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 ℝ, in one dimension. These equations can 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 can 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.