We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying $Z_{q}$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

翻译：我们提出了一种通过Riesz分数阶积分在全实轴上数值计算分数阶导数（或分数阶拉普拉斯算子）的方法。将紧化后的实轴划分为多个区间，从而构成一种多区域方法；依据确保各被积函数解析性的$Z_{q}$曲线进行变换后，采用Clenshaw-Curtis算法计算不同区域上的积分。作为示例，我们考虑了分数阶Korteweg-de Vries方程的孤立波，并将其与通过离散傅里叶变换获得的结果进行了比较。