Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case. (November 2021)

• Record Type: Journal Article
• Authors: Awad, Emad; Sandev, Trifce; Metzler, Ralf; Chechkin, Aleksei
• Is Part Of: Chaos, solitons and fractals. Volume 152(2021)
• Publication Date: 2021-11
• Subjects: Retarding anomalous diffusion -- Caputo fractional derivative -- Bi-fractional diffusion equation -- Fox H-function -- Schneider-Wyss solution; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2021.111357
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: We present higher-dimensional solutions to the distributed-order time-fractional diffusion equation modelling the crossover between two anomalous-diffusion regimes, here the retarding case is analysed. We derive closed-form solutions and demonstrate their positivity. Asymptotic behaviours at short and long times are discussed. 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.