Non-Fickian Transport in Porous Media: Always Temporally Anomalous?

Zhokh, Alexey; Strizhak, P. E.

doi:10.1007/s11242-018-1066-6

Cited by 12 publications

(3 citation statements)

References 69 publications

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“…where p(x, t) is a probability density function and θ is an asymmetry parameter. Fractional derivatives in different models often arise due to fractal properties of systems (see, e.g., [12,14,16,29,[31][32][33]). Here, we argue that the Gangal fractal derivative is the progenitor of the fractional derivative.…”

Section: Stochastic Lévy-lorentz Gas and Fractal Timementioning

confidence: 99%

Fractal Stochastic Processes on Thin Cantor-Like Sets

Golmankhaneh

Sibatov

2021

Mathematics

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We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

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Section: Stochastic Lévy-lorentz Gas and Fractal Timementioning

confidence: 99%

Fractal Stochastic Processes on Thin Cantor-Like Sets

Golmankhaneh

Sibatov

2021

Mathematics

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“…Space-time-fractional equations have also been used to model anomalous diffusion, namely using a Riemann-Liouville time-fractional derivative and a Riesz-Feller space-fractional derivative [77]; space-time Caputo fractional derivatives have proved to fit experimental data on methanol transport through porous media, e.g. pelletized zeolite-based catalyst [78]; space-timefractional differential equations have been investigated to model stochastic advection-diffusion problems in fractal media with long-range, correlated spatial fluctuations [79].…”

Section: Diffusion In Porous Mediamentioning

confidence: 99%

Advanced materials modelling via fractional calculus: challenges and perspectives

Failla

Zingales

2020

Phil. Trans. R. Soc. A.

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Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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“…Aqui, vamos considerar a equação da difusão com derivada fracionária temporal aplicada no termo da difusão, como proposto em dois artigos de revisão de Metzler e Klafter [7,8]. A outra variação da equação da difusão que será trabalhada também com derivada fracionária temporal é a que foi utilizada com relativo sucesso no problema do transporte não fickiano em meios porosos [15], contaminante de fratura em matriz de rocha porosa [4], dentre outros.…”

Section: Introductionunclassified

Uma abordagem numérica sobre o efeito dos operadores fracionários na derivação temporal para a equação de difusão fracionária.

Faria,

Moura,

Negreiros

2023

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Neste trabalho apresentamos um estudo sobre o efeito do operador de derivada fracionária na derivação temporal em duas variações da equação de difusão fracionária. As equações de difusão generalizada com derivadas de ordem fracionária têm se mostrado bastante eficientes para descrever a difusão em sistemas complexos, com a vantagem de produzir em alguns casos modelos mais precisos. A abordagem numérica escolhida é o método das diferenças finitas, em um esquema implícito. As derivadas fracionárias adotadas são a de Riemann-Liouville, a de Caputo-Fabrizio e a de Katugampola. Um experimento numérico será apresentado com os resultados exibidos com o intuito de comparar os modelos apresentados.Palavras-chave. Equação da difusão fracionária, Aproximações por diferenças finitas, Derivada fracionária de Riemann-Liouville, Caputo-Fabrizio e Katugampola.

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Non-Fickian Transport in Porous Media: Always Temporally Anomalous?

Cited by 12 publications

References 69 publications

Fractal Stochastic Processes on Thin Cantor-Like Sets

Fractal Stochastic Processes on Thin Cantor-Like Sets

Advanced materials modelling via fractional calculus: challenges and perspectives

Uma abordagem numérica sobre o efeito dos operadores fracionários na derivação temporal para a equação de difusão fracionária.

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