New Solutions for Singular Lane-Emden Equations Arising in Astrophysics Based on Shifted Ultraspherical Operational Matrices of Derivatives

In this paper, the ultraspherical operational matrices of derivatives are constructed. Based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems. The basic idea behind the suggested algorithms is basically built on transforming the equations with their initial conditions into systems of linear or nonlinear algebraic equations which can be solved by using suitable numerical solvers. The Legendre and first and second kind Chebyshev operational matrices of derivatives can be deduced as special cases of the constructed operational matrices. For the sake of testing the validity and applicability of the suggested numerical algorithms, three illustrative examples are presented.