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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.
This paper presents a novel semi-analytical collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method it employs the Fourier series expansion for spatial discretization, wh
The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional prob
We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due both to the kernel singularity in the
Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed c
We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients o