  In this paper, we develop two fast implicit difference schemes for solving a
class of variable-coefficient time-space fractional diffusion equations with
integral fractional Laplacian (IFL). The proposed schemes utilize the graded
$L1$ formula for the Caputo fractional derivative and a special finite
difference discretization for IFL, where the graded mesh can capture the model
problem with a weak singularity at initial time. The stability and convergence
are rigorously proved via the $M$-matrix analysis, which is from the spatial
discretized matrix of IFL. Moreover, the proposed schemes use the fast
sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the
computational cost for the nonlocal property of time and space fractional
derivatives, respectively. The fast schemes greatly reduce the computational
work of solving the discretized linear systems from $\mathcal{O}(MN^3 + M^2N)$
by a direct solver to $\mathcal{O}(MN(\log N + N_{exp}))$ per preconditioned
Krylov subspace iteration and a memory requirement from $O(MN^2)$ to
$O(NN_{exp})$, where $N$ and $(N_{exp} \ll)~M$ are the number of spatial and
temporal grid nodes. The spectrum of preconditioned matrix is also given for
ensuring the acceleration benefit of circulant preconditioners. Finally,
numerical results are presented to show the utility of the proposed methods.
