We address two problems in the study of fractional diffusion in finite size domains. The first problem focusses on the physics limitations of the standard fractional diffusion model based on Levy flights driven by alpha-stable distributions. It is shown that tempered Levy distributions, and their corresponding tempered fractional diffusion operators, offer a more realistic model to incorporate finite size effects. The second problem deals with the incorporation of physically meaningful and mathematically well-posed boundary conditions in fractional diffusion. The standard fractional diffusion model in a bounded domain is in general singular. Our approach is based on the regularization of the singularities at the boundary. Following the formal construction of the regularized fractional diffusion model we present numerical solutions.

In astrophysical and other applications, plasma is often in a
collisionless state, where resistive and viscous effects are not
important on the time/spatial scales of interest. Extended MHD (XMHD) is
a one-fluid Hamiltonian theory endowed with 2-fluid effects, which are
thought to be important for the formation of relativistic jets from
active galactic nuclei, micro-quasars, and gamma-ray bursts. We
consider the relativistic [1] generalization of XMHD, obtained via an
action principle (AP). We describe covariant Poisson bracket in terms
of Eulerian variables, with constraints implemented via the degeneracy
of the Poisson bracket.

Upon taking appropriate limits, AP leads to relativistic Hall MHD.
While nonrelativistic HMHD does not have a direct mechanism for
collisionless reconnection, relativistic HMHD allows the violation of
the frozen-in magnetic flux condition via an electron thermal inertia
effect. An alternative frozen-in flux has also been found in a manner
similar to that for nonrelativistic IMHD. The scale length of the
collisionless reconnection is shown to correspond to the reconnection
layer width estimated by the Sweet-parker model [2].

Finally we perform a 3+1 decomposition. This results in a bracket that
is more general than the one derived earlier by Abdelhamid at al. [3],
as it is valid for arbitrary electron to ion mass ratio and can be
applied, for instance, to electron-positron plasmas. Further, the
Casmir invariants one obtains this way turn out to be precisely the
generalized helicities, which topologically constrain possible evolution
of the plasma.

[1] Y. Kawazura, G. Miloshevich, P. J. Morrison, Phys. Plasmas 24,
022103 (2017)

[2] L. Comisso and F. A. Asenjo, PRL 113, 045001 (2014)

[3] H. M. Abdelhamid, Y. Kawazura, and Z. Yoshida, J. Phys.A 48, 235502
(2015)