Preconditioning fractional spectral collocation

TL;DR

This paper introduces fractional integration preconditioning matrices for fractional spectral collocation methods, significantly improving the conditioning of linear systems independent of collocation points, thus enhancing numerical stability.

Contribution

It proposes a novel preconditioning approach based on fractional Birkhoff interpolation to address ill-conditioning in FSC linear systems.

Findings

Condition numbers become independent of collocation points.

Preconditioning improves numerical stability.

Numerical examples validate effectiveness.

Abstract

Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely ill-conditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the ill-conditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.