A sparse spectral method for fractional differential equations in one-spatial dimension

Abstract

We develop a sparse spectral method for a class of fractional differential equations, posed on R, in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on [−1, 1] whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping [−1, 1] to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size O(n) × O(n), with O(n) nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an O(n) complexity solve. Applications to fractional heat and wave equations are considered.