  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.

Numerical Analysis