2017/09/29 by Beichuan Deng, Zhimin Zhang, Deng, Beichuan +3 · 1 citation
Mathematics · #26A33 #41A05 #41A10 #65M15 #65N35 #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1709.10223
openalex publication_date 2017/09/29 · openalex created_date 2017/10/06 · openalex updated_date 2026/07/28
In this paper, superconvergence points are located for the approximation of the Riesz derivative of order α using classical Lobatto-type polynomials when α∈ (0,1) and generalized Jacobi functions (GJF) for arbitrary α> 0, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different α at the superconvergence points is at least O(N-2) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order α> 1 with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be O(N-(α+3)/2) for α∈ (0,1) and O(N-2) for α> 1. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.