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.
翻译:在本文中,我们为移动-移动可变序(VO)时间折射式扩散方程式(tFDEs)开发了一种强大的快速方法,该方程式优于处理小的或消失的VO函数下下限的情况。VO Caputo分解衍生物的有效快速近似值分别使用部件集成法和指数-总和(指数-总和)法获得。与一般直接法相比,拟议的算法($-L1美元公式)将动作内存从$\mathcal{O}(log_2 n)减到$\mathcal{O}(n_2美元)和计算成本从$\mathcal{O}(n_2美元)到$\mathcal{O}(n\log%2 n),而美元是按时间水平乘以美元获得的。与一般直接法($-L1美元公式)相比,为VO tFDE制定快速限差法(快速缩小记忆要求和计算复杂性。拟议办法的计算方法的误差数估计数仅根据常规方法进行研究,但不作假设,只是对数值的假设。