Quasilinearization method to solve several classes of nonlinear Lane-Emden equations by Gegenbauer polynomials

This paper presents a comprehensive numerical approach for solving various types of singular nonlinear Lane-Emden equations. The proposed method begins by applying the Quasilinearization Method (QLM), to transform the nonlinear differential equation into a sequence of linear equations. Subsequently, these linear equations are solved using the collocation method with Gegenbauer functions. A notable advantage of Gegenbauer polynomials lies in their adjustable parameter, when appropriately selected, this parameter enhances their effectiveness in obtaining numerical solutions for specific problems. After establishing the convergence of the method, we present numerical examples to demonstrate its effectiveness. A comparison of the results with those obtained using other basis functions highlights the superior performance and applicability of these polynomials.