Approximate polynomial solutions of the nonlinear Lane–Emden type equations arising in astrophysics using the squared remainder minimization method

Laguerre polynomial approach for solving Lane–Emden type functional differential equations

“…Consequently, the obtained solutions for various particular cases demonstrate the validity and applicability of the method compare to the other existing methods [11][12][13][14][15][16][17][18]. We predict that the Laguerre expansion method will be a promising method for investigating analytic solutions to Lane-Emden equation.…”

Constructing analytic approximate solutions to the Lane–Emden equation

“…It is therefore unclear why so many papers in recent years have focused on using more complicated approaches (Adomian decomposition, Homotopy analysis method, Boubaker polynomials, among others; see [9] and references therein) to derive alternative approximations over [0,1]. Often in these papers important works on the properties of series solutions to the LEE, such as [10] and [11], are not cited, and a detailed comparison with relevant previous work is lacking.…”

“…In the last decade, the search for approximate solutions to the LEE has produced many papers (see, e.g., the list given in the introduction of [9]), but, apparently, few useful results. A problem is that most of these works restrict to the interval [0, 1], denoted as the "core region" in astrophysical contexts, even though the radial ranges of interest are typically much larger.…”

A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

“…Due to the nonlinear term e −y the above equation was solved by different methods such as SRM [32], ADM [1] and DTM [13] as the equation has no exact solution so we compared our results with ADM [1], HNN [22] and ANN [33] to demonstrate the strength of proposed method and results are shown in Figure 4 and Table 5. Table 6 displayed the value of weights after training.…”

“…The results are compared with ADM [1], squared remainder minimization method (SRM) [32], DTM [13], HNN [22] and ANN [33] that can be seen in Table 3. Graphical comparison of LSANN with ADM [1], HNN [22] and ANN [33] can be visualized in Figure 3.…”