Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations (Communications in Computational Physics, 2018)
The computational work and storage of numerically solving the time fractionalPDEs are generally huge for the traditional direct methods since they requiretotal $\mathcal{O}$($N_S$$N_T$) memory and $\mathcal{O}$(($N_S$$N^2_T$) work, where $N_T$ and $N_S$ represent the totalnumber of time steps and grid points in space, respectively. To overcome this difficulty,we present an efficient algorithm for the evaluation of the Caputo fractional derivative $^C_0$$D^α_t$$f(t)$ of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials(SOE) approximation for the kernel $t^{−1−α}$ on the interval [∆t,T] with a uniform absoluteerror ε. We give the theoretical analysis to show that the number of exponentials $N_{exp}$ needed is of order $\mathcal{O}$($log$$N_T$) for T≫1 or $\mathcal{O}$($log^2N_T$) for T≈1 for fixed accuracyε. The resulting algorithm requires only $\mathcal{O}$($N_SN_{exp}$) storage and $\mathcal{O}$($N_SN_TN_{exp}$) workwhen numerically solving the time fractional PDEs. Furthermore, we also give the stabilityand error analysis of the new scheme, and present several numerical examplesto demonstrate the performance of our scheme.