Numerical Solution of Singular Lane-Emden Equation

A new approach for solving the nonlinear Lane-Emden type equations has been proposed. The method is based on Legendre wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution. 1. Introduction The Lane-Emden type equations are nonlinear ordinary differential equations on semi-infinite domain. They are categorized as singular initial value problems. These equations describe the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. The polytrophic theory of stars essentially follows out of thermodynamic considerations that deal with the issue of energy transport, through the transfer of material between different levels of the star. These equations are one of the basic equations in the theory of stellar structure and have been the focus of many studies. The general form of the Lane-Emden equations is the following form: with the following initial conditions: where is a continuous real-value function and ？？is an analytical function. Equation (1) was 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 sphere, and theory of thermionic currents [1, 2]. The solution of the Lane-Emden equation, as well as those of a variety of nonlinear problems in quantum mechanics and astrophysics such as the scattering length calculations in the variable phase approach, is numerically challenging because of the singular point at the origin. Bender et al. [3] proposed a new perturbation technique based on an artificial parameter？？ ; the method is often called -method. El-Gebeily and O’Regan [4] used the quasilinearization approach to solve the standard Lane-Emden equation. This method approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories, it is not based on the existence of some small parameters. Approximate solutions to the above problems were presented by Shawagfeh [5] and Wazwaz [6, 7] by applying the Adomian method which provides a convergent series solution. Nouh [8] accelerated the convergence of a power series solution of the Lane-Emden equation by using an Euler-Abel transformation and Padé approximation. Mandelzweig and Tabakin [9] applied Bellman and Kalaba’s quasilinearization method, and Ramos [10] used a