A neuro-swarming intelligent heuristic for second-order nonlinear Lane–Emden multi-pantograph delay differential system

Li W, Chen B, Meng C, Fang W, Xiao Y, Li X, Hu Z, Xu Y, Tong L, Wang H, Liu W (2014) Ultrafast all-optical graphene modulator. Nano Lett 14(2):955–959 Kuang Y (ed) (1993) Delay differential equations: with applications in population dynamics, vol 191. Academic Press, Cambridge Niculescu SI (2001) Delay effects on stability: a robust control approach, vol 269. Springer Science & Business Media, Berlin Li DS, Liu MZ (2000) Exact solution properties of a multi-pantograph delay differential equation. J Harbin Inst Technol 32(3):1–3 Beretta E, Kuang Y (2002) Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J Math Anal 33(5):1144–1165 Frazier MW (1999) Background: complex numbers and linear algebra. An introduction to wavelets through linear algebra, pp 7–100 Rangkuti YM, Noorani MSM (2012) The exact solution of delay differential equations using coupling variational iteration with Taylor series and small term. Bull Math 4(01):1–15 Chapra SC (2012) Applied numerical methods. McGraw-Hill, Columbus Bogachev L, Derfel G, Molchanov S, Ochendon J (2008) On bounded solutions of the balanced generalized pantograph equation. In: Chow P-L, Yin G, Mordukhovich B (eds) Topics in stochastic analysis and nonparametric estimation, vol 145. The IMA volumes in mathematics and its applications. Springer, New York, pp 29–49 Soleymani KV, Sedighi HK (2011) On the numerical solution of generalized pantograph equation. World Appl Sci J 13(12):2531–2535 Liu MZ, Li D (2004) Properties of analytic solution and numerical solution of multi-pantograph equation. Appl Math Comput 155(3):853–871 Koroma MA, Zhan C, Kamara AF, Sesay AB (2013) Laplace decomposition approximation solution for a system of multi-pantograph equations. Int J Math Comput Sci Eng 7(7):39–44 Sezer M, Şahin N (2008) Approximate solution of multi-pantograph equation with variable coefficients. J Comput Appl Math 214(2):406–416 Zhu Q (2019) Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Trans Autom Control 64(9):3764–3771 Zhu Q (2018) Stability analysis of stochastic delay differential equations with Lévy noise. Syst Control Lett 118:62–68 Wang H, Zhu Q (2020) Global stabilization of a class of stochastic nonlinear time-delay systems with SISS inverse dynamics. IEEE Trans Autom Control 65(10):4448–4455 Zhu Q, Huang T (2020) Stability analysis for a class of stochastic delay nonlinear systems driven by G-Brownian motion. Syst Control Lett 140:104699 Flockerzi D, Sundmacher K (2011) On coupled Lane–Emden equations arising in dusty fluid models. In: Journal of physics: conference series, vol 268, no 1. IOP Publishing, p 012006 Mandelzweig VB, Tabakin F (2001) Quasi linearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141(2):268–281 Luo T, Xin Z, Zeng H (2016) Nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities. Commun Math Phys 347(3):657–702 Rach R, Duan JS, Wazwaz AM (2014) Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J Math Chem 52(1):255–267 Abbas F, Kitanov P, Chimene S, Rehmani A (2020) Analytical approach to study the generalized Lane–Emden model arises in the study of stellar configuration. Appl Math 14(3):1–10 Khan JA, Raja MAZ, Rashidi MM, Syam MI, Wazwaz AM (2015) Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory. Connect Sci 27(4):377–396 Bhrawy AH, Alofi AS, Van Gorder RA (2014) An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry. In: Abstract and applied analysis, vol 2014. Hindawi Publishing Corporation Ramos JI (2003) Linearization methods in classical and quantum mechanics. Comput Phys Commun 153(2):199–208 Dehghan M, Shakeri F (2008) Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method. Prog Electromagn Res 78:361–376 Radulescu V, Repovs D (2012) Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal Theory Methods Appl 75(3):1524–1530 Taghavi A, Pearce S (2013) A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method. Math Methods Appl Sci 36(10):1240–1247 Wazwaz AM (2001) A new algorithm for solving differential equations of Lane–Emden type. Appl Math Comput 118(2):287–310 Shawagfeh NT (1993) Non-perturbative approximate solution for Lane–Emden equation. J Math Phys 34(9):4364–4369 Liao S (2003) A new analytic algorithm of Lane–Emden type equations. Appl Math Comput 142(1):1–16 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 Umar M et al (2020) Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Eur Phys J Plus 135(6):403 Hassan A et al (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 Raja MAZ et al (2018) 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 Sabir Z et al (2020) Design of stochastic numerical solver for the solution of singular three-point second-order boundary value problems. Neural Comput Appl 33(7):2427-2443 Sabir Z et al (2018) Neuro-heuristics for nonlinear singular Thomas-Fermi systems. Appl Soft Comput 65:152–169 Raja MAZ et al (2019) Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput Appl 31(3):793–812 Umar M et al (2020) A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment. Eur Phys J Plus 135(7):1–23 Sabir Z, Raja MAZ, Guirao JL, Shoaib M (2020) A neuro-swarming intelligence-based computing for second order singular periodic non-linear boundary value problems. Front Phys 8:224 Umar M et al (2019) Intelligent computing for numerical treatment of nonlinear prey–predator models. Appl Soft Comput 80:506–524 Sabir Z et al (2020) Neuro-swarm intelligent computing to solve the second-order singular functional differential model. Eur Phys J Plus 135(6):474 Sabir Z et al (2019) Stochastic numerical approach for solving second order nonlinear singular functional differential equation. Appl Math Comput 363:124605 Adel W et al (2020) Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method. Eur Phys J Plus 135(6):427 Shi Y, Eberhart RC (1999) Empirical study of particle swarm optimization. In: Proceedings of the 1999 congress on evolutionary computation-CEC99 (Cat. No. 99TH8406), vol 3. IEEE, pp 1945–1950 Shi Y (2001) Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No. 01TH8546), vol 1. IEEE, pp 81–86 Engelbrecht AP (2007) Computational intelligence: an introduction. Wiley, New York Raja MAZ (2014) Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP. Connect Sci 26(3):195–214 Aydoğan EK, Delice Y, Özcan U, Gencer C, Bali Ö (2019) Balancing stochastic U-lines using particle swarm optimization. J Intell Manuf 30(1):97–111 Raja MAZ, Zameer A, Kiani AK, Shehzad A, Khan MAR (2018) Nature-inspired computational intelligence integration with Nelder-Mead method to solve nonlinear benchmark models. Neural Comput Appl 29(4):1169–1193 Ibrahim RA, Ewees AA, Oliva D, Elaziz MA, Lu S (2019) Improved salp swarm algorithm based on particle swarm optimization for feature selection. J Ambient Intell Humaniz Comput 10(8):3155–3169 Takano H, Asano H, Gupta N (2020) Application example of particle swarm optimization on operation scheduling of microgrids. In: Frontier applications of nature inspired computation. Springer, Singapore, pp 215–239 Raja MAZ, Aslam MS, Chaudhary NI, Khan WU (2018) Bio-inspired heuristics hybrid with interior-point method for active noise control systems without identification of secondary path. Front Inf Technol Electron Eng 19(2):246–259 Sicre MR, Svaiter BF (2018) A $$\mathcal {O}$$(1/k3/2) hybrid proximal extragradient primal–dual interior point method for nonlinear monotone mixed complementarity problems. Comput Appl Math 37(2):1847–1876 Stefanova M, Yakunin S, Petukhova M, Lupuleac S, Kokkolaras M (2018) An interior-point method-based solver for simulation of aircraft parts riveting. Eng Optim 50(5):781–796 Umenberger J, Manchester IR (2018) Specialized interior-point algorithm for stable nonlinear system identification. IEEE Trans Autom Control 64(6):2442–2456 Raja MAZ, Ahmed U, Zameer A, Kiani AK, Chaudhary NI (2019) Bio-inspired heuristics hybrid with sequential quadratic programming and interior-point methods for reliable treatment of economic load dispatch problem. Neural Comput Appl 31(1):447–475 Hu W, Zhu Q, Karimi HR (2019) Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems. IEEE Trans Autom Control 64(12):5207–5213 Sabir Z et al (2020) Integrated intelligent computing paradigm for nonlinear multi-singular third-order Emden-Fowler equation. Neural Comput Appl. https://doi.org/10.1007/s00521-020-05187-w Sabir Z et al (2020) Heuristic computing technique for numerical solutions of nonlinear fourth order Emden-Fowler equation. Math Comput Simul 178:534–548 Shoaib M et al (2021) A stochastic numerical analysis based on hybrid NAR-RBFs networks nonlinear SITR model for novel COVID-19 dynamics. Comput Methods Programs Biomed 202:105973 Cheema TN et al (2020) Intelligent computing with Levenberg–Marquardt artificial neural networks for nonlinear system of COVID-19 epidemic model for future generation disease control. Eur Phys J Plus 135(11):1–35 Sabir Z et al (2020) Design of a novel second-order prediction differential model solved by using Adams and explicit Runge–Kutta numerical methods. Math Probl Eng 2020 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 Ilyas H et al (2021) Intelligent computing for the dynamics of fluidic system of electrically conducting Ag/Cu nanoparticles with mixed convection for hydrogen possessions. Int J Hydrogen Energy 46(7):4947–4980 Awan SE et al (2021) Numerical computing paradigm for investigation of micropolar nanofluid flow between parallel plates system with impact of electrical MHD and Hall current. Arab J Sci Eng 46(1):645–662 Ilyas H et al (2021) A novel design of Gaussian WaveNets for rotational hybrid nanofluidic flow over a stretching sheet involving thermal radiation. Int Commun Heat Mass Transf 123:105196 Sabir Z et al (2019) A computational analysis of two-phase casson nanofluid passing a stretching sheet using chemical reactions and gyrotactic microorganisms. Math Probl Eng 2019 Premkumar M, Sowmya R, Jangir P, Nisar KS, Aldhaifallah M (2021) A new metaheuristic optimization algorithms for brushless direct current wheel motor design problem. https://doi.org/10.32604/cmc.2021.015565 Jadoon I et al (2021) Design of evolutionary optimized finite difference based numerical computing for dust density model of nonlinear Van-der Pol Mathieu’s oscillatory systems. Math Comput Simul 181:444–470 Jumani TA, Mustafa MW, Hussain Z, Rasid MM, Saeed MS, Memon MM, Khan I, Nisar KS (2020) Jaya optimization algorithm for transient response and stability enhancement of a fractional-order PID based automatic voltage regulator system. Alex Eng J 59(4):2429–2440 Sabir Z et al (2020) Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden-Fowler equation. Eur Phys J Plus 135(6):1–17 Palmer JM et al (2010) Novel mechanism of rapamycin in GVHD: increase in interstitial regulatory T cells. Bone Marrow Transplant 45(2):379–384 Muhammad U et al (2021) Computational intelligent paradigms to solve the nonlinear SIR system for spreading infection and treatment using Levenberg–Marquardt backpropagation. Symmetry 13(4):618. https://doi.org/10.3390/sym13040618 Sabir Z et al (2021) Integrated intelligence of neuro-evolution with sequential quadratic programming for second-order Lane–Emden pantograph models. Math Comput Simul 188:87–101 Mehmood A et al (2019) Nature-inspired heuristic paradigms for parameter estimation of control autoregressive moving average systems. Neural Comput Appl 31(10):5819–5842