Fractional differential problems with numerical anti-reflective boundary conditions from a numerical linear algebra perspective: A computational study with an extensive numerical validation

In the current work, we propose numerical anti-reflective boundary conditions (BCs) in the context of nonlocal problems of fractional differential type: the numerical linear algebra goal is a O(NlogN) complexity of the resulting direct and iterative algorithms, accompanied by a qualitative better approximation, with the mitigation of boundary artifacts. In fact, for showing the quality of the numerical anti-reflective BCs, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflective BCs, we compare them from the perspective of computational efficiency, by considering nontruncated and truncated versions, and also other standard numerical BCs such as periodic BCs or reflective/Neumann BCs. A short theoretical analysis and several numerical tests, tables, and visualizations are provided and critically discussed. The conclusion is that the truncated numerical anti-reflective BCs perform better, both in terms of low computational cost and accuracy.