Wellposedness and regularity of a variable-order space-time fractional diffusion equation

Zheng, Xiangcheng; Wang, Hong

doi:10.1142/s0219530520500013

Cited by 17 publications

(4 citation statements)

References 35 publications

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“…It is straightforward to extend the current pseudo-likelihood approach to two-and three-dimensional problems. However, extending the approach to other types of equations, such as space-time fractional stochastic PDEs (1) and even the stochastic versions of variable-order fractional PDEs [27,28], is not that trivial. Reliable discretization schemes for target equations are needed to be developed for those equations, and proper parametrization is required for the variable fractional orders.…”

Section: Discussionmentioning

confidence: 99%

Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Pang

Cao

2021

Fractal Fract

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Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation.

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Section: Discussionmentioning

confidence: 99%

Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Pang

Cao

2021

Fractal Fract

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“…While models with constant fractional order are the simplest and most widely used, some of the model descriptions we discuss in the following sections are improved by the use of a variable fractional order. In recent years, with the purpose of increasing the descriptive power of fractional operators, new models characterized by a variable fractional order have been introduced for both space-and time-fractional differential operators [7,51,55,181,266] and several discretization methods have been designed [42,201,257,265,270]. The improved descriptive power of variable-order fractional operators has been demonstrated in some recent works on parameter estimation [172,170,267].…”

Section: Variable-order Fractional Derivativesmentioning

confidence: 99%

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

Suzuki¹,

Gulian²,

Zayernouri³

et al. 2021

Preprint

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Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas -subsurface transport, turbulence, and anomalous materials -in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.

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“…In recent years, with the purpose of increasing the descriptive power of fractional operators, new models characterized by a variable fractional order have been introduced for both space‐ and time‐fractional differential operators [21–23] and several discretization methods have been designed [24–28]. However, the analysis of variable‐order models is still in its infancy, with [29] and [30] being perhaps the only relevant works that address theoretical questions such as well‐posedness for space‐fractional differential operators with variable order

s = s false(boldx false)

.…”

Section: Introductionmentioning

confidence: 99%

A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

D’Elia

Glusa

2022

Numerical Methods Partial

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Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly‐variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well‐posed. We also introduce a finite element discretization and describe an efficient implementation of the finite‐element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one‐dimensional and two‐dimensional h‐convergence results that show that the variable‐order model has the same convergence behavior as the constant‐order model.

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Wellposedness and regularity of a variable-order space-time fractional diffusion equation

Cited by 17 publications

References 35 publications

Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

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