![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;](https://image.slidesharecdn.com/n050602081085-160908093937/85/Numerical-Solution-of-Third-Order-Time-Invariant-Linear-Differential-Equations-by-Adomian-Decomposition-Method-1-320.jpg)
![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
](https://image.slidesharecdn.com/n050602081085-160908093937/85/Numerical-Solution-of-Third-Order-Time-Invariant-Linear-Differential-Equations-by-Adomian-Decomposition-Method-2-320.jpg)
![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.](https://image.slidesharecdn.com/n050602081085-160908093937/85/Numerical-Solution-of-Third-Order-Time-Invariant-Linear-Differential-Equations-by-Adomian-Decomposition-Method-5-320.jpg)
Numerical Solution of Third Order Time-Invariant Linear Differential Equations by Adomian Decomposition Method
- 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.