Artificial Bee Colony Algorithm for Solving Initial Value Problems

L. Djerou, N. Khelil, S. Aichouche

Abstract


A novel numerical differential equation method is presented to solve approximately an initial-value problem (IVP). The IVP is formulated as an optimization problem and the artificial bee colony algorithm (ABC) is used in order to find numerical solutions for this problem. Finally, we use an initial value problem example as illustration to testify the efficiency of the proposed method. The computational results showed that the proposed new method is quite promising, and the potential of the algorithm could be applied successfully in near future to other numerical method as well.

Keywords


Initial-value problem; Optimization problem; Artificial bee colony algorithm

Full Text:

PDF

References


U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations, SIAM, Philadelphia (1998).

W. Bin and C.H. Qian, Differential artificial bee colony algorithm for global numerical optimization, Journal of Computers 6 (5) (2011), 841 – 848.

P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev. 27 (4) (1985), 505 – 535.

C.W. Gear, Numerical Initial-Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971).

P. Henrici, Elements of Numerical Analysis, McGraw-Hill, New York (1964).

D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, New Jersey (1989).

D. Karaboga, An Idea Based On Honey Bee Swarm For Numerical Optimization, Technical Report-TR06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005).

D. Karaboga and B. Basturk, On the performance of artificial bee colony (ABC) algorithm, Applied Soft Computing 8 (1) (2008), 687 – 697.

D. Karaboga and B. Akay, A modified Artificial Bee Colony (ABC) algorithm for constrained optimization problems, Applied Soft Computing 11 (2011), 3021 – 3031.

D. Karaboga, B. Gorkemli, C. Ozturk and N. Karaboga, A comprehensive survey: artificial bee colony (ABC) algorithm and applications, Artificial Intelligence Review 42 (1) (2012), 21 – 57.

J.A. Khan, R.M.A. Zahoor and I.M. Qureshi, Swarm Intelligence for the problems of non-linear ordinary differential equations and its application to well known Wessinger’s equation, European Journal of Scientific Research 34 (4) (2009), 514 – 525.

M.S. Kiran and M. Gunduz, A novel artificial bee colony-based algorithm for solving the numerical optimization problems, International Journal of Innovative Computing, Information and Control ICIC 8 (9) (2012), 6107 – 6121.

I.E. Lagaris, A. Likas and D.I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 9 (5) (1998), 987 – 1000.

J.D. Lambert, Computational Methods in Ordinary Differential Equations, J. Wiley, New York (1973).

Q.K. Pan, M. Fatih Tasgetiren, P.N. Suganthan and T.J. Chua, A discrete artificial bee colony algorithm for the lot-streaming flow shop scheduling problem, Information Sciences 181 (2011), 2455 – 2468.

P.W. Tsai, J.S. Pan, B.Y. Liao and S.C. Chu, Enhanced artificial bee colony optimization, International Journal of Innovative Computing, Information and Control 5 (12) (2009), 5092 – 5092.

A. Yekrangi, M. Ghalambaz and A. Noghrehabadi et al., Approximate solution for a simple pendulum beyond the small angles regimes using hybrid artificial neural network and particle swarm optimization algorithm, Procedia Engineering 10 (2011), 3742 – 3748.

W. Zou, Y. Zhu, H. Chen and B. Zhang, Solving multiobjective optimization problems using artificial bee colony algorithm, Discrete Dynamics in Nature and Society 2011 (2011), Article ID 569784, 37

pages.


Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905