Robust fast method for variable-order time-fractional diffusion equations without regularity assumptions

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