Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison

Yu, Xiangnan; Zhang, Yong; Sun, Hongwei; Chen, Kewei

doi:10.1016/j.chaos.2018.08.026

Cited by 28 publications

(19 citation statements)

References 29 publications

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“…In particular, Magin et al [17,18] solved the fractional-derivative equations in space with the fractionalorder time α = 1 and the fractional-order in space 0.5 < β < 1 and in time with 0 < α < 1 and β = 1. Successively, Mittag-Leffler type function [21], including both variable α and β were used [22,23].…”

Section: Ad By Mri: Mathematical and Physical Effective Approachesmentioning

confidence: 99%

Mini Review on Anomalous Diffusion by MRI: Potential Advantages, Pitfalls, Limitations, Nomenclature, and Correct Interpretation of Literature

Capuani

Palombo

2020

Front. Phys.

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In this mini-review, we addressed the transient-anomalous diffusion by MRI, starting from the assumption that transient-anomalous diffusion is ubiquitously observed in biological tissues, as demonstrated by different single-particle-tracking optical experiments. The purpose of this review is to identify the main pitfalls that can be encountered when venturing into the field of anomalous diffusion quantified by diffusion-MRI methods. Therefore, the theory of anomalous diffusion deriving from its mathematical definition was reported and connected with the consolidated description and the established procedures of conventional diffusion-MRI of tissues. We highlighted the two different modalities for quantifying subdiffusion and superdiffusion parameters of anomalous diffusion. Then we showed that most of the papers concerning anomalous diffusion, actually deal with pseudo-superdiffusion due to the use of a superdiffusion signal representation. Pseudo-superdiffusion depends on water diffusion multi-compartmentalization and local magnetic in-homogeneities that mimic the superdiffusion of spins. In addition to the relatively large production of pseudosuperdiffusion images, anomalous diffusion research is still in its early stages due to the limited flexibility of conventional clinical MRI scanners that currently prevent the acquisition of diffusion-weighted images by varying the diffusion time (the necessary acquisition modality to quantify transient-subdiffusion in human tissues). Moreover, the wide diffusion gradient pulses complicates the definition of a reliable function representative of anomalous diffusion signal behavior to fit data. Nevertheless, it is important and possible to address these limitations, as one of the potentialities of anomalous diffusion imaging is to increase the resolution, sensitivity, and specificity of MRI.

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Section: Ad By Mri: Mathematical and Physical Effective Approachesmentioning

confidence: 99%

Mini Review on Anomalous Diffusion by MRI: Potential Advantages, Pitfalls, Limitations, Nomenclature, and Correct Interpretation of Literature

Capuani

Palombo

2020

Front. Phys.

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“…In a similar way, the space-dependent variable-order models can effectively capture the space-dependent anomalous diffusion. Using the Mittag-Leffler function to replace the traditional power-law kernel, Yu et al (2018) derived the analytical solutions for many time-fractional models in a bounded domain. Their solutions overcome the singularity problem caused by the traditional power-law function kernel.…”

Section: General Form Ftadesmentioning

confidence: 99%

“…The most important assumption underlying the process of Fickian diffusion or Brownian motion is that particle diffusion is driven by the concentration gradient that is short‐range relevant (Paradisi, Cesari, Mainardi, Maurizi, & Tampieri, 2001). When the transport of particles is viewed as a series of random jumps, the jump lengths follow a Gaussian distribution in which the significant long‐distance jumps that occur at a single step are negligible (Bradley, Tucker, & Benson, 2010; Yu et al, 2018). Thus, in the ADE, there are no large deviations for the particles from the average transport velocity or solute mass center.…”

Section: Applications Of Fadesmentioning

confidence: 99%

“…T. Liu, Sun, Zhang, & Fu, 2019;X. T. Liu, Sun, Zhang, Zheng, & Yu, 2019;Wei, Chen, Zhang, Wei, & Garrard, 2018;Yi & Sun, 2019;Yu et al, 2018).…”

Section: Challenges and Suggestions For Future Workmentioning

confidence: 99%

“…Time (T) McGann, 2019; Bohaienko, 2019;Cartalade, Younsi, & Neel & M. C., 2019;Cheng, Duan, & Li, 2019;Hussain & Haq, 2019;Jannelli et al, 2019;Yu, Zhang, Sun, & Zheng, 2018). However, there is still a certain gap for FADEs to be an effective prediction tool.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Sun

Qiu

et al. 2020

WIREs Water

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Fractional advection–dispersion equations (FADEs) have been widely used in hydrological research to simulate the anomalous solute transport in surface and subsurface water. However, a large gap still exists between real‐world application (i.e., being a prediction tool) and theoretical FADEs. To better understand this disparity, the FADEs are firstly reviewed from the perspective of fractional‐in‐time and fractional‐in‐space, as well as the anomalous characteristics described by those functions. Then, challenges for the application of FADEs are summarized, including the theoretical gap of FADEs that needs multidisciplinary efforts to fill, extensive requirements for computation techniques and mathematical knowledge to apply the FADEs, the poor predictability for most parameters in the FADEs, and the limitations for collecting geologic information of flow fields. Then, some suggestions are given for future work, such as developing excellent code sets and mature simulation software with a friendly interface. This kind of work would alleviate the computation workload of hydrologists, especially those without coding expertise. Summarizing and determining the value ranges of the important parameters are needed (e.g., the order of the fractional derivative), through the extensive field, laboratory, and numerical experiments rather than blindly using their mathematical ranges. It is also needed to perform global sensitivity analysis for the FADEs. Meanwhile, comprehensive comparison work is necessary for suggesting a model suitable for a specific problem. Last but not least, further research is clearly needed to establish a link between nonlocal parameters and the heterogeneity property to develop more efficient fractional order partial differential equations. This article is categorized under: Science of Water > Methods Science of Water > Water Quality

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Synchronization and Fractional-Order Systems

Martínez-Guerra

Flores-Flores

2023

Understanding Complex Systems

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Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison

Cited by 28 publications

References 29 publications

Mini Review on Anomalous Diffusion by MRI: Potential Advantages, Pitfalls, Limitations, Nomenclature, and Correct Interpretation of Literature

Mini Review on Anomalous Diffusion by MRI: Potential Advantages, Pitfalls, Limitations, Nomenclature, and Correct Interpretation of Literature

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Synchronization and Fractional-Order Systems

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