Chelyshkov polynomials approximation for solving a class of linear and nonlinear equations arising in astrophysics

The aim of this paper is to present a numerical approach for solving linear and nonlinear differential equations arising in astrophysics, commonly known as Lane-Emden equations. The proposed method is based on the collocation method and first involves taking the truncated Chelyshkov series of the function in the equation. The computational cost is reduced due to the orthogonality of Chelyshkov polynomials, and the solution of a linear or nonlinear Lane-Emden equation is reduced to solving a system of linear or nonlinear algebraic equations. Several test examples of these types of differential equations, modeling different physical problems with initial and boundary conditions, are solved to demonstrate the reliability of the method. To demonstrate its effectiveness, absolute error tables and graphs are presented, and the numerical results are compared with other methods and exact solutions. It is observed that when the exact solution has a polynomial form, the proposed method proves to be highly accurate and effective.