Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type

A modified Adomian decomposition method for singular initial value Emden-Fowler type equations

“…Parand et al applied an approximation algorithm for the solution of this problem using Hermite functions, as basis functions, and collocation method [2]. Parand et al also adopted a pseudospectral technique based on the rational Legendre functions and Gauss-Radau integration to handle this problem [3]. For further methods, the reader is referred to the references [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations

“…This equation is one of the basic equations in the theory of stellar structure and has been the focus of many studies. In recent years, the approximate solutions to the Lane-Emden equation were given by homotopy perturbation method [15,23], the Legendre wavelets [24], perturbation method [10], the Adomian decomposition method [21], the Bessel collocation method [25], the Pade series method [19], the rational Legendre pseudospectral method [14], the Taylor series method [11], the nonperturbative approximate method [18], and the Hermite functions collocation method [13]. The numerical solving of the Lane-Emden problem, is challenging because of the nonlinearity and singular behavior at the origin.…”

Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

“…In recent years, Many powerful methods have been presented for solving of Lane-Emden type equations. For instance, the homotopy perturbation method [4,6], the Legendre wavelets [7], the variational iteration method [8,9], the B-spline method [10], the Adomian decomposition method [11], the Bessel collocation method [12], the Pade series method [13], the rational Legendre pseudospectral method [14], the nonperturbative approximate method [15], the Hermite functions collocation method [16], and the variational approach method [17]. Continuous or piecewise polynomials are incredibly useful mathematical tools as they are precisely defined, calculated rapidly on a modern computer system and can represent a great variety of functions.…”