dc.contributor.author Huang, Can dc.contributor.author Jiao, Yujian dc.contributor.author Wang, Li-Lian dc.contributor.author Zhang, Zhimin dc.date.accessioned 2017-07-18T04:50:04Z dc.date.available 2017-07-18T04:50:04Z dc.date.issued 2016 dc.identifier.citation 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. en_US dc.identifier.issn 0036-1429 en_US dc.identifier.uri http://hdl.handle.net/10220/42900 dc.description.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. en_US dc.description.sponsorship MOE (Min. of Education, S’pore) en_US dc.format.extent 31 p. en_US dc.language.iso en en_US dc.relation.ispartofseries SIAM Journal on Numerical Analysis en_US dc.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: [http://dx.doi.org/10.1137/16M1059278]. 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. en_US dc.subject Fractional Differential Equations en_US dc.subject Riemann–Liouville Fractional Derivative en_US dc.title Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions en_US dc.type Journal Article dc.contributor.school School of Physical and Mathematical Sciences en_US dc.identifier.doi http://dx.doi.org/10.1137/16M1059278 dc.description.version Published version en_US
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