On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol vanishes at 0 with order alpha, the fractional derivative order that ranges from 1 to 2, and it decays exponentially to zero at pi for increasing p at a rate that becomes faster as alpha approaches 1. This translates into a mitigated conditioning in the low frequencies and into a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with nonfractional diffusion problems, the approximation order for smooth solutions in the fractional case is p+2-alpha for even p, and p+1-alpha for odd p.