Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method

Solution of an Integro-Differential Equation Arising in Oscillating Magnetic Fields Using He's Homotopy Perturbation Method

“…The results are compared with fourth-order Runge-Kutta method. The homotopy perturbation method is compared in [27] to some series solution of the Lane-Emden equation. Also He's homotopy perturbation technique and Wazwaz's two implementations of the Adomian method based on either the introduction of a new differential operator that overcomes the singularity of the Lane-Emden at the origin or the elimination of the first-order derivative term of the original equations.…”

An Iterative Method for Solving Two Special Cases of Lane-Emden Type Equation

“…The series often coincides with the Taylor expansion of the true solution at point 0 0 x = , in the value case, although the series can be rapidly convergent in a very small region. Many numerical methods were developed for this type of nonlinear ordinary differential equations, specifically on Lane-Emden type equations such as the Adomian Decomposition Method (ADM) [5], the Homotopy Perturbation Method (HPM) [6] [7], the Homotopy Analysis Method (HAM) [8] and Berstein Operational Matrix of Integration [9]. In this paper, we show superiority of DTM by applying them on the some type LaneEmden type equations.…”

Perturbative solution to the Lane–Emden equation: an eigenvalue approach

“…In addition to its application in astrophysics as mentioned above, the LEE is also interesting in its own right (see, e.g., Ramos 2008, and references therein). Except for some special values of the polytropic index, namely n = 0, 1 and 5, the LEE does not have any known analytical solutions which are expressible in terms ⋆ Email address: yiplongsang@gmail.com † Email address: tkc004@physics.ucsd.edu ‡ Email address: ptleung@phy.cuhk.edu.hk of elementary functions.…”

“…On the other hand, even for 0 < n < 1.9121, the convergence rate of a formal series solution of the LEE could be slow and is in general dependent on both n and the position variable. In order to remedy the problem of divergence or the slow convergence of the series solution, methods of Padé resummation and variable transformation have been employed to extend the interval of validity and accelerate the convergence of the series (Pascual 1977;Iacono & De Felice 2015;Ramos 2008). For example, in order to extend the radius of convergence of the series solution, Roxburgh & Stockman (1999) used the polytropic mass as the independent variable, in place of the radial distance, to extend the interval of convergence down to the surface of polytropes but thousands of terms are needed in order to achieve satisfactory accuracy near the stellar surface.…”