In recent literature, for modeling reasons, fractional differential problems have been considered equipped with anti-symmetric boundary conditions. Twenty years ago the anti-reflective boundary conditions were introduced in a context of signal processing and imaging for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here we consider the anti-symmetric boundary conditions and we introduce the anti-reflective boundary conditions in the context of nonlocal problems of fractional differential type. In the latter context, we study both types of boundary conditions, which in reality are similar in the essentials, from the perspective of computational efficiency, by considering nontruncated and truncated versions. Several numerical tests, tables, and visualizations are provided and critically discussed.

翻译：近期文献中，出于建模需要，分数阶微分问题被假设具有反对称边界条件。二十年前，反反射边界条件在信号处理与成像领域被提出，旨在提高受噪声污染的模糊信号/图像重建质量，并将整体复杂度降低至仅需若干快速正弦变换，即$O(N\log N)$次实数算术运算，其中$N$为像素数。本文考虑反对称边界条件，并在分数阶微分型非局域问题背景下引入反反射边界条件。在后一背景下，我们从计算效率角度研究这两种本质相似的边界条件，涵盖非截断与截断版本。文中提供了大量数值测试、表格与可视化结果，并进行批判性讨论。