![International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 4 Issue 11|| November 2015 || PP.60-65
www.ijesi.org 60 | Page
Application of Adomian Decomposition Method in Solving
Second Order Nonlinear Ordinary Differential Equations
1
E. U. Agom MSc , 2
A. M. Badmus PhD
1
(Department of Mathematics, University of Calabar, Calabar, Nigeria)
2
(Department of Mathematics and Computer Science, Nigerian Defence Academy, Kaduna, Nigeria)
ABSTRACT: In this paper, we use Adomian Decomposition Method to numerically analyse second order
nonlinear ordinary di_erential equations and implement the continuous algorithm in a discrete domain. This is
facilitated by Maple package. And, the results from the two test problems used shows that the Adomian
Decomposition Method is almost as the classical solutions.
Key words: Adomian Decomposition Method; Nonlinear Di_erential Equations.
I. INTRODUCTION
Nonlinear Di_erential Equations (NDE) arise in the study of many branches of Applied Math-ematics. Like
Rheology, Quantitative Biology, Physiology,Electrochemistry, scattering theory, Di_usion Transport theory,
Potential theory and Elasticity. In the late 20th century, George Adomian [1] introduced a new method to solve
NDE. Many of these NDE, in fact a tiny frac- tion, can be solved by analytical or closed form method. Many a
times the classical method are complicated; requiring use of advanced Mathematical technique which are di_cult
to un- derstand. Of all the the numerical methods available for the solution of NDE, the method of Finite
Di_erence is most commonly used followed by Finite Element method. All of which are based on linearisation.
The Adomian Decomposition Method (ADM) which has been subject to much investiga- tion [1],[2], [3], [4],[6]
avoids arti_cial boundary conditions, linearisation and yields an e_cient numerical solution with high degree
accuracy. It enables the accurate and e_cient analytical solution of NDE without the need to resort to
linearisation or perturbation approaches.
II. THE ADOMIAN DECOMPOSITION METHOD
The ADM involves separating the equation under investigation into linear and nonlinear por- tion. The linear
operator representing the linear portion of the equation is inverted and the linear the linear operator is then
applied to the equation. Any given conditions are taken into consideration. The nonlinear portion is decomposed
into a series of what is called Adomian Polynomials. The method generates a solution in the form of a series
whose terms are de- termined by a recursive relationship using the Adomian Polynomials. A brief outline of the
method is a follows. Consider a general nonlinear di_erential equation as.
where F is the nonlinear di_erential operator, y and f are functions of t. In operator form
equation (2.1) is
where L is an operator representing the linear portion of F which is easily invertible. R is a linear operator for
the remainder of the linear portion, and N is a nonlinear operator representing](https://image.slidesharecdn.com/k411060065-151205081933-lva1-app6892/85/Application-of-Adomian-Decomposition-Method-in-Solving-Second-Order-Nonlinear-Ordinary-Differential-Equations-1-320.jpg)
![Application of Adomian Decomposition Method in Solving Second…
www.ijesi.org 65 | Page
IV. CONCLUSION
Unlike linear Di_erential Equations, there are limited methods for obtaining classical solutions to NDE. In most
cases, qualitative methods are often used in obtaining qualitative information on solution of NDE without
actually solving the problem, like Phase plane method. In this paper, considering the round o_ errors inherited
by taking a _nite series from an in_nite series, the result of ADM and exact solution are in strong agreement
with each other. In comparison with several other methods that have been advanced for solving NDE, the result
from the two test problems in this paper shows that the ADM is reliable powerful and very promising. We
believe that the e_ciency of ADM gives it much wider applicability which needs to be explored further.
REFERENCE
[1] A. M. Wazwaz, A New Algorithm for Calculating Adomian Polynomials for Nonlinear Op- erator, Applied Mathematical
Computation, 111(2000), 53-69. MR1745908.
[2] E. A. Ibijola and B. J. Adegboyegun, On Adomian Decomposition Method for Numerical Solution of Ordinary Di_erential
Equations. Advances in Natural Applied Science, 2(2008): 165-169.
[3] G. Adomian, Solving Frontier Problem of Physics: The Decomposition Method, Boston: Kluwer Academic Publishers, (1994).
[4] G. Adomian and R. Rach, Analytic solution of Nonlinear Boundary-value Problems in Several Dimensions by Decomposition,
Journal of Mathematical Analysis and Application. 174 (1993).
[5] M. J. Ablowitz and J. F. Ladik, Nonlinear Di_erential-Di_erence Equation and Fourier Analysis. Journal of Mathematics and
Physics, 17(1976), 1011-1013.
[6] Q. H. Yahya and M. Z. Liu,Modi_ed Adomian Decomposition Method for Singular Initial Value Problems in Second order
Ordinary Di_erential Equations, Survey in Mathematics and its Application. 3(2000), 183-193.
[7] T. Mavoungou and Y. Cherrault, Convergence of Adomian's Decomposition Method and Application to Nonlinear Di_erential
Equations Kybernetes, 21(6) (1992)](https://image.slidesharecdn.com/k411060065-151205081933-lva1-app6892/85/Application-of-Adomian-Decomposition-Method-in-Solving-Second-Order-Nonlinear-Ordinary-Differential-Equations-6-320.jpg)
Application of Adomian Decomposition Method in Solving Second Order Nonlinear Ordinary Differential Equations
- 1. International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org ||Volume 4 Issue 11|| November 2015 || PP.60-65 www.ijesi.org 60 | Page Application of Adomian Decomposition Method in Solving Second Order Nonlinear Ordinary Differential Equations 1 E. U. Agom MSc , 2 A. M. Badmus PhD 1 (Department of Mathematics, University of Calabar, Calabar, Nigeria) 2 (Department of Mathematics and Computer Science, Nigerian Defence Academy, Kaduna, Nigeria) ABSTRACT: In this paper, we use Adomian Decomposition Method to numerically analyse second order nonlinear ordinary di_erential equations and implement the continuous algorithm in a discrete domain. This is facilitated by Maple package. And, the results from the two test problems used shows that the Adomian Decomposition Method is almost as the classical solutions. Key words: Adomian Decomposition Method; Nonlinear Di_erential Equations. I. INTRODUCTION Nonlinear Di_erential Equations (NDE) arise in the study of many branches of Applied Math-ematics. Like Rheology, Quantitative Biology, Physiology,Electrochemistry, scattering theory, Di_usion Transport theory, Potential theory and Elasticity. In the late 20th century, George Adomian [1] introduced a new method to solve NDE. Many of these NDE, in fact a tiny frac- tion, can be solved by analytical or closed form method. Many a times the classical method are complicated; requiring use of advanced Mathematical technique which are di_cult to un- derstand. Of all the the numerical methods available for the solution of NDE, the method of Finite Di_erence is most commonly used followed by Finite Element method. All of which are based on linearisation. The Adomian Decomposition Method (ADM) which has been subject to much investiga- tion [1],[2], [3], [4],[6] avoids arti_cial boundary conditions, linearisation and yields an e_cient numerical solution with high degree accuracy. It enables the accurate and e_cient analytical solution of NDE without the need to resort to linearisation or perturbation approaches. II. THE ADOMIAN DECOMPOSITION METHOD The ADM involves separating the equation under investigation into linear and nonlinear por- tion. The linear operator representing the linear portion of the equation is inverted and the linear the linear operator is then applied to the equation. Any given conditions are taken into consideration. The nonlinear portion is decomposed into a series of what is called Adomian Polynomials. The method generates a solution in the form of a series whose terms are de- termined by a recursive relationship using the Adomian Polynomials. A brief outline of the method is a follows. Consider a general nonlinear di_erential equation as. where F is the nonlinear di_erential operator, y and f are functions of t. In operator form equation (2.1) is where L is an operator representing the linear portion of F which is easily invertible. R is a linear operator for the remainder of the linear portion, and N is a nonlinear operator representing
- 2. Application of Adomian Decomposition Method in Solving Second… www.ijesi.org 61 | Page
- 3. Application of Adomian Decomposition Method in Solving Second… www.ijesi.org 62 | Page III. APPLICATION AND RESULT
- 4. Application of Adomian Decomposition Method in Solving Second… www.ijesi.org 63 | Page
- 5. Application of Adomian Decomposition Method in Solving Second… www.ijesi.org 64 | Page
- 6. Application of Adomian Decomposition Method in Solving Second… www.ijesi.org 65 | Page IV. CONCLUSION Unlike linear Di_erential Equations, there are limited methods for obtaining classical solutions to NDE. In most cases, qualitative methods are often used in obtaining qualitative information on solution of NDE without actually solving the problem, like Phase plane method. In this paper, considering the round o_ errors inherited by taking a _nite series from an in_nite series, the result of ADM and exact solution are in strong agreement with each other. In comparison with several other methods that have been advanced for solving NDE, the result from the two test problems in this paper shows that the ADM is reliable powerful and very promising. We believe that the e_ciency of ADM gives it much wider applicability which needs to be explored further. REFERENCE [1] A. M. Wazwaz, A New Algorithm for Calculating Adomian Polynomials for Nonlinear Op- erator, Applied Mathematical Computation, 111(2000), 53-69. MR1745908. [2] E. A. Ibijola and B. J. Adegboyegun, On Adomian Decomposition Method for Numerical Solution of Ordinary Di_erential Equations. Advances in Natural Applied Science, 2(2008): 165-169. [3] G. Adomian, Solving Frontier Problem of Physics: The Decomposition Method, Boston: Kluwer Academic Publishers, (1994). [4] G. Adomian and R. Rach, Analytic solution of Nonlinear Boundary-value Problems in Several Dimensions by Decomposition, Journal of Mathematical Analysis and Application. 174 (1993). [5] M. J. Ablowitz and J. F. Ladik, Nonlinear Di_erential-Di_erence Equation and Fourier Analysis. Journal of Mathematics and Physics, 17(1976), 1011-1013. [6] Q. H. Yahya and M. Z. Liu,Modi_ed Adomian Decomposition Method for Singular Initial Value Problems in Second order Ordinary Di_erential Equations, Survey in Mathematics and its Application. 3(2000), 183-193. [7] T. Mavoungou and Y. Cherrault, Convergence of Adomian's Decomposition Method and Application to Nonlinear Di_erential Equations Kybernetes, 21(6) (1992)