A reliable algorithm for the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics

In this paper, we present a reliable algorithm to obtain the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics. The suggested algorithm is based upon the operational matrix of integration for Jacobi polynomials and the collocation method. Convergence analysis and numerical stability of the suggested method are provided. Numerical results for several interesting nonlinear cases of the Lane‐Emden type equations such as the standard Lane‐Emden equation, the white‐dwarf equation, and the isothermal gas spheres equation, as well as Richardson's theory of thermionic current are discussed. These numerical results are shown in the form of tables and figures for the particular cases of Jacobi polynomials such as the Legendre polynomial (P1), the Chebyshev polynomials of the second kind (P2), the Chebyshev polynomials of the third kind (P3), the Chebyshev polynomials of the fourth kind (P4), and the Gegenbauer (or ultraspherical) polynomials (P5). Numerical results are also compared with those that were derived earlier by applying some well‐known and recently developed numerical methods and it is observed that our numerical results are more accurate. The maximum absolute errors and the root mean square errors are calculated by using P1, P2, P3, P4, and P5 for comparison purposes.