  In this paper, we develop a robust fast method for mobile-immobile
variable-order (VO) time-fractional diffusion equations (tFDEs), superiorly
handling the cases of small or vanishing lower bound of the VO function. The
valid fast approximation of the VO Caputo fractional derivative is obtained
using integration by parts and the exponential-sum-approximation method.
Compared with the general direct method, the proposed algorithm ($RF$-$L1$
formula) reduces the acting memory from $\mathcal{O}(n)$ to $\mathcal{O}(\log^2
n)$ and computational cost from $\mathcal{O}(n^2)$ to $\mathcal{O}(n \log^2
n)$, respectively, where $n$ is the number of time levels. Then $RF$-$L1$
formula is applied to construct the fast finite difference scheme for the VO
tFDEs, which sharp decreases the memory requirement and computational
complexity. The error estimate for the proposed scheme is studied only under
some assumptions of the VO function, coefficients, and the source term, but
without any regularity assumption of the true solutions. Numerical experiments
are presented to verify the effectiveness of the proposed method.
