Robust fast method for variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

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 VO function. The valid approximation Caputo fractional derivative is obtained using integration by parts and exponential-sum-approximation method. Compared with general direct method, proposed algorithm ($RF$-$L1$ formula) reduces acting memory from $\mathcal{O}(n)$ to $\mathcal{O}(\log^2 n)$ computational cost $\mathcal{O}(n^2)$ $\mathcal{O}(n \log^2 n)$, respectively, where $n$ number time levels. Then $RF$-$L1$ formula applied construct finite difference scheme tFDEs, which sharp decreases requirement complexity. error estimate studied only under some assumptions function, coefficients, source term, but without any regularity assumption true solutions. Numerical experiments are presented verify effectiveness