有理数阶分数阶导数的多区域谱方法
Multi-domain spectral approach to rational-order fractional derivatives
AI总结：
提出基于Riesz积分的多区域谱方法，通过$Z_{q}$曲线变换和Clenshaw-Curtis算法计算实轴上的分数阶导数，并验证于分数阶KdV方程孤立波求解。
摘要：
我们提出了一种通过Riesz分数阶积分在整个实数轴上数值计算分数阶导数（或分数阶拉普拉斯算子）的方法。紧致化的实数轴被划分为多个区间，从而构成多区域方法；在根据确保相应被积函数解析性的底层$Z_{q}$曲线进行变换后，使用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，并将其与离散傅里叶变换获得的结果进行了比较。
英文摘要：
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.
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