A. Shoja‎ - Department of Mathematics‎, ‎Science and Research Branches‎, ‎Islamic Azad University‎, ‎Tehran‎, ‎Iran.
E. Babolian - Department of Mathematics‎, ‎Science and Research Branches‎, ‎Islamic Azad University‎, ‎Tehran‎, ‎Iran.
A. R. Vahidi - Department of Mathematics‎, ‎Science and Research Branches‎, ‎Islamic Azad University‎, ‎Tehran‎, ‎Iran.

In this paper, a new spectral-iterative method is employed to give approximate solutions of fractional logistic differential equation. This approach is based on combination of two different methods, i.e. the iterative method \cite{۳۵} and the spectral method. The method reduces the differential equation to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. The solutions obtained are compared with Adomian decomposition method and iterative method used in \cite{۳۵‎} and Adams method \cite{۳۶}.‎