Exponential-sum-approximation technique for variable-order time-fractional diffusion equations

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 $\mathcal{O}(\log² n)$ and the computational cost from $\mathcal{O}(n^2)$ to $\mathcal{O}(n\log² n)$, 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 $\mathcal{O}(mn\log² n)$ 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.