2017/05/17 by Kun Wang, Wang, Kun, Jizu Huang +1
Engineering · Mathematics · #Advanced Control Systems Design #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1705.06101
openalex publication_date 2017/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In the paper, we present a high order fast algorithm with almost optimum memory for the Caputo fractional derivative, which can be expressed as a convolution of u'(t) with the kernel (tn-t)-α. In the fast algorithm, the interval [0,tn-1] is split into nonuniform subintervals. The number of the subintervals is in the order of log n at the n-th time step. The fractional kernel function is approximated by a polynomial function of K-th degree with a uniform absolute error on each subinterval. We save K+1 integrals on each subinterval, which can be written as a convolution of u'(t) with a polynomial base function. As compared with the direct method, the proposed fast algorithm reduces the storage requirement and computational cost from O(n) to O((K+1)log n) at the n-th time step. We prove that the convergence rate of the fast algorithm is the same as the direct method even a high order direct method is considered. The convergence rate and efficiency of the fast algorithm are illustrated via several numerical examples.