A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics

Zulqurnain Sabir; Salem Ben Said; Qasem Al-Mdallal; Mohamed R Ali

doi:10.1038/s41598-022-26566-4

A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics

Sci Rep. 2022 Dec 30;12(1):22607. doi: 10.1038/s41598-022-26566-4.

Authors

Zulqurnain Sabir¹, Salem Ben Said², Qasem Al-Mdallal¹, Mohamed R Ali^{3

4}

Affiliations

¹ Department of Mathematical Science, College of Science, United Arab Emirates University, P. O. Box 15551, Al Ain, United Arab Emirates.
² Department of Mathematical Science, College of Science, United Arab Emirates University, P. O. Box 15551, Al Ain, United Arab Emirates. salem.bensaid@uaeu.ac.ae.
³ Faculty of Engineering and Technology, Future University in Egypt, New Cairo, 11835, Egypt.
⁴ Basic Engineering Science Department, Benha Faculty of Engineering, Benha University, Benha, Egypt.

Abstract

The current work provides a mathematical second order perturbed singular delay differential model (SO-PSDDM) by using the standard form of the Lane-Emden model. The inclusive structures based on the delay terms, singular-point and perturbation factor and shape forms of the SO-PSDDM are provided. The novel form of the SO-PSDDM is numerically solved by using the procedures of artificial neural networks (ANNs) along with the optimization measures based on the swarming procedures (PSO) and interior-point algorithm (IPA). An error function is optimized through the swarming PSO procedure along with the IPA to solve the SO-PSDDM. The precision, substantiation and validation are observed for three problems of the SO-PSDDM. The exactness of the novel SO-PSDDM is observed by comparing the obtained and exact solutions. The reliability, stability and convergence of the proposed stochastic algorithms are observed for 30 independent trials to solve the novel SO-PSDDM.