An advance computational intelligent approach to solve the third kind of nonlinear pantograph Lane–Emden differential system

This study presents an advanced computational Levenberg–Marquardt backpropagation (LMB) neural network for the novel third order (NTO) pantograph Emden–Fowler system (PEFS), i.e., (NTO-PEFS) together with its two forms. The designed novel NTO-PEFS is achieved using the pantograph system and standard form of the Emden–Fowler system. The detail of each form of the NTO-PEFS based on the singular points, pantographs and shape factors is also provided. The numerical performance using the LMB neural network is tested for three different variants of the model and obtained results will be compared through the designed dataset based exact solutions. To assess the approximate solutions of the NTO-PEFS for both forms of each example, the process of testing, authentication and training are implemented to reduce the mean square error (MSE) based on the LMB. One can find the values based absolute error are close to 10–04 to 10–08 for each problem to solve the NTO-PEFS using the stochastic computing paradigms. The relative studies and performance investigations for the error histograms, regression, correlation and MSE enhance the effectiveness as well as the exactness of the designed LMB neural network scheme.