Solution of the Volterra Integral Form of the Lane-Emden Equation Using the Variational Iteration Method

Abstract

The Lane-Emden equation plays a pivotal role in modeling astrophysical and fluid dynamic phenomena, such as stellar structures and isothermal gas spheres. Solving this equation poses significant challenges due to its nonlinearity and singularities. While existing literature has addressed these models using the Adomian Decomposition Method, such frameworks require the complex and computationally intensive formulation of Adomian polynomials. This research presents practical ways of applying the Variational Iteration Method as a semi-analytic approach to address these challenges. By obtaining equivalent Volterra integral forms of the Lane-Emden equation and systematically applying Variational Iteration Method, efficient approximate solutions were obtained for five different problems across various parameter cases of shape factor  at and . A favourable series solution was obtained for each of the five problems, and each was independently verified against its closed form exact solution by direct substitution into the given equation and initial conditions. Comparison with these exact solutions showed strong agreement, with absolute errors as small as 10⁻⁷ to 10⁻⁸ for the polynomial case, and 10⁻³ to 10⁻⁵ for the transcendental cases over an identified domain of validity for each problem. These approaches contribute to advancing mathematical modeling in astrophysics and provide a good framework for solving nonlinear differential equations, as shown with the tested problems.