Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method

Abstract

Series solutions of the Lane–Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables. It is shown that these four series solutions are the same as those obtained by a direct application of Adomian’s decomposition method to the original differential equation, He’s homotopy perturbation technique, and Wazwaz’s two implementations of the Adomian method based on either the introduction of a new differential operator that overcomes the singularity of the Lane–Emden equation at the origin or the elimination of the first-order derivative term of the original equation. It is also shown that Adomian’s decomposition technique can be interpreted as a perturbative approach which coincides with He’s homotopy perturbation method. An iterative technique based on Picard’s fixed-point theory is also presented and its convergence is analyzed. The convergence of this iterative approach depends on the independent variable and, therefore, this technique is not as convenient as the series solutions derived by the four methods presented in this paper, He’s homotopy perturbation technique, and Adomian’s decomposition method.