Nonlocal modeling of spatial fractional diffusion with truncated interaction domains and truncated kernel function singularity

Parabolic partial differential equations (PDEs) are in ubiquitous, very effective use to model diffusion processes. However, there are many applications (e.g., such as in hydrology, animal foraging, biology, and light diffusion just do name a few) for which results obtained through the use of parabolic PDEs do not agree with observations. In many situations the use of fractional diffusion models has been found to be more faithful to that which is observed. Specifically, we replace the Laplacian operator in the PDE by a fractional Laplacian operator ${\mathcal L}$ which is an integral operator for which solutions are sought for on all of space, has an unbounded domain of integration, and for a given point $x$ the integrand contains a kernel function $\phi(y-x)$ that is infinite whenever $y=x$. These three features pose impediments not only for the construction of efficient discretization methods but also because all three involve one or more sort of "infinity''. To overcome these impediments we choose to invoke one or more of the following strategies. (a) We seek solutions only within a chosen bounded domain $\Omega$. (b) For every $x\in\Omega$, we choose a bounded domain of integration such as, e.g., an Euclidean ball $B_\delta(x)$ having finite radius $\delta$. (c) We truncate the singularity of $\phi(y-x)$ by setting, for a given constant $\varepsilon>0$, $\phi(y-x)= \phi(\varepsilon)$ whenever $|y-x|\le\varepsilon$. We then provide extensive illustrations of the possible combinations chosen from among (a), (b), and (c). We also illustrate, for the various models defined for each of these combinations, their limiting behavior of solutions such as showing that as $\delta\to0$ we recover the PDE model and also showing that in the limit of some other parameters we recover the fractional Laplacian model.