Analytical approximate solutions for linear and nonlinear Volterra integral and integrodifferential equations and some applications for the Lane-Emden equations using a power series method

Abstract

In the present paper, we have presented a recursive method namely the Power Series Method (PSM)to solve first the linear and nonlinear Volterra integral and integro-differential equations. The PSM isemployed then to solve resulting equations of the nonlinear Volterra integral and integro-differentialforms of the Lane-Emden equations. The Volterra integral and integro-differential equations formsof the Lane-Emden equation overcome the singular behavior at the origin x = 0. Some examplesare solved and different cases of the Lane-Emden equations of first kind are presented. The resultsdemonstrate that the method has many merits such as being derivative-free, and overcoming thedifficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian DecompositionMethod (ADM). It does not require to calculate Lagrange multiplier as in VariationalIteration Method (VIM) and no need to construct a homotopy as in Homotopy Perturbation Method(HPM). The results prove that the present method is very effective and simple and does not requireany restrictive assumptions for nonlinear terms. The software used for the calculations in this studywas MATHEMATICA.