Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies

Najeeb Alam Khan; Samreen Ahmed; Tooba Hameed; Muhammad Asif Zahoor Raja; Najeeb Alam Khan; Samreen Ahmed; Tooba Hameed; Muhammad Asif Zahoor Raja

doi:10.3934/math.2019.4.1114

AIMS Mathematics

2019, Volume 4, Issue 4: 1114-1132. doi: 10.3934/math.2019.4.1114

Previous Article Next Article

Research article Special Issues

Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies

1.
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
2.
Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, Pakistan

Received: 03 April 2019 Accepted: 03 July 2019 Published: 13 August 2019
MSC : 34A08, 65D20

In this work, an endeavor is made for assessing the solutions of nonlinear fractional differential equations, by taking into account the derivative and integral operator in the Caputo sense. The proposed technique is developed by the merger of classical and modern ideas of mathematical analysis. The approximate solution of the fractional differential equation (FDE) accomplished by the careful and profitable implementation of the homotopy perturbation method is fast track by using a powerful and proficient metaheuritic technique, which mimics the flashing pattern and behavior of the fireflies. Accuracy and accelerated convergence are the main attributes of the proposed technique, which are attained by using the firefly algorithm (FA) for the optimization of the fitness function constructed in the mean square sense. Numerical experiments are performed to illustrate the worth mentioning performance of the design methodology in term of accuracy, convergence and competency. The solutions found by the deliberated scheme are far superior to the former results emphasized in the literature.
- Riccati equation,
- fractional derivative,
- firefly algorithm,
- homotopy perturbation method,
- Adam bashforth method
Citation: Najeeb Alam Khan, Samreen Ahmed, Tooba Hameed, Muhammad Asif Zahoor Raja. Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies[J]. AIMS Mathematics, 2019, 4(4): 1114-1132. doi: 10.3934/math.2019.4.1114

Related Papers:

Abstract

In this work, an endeavor is made for assessing the solutions of nonlinear fractional differential equations, by taking into account the derivative and integral operator in the Caputo sense. The proposed technique is developed by the merger of classical and modern ideas of mathematical analysis. The approximate solution of the fractional differential equation (FDE) accomplished by the careful and profitable implementation of the homotopy perturbation method is fast track by using a powerful and proficient metaheuritic technique, which mimics the flashing pattern and behavior of the fireflies. Accuracy and accelerated convergence are the main attributes of the proposed technique, which are attained by using the firefly algorithm (FA) for the optimization of the fitness function constructed in the mean square sense. Numerical experiments are performed to illustrate the worth mentioning performance of the design methodology in term of accuracy, convergence and competency. The solutions found by the deliberated scheme are far superior to the former results emphasized in the literature.

References

[1]	X. J. Yang, D. Baleanu and J. H. He, Transport equations in fractal porous media within fractional complex transform method, Proc. Rom. Acad. Ser. A, 14 (2013), 287-292.
[2]	J. H. He, A tutorial review on fractal spacetime and fractional calculus, Int. J. Theor. Phys., 53 (2014), 3698-3718. doi: 10.1007/s10773-014-2123-8
[3]	J. H. He, Fractal calculus and its geometrical explanation. Results Phys., 10 (2018), 272-276.
[4]	N. A. Pirim and F. Ayaz, Hermite collocation method for fractional order differential equations, Int. J. Optim. Control Theor. Appl. (IJOCTA), 8 (2018), 228-236.
[5]	S. V. Lototsky and B. L. Rozovsky, Classical and generalized solutions of fractional stochastic differential equations, arXiv preprint arXiv:1810.12951, 2018.
[6]	A. Ara, O. A. Razzaq and N. A. Khan, A single layer functional link artificial neural network based on chebyshev polynomials for neural evaluations of nonlinear nth order fuzzy differential equations, Ann. West Univ. Timisoara Math. Comput. Sci., 56 (2018), 3-22. doi: 10.2478/awutm-2018-0001
[7]	H. Aminikhah, Approximate analytical solution for quadratic Riccati differential equation, Iran. J. Numer. Anal. Optim., 3 (2013), 21-31.
[8]	R. Dascaliuc, E. A. Thomann and E. C. Waymire, Stochastic explosion and non-uniqueness for α-Riccati equation, J. Math. Anal. Appl., 476 (2019), 53-85. doi: 10.1016/j.jmaa.2018.11.064
[9]	R. Guglielmi and K. Kunisch, Sensitivity analysis of the value function for infinite dimensional optimal control problems and its relation to Riccati equations, Optimization, 67 (2018), 1461-1485. doi: 10.1080/02331934.2018.1476514
[10]	E. Pereira, E. Suazo and J. Trespalacios, Riccati-Ermakov systems and explicit solutions for variable coefficient reaction-diffusion equations, Appl. Math. Comput., 329 (2018), 278-296.
[11]	D Schuch, Analytical solutions for the quantum parametric oscillator from corresponding classical dynamics via a complex Riccati equation, J. Phys. Conf. Ser., 1071 (2018), 012020.
[12]	S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Appl. Math. Comput., 172 (2006), 485-490.
[13]	B. S. Kashkari and M. I. Syam, Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order, Appl. Math. Comput., 290 (2016), 281-291.
[14]	M. G. Sakar, Iterative reproducing kernel Hilbert spaces method for Riccati differential equations, J. Comput. Appl. Math., 309 (2017), 163-174. doi: 10.1016/j.cam.2016.06.029
[15]	M. Hamarsheh, A. I. Ismail and Z. Odibat, An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Appl. Math. Sci., 10 (2016), 1131-1150.
[16]	V. Mishra and D. Rani. Newton-Raphson based modified Laplace Adomian decomposition method for solving quadratic Riccati differential equations, MATEC Web of Conferences, 57 (2016), 05001.
[17]	H. Jafari, H. Tajadodi and D. Baleanu, A numerical approach for fractional order Riccati differential equation using B-spline operational matrix, Fract. Calc. Appl. Anal., 18 (2015), 387-399.
[18]	S. Shiralashetti and A. Deshi, Haar wavelet collocation method for solving riccati and fractional Riccati differential equations, Bull. Math. Sci. Appl., 17 (2016), 46-56.
[19]	D. N. Yu, J. H. He and A. G. Garcıa, Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators, J. Low Freq. Noise V. A., 2018. Available from: https://doi.org/10.1177/1461348418811028.
[20]	J. H. He, Homotopy perturbation method with an auxiliary term, Abstr. Appl. Anal., 2012 (2012), Article ID 857612.
[21]	J. H. He, Asymptotic methods for solitary solutions and compactons, Abstr. appl. anal., 2012 (2012), Article ID 916793.
[22]	M. Francisco, S. Revollar, P. Vega, et al. A comparative study of deterministic and stochastic optimization methods for integrated design of processes, IFAC Proc. Vol., 38 (2005), 335-340.
[23]	D. B. Shmoys and C. Swamy, Stochastic optimization is (almost) as easy as deterministic optimization. 45th Annual IEEE Symposium on Foundations of Computer Science, 2004.
[24]	R. Dhar and N. Doshi, Simulation for variable transmission using mono level genetic algorithm, In: Proceedings of International Conference on Intelligent Manufacturing and Automation, Springer, 2019.
[25]	W. Zhang, A. Maleki, M. A. Rosen, et al. Optimization with a simulated annealing algorithm of a hybrid system for renewable energy including battery and hydrogen storage, Energy, 163 (2018), 191-207. doi: 10.1016/j.energy.2018.08.112
[26]	A. Ara, N. A. Khan, F. Maz, et al. Numerical simulation for Jeffery-Hamel flow and heat transfer of micropolar fluid based on differential evolution algorithm, AIP Adv., 8 (2018), 015201.
[27]	Y. Mokhtari and D. Rekioua, High performance of maximum power point tracking using ant colony algorithm in wind turbine, Renewable energy, 126 (2018), 1055-1063. doi: 10.1016/j.renene.2018.03.049
[28]	X. S. Yang, Nature-Inspired Metaheuristic Algorithms. Luniver press, 2010.
[29]	E. Fouladi and H. Mojallali, Design of optimised backstepping controller for the synchronisation of chaotic Colpitts oscillator using shark smell algorithm, Pramana, 90 (2018), 1-6. doi: 10.1007/s12043-017-1492-y
[30]	N. A. Khan, T. Hameed, O. A. Razzaq, et al. Intelligent computing for Duffing-Harmonic oscillator equation via the bio-evolutionary optimization algorithm, J. Low Freq. Noise V. A., 2018.
[31]	D. K. Bui, T. Nguyen, J. S. Chou, et al. A modified firefly algorithm-artificial neural network expert system for predicting compressive and tensile strength of high-performance concrete, Constr. Build. Mater., 180 (2018), 320-333. doi: 10.1016/j.conbuildmat.2018.05.201
[32]	D. T. Le, D. K. Bui, T. D. Ngo, et al. A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures, Comput. Struct., 212 (2019), 20-42. doi: 10.1016/j.compstruc.2018.10.017
[33]	X. S. Yang, Firefly algorithms for multimodal optimization, In: InternationalSymposium on Stochastic Algorithms, Springer, 2009, 169-178.
[34]	S. Arora and S. Singh, A conceptual comparison of firefly algorithm, bat algorithm and cuckoo search, In: 2013 International Conference on Control, Computing, Communication and Materials (ICCCCM), 2013, IEEE.
[35]	N. A. Khan, A. Ara and M. Jamil, An efficient approach for solving the Riccati equation with fractional orders, Comput. Math. Appl., 61 (2011), 2683-2689. doi: 10.1016/j.camwa.2011.03.017
[36]	M. Ali, I. Jaradat and M. Alquran, New computational method for solving fractional Riccati equation, J. Math. Comput. Sci., 17 (2017), 106-114. doi: 10.22436/jmcs.017.01.10
[37]	K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22. doi: 10.1023/A:1016592219341

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)