Abstract
In the present study, a novel fractional Meyer neuro-evolution-based intelligent computing solver (FMNEICS) is presented for numerical treatment of doubly singular multi-fractional Lane–Emden system (DSMF-LES) using combined heuristics of Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The design of novel FMNEICS for DSMF-LES is presented after derivation from standard Lane–Emden equation, and the singular points and shape factors along with fractional-order terms are analyzed. The MWNN modeling strength is used to represent the system model DSMF-LES in the mean-squared error-based merit function and optimization of the networks is carried out with integrated optimization ability of GASQP. The verification, validation, and perfection of the FMNEICS for three different cases of DSMF-LES are established through comparative studies from reference solutions on convergence, robustness, accuracy, and stability measures. Moreover, the observations through the statistical analysis further authenticate the worth of proposed fractional MWNN-GASQP-based stochastic solver.
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Ahmad I et al (2016) Bio-inspired computational heuristics to study Lane–Emden systems arising in astrophysics model. SpringerPlus 5(1):1866
Ahmad I et al (2018a) Neuro-evolutionary computing paradigm for Painlevé equation-II in nonlinear optics. Eur Phys J Plus 133(5):184
Ahmad I et al (2018b) Intelligent computing to solve fifth-order boundary value problem arising in induction motor models. Neural Comput Appl 29(7):449–466
Ahmad I et al (2019) Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels. Neural Comput Appl 31(12):9041–9059
Ahmad SUI et al (2020) A new heuristic computational solver for nonlinear singular Thomas-Fermi system using evolutionary optimized cubic splines. Eur Phys J Plus 135:1–29
Akbar S et al (2017) Design of bio-inspired heuristic techniques hybridized with sequential quadratic programming for joint parameters estimation of electromagnetic plane waves. Wirel Pers Commun 96(1):1475–1494
Aman S, Khan I, Ismail Z, Salleh MZ (2018) Applications of fractional derivatives to nanofluids: exact and numerical solutions. Math Model Natural Phenomena 13(1):2
Ara A et al (2018) Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling. Adv Differ Equ 2018(1):8
Asadpour S, Hosseinzadeh H, Yazdani A (2019) Numerical solution of the Lane–Emden equations with moving least squares method. Appl Appl Math 14:2
Baleanu D, Machado JAT, Luo AC (eds) (2011) Fractional dynamics and control. Springer Science & Business Media, Berlin
Bǎleanu D, Lopes AM (eds) (2019) Applications in engineering, life and social sciences. Walter de Gruyter GmbH & Co KG. https://doi.org/10.1515/9783110571905
Bonilla B, Rivero M, Trujillo JJ (2007) On systems of linear fractional differential equations with constant coefficients. Appl Math Comput 187(1):68–78
Bukhari AH et al (2020) Neuro-fuzzy modeling and prediction of summer precipitation with application to different meteorological stations. Alexandria Engineering Journal 59(1):101–116
Căruntu B, Bota C, Lăpădat M, Paşca MS (2019) Polynomial least squares method for fractional Lane–Emden equations. Symmetry 11(4):479
Dabiri A, Butcher EA (2019) Optimal observer-based feedback control for linear fractional-order systems with periodic coefficients. J Vib Control 25(7):1379–1392
Dabiri A, Butcher EA, Poursina M, Nazari M (2017a) Optimal periodic-gain fractional delayed state feedback control for linear fractional periodic time-delayed systems. IEEE Trans Autom Control 63(4):989–1002
Dabiri A, Butcher EA, Nazari M (2017b) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. J Sound Vib 388:230–244
Dabiri A, Moghaddam BP, Machado JT (2018) Optimal variable-order fractional PID controllers for dynamical systems. J Comput Appl Math 339:40–48
Daou RAZ, El Samarani F, Yaacoub C, Moreau X (2020) Fractional derivatives for edge detection: application to road obstacles. In: Smart cities performability, cognition, & security (pp 115–137). Springer, Cham.
Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265(2):229–248
Diethelm K and Freed AD (1999) On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Scientific computing in chemical engineering II (pp 217–224). Springer, Berlin, Heidelberg
Engheia N (1997) On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag Mag 39(4):35–46
Evans RM, Katugampola UN, Edwards DA (2017) Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem. Comput Math Appl 73(6):1346–1362
Farooq MU (2019) Noether-like operators and first integrals for generalized systems of Lane–Emden equations. Symmetry 11(2):162
Hadian-Rasanan AH, Rahmati D, Gorgin S, Parand K (2020) A single layer fractional orthogonal neural network for solving various types of Lane–Emden equation. New Astron 75:101307
Hassan A, Kamran M, Illahi A, Zahoor RMA (2019) Design of cascade artificial neural networks optimized with the memetic computing paradigm for solving the nonlinear Bratu system. Eur Phys J Plus 134(3):122
He JH, Ji FY (2019) Taylor series solution for Lane–Emden equation. J Math Chem 57(8):1932–1934
Hilfer R (ed) (2000) Applications of fractional calculus in physics (Vol. 35, no. 12). World scientific, Singapore, pp 87–130
Ibrahim RW, Momani S (2007) On the existence and uniqueness of solutions of a class of fractional differential equations. J Math Anal Appl 334(1):1–10
Jamal R et al (2019) Hybrid bio-inspired computational Heuristic paradigm for integrated load dispatch problems involving Stochastic wind. Energies 12(13):2568
Jaradat I, Al-Dolat M, Al-Zoubi K, Alquran M (2018) Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fract 108:107–110
Khalifa AS, Hassan HN (2019) Approximate solution of Lane–Emden Type equations using variation of parameters method with an auxiliary parameter. J Appl Math Phys 7(04):921
Khan WU et al (2018) Backtracking search integrated with sequential quadratic programming for nonlinear active noise control systems. Appl Soft Comput 73:666–683
Lodhi S et al (2019) Fractional neural network models for nonlinear Riccati systems. Neural Comput Appl 31(1):359–378
Majeed K et al (2017) A genetic algorithm optimized Morlet wavelet artificial neural network to study the dynamics of nonlinear Troesch’s system. Appl Soft Comput 56:420–435
Masood Z et al (2020) Design of fractional order epidemic model for future generation tiny hardware implants. Future Gener Comput Syst 106:43–54
Matlob MA, Jamali Y (2019) The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: a primer. Crit Rev Biomedl Eng 47:4
Mehmood A et al (2019) Design of nature-inspired heuristic paradigm for systems in nonlinear electrical circuits. Neural Comput Appl 2019:1–17
Mishra SUCHANA, Mishra LN, Mishra RK, Patnaik SRIKANTA (2019) Some applications of fractional calculus in technological development. J Fract Calcul Appl 10(1):228–235
Moghadam BP, Dabiri A, Machado JT (2019) Applications in engineering, life and social sciences. In: Machado JT, Baleanu D, Lopes AM, Chen Y (eds) Handbook of fractional calculus with applications. Springer, New York
Momani S, Ibrahim RW (2008) On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras. J Math Anal Appl 339(2):1210–1219
Raja MAZ, Manzar MA, Samar R (2015) An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl Math Model 39(10–11):3075–3093
Raja MAZ, Samar R, Manzar MA, Shah SM (2017a) Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley-Torvik equation. Math Comput Simul 132:139–158
Raja MAZ, Shah FH, Alaidarous ES, Syam MI (2017b) Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model. Appl Soft Comput 52:605–629
Raja MAZ, Shah Z, Manzar MA, Ahmad I, Awais M, Baleanu D (2018a) A new stochastic computing paradigm for nonlinear Painlevé II systems in applications of random matrix theory. Eur Phys J Plus 133(7):254
Raja MAZ, Shah FH, Syam MI (2018b) Intelligent computing approach to solve the nonlinear Van der Pol system for heartbeat model. Neural Comput Appl 30(12):3651–3675
Raja MAZ, Umar M, Sabir Z, Khan JA, Baleanu D (2018c) A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur Phys J Plus 133(9):364
Raja MAZ, Shah FH, Tariq M, Ahmad I (2018d) Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch’s problem arising in plasma physics. Neural Comput Appl 29(6):83–109
Raja MAZ, Mehmood J, Sabir Z, Nasab AK, Manzar MA (2019) Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput Appl 31(3):793–812
Raja MAZ et al (2020) Integrated intelligence of fractional neural networks and sequential quadratic programming for Bagley-Torvik systems arising in fluid mechanics. J Comput Nonlinear Dyn 15:5
Sabir Z et al (2018) Neuro-heuristics for nonlinear singular Thomas-Fermi systems. Appl Soft Comput 65:152–169
Sabir Z et al (2020) Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation. Math Comput Simul 172:1–14
Singh R, Garg H, Guleria V (2019a) Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. J Comput Appl Math 346:150–161
Singh R, Shahni J, Garg H, Garg A (2019b) Haar wavelet collocation approach for Lane–Emden equations arising in mathematical physics and astrophysics. Eur Phys J Plus 134(11):548
Sumelka W (2014) Fractional viscoplasticity. Mech Res Commun 56:31–36
Sun Z, Tian Y, Li H, Wang J (2016) A superlinear convergence feasible sequential quadratic programming algorithm for bipedal dynamic walking robot via discrete mechanics and optimal control. Optimal Control Appl Methods 37(6):1139–1161
Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231
Torvik PJ, Bagley RL (1984) On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech 51(2):294–298
Uchaikin VV (2013) Fractional derivatives for physicists and engineers, vol 2. Springer, Berlin
Umar M et al (2019) Intelligent computing for numerical treatment of nonlinear prey–predator models. Appl Soft Comput 80:506–524
Waseem W et al (2020) A study of changes in temperature profile of porous fin model using cuckoo search algorithm. Alexandria Eng J 59(1):11–24
Wazwaz AM (2001) A new algorithm for solving differential equations of Lane–Emden type. Appl Math Comput 118(2–3):287–310
Wazwaz AM (2015) Solving two Emden-Fowler type equations of third order by the variational iteration method. Appl Math Inf Sci 9(5):2429
Yang XJ, Machado JT, Cattani C, Gao F (2017) On a fractal LC-electric circuit modeled by local fractional calculus. Commun Nonlinear Sci Numer Simul 47:200–206
Yin KL, Pu YF, Lu L (2020) Combination of fractional FLANN filters for solving the Van der Pol-Duffing oscillator. Neurocomputing 399:183–192
Yu F (2009) Integrable coupling system of fractional soliton equation hierarchy. Phys Lett A 373(41):3730–3733
Zameer A et al (2019) Bio-inspired heuristics for layer thickness optimization in multilayer piezoelectric transducer for broadband structures. Soft Comput 23(10):3449–3463
Zhang Y, Sun H, Stowell HH, Zayernouri M, Hansen SE (2017a) A review of applications of fractional calculus in Earth system dynamics. Chaos Solitons Fract 102:29–46
Zhang Y, Yang X, Cattani C, Dong Z, Yuan T, Han L (2017b) Theory and applications of fractional fourier transform and its variants. Fund Inf 151:1–4
Zhao D, Pan X, Luo M (2018) A new framework for multivariate general conformable fractional calculus and potential applications. Phys A 510:271–280
Zúñiga-Aguilar CJ, Romero-Ugalde HM, Gómez-Aguilar JF, Escobar-Jiménez RF, Valtierra-Rodríguez M (2017) Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks. Chaos Solitons Fract 103:382–403
Acknowledgements
J.F. Gómez-Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.
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Sabir, Z., Raja, M.A.Z., Shoaib, M. et al. FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system. Comp. Appl. Math. 39, 303 (2020). https://doi.org/10.1007/s40314-020-01350-0
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DOI: https://doi.org/10.1007/s40314-020-01350-0
Keywords
- Multi-fractional Lane–Emden model
- Multi-singular systems
- Artificial neural networks
- Meyer wavelet neural networks
- Sequential quadratic programming
- Genetic algorithms