Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions

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.