Eur. Phys. J. Plus (2021) 136: 195
https://doi.org/10.1140/epjp/s13360-021-01171-y

Regular Article

Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm

¹ Department of Mathematics and Statistics, Hazara University Mansehra, Mansehra, Pakistan
² Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Private Bag X2046, Mafikeng Campus, 2735, Mmabatho, Republic of South Africa
³ College of Mathematics and Systems Science, Shandong University of Science and Technology, 266590, Qingdao, Shandong, China
⁴ Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, 64002, Douliou, Yunlin, Taiwan, ROC
⁵ Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, 43600, Attock, Pakistan
⁶ Department of Mathematics, Cankaya University, Ankara, Turkey
⁷ Institute of Space Science, Magurele-Bucharest, Romania

^a zulqurnain_maths@hu.edu.pk

Received: 21 August 2020
Accepted: 29 January 2021
Published online: 9 February 2021

Abstract

In this numerical study, a class of nonlinear singular boundary value problem is solved by implementation of a novel meta-heuristic computing tool based on the artificial neural networks (ANNs) modeling of system and the optimization of decision variable of ANNs through the combined strength of global search via genetic algorithms (GA) and local search ability of active-set algorithm (ASA), i.e., ANN–GA–ASA. The proposed intelligent computing solver ANN–GA–ASA exploits the input, hidden, and output layers’ structure of ANNs. This is to represent the differential model in the nonlinear singular second-order periodic boundary value problems, which are connected to form an error-based objective function (OF) and optimize the OF by the integrated heuristics of GA–ASA. The purpose to present this research is to associate the operational legacy of neural networks and to challenge such kinds of inspiring models. Two different examples of the singular periodic model have been investigated to observe the robustness, proficiency and stability of the ANN–GA–ASA. The proposed outcomes of ANN–GA–ASA are compared with reference to true results so as to establish the value of the designed scheme. Exhaustive comparison has been made and presented between the Log-sigmoidal ANNs results and the radial basis ANNs outcomes. The reliability of the results obtained is endorsed by using both types of networks as well as the value of designed schemes.