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Volume 20, Issue 3
Nonlocal-in-Time Dynamics and Crossover of Diffusive Regimes

Qiang Du & Zhi Zhou

DOI: 10.4208/ijnam2023-1014

Int. J. Numer. Anal. Mod., 20 (2023), pp. 353-370.

Published online: 2023-03

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  • Abstract

The aim of this paper is to 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 sub-diffusion and the long time normal diffusion.

  • Keywords

Nonlocal model, nonlocal operators, mean square displacement, sub-diffusion, numerical methods.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-353, author = {Du , Qiang and Zhou , Zhi}, title = {Nonlocal-in-Time Dynamics and Crossover of Diffusive Regimes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {3}, pages = {353--370}, abstract = {

The aim of this paper is to 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 sub-diffusion and the long time normal diffusion.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1014}, url = {http://global-sci.org/intro/article_detail/ijnam/21537.html} }
TY - JOUR T1 - Nonlocal-in-Time Dynamics and Crossover of Diffusive Regimes AU - Du , Qiang AU - Zhou , Zhi JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 353 EP - 370 PY - 2023 DA - 2023/03 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1014 UR - https://global-sci.org/intro/article_detail/ijnam/21537.html KW - Nonlocal model, nonlocal operators, mean square displacement, sub-diffusion, numerical methods. AB -

The aim of this paper is to 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 sub-diffusion and the long time normal diffusion.

Du , Qiang and Zhou , Zhi. (2023). Nonlocal-in-Time Dynamics and Crossover of Diffusive Regimes. International Journal of Numerical Analysis and Modeling. 20 (3). 353-370. doi:10.4208/ijnam2023-1014
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