An novel efficient and high accuracy numerical method for the fractional differential equations(FDEs) is proposed in this paper. We establish the equivalence between FDEs and the extended problem of fractional differential equations(EFDEs), and prove the stability of EFDEs . In the discretization, we apply BDF-$k$ formula to approximate temporal derivative, and analyze the stability of the temporal discretization scheme for EFDEs. Due to the fact that the solution of EFDEs perform well in the regularity of the expanded direction, the spectral collocation method for spatial is presented. We also provide the error estimate which shows that our $k$ order in temporal and collocation spectral method in spatial converges with order $O(\Delta t^{k} + M^{-m})$, where $\Delta t$, $M$ and $m$ are time step size, number of collocation nodes in $\theta$ direction and regularity index in spatial, respectively.The computational cost and the storage requirement of our scheme are $O(M)$. We demonstrated the effectiveness of our method by linear and non-linear examples with singular solution.