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

Abstract

AbstractIn 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 , 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 . More precisely, in the fractional scenario the symbol vanishes at 0 with order , the fractional derivative order that ranges from 1 to 2, and it decays exponentially to zero at for increasing at a rate that becomes faster as 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 for even , and for odd .