Transparent boundary conditions for a diffusion problem modified by hilfer derivative

We consider a homogeneous fractional diffusion problem in an infinite reservoir sometimes called a "modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate "artificial" boundary conditions. This is to avoid the effect of reflected waves in case of a "solid" standard boundary. Then, an equivalent problem is studied in this bounded domain. To this end we use the Laplace-Fourier transform, the two-parameter Mittag-Leffler function and some properties of fractional derivatives.