{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T16:01:40Z","timestamp":1768838500359,"version":"3.49.0"},"reference-count":74,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Evolving Systems"],"published-print":{"date-parts":[[2023,6]]},"DOI":"10.1007\/s12530-022-09469-7","type":"journal-article","created":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T10:04:07Z","timestamp":1664359447000},"page":"393-412","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["An advance computational intelligent approach to solve the third kind of nonlinear pantograph Lane\u2013Emden differential system"],"prefix":"10.1007","volume":"14","author":[{"given":"Zulqurnain","family":"Sabir","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Asif Zahoor","family":"Raja","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed R.","family":"Ali","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Sadat","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,28]]},"reference":[{"key":"9469_CR1","unstructured":"Abazari, N. and Abazari, R., 2009, October. Solution of nonlinear second-order pantograph equations via differential transformation method. In Proceedings of World Academy of Science, Engineering and Technology (Vol. 58, pp. 1052\u20131056)."},{"key":"9469_CR2","doi-asserted-by":"publisher","first-page":"143","DOI":"10.18576\/amis\/130201","volume":"13","author":"F Abbas","year":"2019","unstructured":"Abbas F et al (2019) Approximate solutions to Lane\u2013Emden equation for stellar configuration. Appl Math Inform Sci 13:143\u2013152","journal-title":"Appl Math Inform Sci"},{"issue":"6","key":"9469_CR3","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1140\/epjp\/s13360-020-00449-x","volume":"135","author":"W Adel","year":"2020","unstructured":"Adel W et al (2020) Solving a new design of nonlinear second-order Lane\u2013Emden pantograph delay differential model via Bernoulli collocation method. Euro Phys J plus 135(6):427","journal-title":"Euro Phys J plus"},{"issue":"1","key":"9469_CR4","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1007\/s00521-016-2400-y","volume":"28","author":"I Ahmad","year":"2017","unstructured":"Ahmad I et al (2017) Neural network methods to solve the Lane\u2013Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput Appl 28(1):929\u2013944","journal-title":"Neural Comput Appl"},{"key":"9469_CR5","doi-asserted-by":"crossref","unstructured":"Ahmad I et al. (2020) Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically. Neural Comput Appl pp 1\u201317.","DOI":"10.1007\/s00521-020-05355-y"},{"key":"9469_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-02384-3","volume-title":"Empirical approach to machine learning","author":"PP Angelov","year":"2019","unstructured":"Angelov PP, Gu X (2019) Empirical approach to machine learning. Springer, New York"},{"issue":"10","key":"9469_CR7","doi-asserted-by":"publisher","first-page":"2981","DOI":"10.1109\/TCYB.2017.2753880","volume":"48","author":"PP Angelov","year":"2018","unstructured":"Angelov PP, Gu X, Pr\u00edncipe JC (Oct. 2018) A generalized methodology for data analysis. IEEE Trans Cybernet 48(10):2981\u20132993. https:\/\/doi.org\/10.1109\/TCYB.2017.2753880","journal-title":"IEEE Trans Cybernet"},{"issue":"7","key":"9469_CR8","doi-asserted-by":"publisher","first-page":"1447","DOI":"10.1063\/1.528326","volume":"30","author":"CM Bender","year":"1989","unstructured":"Bender CM, Milton KA, Pinsky SS, Simmons LM Jr (1989) A new perturbative approach to nonlinear problems. J Math Phys 30(7):1447\u20131455","journal-title":"J Math Phys"},{"key":"9469_CR9","doi-asserted-by":"crossref","unstructured":"Benhammouda, B., Vazquez-Leal, H. and Hernandez-Martinez, L., 2014. Procedure for exact solutions of nonlinear pantograph delay differential equations. Journal of Advances in Mathematics and Computer Science, pp.2738\u20132751.","DOI":"10.9734\/BJMCS\/2014\/11839"},{"issue":"3","key":"9469_CR10","doi-asserted-by":"publisher","first-page":"139","DOI":"10.3390\/fractalfract6030139","volume":"6","author":"T Botmart","year":"2022","unstructured":"Botmart T, Sabir Z, Raja MAZ, Weera W, Sadat R, Ali MR (2022) A numerical study of the fractional order dynamical nonlinear susceptible infected and quarantine differential model using the stochastic numerical approach. Fractal Fractional 6(3):139","journal-title":"Fractal Fractional"},{"issue":"5","key":"9469_CR11","doi-asserted-by":"publisher","first-page":"3325","DOI":"10.1016\/j.aej.2020.04.051","volume":"59","author":"AH Bukhari","year":"2020","unstructured":"Bukhari AH et al (2020) Design of a hybrid NAR-RBFs neural network for nonlinear dusty plasma system. Alex Eng J 59(5):3325\u20133345","journal-title":"Alex Eng J"},{"key":"9469_CR12","volume-title":"An Introduction to the study of stellar structure","author":"S Chandrasekhar","year":"1967","unstructured":"Chandrasekhar S (1967) An Introduction to the study of stellar structure. Dover Publications, New York"},{"key":"9469_CR13","doi-asserted-by":"publisher","first-page":"361","DOI":"10.2528\/PIER07090403","volume":"78","author":"M Dehghan","year":"2008","unstructured":"Dehghan M, Shakeri F (2008) Solution of an integro-differential equation arising in oscillating magnetic fields using He\u2019s homotopy perturbation method. Progress Electromagnetics Res 78:361\u2013376","journal-title":"Progress Electromagnetics Res"},{"issue":"1","key":"9469_CR14","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1006\/jmaa.1997.5483","volume":"213","author":"G Derfel","year":"1997","unstructured":"Derfel G, Iserles A (1997) The pantograph equation in the complex plane. J Math Anal Appl 213(1):117\u2013132","journal-title":"J Math Anal Appl"},{"key":"9469_CR15","unstructured":"Dizicheh, A.K., Salahshour, S., Ahmadian, A. and Baleanu, D., 2020. A novel algorithm based on the Legendre wavelets spectral technique for solving the Lane\u2013Emden equations. Applied Numerical Mathematics."},{"key":"9469_CR16","doi-asserted-by":"publisher","unstructured":"J. D\\v{z}urina, S.R. Grace, I. Jadlovsk\\'{a}, and T. Li, Oscillation criteria for second-order Emden--Fowler delay differential equations with a sublinear neutral term, Math. Nachr. 293 (2020), 1--13. https:\/\/doi.org\/10.1002\/mana.201800196.","DOI":"10.1002\/mana.201800196"},{"issue":"10","key":"9469_CR17","doi-asserted-by":"publisher","first-page":"1445","DOI":"10.1631\/FITEE.1900221","volume":"20","author":"MF Fateh","year":"2019","unstructured":"Fateh MF et al (2019) Differential evolution based computation intelligence solver for elliptic partial differential equations. Front Inform Technol Electron Eng 20(10):1445\u20131456","journal-title":"Front Inform Technol Electron Eng"},{"key":"9469_CR18","unstructured":"Feng, X., 2013. An analytic study on the multi-pantograph delay equations with variable coefficients. Bulletin math\u00e9matique de la Soci\u00e9t\u00e9 des Sciences Math\u00e9matiques de Roumanie, pp.205\u2013215."},{"key":"9469_CR19","doi-asserted-by":"crossref","unstructured":"Flockerzi, D. and Sundmacher, K., 2011. On coupled Lane\u2013Emden equations arising in dusty fluid models. In Journal of Physics: Conference Series (Vol. 268, No. 1, p. 012006). IOP Publishing.","DOI":"10.1088\/1742-6596\/268\/1\/012006"},{"issue":"3","key":"9469_CR20","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/j.crma.2006.12.008","volume":"344","author":"M Ghergu","year":"2007","unstructured":"Ghergu M, Radulescu V (2007) On a class of singular Gierer-Meinhardt systems arising in morphogenesis. Comptes Rendus Math\u00e9matique 344(3):163\u2013168","journal-title":"Comptes Rendus Math\u00e9matique"},{"key":"9469_CR21","doi-asserted-by":"crossref","unstructured":"Guirao JL, Sabir Z, Saeed T (2020) Design and numerical solutions of a novel third-order nonlinear emden\u2013fowler delay differential model. Math Prob Eng 2020.","DOI":"10.1155\/2020\/7359242"},{"issue":"4","key":"9469_CR22","doi-asserted-by":"publisher","first-page":"77","DOI":"10.3390\/math5040077","volume":"5","author":"MS Hashemi","year":"2017","unstructured":"Hashemi MS, Akg\u00fcl A, Inc M, Mustafa IS, Baleanu D (2017) Solving the Lane\u2013Emden equation within a reproducing kernel method and group preserving scheme. Mathematics 5(4):77","journal-title":"Mathematics"},{"key":"9469_CR23","doi-asserted-by":"crossref","unstructured":"Jadoon I et al. (2020) Design of evolutionary optimized finite difference based numerical computing for dust density model of nonlinear Van-der Pol Mathieu\u2019s oscillatory systems. Math Comput Simul","DOI":"10.1016\/j.matcom.2020.10.004"},{"key":"9469_CR24","doi-asserted-by":"crossref","unstructured":"Jadoon I et al. (2020) Integrated meta-heuristics finite difference method for the dynamics of nonlinear unipolar electrohydrodynamic pump flow model. Appl Soft Comput, p 106791.","DOI":"10.1016\/j.asoc.2020.106791"},{"issue":"2","key":"9469_CR25","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1515\/IJNSNS.2007.8.2.159","volume":"8","author":"Y Keskin","year":"2007","unstructured":"Keskin Y, Kurnaz A, Kiris \u039cE, Oturanc G (2007) Approximate solutions of generalized pantograph equations by the differential transform method. Int J Nonlinear Sci Numer Simul 8(2):159\u2013164","journal-title":"Int J Nonlinear Sci Numer Simul"},{"issue":"4","key":"9469_CR26","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1080\/09540091.2015.1092499","volume":"27","author":"JA Khan","year":"2015","unstructured":"Khan JA et al (2015) Nature-inspired computing approach for solving non-linear singular Emden-Fowler problem arising in electromagnetic theory. Connect Sci 27(4):377\u2013396","journal-title":"Connect Sci"},{"key":"9469_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2020.103703","volume":"20","author":"A Khan","year":"2021","unstructured":"Khan A, Zarin R, Hussain G, Ahmad NA, Mohd MH, Yusuf A (2021a) Stability analysis and optimal control of covid-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan). Results Phys 20:103703","journal-title":"Results Phys"},{"issue":"1","key":"9469_CR28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03162-2","volume":"2021","author":"K Khan","year":"2021","unstructured":"Khan K, Zarin R, Khan A, Yusuf A, Al-Shomrani M, Ullah A (2021b) Stability analysis of five-grade Leishmania epidemic model with harmonic mean-type incidence rate. Adv Differ Equ 2021(1):1\u201327","journal-title":"Adv Differ Equ"},{"issue":"7","key":"9469_CR29","first-page":"39","volume":"7","author":"MA Koroma","year":"2013","unstructured":"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\u201344","journal-title":"Int J Math Comput Sci Eng"},{"key":"9469_CR30","unstructured":"Kuang, Y. ed., 1993. Delay differential equations: with applications in population dynamics (Vol. 191). Academic press."},{"issue":"3","key":"9469_CR31","first-page":"1","volume":"32","author":"DS Li","year":"2000","unstructured":"Li DS, Liu MZ (2000) Exact solution properties of a multi-pantograph delay differential equation. J Harbin Inst Technol 32(3):1\u20133","journal-title":"J Harbin Inst Technol"},{"issue":"2","key":"9469_CR32","doi-asserted-by":"publisher","first-page":"955","DOI":"10.1021\/nl404356t","volume":"14","author":"W Li","year":"2014","unstructured":"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\u2013959","journal-title":"Nano Lett"},{"issue":"3","key":"9469_CR33","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/s00605-017-1039-9","volume":"184","author":"T Li","year":"2017","unstructured":"Li T et al (2017) Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatshefte F\u00fcr Mathematik 184(3):489\u2013500","journal-title":"Monatshefte F\u00fcr Mathematik"},{"issue":"2","key":"9469_CR34","doi-asserted-by":"publisher","first-page":"2441","DOI":"10.1093\/mnras\/stx1925","volume":"473","author":"D Li","year":"2018","unstructured":"Li D et al (2018) Lane-Emden equation with inertial force and general polytropic dynamic model for molecular cloud cores. Mon Not R Astron Soc 473(2):2441\u20132464","journal-title":"Mon Not R Astron Soc"},{"issue":"3","key":"9469_CR35","first-page":"853","volume":"155","author":"MZ Liu","year":"2004","unstructured":"Liu MZ, Li D (2004) Properties of analytic solution and numerical solution of multi-pantograph equation. Appl Math Comput 155(3):853\u2013871","journal-title":"Appl Math Comput"},{"issue":"3","key":"9469_CR36","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1007\/s00220-016-2753-1","volume":"347","author":"T Luo","year":"2016","unstructured":"Luo T et al (2016) Nonlinear asymptotic stability of the Lane\u2013Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities. Commun Math Phys 347(3):657\u2013702","journal-title":"Commun Math Phys"},{"key":"9469_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2019.106161","volume":"102","author":"WX Ma","year":"2020","unstructured":"Ma WX (2020) Inverse scattering for nonlocal reverse-time nonlinear Schr\u00f6dinger equations. Appl Math Lett 102:106161","journal-title":"Appl Math Lett"},{"issue":"1","key":"9469_CR38","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1090\/proc\/15174","volume":"149","author":"WX Ma","year":"2021","unstructured":"Ma WX (2021a) Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schr\u00f6dinger equations. Proc Am Math Soc 149(1):251\u2013263","journal-title":"Proc Am Math Soc"},{"key":"9469_CR39","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/j.matcom.2021.05.020","volume":"190","author":"WX Ma","year":"2021","unstructured":"Ma WX (2021b) N-soliton solution and the Hirota condition of a (2+ 1)-dimensional combined equation. Math Comput Simul 190:270\u2013279","journal-title":"Math Comput Simul"},{"key":"9469_CR40","doi-asserted-by":"crossref","unstructured":"Ma, W.X., 2021c. N-soliton solutions and the Hirota conditions in (1+ 1)-dimensions. International Journal of Nonlinear Sciences and Numerical Simulation.","DOI":"10.1515\/ijnsns-2020-0214"},{"issue":"2","key":"9469_CR41","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1016\/S0010-4655(01)00415-5","volume":"141","author":"VB Mandelzweig","year":"2001","unstructured":"Mandelzweig VB, Tabakin F (2001) Quasi linearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141(2):268\u2013281","journal-title":"Comput Phys Commun"},{"issue":"14","key":"9469_CR42","doi-asserted-by":"publisher","first-page":"10337","DOI":"10.1007\/s00521-019-04573-3","volume":"32","author":"A Mehmood","year":"2020","unstructured":"Mehmood A et al (2020) Integrated computational intelligent paradigm for nonlinear electric circuit models using neural networks, genetic algorithms and sequential quadratic programming. Neural Comput Appl 32(14):10337\u201310357","journal-title":"Neural Comput Appl"},{"key":"9469_CR43","unstructured":"Niculescu, S.I., 2001. Delay effects on stability: a robust control approach (Vol. 269). Springer Science & Business Media."},{"issue":"6","key":"9469_CR44","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1016\/j.newast.2004.02.003","volume":"9","author":"MI Nouh","year":"2004","unstructured":"Nouh MI (2004) Accelerated power series solution of polytropic and isothermal gas spheres. New Astron 9(6):467\u2013473","journal-title":"New Astron"},{"key":"9469_CR45","doi-asserted-by":"crossref","unstructured":"Qureshi S, Yusuf A, Aziz S (2020) On the use of Mohand integral transform for solving fractional-order classical Caputo differential equations. J Appl Math Comput Mech 19(3).","DOI":"10.17512\/jamcm.2020.3.08"},{"issue":"2","key":"9469_CR46","doi-asserted-by":"publisher","first-page":"1409","DOI":"10.1016\/j.jksus.2019.11.035","volume":"32","author":"S Qureshi","year":"2020","unstructured":"Qureshi S, Yusuf A (2020) A new third order convergent numerical solver for continuous dynamical systems. J King Saud Univ-Sci 32(2):1409\u20131416","journal-title":"J King Saud Univ-Sci"},{"issue":"1","key":"9469_CR47","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s10910-013-0260-6","volume":"52","author":"R Rach","year":"2014","unstructured":"Rach R et al (2014) Solving coupled Lane\u2013Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J Math Chem 52(1):255\u2013267","journal-title":"J Math Chem"},{"issue":"3","key":"9469_CR48","doi-asserted-by":"publisher","first-page":"1524","DOI":"10.1016\/j.na.2011.01.037","volume":"75","author":"V Radulescu","year":"2012","unstructured":"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\u20131530","journal-title":"Nonlinear Anal Theory Methods Appl"},{"issue":"9","key":"9469_CR49","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1140\/epjp\/i2018-12153-4","volume":"133","author":"MAZ Raja","year":"2018","unstructured":"Raja MAZ et al (2018) A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Euro Phys J plus 133(9):364","journal-title":"Euro Phys J plus"},{"key":"9469_CR50","doi-asserted-by":"crossref","unstructured":"Raja MAZ, Manzar MA, Shah SM, Chen Y (2020) Integrated intelligence of fractional neural networks and sequential quadratic programming for Bagley\u2013Torvik systems arising in fluid mechanics. J Comput Nonlinear Dyn 15(5).","DOI":"10.1115\/1.4046496"},{"issue":"2","key":"9469_CR51","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/S0010-4655(03)00226-1","volume":"153","author":"JI Ramos","year":"2003","unstructured":"Ramos JI (2003) Linearization methods in classical and quantum mechanics. Comput Phys Commun 153(2):199\u2013208","journal-title":"Comput Phys Commun"},{"issue":"2","key":"9469_CR52","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1016\/j.chaos.2006.11.018","volume":"38","author":"JI Ramos","year":"2008","unstructured":"Ramos JI (2008) Series approach to the Lane\u2013Emden equation and comparison with the homotopy perturbation method. Chaos, Solitons Fractals 38(2):400\u2013408","journal-title":"Chaos, Solitons Fractals"},{"issue":"11\u201312","key":"9469_CR53","doi-asserted-by":"publisher","first-page":"2190","DOI":"10.1016\/j.camwa.2009.03.017","volume":"58","author":"A Saadatmandi","year":"2009","unstructured":"Saadatmandi A, Dehghan M (2009) Variational iteration method for solving a generalized pantograph equation. Comput Math Appl 58(11\u201312):2190\u20132196","journal-title":"Comput Math Appl"},{"key":"9469_CR54","doi-asserted-by":"crossref","unstructured":"Sabir, Z., et al., 2020a. Novel design of Morlet wavelet neural network for solving second order Lane\u2013Emden equation. Mathematics and Computers in Simulation","DOI":"10.1016\/j.matcom.2020.01.005"},{"issue":"4","key":"9469_CR55","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-020-01330-4","volume":"39","author":"Z Sabir","year":"2020","unstructured":"Sabir Z et al (2020a) Integrated intelligent computing with neuro-swarming solver for multi-singular fourth-order nonlinear Emden-Fowler equation. Comput Appl Math 39(4):1\u201318","journal-title":"Comput Appl Math"},{"issue":"5","key":"9469_CR56","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/j.taml.2020.01.049","volume":"10","author":"Z Sabir","year":"2020","unstructured":"Sabir Z, Sakar MG, Yeskindirova M, Saldir O (2020b) Numerical investigations to design a novel model based on the fifth order system of Emden\u2013Fowler equations. Theor Appl Mech Lett 10(5):333\u2013342","journal-title":"Theor Appl Mech Lett"},{"key":"9469_CR57","unstructured":"Sabir, Z., et al., 2020b. Intelligence computing approach for solving second order system of Emden\u2013Fowler model. Journal of Intelligent & Fuzzy SystemsS, pp.1\u201316."},{"key":"9469_CR58","doi-asserted-by":"crossref","unstructured":"Sabir, Z., Wahab, H. A., Nguyen, T. G., Altamirano, G. C., Erdo\u011fan, F., & Ali, M. R. (2022a). Intelligent computing technique for solving singular multi-pantograph delay differential equation. Soft Computing, 1\u201313..","DOI":"10.1007\/s00500-022-07065-0"},{"key":"9469_CR59","doi-asserted-by":"crossref","unstructured":"Sabir, Z., Wahab, H. A., Ali, M. R., & Sadat, R. (2022b). Neuron Analysis of the Two-Point Singular Boundary Value Problems Arising in the Thermal Explosion\u2019s Theory. Neural Processing Letters, 1\u201328.\u200f","DOI":"10.1007\/s11063-022-10809-6"},{"key":"9469_CR60","doi-asserted-by":"crossref","unstructured":"Sabir Z, Ali MR, Fathurrochman I, Raja MAZ, Sadat R, Baleanu D (2022c). Dynamics of multi-point singular fifth-order Lane\u2013Emden system with neuro-evolution heuristics. Evol Syst, 1\u201312.","DOI":"10.1007\/s12530-021-09413-1"},{"key":"9469_CR61","doi-asserted-by":"crossref","unstructured":"Sabir, Z., Ali, M. R., Raja, M. A. Z., Sadat, R., & Baleanu, D. (2022d). Dynamics of three-point boundary value problems with Gudermannian neural networks. Evolutionary Intelligence, 1\u201313.","DOI":"10.1007\/s12065-021-00695-7"},{"key":"9469_CR62","doi-asserted-by":"crossref","unstructured":"Sabir, Z. et al., 2020. On a new model based on third-order nonlinear multi singular functional differential equations. Mathematical Problems in Engineering, 2020.","DOI":"10.1155\/2020\/1683961"},{"issue":"5","key":"9469_CR63","doi-asserted-by":"publisher","first-page":"1175","DOI":"10.11650\/tjm\/7969","volume":"21","author":"U Saeed","year":"2017","unstructured":"Saeed U (2017) Haar Adomian method for the solution of fractional nonlinear Lane\u2013Emden type equations arising in astrophysics. Taiwan J Math 21(5):1175\u20131192","journal-title":"Taiwan J Math"},{"issue":"2","key":"9469_CR64","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1016\/j.cam.2007.03.024","volume":"214","author":"M Sezer","year":"2008","unstructured":"Sezer M, \u015eahin N (2008) Approximate solution of multi-pantograph equation with variable coefficients. J Comput Appl Math 214(2):406\u2013416","journal-title":"J Comput Appl Math"},{"key":"9469_CR65","doi-asserted-by":"crossref","unstructured":"Shah, Z., et al., 2020. Design of neural network based intelligent computing for neumerical treatment of unsteady 3D flow of Eyring-Powell magneto-nanofluidic model.\u00a0Journal of Materials Research and Technology.","DOI":"10.1016\/j.jmrt.2020.09.098"},{"issue":"9","key":"9469_CR66","doi-asserted-by":"publisher","first-page":"4364","DOI":"10.1063\/1.530005","volume":"34","author":"NT Shawagfeh","year":"1993","unstructured":"Shawagfeh NT (1993) Non-perturbative approximate solution for Lane\u2013Emden equation. J Math Phys 34(9):4364\u20134369","journal-title":"J Math Phys"},{"issue":"11","key":"9469_CR67","doi-asserted-by":"publisher","first-page":"548","DOI":"10.1140\/epjp\/i2019-12889-1","volume":"134","author":"R Singh","year":"2019","unstructured":"Singh R et al (2019a) Haar wavelet collocation approach for Lane\u2013Emden equations arising in mathematical physics and astrophysics. Euro Phys J plus 134(11):548","journal-title":"Euro Phys J plus"},{"issue":"11","key":"9469_CR68","doi-asserted-by":"publisher","first-page":"548","DOI":"10.1140\/epjp\/i2019-12889-1","volume":"134","author":"R Singh","year":"2019","unstructured":"Singh R, Shahni J, Garg H, Garg A (2019b) Haar wavelet collocation approach for Lane\u2013Emden equations arising in mathematical physics and astrophysics. European Phys J plus 134(11):548","journal-title":"European Phys J plus"},{"issue":"7","key":"9469_CR69","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjp\/s13360-020-00557-8","volume":"135","author":"M Umar","year":"2020","unstructured":"Umar M et al (2020a) A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment. Euro Phys J plus 135(7):1\u201323","journal-title":"Euro Phys J plus"},{"issue":"6","key":"9469_CR70","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1140\/epjp\/s13360-020-00417-5","volume":"135","author":"M Umar","year":"2020","unstructured":"Umar M, Sabir Z, Amin F, Guirao JL, Raja MAZ (2020b) Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Euro Phys J plus 135(6):403","journal-title":"Euro Phys J plus"},{"issue":"10","key":"9469_CR71","doi-asserted-by":"publisher","first-page":"1628","DOI":"10.3390\/sym12101628","volume":"12","author":"M Umar","year":"2020","unstructured":"Umar M et al (2020c) A stochastic intelligent computing with neuro-evolution heuristics for nonlinear SITR system of novel COVID-19 dynamics. Symmetry 12(10):1628","journal-title":"Symmetry"},{"key":"9469_CR72","unstructured":"Soleymani Karimi Vanani, Sedighi Hafshejani and Khan. On the numerical solution of generalized pantograph equation. World Applied Sciences Journal, 13(12):2531\u20132535, 2011."},{"key":"9469_CR73","doi-asserted-by":"crossref","unstructured":"Widatalla, S. and Koroma, M.A., 2012. Approximation algorithm for a system of pantograph equations. Journal of Applied Mathematics, 2012.","DOI":"10.1155\/2012\/714681"},{"issue":"1","key":"9469_CR74","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1006\/jmaa.1995.1239","volume":"193","author":"T Zhao","year":"1995","unstructured":"Zhao T (1995) Global periodic-solutions for a differential delay system modeling a microbial population in the chemostat. J Math Anal Appl 193(1):329\u2013352","journal-title":"J Math Anal Appl"}],"container-title":["Evolving Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12530-022-09469-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12530-022-09469-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12530-022-09469-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T08:19:20Z","timestamp":1683879560000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12530-022-09469-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,28]]},"references-count":74,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["9469"],"URL":"https:\/\/doi.org\/10.1007\/s12530-022-09469-7","relation":{},"ISSN":["1868-6478","1868-6486"],"issn-type":[{"value":"1868-6478","type":"print"},{"value":"1868-6486","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,28]]},"assertion":[{"value":"1 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 September 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}