Eighth-Order Accurate Methods for Boundary Value Problems Arising from the Lane-Emden Equation

TL;DR

This paper develops and validates eighth-order accurate numerical methods for solving boundary value problems from the Lane-Emden equation, addressing singularities and outperforming existing techniques in accuracy and efficiency.

Contribution

The paper introduces high-order numerical schemes specifically designed for Lane-Emden boundary value problems, including rigorous proofs of convergence and accuracy, and demonstrates their superiority over existing methods.

Findings

Methods achieve eighth-order accuracy and convergence.

Numerical experiments confirm theoretical accuracy and efficiency.

Schemes outperform existing methods in solving Lane-Emden problems.

Abstract

This paper presents high-order numerical methods for solving boundary value problems associated with the Lane-Emden equation, which frequently arises in astrophysics and various nonlinear models. A major challenge in studying this equation lies in its singularity at one endpoint. Prior to constructing the numerical methods, we establish the existence and uniqueness of the solution and propose a continuous iterative method. This continuous method is then discretized using the trapezoidal quadrature rule enhanced with correction terms. As a result, we derive three discrete iterative schemes tailored for three specific cases of the Lane-Emden equation. We rigorously prove that the proposed methods achieve eighth-order accuracy and convergence. A series of numerical experiments is conducted to validate the theoretical findings and demonstrate the accuracy and convergence order of the proposed schemes, which outperform existing methods. These schemes thus provide efficient tools for solving Lane-Emden boundary value problems and can be readily extended to higher-order nonlinear singular models, such as Emden-Fowler equations, which arise in many applications.