Appl. Math. Inf. Sci. 9, No. 5, 2429-2436 (2015) 2429 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090526 Solving Two Emden–Fowler Type Equations of Third Order by the Variational Iteration Method Abdul-Majid Wazwaz ∗ Department of Mathematics, Saint Xavier University, Chicago, IL 60655, U.S.A Received: 27 Jan. 2015, Revised: 28 Apr. 2015, Accepted: 29 Apr. 2015 Published online: 1 Sep. 2015 Abstract: In this paper, we establish two kinds of Emden–Fowler type equations of third order. We investigate the linear and the nonlinear third-order equations, with specified initial conditions, by using the systematic variational iteration method. We corroborate this study by investigating several Emden–Fowler type examples with initial value conditions. Keywords: Emden–Fowler equation; variational iteration method; Lagrange multipliers. 1 Introduction Many scientific applications in the literature of mathematical physics and fluid mechanics can be distinctively described by the Emden–Fowler equation y ′′ + k x y ′ + f (x)g(y)= 0, y(0)= y 0 , y ′ (0)= 0, (1) where f (x) and g(y) are some given functions of x and y, respectively, and k is called the shape factor. The Emden– Fowler equation (1) describes a variety of phenomena in fluid mechanics, relativistic mechanics, pattern formation, population evolution and in chemically reacting systems. For f (x)= 1 and g(y)= y m , Eq. (1) becomes the standard Lane–Emden equation of the first order and index m, given by y ′′ + k x y ′ + y m = 0, y(0)= y 0 , y ′ (0)= 0, (2) The Lane–Emden equation (2) models the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules [1–12] and subject to the classical laws of thermodynamics. Moreover, the Lane–Emden equation of first order is a useful equation in astrophysics for computing the structure of interiors of polytropic stars. On the other side, for f (x)= 1 and g(y)= e y , Eq. (1) becomes the standard Lane–Emden equation of the second order that models the non-dimensional density distribution y(x) in an isothermal gas sphere [9–22] Moreover, the Lane–Emden equation (2) describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of thermodynamics. In addition, the Lane-Emden equation of the first kind appears also in other context such as in the case of radiatively cooling, self-gravitating gas clouds, in the mean-field treatment of a phase transition in critical adsorption and in the modeling of clusters of galaxies [17–19]. The Lane-Emden equation was first studied by astrophysicists Jonathan Homer Lane and Robert Emden, where they considered the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. The well-known Lane-Emden equation has been used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, the theory of thermionic currents, and in the modeling of clusters of galaxies. The Emden–Fowler equation was studied by Fowler [2] to describe a variety of phenomena in fluid mechanics and relativistic mechanics among others. The singular behavior that occurs at x = 0 is the main difficulty of Equations (1) and (2). ∗ Corresponding author e-mail: wazwaz@sxu.edu c  2015 NSP Natural Sciences Publishing Cor.