Eur. Phys. J. Plus (2020) 135: 410
https://doi.org/10.1140/epjp/s13360-020-00424-6

Regular Article

Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden–Fowler equation

¹ Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
² Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, 64002, Yunlin, Taiwan, ROC
³ Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan
⁴ Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan

^* e-mail: zulqurnain_maths@hu.edu.pk

Received: 22 March 2020
Accepted: 28 April 2020
Published online: 19 May 2020

Abstract

In this study, a novel neuro-swarming computing solver is developed for numerical treatment of third-order nonlinear multi-singular Emden–Fowler equation (TONMS-EFE) by using function approximation ability of artificial neural networks (ANNs) modeling and global optimization mechanism of particle swarm optimization (PSO) integrated with local search of interior-point scheme (IPS), i.e., ANN-PSO-IPS. The inspiration for the design of ANN-PSO-IPS-based numerical solver comes with an objective of presenting a reliable, accurate and viable structure that combines the strength of ANNs optimized with the integrated soft computing frameworks to deal with such challenging systems based on TONMS-EFE. The proposed ANN-PSO-IPS is implemented for four variants of TONMS-EFEs, and comparison with exact solutions relieved its robustness, correctness and effectiveness, which is further authenticated through statistical explorations.