  This work explores different particle-based approaches to the simulation of
one-dimensional fractional subdiffusion equations in unbounded domains. We rely
on smooth particle approximations, and consider four methods for estimating the
fractional diffusion term. The first method is based on direct differentiation
of the particle representation, it follows the Riesz definition of the
fractional derivative and results in a non-conservative scheme. The other three
methods follow the particle strength exchange (PSE) methodology and are by
construction conservative, in the sense that the total particle strength is
time invariant. The first PSE algorithm is based on using direct
differentiation to estimate the fractional diffusion flux, and exploiting the
resulting estimates in an integral representation of the divergence operator.
Meanwhile, the second one relies on the regularized Riesz representation of the
fractional diffusion term to derive a suitable interaction formula acting
directly on the particle representation of the diffusing field. A third PSE
construction is considered that exploits the Green's function of the fractional
diffusion equation. The performance of all four approaches is assessed for the
case of a one-dimensional diffusion equation with constant diffusivity. This
enables us to take advantage of known analytical solutions, and consequently
conduct a detailed analysis of the performance of the methods. This includes a
quantitative study of the various sources of error, namely filtering,
quadrature, domain truncation, and time integration, as well as a space and
time self-convergence analysis. These analyses are conducted for different
values of the order of the fractional derivatives, and computational
experiences are used to gain insight that can be used for generalization of the
present constructions.
