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 ${}_2F_1$, 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.

翻译：本文提出一种精确的伪谱方法，用于在实数域$\mathbb R$上数值逼近Riesz-Feller算子$D_\gamma^\alpha$，其中$\alpha\in(0,2)$，且$|\gamma|\le\min\{\alpha, 2 - \alpha\}$。当$\alpha\in(0,1)$时，该算子可表示为Weyl-Marchaud导数$\mathcal{D}^{\alpha}$与$\overline{\mathcal{D}^\alpha}$的线性组合；当$\alpha\in(1,2)$时，则可表示为$\partial_x\mathcal{D}^{\alpha-1}$与$\partial_x\overline{\mathcal{D}^{\alpha-1}}$的线性组合。针对Higgins函数$\lambda_k(x) = ((ix-1)/(ix+1))^k$（其中$k\in\mathbb Z$），我们利用复变技术显式计算了$\mathcal{D}^{\alpha}[\lambda_k](x)$、$\overline{\mathcal{D}^\alpha}[\lambda_k](x)$、$\partial_x\mathcal{D}^{\alpha-1}[\lambda_k](x)$、$\partial_x\overline{\mathcal{D}^{\alpha-1}}[\lambda_k](x)$及$D_\gamma^\alpha[\lambda_k](x)$，这些结果均以高斯超几何函数${}_2F_1$表示，并与分数阶Laplacian的已有研究成果建立了关联。基于此，我们可对连续有界函数$u(x)$的$\mathcal{D}^{\alpha}[u](x)$、$\overline{\mathcal{D}^\alpha}[u](x)$、$\partial_x\mathcal{D}^{\alpha-1}[u](x)$、$\partial_x\overline{\mathcal{D}^{\alpha-1}}[u](x)$及$D_\gamma^\alpha[u](x)$进行数值近似。最后，我们模拟了一个非线性Riesz-Feller分数阶扩散方程，其特征是其传播前沿解的速度随时间呈指数增长。