TY - JOUR
T1 - A Fast Algorithm for the Caputo Fractional Derivative
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 656
EP - 677
PY - 2018
DA - 2018/10
SN - 8
DO - http://doi.org/10.4208/eajam.080418.200618 
UR - https://global-sci.org/intro/article_detail/eajam/12813.html
KW - Caputo derivative, fast algorithm, polynomial approximation, error, graded mesh.
AB - <p style="text-align: justify;">A fast algorithm with almost optimal memory for the computation of Caputo’s
fractional derivative is developed. It is based on a nonuniform splitting of the
time interval [0, $t_n$] and a polynomial approximation of the kernel function $(1−τ)^{−α}$.
Both the storage requirements and the computational cost are reduced from $\mathscr{O}(n)$ to $(K +1)\mathscr{O}(log~n)$ with $K$ being the degree of the approximated polynomial. The algorithm
is applied to linear and nonlinear fractional diffusion equations. Numerical results show
that this scheme and the corresponding direct methods have the same order of convergence
but the method proposed performs better in terms of computational time.</p>