Discrete random walk models for space-time fractional diffusion

By Rudolf Gorenflo, Francesco Mainardi, Daniele Moretti, Gianni Pagnini, Paolo Paradisi. 2002

305 citations recorded by the source corpus.

Abstract A physical–mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space–time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2] and skewness θ (|θ|⩽min{α,2−α}), and the first-order time derivative with a Caputo derivative of order β∈(0,1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/a7734479fb472cdc1c32c535f8a21ab89d505376

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In Applied Mathematics

1. Fawang Liu
2. H. Zhan
3. Quanzhong Huang
4. Vo Anh
5. Ralf Metzler
6. Joseph Klafter
7. Wen Chen
8. H. Zhang

In Modeling and Simulation

1. Fawang Liu
2. Vo Anh
3. H. Zhan
4. Quanzhong Huang
5. Joseph Klafter
6. H. Zhang
7. Ralf Metzler
8. Wen Chen

Citations in the corpus

• A Convergent Algorithm for Solving Higher-Order Nonlinear Fractional Boundary Value Problems (2015, unknown): “see, by way of example not exhaustive enumeration, [1], [2], [7], [9], [12], [13], [16], [20], [21], [22], [24], [28], [31], [32], [41], [43], [37], [44] and the many references therein.”
• Weighted average finite difference methods for fractional diffusion equations (2004, unknown): “[ 20 ,21], but for a fractional diffusion equation formally different from the one considered in this paper, Eq. (1).”
• The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey (2010, unknown): “[7, 8, 9], [20, 21], [29, 30], [71].”
• A speculative study of 23-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. (2006, background): “( 16 ) can be considered the turbulence diffusivity which leads to the enhanced”
• An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations (2003, unknown): “[20], who presented a scheme for building discrete models of random walks suitable for the Monte Carlo simulation of random variables with a probability density governed by fractional diffusion equations.”
• Anomalous diffusion of proteins due to molecular crowding. (2005, background): “However, whereas Fick’s law is an established phenomenological law for diffusion in isotropic fluids, there is no physical reason why it should always apply to more complex systems (20).”
• Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation (2007, background): “[5–9] considered the numerical approximation for the following cases: (1) β = 1, α ∈ (0, 2] (α 6= 1), θ = 0, (2) β = 1, α ∈ (0, 2] (α 6= 1), | θ |≤ min{α, 2−α}, (3) α = 2, β ∈ (0, 1], θ = 0, and (4) α ∈ (0, 2] (α 6= 1), β ∈ (0, 1], θ = 0, respectively.”
• A finite element solution for the fractional advection–dispersion equation (2008, methodology): “Efforts have been made for developing numerical solutions of the FADE with the finite difference (FD) method [11,22,23,25,26], the finite volume (FV) method [52,53], the random walk (RW) method [14,15,47,27], the random walk particle tracking (RWPT) method [49,50], as well as the finite element (FEM”
• The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics (2004, methodology): “…we employ for determining the PDF defined by the LFFPE are mainly based on the Grünwald–Letnikov representation of the fractional Riesz derivative (Podlubny 1998, Gorenflo 1997, Gorenflo et al 2002a), details of which are also described in Chechkin et al (2003b); see also Lynch et al (2003)20.”
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• Time-space fabric underlying anomalous diffusion (2005, methodology): “of fractional derivatives is typically used in the corresponding phenomenological continuum modeling, which in isotropic linear media is stated as [ 1 ,3,4,13] oas ota þ cð�r 2Þbs ¼ 0; 0 � a; b 6 1; ð2Þ”