Lane−Emden Equation: Picard vs Pade

Introduction The investigation of Lane Emden HLEEL equation has a long standing history Hsee, e.g., @1 10D, to mention only part of a vast literatureL. Recently Schaudt @6D has shown that Picard type iteration scheme may be used to show the existence, uniqueness and regularity of global solutions of LEE , in particular, for n 3 1. In this paper it was shown that straightforward usage of Picard iteration at general case of arbitrary n Heven for n 3 1L is not possible analytically, and so we apply this procedure for particular case of n = 2. The iterations are converging and allow to obtain the solution up to first zero of solution with great accuracy. Then we consider the modification of iteration scheme allowing to get very accurate Pade approximations and the values of relevant parameter at first zero of Lane Emden function. We conclude that integral form of LEE allows to get HanalyticalL approximations in the terms of economized series Hrather polynomialsL and of Pade approximants in the way which is not straightforward from differential form of LEE. Presented results , for the case of n = 2, are more accurate than ones known in literature.