  In this paper, we study the variable-order (VO) time-fractional diffusion
equations. For a VO function $\alpha(t)\in(0,1)$, we develop an
exponential-sum-approximation (ESA) technique to approach the VO Caputo
fractional derivative. The ESA technique keeps both the quadrature exponents
and the number of exponentials in the summation unchanged at the different time
levels. Approximating parameters are properly selected to achieve efficient
accuracy. Compared with the general direct method, the proposed method reduces
the storage requirement from $\mathcal{O}(n)$ to $\mathcal{O}(\log^2 n)$ and
the computational cost from $\mathcal{O}(n^2)$ to $\mathcal{O}(n\log^2 n)$,
respectively, with $n$ being the number of the time levels. When this fast
algorithm is exploited to construct a fast ESA scheme for the VO
time-fractional diffusion equations, the computational complexity of the
proposed scheme is only of $\mathcal{O}(mn\log^2 n)$ with
$\mathcal{O}(m\log^2n)$ storage requirement, where $m$ denotes the number of
spatial grids. Theoretically, the unconditional stability and error analysis of
the fast ESA scheme are given. The effectiveness of the proposed algorithm is
verified by numerical examples.
