Numerical approximation of Riesz-Feller operators on R

In this paper, we develop an accurate pseudospectral method to approximate numerically the Riesz-Feller operator Dγα on R, where α∈(0,2), and |γ|≤min{α,2−α}. This operator can be written as a linear combination of the Weyl–Marchaud derivatives Dα and Dα¯, when α∈(0,1), and of ∂xDα−1 and ∂xDα−1¯, when α∈(1,2). Given the so-called Higgins functions λk(x)=((ix−1)/(ix+1))k, where k∈Z, we compute explicitly, using complex variable techniques, Dα[λk](x), Dα¯[λk](x), ∂xDα−1[λk](x), ∂xDα−1¯[λk](x) and Dγα[λk](x), in terms of the Gaussian hypergeometric function 2F1, and relate these results to previous ones for the fractional Laplacian. This enables us to approximate Dα[u](x), Dα¯[u](x), ∂xDα−1[u](x), ∂xDα−1¯[u](x) and Dγα[u](x), for bounded continuous functions u(x). Finally, we simulate a nonlinear Riesz-Feller fractional diffusion equation, characterized by having front propagating solutions whose speed grows exponentially in time.