Fractal scaling of fractional diffusion processes

Abstract

The fractal growth of fractional diffusion is analyzed from the viewpoint of the influence of fractional derivative order on scaling exponents. Fractional diffusion is considered here as deterministic with stochastic forcing, and with time and space fractional derivatives defined as the Caputo and Riesz–Feller forms, respectively. Fractal growth, as characterized by a dynamic scaling exponent, z, roughness, α, and growth exponent, β, is model-dependent, and we show that the exponents are distinguished by the order of fractional derivatives and the form of stochastic process. Finally, scaling exponents for nonconservative (or uncorrelated) and correlated stochastic processes are found in the lowest-order linear fractional differential equations, and the prospect for construction of stochastic nonlinear fractional diffusion evolution equations is explored here. In application to signal processing, this amounts to modeling dynamic processes (signal evolution, image growth, surface and interface growth, structural growth, etc.) and their equivalence (or approximation) in fractional differencing models of discrete data.