?There is much literature available regarding the subject of solving the two-dimensional steady-state transport equation that it would be impossible to mention all of them. Nevertheless, the literature concerning convergence and the estimative of the error is scarce. Therefore in this thesis we focus our attention in this direction. In the first part we study the spectral Chebyshev polynomial expansion combined with the Sumudu transform leading to solve, analytically , the neutron transport equation in isotropic one-dimensional media. Next we study the convergence as well as an estimative of error for the spectral solution of the isotropic two-dimenssional discrete ordinates problem where a special quadrature rule is used to discretize in the angular variables, approximating the scalar flux. Finally the spectral equation is derived in a three dimensional setting.