Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions
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 g...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2017
|
| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/83949 http://hdl.handle.net/10220/42900 |
| _version_ | 1811689648971317248 |
|---|---|
| author | Huang, Can Jiao, Yujian Wang, Li-Lian Zhang, Zhimin |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Huang, Can Jiao, Yujian Wang, Li-Lian Zhang, Zhimin |
| author_sort | Huang, Can |
| collection | NTU |
| description | 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. |
| first_indexed | 2024-10-01T05:51:27Z |
| format | Journal Article |
| id | ntu-10356/83949 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:51:27Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/839492023-02-28T19:39:26Z Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions Huang, Can Jiao, Yujian Wang, Li-Lian Zhang, Zhimin School of Physical and Mathematical Sciences Fractional Differential Equations Riemann–Liouville Fractional Derivative 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. MOE (Min. of Education, S’pore) Published version 2017-07-18T04:50:04Z 2019-12-06T15:35:10Z 2017-07-18T04:50:04Z 2019-12-06T15:35:10Z 2016 Journal Article 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. 0036-1429 https://hdl.handle.net/10356/83949 http://hdl.handle.net/10220/42900 10.1137/16M1059278 en SIAM Journal on Numerical Analysis © 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. 31 p. application/pdf |
| spellingShingle | Fractional Differential Equations Riemann–Liouville Fractional Derivative Huang, Can Jiao, Yujian Wang, Li-Lian Zhang, Zhimin Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title | Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title_full | Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title_fullStr | Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title_full_unstemmed | Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title_short | Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions |
| title_sort | optimal fractional integration preconditioning and error analysis of fractional collocation method using nodal generalized jacobi functions |
| topic | Fractional Differential Equations Riemann–Liouville Fractional Derivative |
| url | https://hdl.handle.net/10356/83949 http://hdl.handle.net/10220/42900 |
| work_keys_str_mv | AT huangcan optimalfractionalintegrationpreconditioninganderroranalysisoffractionalcollocationmethodusingnodalgeneralizedjacobifunctions AT jiaoyujian optimalfractionalintegrationpreconditioninganderroranalysisoffractionalcollocationmethodusingnodalgeneralizedjacobifunctions AT wanglilian optimalfractionalintegrationpreconditioninganderroranalysisoffractionalcollocationmethodusingnodalgeneralizedjacobifunctions AT zhangzhimin optimalfractionalintegrationpreconditioninganderroranalysisoffractionalcollocationmethodusingnodalgeneralizedjacobifunctions |