1
t
72576
1
t
40320
1
x 
17161514
t
960003556874280
1
t
6004184557977
1
t
09340531200
1
t
3353011200
1

14131211109
6
t
908107200
1
t
83026944
1
t
10644480
1
t
1995840
1
t
604800
1
t
362880
1
x 
18171615
t
864003201186852
1
t
8007904165068
1
t
402789705318
1
t
01362160800
1

2019
t
1766400002432902008
1
t
14720002027418340
1

151413121110
7
t
6054048000
1
t
622702080
1
t
6227020800
71
t
17740800
1
t
5702400
1
t
362880
1
x 
19181716
t
488008688935743
1
t
80002964061900
1
t
320001185624760
89
t
0006974263296
89


3. Solution Of Third Order Time-Invariant Linear Differential Equations By Adomian Decomposition…
www.theijes.com The IJES Page 83
222120
t
039658240001605715325
1
t
74707200001892257711
1
t
1766400002432902008
71

23
t
00088849766402585201673
1

Proceeding in this order, we have that;



10
0n
108642
n
t
3628800
1
t
40320
1
t
720
1
t
24
1
t
2
1
1x . . . (9)
Comparing equations (8) and (9), we find that they are the same. The similarities between the exact solution and
ADM solution of problem 1 is further shown in Fig. 1 and Fig. 2.
Problem 2
0xx5x4x   (10)
With initial conditions given as
23)0(xand7)0(x,4)0(x  
The exact solution of (10) is;
t5t
ee25x

 (11)
The series form of (10) is ;
8765432
t
40320
390623
t
720
11161
t
720
15623
t
120
3127
t
24
623
t
6
127
t
2
23
t74x  - . . . (12)
Similarly, applying equations (2) to (5) on (10), we obtain;
2
0
t
2
23
t74x 
43
1
t
24
115
t
6
127
x 
654
2
t
144
115
t
8
73
t
6
127
x 
8765
3
t
8064
575
t
1008
1555
t
18
173
t
15
254
x 
109876
4
t
145152
575
t
72576
10075
t
224
365
t
63
473
t
45
508
x 
121110987
5
t
19160064
2875
t
16128
125
t
9072
1325
t
2268
2825
t
21
100
t
315
2032
x 

4. Solution Of Third Order Time-Invariant Linear Differential Equations By Adomian Decomposition…
www.theijes.com The IJES Page 84
141312111098
6
t
3487131648
14375
t
249080832
73375
t
4790016
38875
t
16632
825
t
1134
865
t
567
1454
t
315
1016
x 
1514131211109
7
t
1494484992
12125
t
69118912
2125
t
2223936
13375
t
2673
175
t
33
13
t
405
488
t
2835
4064
x 
16
t
041673823191
14375

Continuing in this order, we have;


765432
12
0n
n
t
720
11161
t
720
15623
t
120
3127
t
24
623
t
6
127
t
2
23
t74x (13)
Where

12111098
t
479001600
244140623
t
39916800
48828127
t
518400
1395089
t
362880
1953127
t
40320
390623
. . .
Equation (13) is the ADM solution of problem 2 i.e. equation (10). The terms of equation (13) are exactly the
same as those of equation (12), which is the classical solution of problem 2. The similarity between the exact
solution and that of ADM is further depicted in Fig. 3 and Fig. 4.
Problem 3
0x25x   (14)
With initial conditions given as
5)0(xand5)0(x,1)0(x  
The exact solution of (14) is;
t5sint5cos
5
1
5
4
x  (15)
The series form of (15) is ;

765432
t
1008
15625
t
144
625
t
24
625
t
24
125
t
6
125
t
2
5
t51x . . . (16)
Similarly, applying equations (2) to (5) on (14), we obtain;
2
0
t
2
5
t51x 
43
1
t
24
125
t
6
125
x 
65
2
t
144
625
t
24
625
x 

5. Solution Of Third Order Time-Invariant Linear Differential Equations By Adomian Decomposition…
www.theijes.com The IJES Page 85
87
3
t
8064
15625
t
1008
15625
x 
Continuing in this order, we have;



8765432
0n
n
t
8064
15625
t
1008
15625
t
144
625
t
24
625
t
24
125
t
6
125
t
2
5
t51x (17)
Equation (17) is the ADM solution of problem 3 i.e. equation (14). The terms of equation (17) are exactly the
same as those of equation (16), which is the classical solution of problem 3. The similarity between the exact
solution and that of ADM is further depicted in Fig. 5 and Fig. 6.
IV. CONCLUSION
We have successfully applied ADM to third order linear time-invariant differential equations. Although in the
ADM we considered only finite terms of and an infinite series, nonetheless, the result obtained by this method
are in total agreement with their exact counterparts. This consideration is some worth obvious in Fig. 2, Fig. 6
and not in any way obvious in Fig. 4. Possible extension of the method to 4th
order linear differential equations
can be investigated.
REFERENCES
[1]. E. U. Agom and F. O. Ogunfiditimi, Adomian Decomposition Method for Bernoulli Differential Equations, International Journal
of Science and Research, 4(12), 2015, 1581-1584.
[2]. F. O. Ogunfiditimi, Numerical Solution of Delay Differential Equations Using Adomian Decomposition Method (ADM), The
International Journal of Engineering and Science 4(5), 2015, 18-23.
[3]. E. U. Agom and A. M. Badmus, A Concrete Adomian Decomposition Method for Quadratic Riccati’s Differential Equations,
Pacific Journal of Science and Technology, 16(2), 2015, 57-62.
[4]. M. Almazmumy, F. A. Hendi, H. O. Bokodah and H. Alzumi, Recent Modifications of Adomian Decomposition Method for
Initial Value Problems in Ordinary Differential Equations, American Journal of Computational Mathematics, 4, 2012, 228-234.
[5]. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method (Springer, New York, 1993).
[6]. E. U. Agom and F. O. Ogunfiditimi , Modified Adomian Polynomial for Nonlinear Functional with Integer Exponent,
International Organization Scientific Research - Journal of Mathematics, 11(6), version 5, (2015), 40 – 45. DOI: 10.9790/5728-
11654045.
Biography
[7]. E. U. Agom is a Lecturer in Department of Mathematics, University of Calabar, Calabar, Nigeria. His research interest is in
Applied Mathematics.