A reliable iterative method for solving Volterra integro-differential equations and some applications for the Lane–Emden equations of the first kind

In this paper, we implement the new iterative method proposed by Daftardar-Gejji and Jafari namely new iterative method (DJM) to solve the linear and non-linear Volterra integro-differential equations and systems of linear and non-linear Volterra integro-differential equations. The applications of the DJM for solving the resulting equations of the non-linear Volterra integro-differential equations forms of the Lane–Emden equations are presented. The Volterra integro-differential equations forms of the Lane–Emden equation overcome the singular behaviour at the origin x = 0 of the original differential equation. Some examples are solved and different cases of the Lane–Emden equations of first kind are presented. Moreover, the DJM is applied to solve the system of the linear and non-linear Volterra integro-differential forms of the Lane–Emden equations. The results demonstrate that the method has many merits such as being derivative-free, and overcoming the difficulty arising in calculating Adomian polynomials to handle the non-linear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM) and no need to construct a homotopy in Homotopy Perturbation Method (HPM) and solve the corresponding algebraic equations.