Mark M. Meerschaert

Mehdi Samiee, Mohsen Zayernouri, Mark M. Meerschaert

We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in \cite{samiee2017Unified}, for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any ($1+d$)-dimensional space-time hypercube, $d = 1, 2, 3, \cdots$, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.

Boris Baeumer, Mihály Kovács, Mark M. Meerschaert et al.

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.