Numerical investigation of third-order Emden–Fowler type equations using unified Bernstein collocation

Abstract The Emden–Fowler (EF) equations are fundamental in modeling complex phenomena across astrophysics, quantum mechanics, and nonlinear science disciplines. This work proposes an efficient computational approach for solving the third-order EF equation with suitable boundary conditions. We transform the differential equation into its integral counterpart to manage the singularity at the origin. The solution is then approximated using a Bernstein polynomial-based collocation technique , with Chebyshev points selected to optimize convergence and stability. This strategy leads to a nonlinear system of equations, which is effectively handled using Newton’s iterative method. We provide a theoretical analysis of the existence and uniqueness of solutions within this framework and establish error estimates for the proposed numerical scheme. Numerical experiments, including comparisons with established techniques, demonstrate that our approach delivers high accuracy and computational efficiency for benchmark problems. These results highlight the method’s potential for reliably addressing singular and nonlinear boundary value problems in scientific modeling.