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, ${\mathcal{D}}^{\alpha }[{\lambda }_k](x)$, $\overline{{\mathcal{D}}^{\alpha }}[{\lambda }_k](x)$, ${\partial }_x{\mathcal{D}}^{\alpha -1}[{\lambda }_k](x)$, ${\partial }_x\overline{{\mathcal{D}}^{\alpha -1}}[{\lambda }_k](x)$ and $D_{\gamma }^{\alpha }[{\lambda }_k](x)$, in terms of the Gaussian hypergeometric function ${}_2{F}_1$, and relate these results to previous ones for the fractional Laplacian. This enables us to approximate ${\mathcal{D}}^{\alpha }[u](x)$, $\overline{{\mathcal{D}}^{\alpha }}[u](x)$, ${\partial }_x{\mathcal{D}}^{\alpha -1}[u](x)$, ${\partial }_x\overline{{\mathcal{D}}^{\alpha -1}}[u](x)$ and $D_{\gamma }^{\alpha }[u](x)$, 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.

28 pages, 4 figures, 2 Matlab listings