A robust Legendre collocation scheme and convergence analysis for Lane-Emden multi-pantograph delay differential equations arising in stellar physics including Chandrasekhar’s white dwarf problem

In recent years, Lane-Emden pantograph delay differential equations (PDDEs) have become a vital focus of research due to their extensive applications in astrophysics and various engineering fields. These equations serve as powerful tools for modelling complex phenomena across diverse domains, including astronomical systems, nonlinear wave propagation, control engineering, neural networks, and cognitive science. The distinctive features of PDDEs make them especially effective for capturing and predicting the dynamics of systems characterised by inherent time delays, nonlinear responses, time-dependent behaviors, and feedback mechanisms. Developing efficient solutions for PDDEs poses significant challenges due to their complex nonlinearity and singularity. This article introduces an effective numerical approach that integrates Legendre polynomials with the collocation technique to solve a specific class of nonlinear Lane-Emden PDDEs efficiently. The mathematical formulation is reinforced through a comprehensive convergence analysis, demonstrating the reliability and accuracy of the proposed method. Several examples related to astrophysics, including Chandrasekhar’s white dwarf problems, are presented to demonstrate the practical relevance and efficiency of the method. The proposed approach exhibits faster convergence compared to existing wavelet and polynomial-based techniques, highlighting its ability in accurately capturing solutions with reduced computational effort (takes ≈14 times less CPU time). The method demonstrates exceptional precision, yielding significantly lower absolute and maximum absolute errors compared to existing techniques, and closely aligning the numerical results with the exact solutions. These qualities underscore its robust potential for solving complex delay differential equations with superior accuracy and efficiency.