Fractional flux and non-normal diffusion

Abstract

Fractional diffusion equations are widely used for mass spreading in heterogeneous media. The correspondence between fractional equations and random walks based upon stable Levy laws, keeps in analogy with that between heat equation and Brownian motion. Several definitions of fractional derivatives yield operators, which coincide on a wide domain and can be used in fractional partial differential equations. Then, the various definitions are useful in different purposes: they may be very close to some physics, or to numerical schemes, or be based upon important mathematical properties. Here we present a definition, which enables us to describe the flux of particles, performing a random walk. We show that it is a left inverse to fractional integrals. Hence it coincides with Riemann-Liouville and Marchaud's derivatives when applied to functions, belonging to suitable domains.

Keywords

• Diffusion equations
• fractional calculus
• random walks
• stable probability distributions