{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T12:42:42Z","timestamp":1771677762210,"version":"3.50.1"},"reference-count":75,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T00:00:00Z","timestamp":1662076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University","award":["GRANT691"],"award-info":[{"award-number":["GRANT691"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this study is to provide a mathematical construction based on the novel singular perturbed model of the second kind (NSPM-SK) using the standard form of the Lane\u2013Emden. The singular Lane\u2013Emden types of the models have abundant applications in astrophysics. The inclusive features of this model are provided using the perturbed, pantograph, singular point together and the shape factor based on the NSPM-SK. These models become more complicated by using these factors through the artificial neural networks (ANNs) together with the optimization procedures of the swarming particle swarm optimization (PSO) paradigms and the local sequential quadratic programming (SQP). An objective function is provided based on the differential form of the NSPM-SK and then optimization is performed through the hybridization of the PSOSQP. The exactness of the model is attained to solve three different variations of the mathematical NSPM-SK by using the overlapping of the obtained and true results. The stability, robustness, and convergence of the designed numerical approach are perceived by using different statistical performances of the ANNs together with the optimization of the PSOSQP for 30 independent executions.<\/jats:p>","DOI":"10.3390\/axioms11090449","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T20:48:25Z","timestamp":1662410905000},"page":"449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Swarming Approach for the Novel Second Order Perturbed Pantograph Lane\u2013Emden Model Arising in Astrophysics"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9076-9945","authenticated-orcid":false,"given":"Muneerah","family":"Al Nuwairan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia"}]},{"given":"Zulqurnain","family":"Sabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21120, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.cam.2008.11.011","article-title":"The Optimal Exponentially-Fitted Numerov Method for Solving Two-Point Boundary Value Problems","volume":"230","author":"Holevoet","year":"2010","journal-title":"J. Comp. Appl. Math."},{"key":"ref_2","first-page":"341","article-title":"A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers","volume":"4","author":"Phaneendra","year":"2010","journal-title":"Appl. Math. Inf. 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