Sabir, ZulqurnainSalahshour, SoheılUmar, MuhammadSalahshour, SoheilSaeed, Tareq2024-09-112024-09-11202400217-98491793-664010.1142/S02179849245047362-s2.0-85200398229https://doi.org/10.1142/S0217984924504736https://hdl.handle.net/20.500.14517/6247Salahshour, Soheil/0000-0003-1390-3551; Saeed, Tareq/0000-0002-0170-5286; sabir, zulqurnain/0000-0001-7466-6233An innovative singular nonlinear sixth-order (SNSO) pantograph differential 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 difficult 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 efficacy of the innovative SNSO-PDM, the numerical performances of the solutions are depicted in the sense of regression, error histograms and correlation.eninfo:eu-repo/semantics/closedAccessPantographsixth orderEmden-Fowlerneural networkLevenberg-Marquardt backpropagationArticleQ2Q2WOS:001280068500006