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An 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.

The pantograph-catenary system is essential to the operation of modern railways. The overhead catenary system serves to deliver steady electric power to the trains running on the railways. The propagation of waves in the catenary system can be significantly affected by its interaction with the pantograph. To investigate the wave transmission via the contact wires of a catenary system, a simplified model composed of a pre-tensioned wire suspended by periodic spring supports is adopted. For a periodic structure with wider band gaps (or stop bands), a larger cluster of frequencies of the transmitting waves can be filtered out. This is beneficial to the maintenance of the catenary system as unwanted vibrations have been reduced. To widen the band gaps, a resonator can be equipped on each of the spring supports for attenuation of a wider range of transmitting waves in the pre-tensioned wire. In this study, a unit cell conceived as the spring–resonator–wire unit is adopted to formulate the dispersion equation in closed form, from which the critical condition for widening the band gaps is derived. In addition, the moving pantograph is modeled as a moving force. From the exemplar study, it was shown that the installation of proper resonators on a catenary system can increase the gap bandwidth, such that the pantograph-induced wave transmission in the contact wires will be attenuated or filtered out.