Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Abstract. The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order µ ∈ (0, 1) to compute that of any order k + µ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order µ ∈ (0, 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at “interior” JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo ...

• Type: Journal
• Year: 2016
• Where: JCPHY
• Authors: Yujian Jiao, Li-Lian Wang, Can Huang
• Keywords: JCPHY 2016