Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems
-
2178
Downloads
-
2691
Views
Authors
N. R. Anakira
- Department of Mathematics, Faculty of Science and Technology, Irbid National University, 2600 Irbid, Jordan.
A. k. Alomari
- Department of Mathematics, Faculty of Science, Yarmouk University, 211-63 Irbid, Jordan.
A. F. Jameel
- School of Quantitative Sciences, Universiti Utara Malaysia (UUM), Kedah, Sintok, 06010, Malaysia.
I. Hashim
- School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia.
Abstract
In this paper, a new approximate analytical algorithm namely multistage optimal homotopy asymptotic
method (MOHAM) is presented for the first time to obtain approximate analytical solutions for linear,
nonlinear and system of initial value problems (IVPs). This algorithm depends on the standard optimal
homotopy asymptotic method (OHAM), in which it is treated as an algorithm in a sequence of subinterval.
The main advantage of this study is to obtain continuous approximate analytical solutions for a long time
span. Numerical examples are tested to highlight the important features of the new algorithm. Comparison
of the MOHAM results, standard OHAM, available exact solution and the fourth-order Runge Kutta (RK4)
reveale that this algorithm is effective, simple and more impressive than the standard OHAM for solving
IVPs.
Share and Cite
ISRP Style
N. R. Anakira, A. k. Alomari, A. F. Jameel, I. Hashim, Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 4, 1826--1843
AMA Style
Anakira N. R., Alomari A. k., Jameel A. F., Hashim I., Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems. J. Nonlinear Sci. Appl. (2016); 9(4):1826--1843
Chicago/Turabian Style
Anakira, N. R., Alomari, A. k., Jameel, A. F., Hashim, I.. "Multistage Optimal Homotopy Asymptotic Method for Solving Initial-Value Problems." Journal of Nonlinear Sciences and Applications, 9, no. 4 (2016): 1826--1843
Keywords
- Optimal homotopy asymptotic method (OHAM)
- multistage optimal homotopy asymptotic method (MOHAM)
- initial value problems
- series solution
- Mathematica 9.
MSC
References
-
[1]
J. Ali, S. Islam, S. Islam, G. Zaman, The solution of multipoint boundary value problems by the optimal homotopy asymptotic method, Comput. Math. Appl., 59 (2010), 2000-2006.
-
[2]
A. K. Alomari, A novel solution for fractional chaotic Chen system, J. Nonlinear Sci. Appl., 8 (2015), 478-488.
-
[3]
A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W. Shatanawi, Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893-8903.
-
[4]
A. K. Alomari, N. Ratib Anakira, A. S. Bataineh, I. Hashim, Approximate solution of nonlinear system of BVP arising in fluid flow problem, Math. Probl. Eng., 2013 (2013), 7 pages.
-
[5]
A. K. Alomari, N. Ratib Anakira, I. Hashim, Multiple solutions of problems in fluid mechanics by predictor optimal homotopy asymptotic method , Adv. Mech. Eng., 6 (2014), 20 pages.
-
[6]
J. D. Faires, R. Burden, Numerical Methods, Brooks/Cole Publishing Co, Pacific Grove (1993)
-
[7]
M. Ghoreishi, A. Ismail, A. K. Alomari , Application of the homotopy analysis method for solving a model for HIV infection of CD4(+) T-cells, Math. Comput. Model, 54 (2013), 3007-3015.
-
[8]
M. Ghoreishi, A. Ismail, A. K. Alomari, A. S. Bataineh, The comparison between homotopy analysis method and optimal homotopy asymptotic method for nonlinear age-structured population models, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1163-1177.
-
[9]
S. Ghosh, A. Roy, D. Roy, An adaptation of adomian decomposition for numericanalytic integration of strongly nonlinear and chaotic oscillators , Comput. Methods Appl. Mech. Eng., 196 (2007), 1133-1153.
-
[10]
S. Haq, M. Ishaq, Solution of coupled Whitham-Broer-Kaup equations using optimal homotopy asymptotic method, Ocean Eng., 84 (2014), 81-88.
-
[11]
I. Hashim, M. S. H. Chowdhury , Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs , Phys. Lett. A, 372 (2008), 470-481.
-
[12]
M. S. Hashim, N. Khan, S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind , Appl. Math. Comput., 218 (2012), 10982-10989.
-
[13]
I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail, A. M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos Solitons Fractals, 28 (2006), 1149-1158.
-
[14]
N. Herişanu, V. Marinca , Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Comput. Math. Appl., 60 (2010), 1607- 1615.
-
[15]
N. Herişanu, V. Marinca , Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia, Meccanica, 45 (2010), 847-855.
-
[16]
S. Iqbal, A. Javed , Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation , Appl. Math. Comput., 217 (2011), 7753-7761.
-
[17]
M. J. Jang, C. L. Chen, Y. C. Liy, On solving the initial-value problems using the differential transformation method, Appl. Math. Comput., 115 (2000), 145-160.
-
[18]
V. Marinca, R. D. Ene, Dual approximate solutions of the unsteady viscous flow over a shrinking cylinder with optimal homotopy asymptotic method, Adv. Math. Phys., 2014 (2014), 11 pages.
-
[19]
V. Marinca, R. D. Ene, B. Marinca, Analytic approximate solution for Falkner-Skan Equation , Sci. World J., 2014 (2014), 22 pages.
-
[20]
V. Marinca, N. Herişanu , Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer , Int. Commun. Heat Mass, 35 (2008), 710-715.
-
[21]
V. Marinca, N. Herişanu, The optimal homotopy asymptotic method for solving Blasius equation, Appl. Math. Comput., 231 (2014), 134-139.
-
[22]
V. Marinca, N. Herişanu, C. Bota, B. Marinca, An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. Math. Lett., 22 (2009), 245-251.
-
[23]
V. Marinca, N. Herişanu, I. Nemeş , Optimal Homotopy Asymptotic Method with application to thin film flow, Cent. Eur. J. Phys., 6 (2008), 648-653.
-
[24]
N. Ratib Anakira, A. K. Alomari, I. Hashim , Numerical scheme for solving singular two-point boundary value problems, J. Appl. Math., 2013 (2013), 8 pages.
-
[25]
N. Ratib Anakira, A. K. Alomari, I. Hashim, Optimal homotopy asymptotic method for solving delay differential equations, Math. Probl. Eng., 2013 (2013), 11 pages.
-
[26]
N. Shawagfeh, D. Kaya, Comparing numerical method for the solutions of systems of ordinary differenial equations , Appl. Math. Lett., 17 (2004), 323-328.
-
[27]
P. Vadasza, S. Olekb, Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations, Int. J. Heat Mass Transfer, 43 (2000), 1715-1734.