Solution of the Lane-Emden Equation Using the Bernstein Operational Matrix of Integration

Abstract

Lane-Emden's equation has fundamental importance in the recent analysis of many problems in relativity and astrophysics including some models of density profiles for dark matter halos. An efficient numerical method is presented for linear and nonlinear Lane-Emden-type equations using the Bernstein polynomial operational matrix of integration. The proposed approach is different from other numerical techniques as it is based on the Bernstein polynomial integration matrix. Some illustrative examples are given to demonstrate the efficiency and validity of the proposed algorithm.

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.…”

Bernstein Series Solution of a Class of Lane-Emden Type Equations

“…In 2012, Pandey and coworkers [19][20][21][22] studied five methods. First, Pandey et al [19] gave a numerical method for solving linear and nonlinear Lane-Emden type equations using Legendre operational matrix of differentiation.…”

“…Second, Pandey et al [20] studied a numerical method to solve linear and nonlinear Lane-Emden type equations using Chebyshev wavelet operational matrix. Third, Kumar et al [21] presented a method for linear and nonlinear Lane-Emden type equations using the Bernstein polynomial operational matrix of integration. Fourth, Pandey and Kumar [22] proposed a numerical method for solving Lane-Emden type equations arising in astrophysics using Bernstein polynomials.…”

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

“…• Bernstein operational matrix of integration, Kumar et al [20]; • Optimal homotopy asymptotic method, Iqbal and Javed [22]. • Pandey and Kumar [21].…”