  In this paper, two numerical approaches based on the Newton iteration method
with spectral algorithms are introduced to solve the Thomas-Fermi equation.
That Thomas-Fermi equation is a nonlinear singular ordinary differential
equation (ODE) with boundary condition in infinite. In these schemes, the
Newton method is combined with a spectral method where in one of those, by
Newton method we convert nonlinear ODE to a sequence of linear ODE then, solve
them using the spectral method. In another one, by the spectral method the
nonlinear ODE be converted to system of nonlinear algebraic equations, then,
this system is solved by Newton method. In both approaches, the spectral method
is based on fractional order of rational Gegenbauer functions. Finally, the
obtained results of two introduced schemes are compared to each other in
accuracy, runtime and iteration number. Numerical experiments are presented
showing that our methods are as accurate as the best results which obtained
until now.
