Volume 2, Issue 2, December 2018, Page: 31-36
Solving Initial Value Problems by Flower Pollination Algorithm
Fatima Ouaar, Department of Mathematics, Faculty of Exact Sciences, Mohamed Khider University, Biskra, Algeria
Naceur Khelil, Department of Mathematics, Faculty of Exact Sciences, Mohamed Khider University, Biskra, Algeria
Received: Nov. 20, 2018;       Accepted: Dec. 13, 2018;       Published: Jan. 14, 2019
DOI: 10.11648/j.ajece.20180202.14      View  108      Downloads  39
Abstract
Differential equations are very important in modeling many phenomena mathematically. The aim of this paper is to consider Initial Value Problems (IVPs) in ordinary differential equations (ODEs) as an optimization problem, solved by using a meta-heuristic algorithm which is considered as an alternative way to find numerical approximation of (IVPs) since they can almost be solved simply by classical mathematical tools which are not very precise. By selecting a methodical way based on the use of recent and efficient algorithm, that is, Flower Pollination Algorithm (FPA), inspired by the pollination process of flowers plants to solve approximately an (IVP) when a specified example is selected that is the exponential problem which have an imperative role to describe many real problems. The effectiveness of the proposed method is tested via a simulation study between the exact results, the FPA results and Euler method which is considerate as a classical tool to solve numerically an (IVP). The final results and after a comparison between the performance of FPA and Euler method in terms of solution quality shows that FPA yields satisfactorily precise approximation of the solution. That ensures the ability of FPA to solve such important problems and highly complexes problems efficiently with minimal error.
Keywords
Initial Value Problem (IVP), Optimization Problem, Exponential Model, Flower Pollination Algorithm (FPA)
To cite this article
Fatima Ouaar, Naceur Khelil, Solving Initial Value Problems by Flower Pollination Algorithm, American Journal of Electrical and Computer Engineering. Vol. 2, No. 2, 2018, pp. 31-36. doi: 10.11648/j.ajece.20180202.14
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
D. H. Ackley, (1987), “A Connectionist Machine for Genetic Hillclimbing”, Kluwer Academic Publishers.
[2]
A. Y. Abdelaziz., E. S. Ali., S. M. Abd Elazim., (2016), “Flower pollination algorithm and loss sensitivity factors for optimal sizing and placement of capacitors in radial distribution systems”, Int. J. Electrical Power and Energy Systems, 78 (1), 207-214.
[3]
D. F. Alam., D. A. Yousri., M. B. Eteiba., (2015), “Flower pollination algorithm based solar PV parameter estimation”, Energy Conversion and Management, 101, 410-420.
[4]
L. C. Cagnina., S. C. Esquivel., and C. A. Coello, (2008), “Solving engineering optimization problems with the simple constrained particle swarm optimizer”, Informatica, 32, 319- 326.
[5]
H. Chiroma., N. L. M. Shuib., S. A. Muaz., A. I. Abubakar., L. B. Ila., J. Z. Maitama., (2015), “A review of the application of bio-inspired flower pollination algorithm”, Procedia Computer Science, 62, 435-441.
[6]
M. Dorigo., V. Maniezzo., A. Colorni., (1996), “The ant system: optimization by a colony of cooperating agents”, IEEE Trans. Syst. Man Cybern, B 26, 29-41.
[7]
L. Djerou., N. Khelil., S. Aichouche., (2017), “Artificial Bee Colony Algorithm for Solving Initial Value Problems”. Communications in Mathematics and Applications Published by RGN Publications, 8 (2), 119-125.
[8]
B. J. Glover., (2007), “Understanding Flowers and Flowering: An Integrated Approach”. Oxford University Press.
[9]
D. E. Goldberg., (1989), “Genetic Algorithms in Search. Optimization and Machine Learning. Addison Wesley”, Boston.
[10]
J. H. Holland., (1975), “Adaptation in Natural and Artificial Systems”. University of Michigan Press, Ann Arbor.
[11]
X. He., X. S. Yang., M. Karamanoglu., Y. Zhao., (2017), “ Flower pollination algorithm: a discrete-time Markov chain approach‏”, Procedia Computer Science, Elsevier‏, 108, 1354-1363.
[12]
P. Henrici., (1964), “Elements of Numerical Analysis”, Mc Graw-Hill, New York.
[13]
J. Kennedy., R. C. Eberhart., (1995), “Particle swarm optimization”, In: Proceedings of IEEE International Conference on Neural Networks No. IV. 27 Nov-1 Dec, pp. 1942--1948, Perth Australia.
[14]
B. Mahdad., and K. Srairi., (2016), “Security constrained optimal power flow solution using new adaptive partitioning flower pollination algorithm”, Applied Soft Computing, 46, 501-522.
[15]
S. Nakrani., C. Tovey., (2004), “On honey bees and dynamic allocation in an internet server colony”. Adapt. Behav. 12 (3--4). 223-240.
[16]
I. Pavlyukevich., (2007), “Levy flights, non-local search and simulated annealing”, J. Computational Physics, 226, 1830-1844..
[17]
D. Rodrigues., G. F. A. Silva., J. P. Papa., A. N. Marana., X. S. Yang., (2016), “EEG-based person identification through binary flower pollination algorithm”, Expert Systems with Applications, 62 (1), 81-90.
[18]
S. A. Sayed., E. Nabil., A. Badr., (2016), “A binary clonal flower pollination algorithm for feature selection”, Pattern Recognition Letters, 77 (1), 21-27.
[19]
R. Salgotra., and U. Singh., (2017), “Application of mutation operators to flower pollination algorithm”, Expert Systems with Applications, 79 (1), 112-129.
[20]
S. Velamuri., S. Sreejith., P. Ponnambalam., (2016), “Static economic dispatch incorporating wind farm using flower pollination algorithm”, Perspectives in Science, 8, 260-262.
[21]
N. M. Waser., (1986), “Flower constancy: definition, cause and measurement”, The American Naturalist, 127 (5), 596-603.
[22]
P. Willmer., (2011), “Pollination and Floral Ecology”, Princeton University Press.
[23]
X. S. Yang., (2104), “Book Nature Inspired Optimization Algorithm”, Elsevier.
[24]
X. S. Yang., (2013), “Flower Pollination Algorithm for Global Optimization”, arXiv: 1312. 5673v1 [math. OC] 19 Dec.
[25]
X. S. Yang., A. H. Gandomi., (2012), “Bat algorithm: a novel approach for global engineering optimization” Eng. Comput. 29 (5), 464-483.
[26]
X. S. Yang., (2008), “Nature-Inspired Metaheuristic Algorithms”, Luniver Press.
[27]
Y. Q. Zhou., R. Wang., Q. F. Luo., (2016), “Elite opposition-based flower pollination algorithm”, Neurocomputing, 188, 294-310.
Browse journals by subject