The fractional Laplacian in R d, which we write as (− Δ) α/2 with α∈(0, 2), has multiple
equivalent characterizations. Moreover, in bounded domains, boundary conditions must be …

We report on recent progress in the study of nonlinear diffusion equations involving
nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous …

Partial differential equations (PDEs) are used with huge success to model phenomena
across all scientific and engineering disciplines. However, across an equally wide swath …

This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace
equation. For this problem weighted and fractional Sobolev a priori estimates are provided …

Let L=− div x (A (x)∇ x) be a uniformly elliptic operator in divergence form in a bounded
domain Ω. We consider the fractional nonlocal equations {L su= f, in Ω, u= 0, on∂ Ω, and {L …

We present three schemes for the numerical approximation of fractional diffusion, which
build on different definitions of such a non-local process. The first method is a PDE approach …

The fractional Laplacian (-Δ)^α/2 is a nonlocal operator which depends on the parameter α
and recovers the usual Laplacian as α→2. A numerical method for the fractional Laplacian is …

We describe the mathematical theory of diffusion and heat transport with a view to including
some of the main directions of recent research. The linear heat equation is the basic …

We present and study a novel numerical algorithm to approximate the action of $ T^\beta:=
L^{-\beta} $ where $ L $ is a symmetric and positive definite unbounded operator on a …

The method of semigroups is a unifying, widely applicable, general technique to formulate
and analyze fundamental aspects of fractional powers of operators L and their regularity …