Time Fractional Diffusion: A Discrete Random Walk Approach

• Published in: Nonlinear dynamics Vol. 29; no. 1-4; pp. 129 - 143
• Main Authors: Gorenflo, Rudolf, Mainardi, Francesco, Moretti, Daniele, Paradisi, Paolo
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01-07-2002
• Subjects: Cauchy problems; Density; Diffusion rate; Finite difference method; Markov processes; Random variables; Random walk; Self-similarity
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1023/A:1016547232119

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.