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In this paper, we develop fast procedures for solving linear systems arisingfrom discretization of ordinary and partial differential equations with Caputofractional derivative w.r.t time variable. First, we consider a finitedifference scheme to solve a two-sided fractional ordinary equation.Furthermore, we present a fast solution technique to accelerate Toeplitzmatrix-vector multiplications arising from finite difference discretization.This fast solution technique is based on a fast Fourier transform and dependson the special structure of coefficient matrices, and it helps to reduce thecomputational work from $O(N^{3})$ required by traditional methods to$O(Nlog^{2}N)$ and the memory requirement from $O(N^{2})$ to $O(N)$ withoutusing any lossy compression, where $N$ is the number of unknowns. Two finitedifference schemes to solve time fractional hyperbolic equations with differentfractional order $\gamma$ are considered. We present a fast solution techniquedepending on the special structure of coefficient matrices by rearranging theorder of unknowns. It helps to reduce the computational work from $O(N^2M)$required by traditional methods to $O(N$log$^{2}N)$ and the memory requirementfrom $O(NM)$ to $O(N)$ without using any lossy compression, where $N=\tau^{-1}$and $\tau$ is the size of time step, $M=h^{-1}$ and $h$ is the size of spacestep. Importantly, a fast method is employed to solve the classical timefractional diffusion equation with a lower coast at $O(MN$log$^2N)$, where thedirect method requires an overall computational complexity of $O(N^2M)$.Moreover, the applicability and accuracy of the scheme are demonstrated bynumerical experiments to support our theoretical analysis.

Fast procedures for Caputo fractional derivative and its applications to ordinary and partial differential equations