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Title: Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions
Authors: Huang, Can
Jiao, Yujian
Wang, Li-Lian
Zhang, Zhimin
Keywords: Fractional Differential Equations
Riemann–Liouville Fractional Derivative
Issue Date: 2016
Source: Huang, C., Jiao, Y., Wang, L.-L., & Zhang, Z. (2016). Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions. SIAM Journal on Numerical Analysis, 54(6), 3357-3387.
Series/Report no.: SIAM Journal on Numerical Analysis
Abstract: In this paper, a nonpolynomial-based spectral collocation method and its well-conditioned variant are proposed and analyzed. First, we develop fractional differentiation matrices of nodal Jacobi polyfractonomials [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517] and generalized Jacobi functions [S. Chen, J. Shen, and L. L. Wang, Math. Comp., 85 (2016), pp. 1603--1638] on Jacobi--Gauss--Lobatto (JGL) points. We show that it suffices to compute the matrix of order $\mu\in (0,1)$ to compute that of any order $k +\mu$ with integer $k \geq 0$. With a different definition of the nodal basis, our approach also fixes a deficiency of the polyfractonomial fractional collocation method in [M. Zayernouri and G. E. Karniadakis, SIAM J. Sci. Comput., 38 (2014), pp. A40--A62]. Second, we provide explicit and compact formulas for computing the inverse of direct fractional differential collocation matrices at “interior” points by virtue of fractional JGL Birkhoff interpolation. This leads to optimal integration preconditioners for direct fractional collocation schemes and results in well-conditioned collocation systems. Finally, we present a detailed analysis of the singular behavior of solutions to rather general fractional differential equations (FDEs). Based upon the result, we have the privilege to adjust an index in our nonpolynomial approximation. Furthermore, by using the result, a rigorous convergence analysis is conducted by transforming an FDE into a Volterra (or mixed Volterra--Fredholm) integral equation.
ISSN: 0036-1429
DOI: 10.1137/16M1059278
Rights: © 2016 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Numerical Analysis and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The published version is available at: []. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.
Fulltext Permission: open
Fulltext Availability: With Fulltext
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