{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T01:20:09Z","timestamp":1767144009888,"version":"build-2238731810"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T00:00:00Z","timestamp":1730678400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T00:00:00Z","timestamp":1730678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Hum-Cent Intell Syst"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we examine the mathematics that makes it possible to couple a conventional numerical integrator (e.g., Euler, Runge-Kutta, among others) with some universal approximator of functions (e.g., artificial neural networks, fuzzy inference systems, among others). These hybrid structures, known as Universal Numerical Integrators (UNIs), are analyzed through a set of significant properties essential for their proper design. Theoretical foundations are complemented by numerical and computational experiments, validating the proposed UNI models. We also hope that the theoretical content in this article can help guide researchers aiming to computationally design general problems related to some Universal Numerical Integrator (UNI).<\/jats:p>","DOI":"10.1007\/s44230-024-00084-0","type":"journal-article","created":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T15:04:30Z","timestamp":1730732670000},"page":"1-20","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Quantitative Approach to Universal Numerical Integrators (UNIS) with Computational Application"],"prefix":"10.1007","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7069-8430","authenticated-orcid":false,"given":"Paulo M.","family":"Tasinaffo","sequence":"first","affiliation":[]},{"given":"Luiz A. V.","family":"Dias","sequence":"additional","affiliation":[]},{"given":"Adilson M.","family":"da Cunha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,4]]},"reference":[{"key":"84_CR1","first-page":"953","volume":"114","author":"AN Kolmogorov","year":"1957","unstructured":"Kolmogorov AN. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Doklady Akademii Nauk SSR. 1957;114:953\u20136.","journal-title":"Doklady Akademii Nauk SSR"},{"key":"84_CR2","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1038\/323533a0","volume":"323","author":"DE Rumelhart","year":"1986","unstructured":"Rumelhart DE, Hinton GE, Williams RJ. Learning representations by back-propagating errors. Nature. 1986;323:533\u20136. https:\/\/doi.org\/10.1038\/323533a0.","journal-title":"Nature"},{"issue":"4","key":"84_CR3","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"G Cybenko","year":"1989","unstructured":"Cybenko G. Approximation by superpositions of a sigmoidal function. Math Control Signals Syst. 1989;2(4):303\u201314. https:\/\/doi.org\/10.1007\/BF02551274.","journal-title":"Math Control Signals Syst"},{"issue":"5","key":"84_CR4","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K Hornik","year":"1989","unstructured":"Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Netw. 1989;2(5):359\u201366. https:\/\/doi.org\/10.1016\/0893-6080(89)90020-8.","journal-title":"Neural Netw"},{"key":"84_CR5","volume-title":"Neural networks: a comprehensive foundation","author":"S Haykin","year":"1999","unstructured":"Haykin S. Neural networks: a comprehensive foundation. New Jersey: Prentice-Hall Inc; 1999."},{"key":"84_CR6","volume-title":"Elements of numerical analysis","author":"P Henrici","year":"1964","unstructured":"Henrici P. Elements of numerical analysis. New York: John Wiley and Sons; 1964."},{"key":"84_CR7","volume-title":"Numerical solution of ordinary differential equations","author":"L Lapidus","year":"1971","unstructured":"Lapidus L, Seinfeld JH. Numerical solution of ordinary differential equations. New York and London: Academic Press; 1971."},{"key":"84_CR8","volume-title":"Computational methods in ordinary differential equations","author":"JD Lambert","year":"1973","unstructured":"Lambert JD. Computational methods in ordinary differential equations. New York: John Wiley and Sons; 1973."},{"key":"84_CR9","doi-asserted-by":"publisher","unstructured":"Vidyasagar M. Nonlinear systems analysis. Electrical Engineering Series. New Jersey: Prentice-Hall; 1978. https:\/\/doi.org\/10.1137\/1.9780898719185","DOI":"10.1137\/1.9780898719185"},{"key":"84_CR10","doi-asserted-by":"crossref","unstructured":"Ames WF. Numerical methods for parcial differential equations. 2. ed. New York: Academic Press; (1977)","DOI":"10.1016\/B978-0-12-056760-7.50009-8"},{"key":"84_CR11","unstructured":"Rama\u00a0Rao K. A review on numerical methods for initial value problems. In: Internal Report, INPE-3011-RPI\/088, S\u00e3o Jos\u00e9 dos Campos\/SP, Brazil. 1984"},{"issue":"6","key":"84_CR12","doi-asserted-by":"publisher","first-page":"2157","DOI":"10.1080\/00207178908559767","volume":"49","author":"SA Billings","year":"1989","unstructured":"Billings SA, Chen S, Koreberg MJ. Identification of MIMO non-linear systems using forward-regression orthogonal estimator. Int J Control. 1989;49(6):2157\u201389. https:\/\/doi.org\/10.1080\/00207178908559767.","journal-title":"Int J Control"},{"issue":"2","key":"84_CR13","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1080\/00207179208934317","volume":"56","author":"S Chen","year":"1992","unstructured":"Chen S, Billings SA. Neural networks for nonlinear dynamic system modelling and identification. Int J Control. 1992;56(2):319\u201346. https:\/\/doi.org\/10.1080\/00207179208934317.","journal-title":"Int J Control"},{"key":"84_CR14","unstructured":"Tasinaffo PM. Estruturas de integra\u00e7\u00e3o neural feedforward testadas em problemas de controle preditivo. PhD thesis, INPE-10475-TDI\/945, S\u00e3o Jos\u00e9 dos Campos\/SP, Brazil. 2003"},{"key":"84_CR15","unstructured":"Euler LP. Institutiones Calculi Integralis, St. Petersburg. 1768"},{"issue":"2","key":"84_CR16","doi-asserted-by":"publisher","first-page":"98","DOI":"10.21528\/LNLM-vol3-no2-art5","volume":"3","author":"PM Tasinaffo","year":"2005","unstructured":"Tasinaffo PM, Rios Neto A. Mean derivatives based neural Euler integrator for nonlinear dynamic systems modeling. Learn Nonlinear Models. 2005;3(2):98\u2013109.","journal-title":"Learn Nonlinear Models"},{"issue":"5","key":"84_CR17","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.24507\/ijicic.12.05.1703","volume":"12","author":"PM Tasinaffo","year":"2016","unstructured":"Tasinaffo PM, Guimar\u00e3es RS, Dias LAV, Strafacci J\u00fanior V. Discrete and exact general solution for nonlinear autonomous ordinary differential equations. Int J Innov Comput Inf Control (IJICIC). 2016;12(5):1703\u201319. https:\/\/doi.org\/10.24507\/ijicic.12.05.1703.","journal-title":"Int J Innov Comput Inf Control (IJICIC)"},{"issue":"5","key":"84_CR18","doi-asserted-by":"publisher","first-page":"1721","DOI":"10.24507\/ijicic.12.05.1721","volume":"12","author":"MO Figueiredo","year":"2016","unstructured":"Figueiredo MO, Tasinaffo PM, Dias LAV. Modeling autonomous nonlinear dynamic systems using mean derivatives, fuzzy logic and genetic algorithms. Int J Innov Comput Inf Control (IJICIC). 2016;12(5):1721\u201343. https:\/\/doi.org\/10.24507\/ijicic.12.05.1721.","journal-title":"Int J Innov Comput Inf Control (IJICIC)"},{"issue":"6","key":"84_CR19","doi-asserted-by":"publisher","first-page":"881","DOI":"10.24507\/ijicic.12.06.1881","volume":"12","author":"PM Tasinaffo","year":"2016","unstructured":"Tasinaffo PM, Guimar\u00e3es RS, Dias LAV, Strafacci J\u00fanior V. Mean derivatives methodology by using Euler integrator improved to allow the variation in the size of integration step. Int J Innov Comput Inf Control (IJICIC). 2016;12(6):881\u20131891. https:\/\/doi.org\/10.24507\/ijicic.12.06.1881.","journal-title":"Int J Innov Comput Inf Control (IJICIC)"},{"issue":"2","key":"84_CR20","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1109\/72.661124","volume":"9","author":"Y-J Wang","year":"1998","unstructured":"Wang Y-J, Lin C-T. Runge-Kutta neural network for identification of dynamical systems in high accuracy. IEEE Trans Neural Netw. 1998;9(2):294\u2013307. https:\/\/doi.org\/10.1109\/72.661124.","journal-title":"IEEE Trans Neural Netw"},{"key":"84_CR21","doi-asserted-by":"crossref","unstructured":"Rios\u00a0Neto A. Dynamic systems numerical integrators in neural control schemes. In: V Congresso Brasileiro de Redes Neurais, Rio de Janeiro-RJ, Brazil. 2001. pp. 85\u201388","DOI":"10.21528\/CBRN2001-019"},{"issue":"5","key":"84_CR22","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1590\/S0103-17592010000500005","volume":"21","author":"RP Melo","year":"2010","unstructured":"Melo RP, Tasinaffo PM. Uma metodologia de modelagem emp\u00edrica utilizando o integrador neural de m\u00faltiplos passos do tipo Adams-Bashforth. Revista Controle e Automa\u00e7\u00e3o (Sociedade Brasileira de Autom\u00e1tica). 2010;21(5):487\u2013509. https:\/\/doi.org\/10.1590\/S0103-17592010000500005.","journal-title":"Revista Controle e Automa\u00e7\u00e3o (Sociedade Brasileira de Autom\u00e1tica)"},{"key":"84_CR23","unstructured":"Melo RP. Metodologia de modelagem emp\u00edrica utilizando integradores neurais de m\u00faltiplos-passos aplicado a sistemas din\u00f4micos n\u00e3o-lineares. Master\u2019s thesis, Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos\/SP, Brazil. 2008"},{"issue":"1","key":"84_CR24","doi-asserted-by":"publisher","first-page":"383","DOI":"10.24507\/ijicic.15.01.383","volume":"15","author":"PM Tasinaffo","year":"2019","unstructured":"Tasinaffo PM, Gon\u00e7alves GS, Cunha AM, Dias LAV. An introduction to universal numerical integrators. Int J Innov Comput Inf Control (IJICIC). 2019;15(1):383\u2013406. https:\/\/doi.org\/10.24507\/ijicic.15.01.383.","journal-title":"Int J Innov Comput Inf Control (IJICIC)"},{"issue":"5","key":"84_CR25","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"IE Lagaris","year":"1998","unstructured":"Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw. 1998;9(5):987\u20131000. https:\/\/doi.org\/10.1109\/72.712178.","journal-title":"IEEE Trans Neural Netw"},{"issue":"1","key":"84_CR26","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s00521-013-1476-x","volume":"25","author":"AA Anastassi","year":"2014","unstructured":"Anastassi AA. Constructing Runge-Kutta method with the use of artificial neural networks. Neural Comput Appl. 2014;25(1):229\u201336.","journal-title":"Neural Comput Appl"},{"issue":"1","key":"84_CR27","doi-asserted-by":"publisher","first-page":"180","DOI":"10.14419\/jacst.v4i1.4365","volume":"4","author":"M Dehghanpour","year":"2015","unstructured":"Dehghanpour M, Rahati A, Dehghanian E. ANN-based modeling of third order rungekutta method. J Adv Comput Sci Technol. 2015;4(1):180\u20139. https:\/\/doi.org\/10.14419\/jacst.v4i1.4365.","journal-title":"J Adv Comput Sci Technol"},{"issue":"17","key":"84_CR28","doi-asserted-by":"publisher","first-page":"7769","DOI":"10.1007\/s00500-018-3405-5","volume":"23","author":"K U\u00e7ak","year":"2019","unstructured":"U\u00e7ak K. A Runge-Kutta neural network-based control method for nonlinear MIMO systems. Soft Comput. 2019;23(17):7769\u2013803. https:\/\/doi.org\/10.1007\/s00500-018-3405-5.","journal-title":"Soft Comput"},{"key":"84_CR29","doi-asserted-by":"publisher","unstructured":"Chen RTQ, Rubanova Y, Bettencourt J, Duveand D. Neural ordinary differential equations. In: 32nd conference on neural information processing systems (NeurlPS), Montr\u00e9al, Canada. 2018. pp. 1\u201319. https:\/\/doi.org\/10.48550\/arXiv.1806.07366","DOI":"10.48550\/arXiv.1806.07366"},{"key":"84_CR30","doi-asserted-by":"publisher","unstructured":"U\u00e7ak, K.: A Runge-Kutta MLP neural network based control method for nonlinear MIMO systems. In: 6th International Conference on Electrical and Electronics Engineering (ICEEE), Istanbul, Turkey. 2019. pp. 186\u2013192. https:\/\/doi.org\/10.1109\/ICEEE2019.2019.00043","DOI":"10.1109\/ICEEE2019.2019.00043"},{"issue":"1","key":"84_CR31","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1590\/S0103-17592007000100007","volume":"18","author":"PM Tasinaffo","year":"2006","unstructured":"Tasinaffo PM, Rios Neto A. Predictive control with mean derivative based neural Euler integrator dynamic model. Revista Controle e Automa\u00e7\u00e3o (Sociedade Brasileira de Autom\u00e1tica). 2006;18(1):94\u2013105.","journal-title":"Revista Controle e Automa\u00e7\u00e3o (Sociedade Brasileira de Autom\u00e1tica)"},{"issue":"2","key":"84_CR32","doi-asserted-by":"publisher","first-page":"1789","DOI":"10.1007\/s11063-019-10167-w","volume":"51","author":"K U\u00e7ak","year":"2020","unstructured":"U\u00e7ak K. A novel model predictive Runge-Kutta neural network controller for nonlinear MIMO systems. Neural Process Lett. 2020;51(2):1789\u2013833. https:\/\/doi.org\/10.1007\/s11063-019-10167-w.","journal-title":"Neural Process Lett"},{"key":"84_CR33","unstructured":"Rios\u00a0Neto A, Tasinaffo PM. Neural numerical integrators in predictive control tested in na orbit transfer problem. In: DINCON-2003 or II Congresso Tem\u00e1tico de Din\u00f4mica, Controle e Aplica\u00e7\u00f5es, S\u00e3o Jos\u00e9 dos Campos\/SP, Brazil. 2003. pp. 692\u2013702"},{"issue":"2","key":"84_CR34","doi-asserted-by":"publisher","first-page":"445","DOI":"10.24507\/ijicic.15.02.445","volume":"15","author":"PM Tasinaffo","year":"2019","unstructured":"Tasinaffo PM, Rios Neto A. Adams-Bashforth neural networks applied in a predictive control structure with only one horizon. Int J Innov Comput Inf Control (IJICIC). 2019;15(2):445\u201364. https:\/\/doi.org\/10.24507\/ijicic.15.02.445.","journal-title":"Int J Innov Comput Inf Control (IJICIC)"},{"issue":"5","key":"84_CR35","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1109\/72.870037","volume":"11","author":"IE Lagaris","year":"2000","unstructured":"Lagaris IE, Likas A, Fotiadis DI. Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw. 2000;11(5):1041\u20139. https:\/\/doi.org\/10.1109\/72.870037.","journal-title":"IEEE Trans Neural Netw"},{"issue":"19","key":"84_CR36","doi-asserted-by":"publisher","first-page":"2812","DOI":"10.3923\/jas.2007.2812.2817","volume":"7","author":"M Hayati","year":"2007","unstructured":"Hayati M, Karami B. Feedforward neural network for solving partial differential equations. J Appl Sci. 2007;7(19):2812\u20137. https:\/\/doi.org\/10.3923\/jas.2007.2812.2817.","journal-title":"J Appl Sci"},{"key":"84_CR37","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s40304-017-0117-6","volume":"5","author":"E Weinan","year":"2017","unstructured":"Weinan E, Han J, Jentzen A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun Math Stat. 2017;5:349\u201380. https:\/\/doi.org\/10.1007\/s40304-017-0117-6.","journal-title":"Commun Math Stat"},{"issue":"34","key":"84_CR38","doi-asserted-by":"publisher","first-page":"8505","DOI":"10.1073\/pnas.1718942115","volume":"115","author":"J Han","year":"2018","unstructured":"Han J, Arnulf J, Weinan E. Solving high-dimensional partial differential equations using deep learning. Proc Natl Acad Sci. 2018;115(34):8505\u201310. https:\/\/doi.org\/10.1073\/pnas.1718942115.","journal-title":"Proc Natl Acad Sci"},{"key":"84_CR39","doi-asserted-by":"publisher","first-page":"5595","DOI":"10.48550\/arXiv.1502.05767","volume":"18","author":"AG Baydin","year":"2018","unstructured":"Baydin AG, Pearlmutter BA, Radul AA, Siskind JM. Automatic differential in machine learning: a survey. J Mach Learn Res. 2018;18:5595\u2013637. https:\/\/doi.org\/10.48550\/arXiv.1502.05767.","journal-title":"J Mach Learn Res"},{"key":"84_CR40","doi-asserted-by":"publisher","unstructured":"Dockhorn T. A discussion on solving partial differential equations using neural networks. ArXiv:1904.07200. 2019. https:\/\/doi.org\/10.48550\/arXiv.1904.07200","DOI":"10.48550\/arXiv.1904.07200"},{"key":"84_CR41","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686\u2013707. https:\/\/doi.org\/10.1016\/j.jcp.2018.10.045.","journal-title":"J Comput Phys"},{"key":"84_CR42","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109672","volume":"419","author":"J Han","year":"2020","unstructured":"Han J, Nica M, Stinchcombe AR. A derivative-free method for solving elliptic partial differential equations with deep neural networks. J Comput Phys. 2020;419: 109672. https:\/\/doi.org\/10.1016\/j.jcp.2020.109672.","journal-title":"J Comput Phys"},{"issue":"1","key":"84_CR43","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1137\/19M1274067","volume":"63","author":"L Lu","year":"2021","unstructured":"Lu L, Meng X, Mao Z, Karniadakis GE. Deepxde: a deep learning library for solving differential equations. SIAM Rev. 2021;63(1):208\u201328. https:\/\/doi.org\/10.1137\/19M1274067.","journal-title":"SIAM Rev"},{"issue":"1","key":"84_CR44","doi-asserted-by":"publisher","first-page":"42","DOI":"10.48550\/arXiv.2103.08915","volume":"15","author":"J Yang","year":"2022","unstructured":"Yang J, Zhu Q. A local deep learning method for solving high order partial differential equations. Numer Math Theor Meth Appl. 2022;15(1):42\u201367. https:\/\/doi.org\/10.48550\/arXiv.2103.08915.","journal-title":"Numer Math Theor Meth Appl"},{"key":"84_CR45","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/0168-9274(95)00109-3","volume":"20","author":"PJ Houven","year":"1996","unstructured":"Houven PJ. The development of Runge-kutta methods for patial differential equations. Appl Numer Math. 1996;20:261\u201372. https:\/\/doi.org\/10.1016\/0168-9274(95)00109-3.","journal-title":"Appl Numer Math"},{"key":"84_CR46","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0307-904X(77)90006-3","volume":"1","author":"A Kumar","year":"1977","unstructured":"Kumar A, Unny TE. Application of Runge-kutta method for the solution of non-linear partial differential equations. Appl Math Model. 1977;1:199\u2013204. https:\/\/doi.org\/10.1016\/0307-904X(77)90006-3.","journal-title":"Appl Math Model"},{"issue":"4","key":"84_CR47","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1109\/MCSE.2006.73","volume":"8","author":"Z Zhe-Zhao","year":"2006","unstructured":"Zhe-Zhao Z, Yao-Nan W, Hui W. Numerical integration based on neural network algorithm. Comput Sci Eng. 2006;8(4):42\u20138. https:\/\/doi.org\/10.1109\/MCSE.2006.73.","journal-title":"Comput Sci Eng"},{"issue":"6","key":"84_CR48","doi-asserted-by":"publisher","first-page":"112","DOI":"10.11648\/j.mcs.20190406.13","volume":"4","author":"PM Tasinaffo","year":"2019","unstructured":"Tasinaffo PM, Santos AHM, Cavalcante Junior E, Forster CHQ, Shigemura RAL, Jacomel R, Pugliese VU, Iha BKV, Cunha AM, Gon\u00e7alves GS, Dias LAV. Determination of forest reserves area using images processed by drones, neural networks and Monte Carlo method. Math Comput Sci. 2019;4(6):112\u201329.","journal-title":"Math Comput Sci"},{"key":"84_CR49","volume-title":"Calculus with analytic geometry (Volumes I and II)","author":"MA Munem","year":"1978","unstructured":"Munem MA, Foulis DJ. Calculus with analytic geometry (Volumes I and II). New York: Worth Publishers Inc; 1978."},{"key":"84_CR50","volume-title":"Advanced calculus","author":"E Wilson","year":"1958","unstructured":"Wilson E. Advanced calculus. New York, USA: Dover Publications; 1958."},{"key":"84_CR51","unstructured":"Hy\u00f6tyniemi H: Turing machines are recurrent neural networks. In: The symposium on artificial networks, Vaasa, Finland. 1996. pp. 1\u201313. https:\/\/api.semanticscholar.org\/CorpusID:837336"},{"key":"84_CR52","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1016\/0022-247X(72)90062-5","volume":"40","author":"D Sarafyan","year":"1972","unstructured":"Sarafyan D. Improved sixth-order Runge-Kutta formulas and approximate continuous solution of ordinary differential equations. J Math Anal Appl. 1972;40:436\u201345. https:\/\/doi.org\/10.1016\/0022-247X(72)90062-5.","journal-title":"J Math Anal Appl"},{"key":"84_CR53","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/s11075-010-9437-2","volume":"57","author":"E Misili","year":"2011","unstructured":"Misili E, Gurefe Y. Multiplicative Adams-Bashforth-Moulton methods. Numer Algor. 2011;57:425\u201339. https:\/\/doi.org\/10.1007\/s11075-010-9437-2.","journal-title":"Numer Algor"},{"issue":"103","key":"84_CR54","doi-asserted-by":"publisher","first-page":"5115","DOI":"10.12988\/ams.2013.36314","volume":"7","author":"G Polla","year":"2013","unstructured":"Polla G. Comparing accuracy of differential equation results between Runge-Kutta Fehlberg methods and adams-moulton methods. Appl Math Sci. 2013;7(103):5115\u201327. https:\/\/doi.org\/10.12988\/ams.2013.36314.","journal-title":"Appl Math Sci"},{"key":"84_CR55","volume-title":"Numerical methods using Matlab","author":"JH Mathews","year":"2004","unstructured":"Mathews JH, Fink KD. Numerical methods using Matlab. New Jersey, USA: Prentice-Hall Inc; 2004."}],"updated-by":[{"DOI":"10.1007\/s44230-025-00101-w","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T00:00:00Z","timestamp":1751414400000}}],"container-title":["Human-Centric Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44230-024-00084-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s44230-024-00084-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44230-024-00084-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T02:14:30Z","timestamp":1751508870000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s44230-024-00084-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,4]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["84"],"URL":"https:\/\/doi.org\/10.1007\/s44230-024-00084-0","relation":{},"ISSN":["2667-1336"],"issn-type":[{"value":"2667-1336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,4]]},"assertion":[{"value":"17 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2025","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The original online version of this article has been revised to reflect a title change as well as updates to the abstract and several paragraphs.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2025","order":7,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":8,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":9,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s44230-025-00101-w","URL":"https:\/\/doi.org\/10.1007\/s44230-025-00101-w","order":10,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that no financial interests or social relationships directly affect the work reported in this article. The authors also declare no Conflict of interest regarding the submitted article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}},{"value":"The authors declare that they followed all the rules of good scientific practice. An approval Committee for our article is not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"value":"All authors approve of their participation in this work.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to Participate"}},{"value":"The authors approve the publication of this research.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for Publication"}},{"value":"This article contains no studies performed by authors with human participants or animals.","order":6,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and Animal Ethics"}}]}}