Stochastic representation of solution to nonlocal-in-time diffusion

Du, Qiang; Toniazzi, Lorenzo

doi:10.1016/j.spa.2019.06.011

Cited by 18 publications

(9 citation statements)

References 36 publications

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“…Similar studies have also been made for models that are nonlocal in time and space (Chen, Du, Li and Zhou 2017), which are generalizations of models developed in Du, Yang and Zhou (2017b) and Du, Toniazzi and Zhou (2020b). Du et al (2017bDu et al ( , 2020b only considered nonlocal memory/history effects in time but the spatial interactions remained local. Chen et al (2017) also replaced the local spatial differential operators with nonlocal operators.…”

Section: Finite Difference Methods For the Strong Form Of Nonlocal DImentioning

confidence: 79%

Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Section: Finite Difference Methods For the Strong Form Of Nonlocal DImentioning

confidence: 79%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…The proofs there are different from ours and scope of that two papers are restricted to Feller generators in Euclidean spaces. See also [14] for a recently related work where the time fractional derivative having a possibly time-dependent kernel and the spatial operator is a Dirichlet Laplacian in a regular Euclidean domain.…”

Section: Introductionmentioning

confidence: 99%

Time fractional Poisson equations: Representations and estimates

Chen

Kim

Kumagai

et al. 2020

Journal of Functional Analysis

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In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel q(t, x, y), which we call the fundamental solution for the time fractional Poisson equation, if the semigroup for L has an integral kernel. We further show that q(t, x, y) can be expressed as a time fractional derivative of the fundamental solution for the homogenous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution q(t, x, y) through explicit estimates of transition density functions of subordinators. ∞ 0 min{1, x}(−dw(x)) < ∞, a generalized time fractional derivative with weight w is defined by∂ w t f (t) := d dt t 0 whenever the right hand side is well defined. See [8, 13, 24, 27]. In particular, when w(s) = 1 Γ(1−β) s −β for β ∈ (0, 1) (where Γ(t) = ∞ 0 s t−1 e −s ds is the Gamma function), ∂ w t f is just the Caputo derivative of order β, i.e., ∂ β t f (t) := 1 Γ(1 − β) d dt t 0 Zhen-Qing Chen

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“…In case that s −1 ρ δ (s) is unbounded only at the origin, the integral in (1.2) should be interpreted as the limit of the integral of the same integrand over ( , δ) for > 0 as → 0, where such a limit exists in an appropriate mathematical sense. The nonlocal operator G δ [11,8,9] forms part of the nonlocal vector calculus [6,21]. The positive nonlocal horizon parameter δ appearing in (1.2) represents the range of nonlocal interactions or memory span.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of this paper is to present the modeling capability and the behavior of the solutions of the nonlocal-intime dynamics (1.1), so as to demonstrate how the simple PDE model can serve as a bridge linking the standard diffusion with the fractional sub-diffusion. We refer to [11,8,21,10] for more development on the mathematical background and numerical analysis of the nonlocal operators and the nonlocal-in-time dynamic systems.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal-in-time dynamics and crossover of diffusive regimes

Du¹,

Zhou²

2020

Preprint

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We study a simple nonlocal-in-time dynamic system proposed for the effective modeling of complex diffusive regimes in heterogeneous media. We present its solutions and their commonly studied statistics such as the mean square distance. This interesting model employs a nonlocal operator to replace the conventional first-order time-derivative. It introduces a finite memory effect of a constant length encoded through a kernel function. The nonlocal-in-time operator is related to fractional time derivatives that rely on the entire time-history on one hand, while reduces to, on the other hand, the classical time derivative if the length of the memory window diminishes. This allows us to demonstrate the effectiveness of the nonlocal-in-time model in capturing the crossover widely observed in nature between the initial subdiffusion and the long time normal diffusion.

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Stochastic representation of solution to nonlocal-in-time diffusion

Cited by 18 publications

References 36 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

Time fractional Poisson equations: Representations and estimates

Nonlocal-in-time dynamics and crossover of diffusive regimes

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