A Reliable Neural Network Procedure for the Novel Sixth-Order Nonlinear Singular Pantograph Differential Model

Abstract

An innovative singular nonlinear sixth-order (SNSO) pantograph di®erential model (PDM), known as the SNSO-PDM, is the subject of this novel study along with its numerical investigation. The concepts of pantograph and conventional Emden-Fowler have been presented in the design of the novel SNSO-PDM. The models based on Emden{Fowler have huge applications in mathematics and engineering and are always di±cult to solve due to singularity. For each class of the innovative SNSO-PDM, the singularity, shape and pantograph factors are described. A reliable stochastic Levenberg-Marquardt backpropagation neural network (LMBPNN) procedure is designed for the SNSO-PDM. The correctness of the SNSOs-PDM is observed through the comparison performances of the achieved and reference outputs. The obtained results of the SNSO-PDM are considered by applying the process of training, certification, and testing to reduce the mean square error. To authenticate the e±cacy of the innovative SNSO-PDM, the numerical performances of the solutions are depicted in the sense of regression, error histograms and correlation. © 2026 World Scientific Publishing Company.