Design of Backpropagated Intelligent Networks for Nonlinear Second-Order Lane–Emden Pantograph Delay Differential Systems

In physical science, nonlinear singular Lane–Emden and pantograph delay differential equations (LE–PDDEs) have abundant applications and thus are of great interest for the researchers. The presented investigation is related to the development of a new application of intelligent computing for the solution of the LE–PDDEs-based system introduced recently by merging the essence of delay differential equation of Pantograph type and standard second-order Lane–Emden equation. Intelligent computing is exploited through Levenberg–Marquardt backpropagation networks (LMBNs) and Bayesian regularization backpropagation networks (BRBNs) to provide the solutions to nonlinear second-order LE–PDDEs. The performance of design LMBNs and BRBNs is substantiated on three different case studies through comparative analysis from known exact/explicit solutions. The correctness of the designed solvers for LE–PDDEs is further certified by accomplishing through assessment on error histograms, regression measures and index of mean squared error.