Design of Artificial Neural Networks for the Novel Applications of the Sixth Order Singular Nonlinear Pantograph Engineering Model

Abstract

This aim of this study is to design and numerically investigate a novel nonlinear singular sixth order (SSO) pantograph differential system (PDS), i.e., SSO-PDS along with its five categories. The design of the novel nonlinear SSO-PDS is presented using the concepts of the standard Emden-Fowler along with the pantograph differential system. The singular systems are applied in many engineering and mathematical applications, like inverse models and viscoelasticity systems. The shape factor, singular point and pantograph terms are provided for each category of the novel nonlinear SSO-PDS. The numerical investigations of the nonlinear SSO-PDS are presented through the artificial neural networks (ANNs) using the Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The proposed results have been compared with the reference solutions to check the correctness of the designed novel model. The proposed solutions of the novel SSO-PDS are assessed using the training, testing and verification procedures in the form of mean square error (MSE). For the capability, efficacy and accuracy of the novel SSO-PDS through the ANNs, the numerical results are drawn using the MSE, error histograms and regression as well as correlation graphs.

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

“…Recently, many systems that have been designed with the terminology of the Lane-Emden model. Few of them are the singular 2nd and 3rd pantograph model, 4th, 5th order and 6th kinds of functional singular systems, prediction, delay and pantograph form of the singular models [36][37][38][39][40][41][42][43][44] . Based on these applications, the authors are inspired to construct a novel SO-PSDDM.…”