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

We develop a sparse spectral method for a class of fractional differential equations, posed on \(\mathbb {R}\), 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 \(\mathcal {O}(n)\times \mathcal {O}(n)\), with \(\mathcal {O}(n)\) 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 \(\mathcal {O}(n)\) complexity solve. Applications to fractional heat and wave equations are considered.