Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case

Emad Awad, Trifce Sandev, Ralf Metzler and Aleksei Chechkin

Chaos, Solitons & Fractals, 2021, vol. 152, issue C

Abstract: Numerous anomalous diffusion processes are characterized by crossovers of the scaling exponent in the mean squared displacement at some correlations time. The bi-fractional diffusion equation containing two time-fractional derivatives is a versatile mathematical tool describing specifically retarded subdiffusive transport, in which the scaling exponents acquires a smaller value, i.e., the diffusion becomes even slower after the crossover. We here derive closed-form multi-dimensional solutions for this integro-differential equation in n spatial dimensions by generalizing the classical Schneider-Wyss solution of the fractional diffusion equation with a single fractional derivative. In the two-dimensional case we develop a limiting approach based on the solution of the space-time fractional diffusion equation. The probabilistic interpretation in higher dimensions is discussed. The asymptotic long- and short-time behaviors are derived. It is shown that the solution of the bi-fractional diffusion equation can be interpreted in terms of the Fox H-transform of the Gaussian distribution.

Keywords: Retarding anomalous diffusion; Caputo fractional derivative; Bi-fractional diffusion equation; Fox H-function; Schneider-Wyss solution (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:152:y:2021:i:c:s0960077921007116

DOI: 10.1016/j.chaos.2021.111357

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