Hybrid Methods of Series Decomposition and Legendre Collocation for the Emden‐Fowler Equation With Blow‐Up

This paper proposes a series decomposition algorithm and two Legendre collocation methods to solve the Emden‐Fowler equation with probably non‐smooth or blow‐up solution. By reformatting the equation into an equivalent integral form, we derive a psi‐series solution that includes logarithmic polynomials via successive iteration, which achieves high accuracy near the origin. By cutting off a small interval containing the singularity, we develop the Legendre collocation methods in both integral form with Legendre‐Gauss points and differential form using Legendre‐Gauss‐Radau points to solve the problem on the regular interval. We also prove the convergence of the Legendre‐Gauss method in integral form. For the blow‐up problem, we apply the Padé technique to the series solution to evaluate the blow‐up time and discuss the blow‐up behavior of the solution. We then improve the accuracy of the Legendre collocation methods by removing the blow‐up singularity from the equation. Several examples illustrate that the hybrid methods in this paper can achieve high accuracy for the Emden‐Fowler equation with initial conditions.