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Time Fractional Diffusion: A Discrete Random Walk Approach

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Abstract

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.

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Gorenflo, R., Mainardi, F., Moretti, D. et al. Time Fractional Diffusion: A Discrete Random Walk Approach. Nonlinear Dynamics 29, 129–143 (2002). https://doi.org/10.1023/A:1016547232119

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