Eur. Phys. J. Plus (2022) 137: 638
https://doi.org/10.1140/epjp/s13360-022-02869-3

Review

Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion

¹ Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, UAE
² Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

^a zulqurnain_maths@hu.edu.pk

Received: 2 May 2022
Accepted: 23 May 2022
Published online: 30 May 2022

Abstract

The motive of this work is to provide the neural investigations using the artificial neural networks (ANNs) through the particle swarm optimization for the singular two-point (STP) boundary value problems (BVPs), i.e., STP-BVPs arising in the theory of thermal explosion. The main purpose of this work is to perform the neural studies based on the large and small (45, 15, 3) neurons together with the complexity cost. The neuron performances have been designated in the form of absolute error. The best results have been achieved in case of large neurons as compared to small neurons, but the complexity cost gets high. The optimization measures of an error function are performed by using the swarming computational global search scheme along with the local search interior-point algorithms (IPA) for the STP-BVPs arising in the theory of thermal explosion. The exactness of the proposed scheme is approved by using the comparison of the obtained and reference solutions. Moreover, the generalization of the data is proposed in terms of statistical analysis to substantiate the capability and trustworthiness of the designed approach for the STP-BVPs.