Homotopy analysis method for singular IVPs of Emden–Fowler type

Optimum Path of a Flying Object with Exponentially Decaying Density Medium

“…It is found that, when k ≤ −1, the solution of the high-order deformation (19) or (24) contains the terms ln(x) or 1 x s (s ≥ 1), which incidentally disobeys the rule of solution expression (13). When k ≥ 1, the base x 2 always disappears in the solution expression of the high-order deformation (19) or (24), so that the coefficient of the term x 2 cannot be modified even if the order of approximation tends to infinity. Thus, we had to set k = 0, which uniquely determines the corresponding auxiliary function H(τ) = 1.…”

“…According to the rule of solution expression denoted by (13) and from (19), the auxiliary function H(τ) should be in the form H(τ) = τ k , where k is an integer. It is found that, when k ≤ −1, the solution of the high-order deformation (19) or (24) contains the terms ln(x) or 1 x s (s ≥ 1), which incidentally disobeys the rule of solution expression (13).…”

“…According to the rule of solution expression denoted by (13) and from (19), the auxiliary function H(τ) should be in the form H(τ) = τ k , where k is an integer. It is found that, when k ≤ −1, the solution of the high-order deformation (19) or (24) contains the terms ln(x) or 1 x s (s ≥ 1), which incidentally disobeys the rule of solution expression (13). When k ≥ 1, the base x 2 always disappears in the solution expression of the high-order deformation (19) or (24), so that the coefficient of the term x 2 cannot be modified even if the order of approximation tends to infinity.…”

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

“…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]. In the present article, ADM is modified effectively to solve presented singular initial value problem.…”

Haar wavelet collocation method for the approximate solutions of Emden-Fowler type equations

“…Many methods havebeen developed for solving these equations numerically. Some numerical methods in the literature include the following: the homotopy perturbation method (Chowdhury & Hashim, 2009, Ahamed et al, 2017, Adomian decomposition method Wazwaz et al, 2014) the variational iteration method (Wazwaz, 2015), differential transform method (Ibis, 2012), homotopy analysis method (Bataineh et al, 2009), hybrid functions method (Tabrizidooz et al, 2017).…”