New efficient numerical methods for solving boundary value problems of the Lane–Emden equation

This paper presents new eighth-order numerical methods for solving singular boundary value problems arising from the Lane–Emden equation, which appears in theoretical chemistry, chemical physics, and biophysics. A major difficulty in solving this equation is the singularity at the origin. Consequently, existing numerical methods typically achieve at most seventh-order accuracy. To overcome this accuracy limitation, we propose a novel approach for this singular problem. We first establish the existence and uniqueness of the solution by reformulating the problem as a nonlinear operator equation and constructing a continuous iterative scheme. Based on this framework, we develop three eighth-order schemes using corrected trapezoidal quadrature formulas derived from the Euler–Maclaurin expansion. Rigorous theoretical analysis confirms convergence and establishes the eighth-order accuracy of the proposed methods. Extensive numerical experiments validate the theoretical results and demonstrate the eighth-order convergence of the proposed schemes compared with existing methods.