Numerical Approximation of Riesz-Feller Operators on $\mathbb R$

Abstract: In this paper, we develop an accurate pseudospectral method to approximate numerically the Riesz-Feller operator $D_\gamma^\alpha$ on $\mathbb R$, where $\alpha\in(0,2)$, and $|\gamma|\le\min{\alpha, 2 - \alpha}$. This operator can be written as a linear combination of the Weyl-Marchaud derivatives $\mathcal{D}^{\alpha}$ and $\overline{\mathcal{D}^\alpha}$, when $\alpha\in(0,1)$, and of $\partial_x\mathcal{D}^{\alpha-1}$ and $\partial_x\overline{\mathcal{D}^{\alpha-1}}$, when $\alpha\in(1,2)$. Given the so-called Higgins functions $\lambda_k(x) = ((ix-1)/(ix+1))^k$, where $k\in\mathbb Z$, we compute explicitly, using complex variable techniques, $\mathcal{D}^{\alpha}\lambda_k$, $\overline{\mathcal{D}^\alpha}\lambda_k$, $\partial_x\mathcal{D}^{\alpha-1}\lambda_k$, $\partial_x\overline{\mathcal{D}^{\alpha-1}}\lambda_k$ and $D_\gamma^\alpha\lambda_k$, in terms of the Gaussian hypergeometric function ${}2F_1$, and relate these results to previous ones for the fractional Laplacian. This enables us to approximate $\mathcal{D}^{\alpha}u$, $\overline{\mathcal{D}^\alpha}u$, $\partial_x\mathcal{D}^{\alpha-1}u$, $\partial_x\overline{\mathcal{D}^{\alpha-1}}u$ and $D\gamma^\alphau$, 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.