@Article{IJNAM-15-628,
author = {Lyngaas , Isaac and Peterson , Janet},
title = {Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2018},
volume = {15},
number = {4-5},
pages = {628--648},
abstract = {<p style="text-align: justify;">The goal of this work is to solve nonlocal diffusion and anomalous diffusion problems
by approximating the nonlocal integral appearing in the integro-differential equation by novel
quadrature rules. These quadrature rules are derived so that they are exact for a nonlocal integral
evaluated at translations of a given radial basis function (RBF). We first illustrate how to derive
RBF-generated quadrature rules in one dimension and demonstrate their accuracy for approximating
a nonlocal integral. Once the quadrature rules are derived as a preprocessing step, we
apply them to approximate the nonlocal integral in a nonlocal diffusion problem and when the
temporal derivative is approximated by a standard difference approximation a system of difference
equations are obtained. This approach is extended to two dimensions where both a circular
and rectangular nonlocal neighborhood are considered. Numerical results are provided and we
compare our results to published results solving nonlocal problems using standard finite element
methods.</p>},
issn = {2617-8710},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnam/12535.html}
}