Solving Initial Value Problems by Flower Pollination Algorithm

Fatima Ouaar; Naceur Khelil

doi:doi:10.11648/j.ajece.20180202.14

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Solving Initial Value Problems by Flower Pollination Algorithm

Fatima Ouaar, Naceur Khelil

Published in American Journal of Electrical and Computer Engineering (Volume 2, Issue 2)

Received: 20 November 2018 Accepted: 13 December 2018 Published: 14 January 2019

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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.

Published in	American Journal of Electrical and Computer Engineering (Volume 2, Issue 2)
DOI	10.11648/j.ajece.20180202.14
Page(s)	31-36
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Initial Value Problem (IVP), Optimization Problem, Exponential Model, Flower Pollination Algorithm (FPA)

References

[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.

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APA Style

Fatima Ouaar, Naceur Khelil. (2019). Solving Initial Value Problems by Flower Pollination Algorithm. American Journal of Electrical and Computer Engineering, 2(2), 31-36. https://doi.org/10.11648/j.ajece.20180202.14

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ACS Style

Fatima Ouaar; Naceur Khelil. Solving Initial Value Problems by Flower Pollination Algorithm. Am. J. Electr. Comput. Eng. 2019, 2(2), 31-36. doi: 10.11648/j.ajece.20180202.14

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AMA Style

Fatima Ouaar, Naceur Khelil. Solving Initial Value Problems by Flower Pollination Algorithm. Am J Electr Comput Eng. 2019;2(2):31-36. doi: 10.11648/j.ajece.20180202.14

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@article{10.11648/j.ajece.20180202.14,
  author = {Fatima Ouaar and Naceur Khelil},
  title = {Solving Initial Value Problems by Flower Pollination Algorithm},
  journal = {American Journal of Electrical and Computer Engineering},
  volume = {2},
  number = {2},
  pages = {31-36},
  doi = {10.11648/j.ajece.20180202.14},
  url = {https://doi.org/10.11648/j.ajece.20180202.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajece.20180202.14},
  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.},
 year = {2019}
}

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TY  - JOUR
T1  - Solving Initial Value Problems by Flower Pollination Algorithm
AU  - Fatima Ouaar
AU  - Naceur Khelil
Y1  - 2019/01/14
PY  - 2019
N1  - https://doi.org/10.11648/j.ajece.20180202.14
DO  - 10.11648/j.ajece.20180202.14
T2  - American Journal of Electrical and Computer Engineering
JF  - American Journal of Electrical and Computer Engineering
JO  - American Journal of Electrical and Computer Engineering
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2640-0502
UR  - https://doi.org/10.11648/j.ajece.20180202.14
AB  - 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.
VL  - 2
IS  - 2
ER  -

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Author Information

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

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APA Style

Fatima Ouaar, Naceur Khelil. (2019). Solving Initial Value Problems by Flower Pollination Algorithm. American Journal of Electrical and Computer Engineering, 2(2), 31-36. https://doi.org/10.11648/j.ajece.20180202.14

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ACS Style

Fatima Ouaar; Naceur Khelil. Solving Initial Value Problems by Flower Pollination Algorithm. Am. J. Electr. Comput. Eng. 2019, 2(2), 31-36. doi: 10.11648/j.ajece.20180202.14

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AMA Style

Fatima Ouaar, Naceur Khelil. Solving Initial Value Problems by Flower Pollination Algorithm. Am J Electr Comput Eng. 2019;2(2):31-36. doi: 10.11648/j.ajece.20180202.14

Copy | Download

@article{10.11648/j.ajece.20180202.14,
  author = {Fatima Ouaar and Naceur Khelil},
  title = {Solving Initial Value Problems by Flower Pollination Algorithm},
  journal = {American Journal of Electrical and Computer Engineering},
  volume = {2},
  number = {2},
  pages = {31-36},
  doi = {10.11648/j.ajece.20180202.14},
  url = {https://doi.org/10.11648/j.ajece.20180202.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajece.20180202.14},
  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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Solving Initial Value Problems by Flower Pollination Algorithm
AU  - Fatima Ouaar
AU  - Naceur Khelil
Y1  - 2019/01/14
PY  - 2019
N1  - https://doi.org/10.11648/j.ajece.20180202.14
DO  - 10.11648/j.ajece.20180202.14
T2  - American Journal of Electrical and Computer Engineering
JF  - American Journal of Electrical and Computer Engineering
JO  - American Journal of Electrical and Computer Engineering
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2640-0502
UR  - https://doi.org/10.11648/j.ajece.20180202.14
AB  - 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.
VL  - 2
IS  - 2
ER  -

Copy | Download