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Volume 15, Issue 4
Analysis of Anisotropic Nonlocal Diffusion Models: Well-Posedness of Fractional Problems for Anomalous Transport

Marta D’Elia & Mamikon Gulian

DOI: 10.4208/nmtma.OA-2022-0001s

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 851-875.

Published online: 2022-10

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  • Abstract

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders $s ∈ [0.5, 1).$ We also present an application of the advection-diffusion equation to anomalous transport of solutes.

  • Keywords

Nonlocal models, fractional models, anomalous diffusion, anisotropic diffusion, solute transport.

  • AMS Subject Headings

4B10, 35R11, 35B40, 26B12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-851, author = {D’Elia , Marta and Gulian , Mamikon}, title = {Analysis of Anisotropic Nonlocal Diffusion Models: Well-Posedness of Fractional Problems for Anomalous Transport}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {851--875}, abstract = {

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders $s ∈ [0.5, 1).$ We also present an application of the advection-diffusion equation to anomalous transport of solutes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0001s}, url = {http://global-sci.org/intro/article_detail/nmtma/21083.html} }
TY - JOUR T1 - Analysis of Anisotropic Nonlocal Diffusion Models: Well-Posedness of Fractional Problems for Anomalous Transport AU - D’Elia , Marta AU - Gulian , Mamikon JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 851 EP - 875 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0001s UR - https://global-sci.org/intro/article_detail/nmtma/21083.html KW - Nonlocal models, fractional models, anomalous diffusion, anisotropic diffusion, solute transport. AB -

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders $s ∈ [0.5, 1).$ We also present an application of the advection-diffusion equation to anomalous transport of solutes.

D’Elia , Marta and Gulian , Mamikon. (2022). Analysis of Anisotropic Nonlocal Diffusion Models: Well-Posedness of Fractional Problems for Anomalous Transport. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 851-875. doi:10.4208/nmtma.OA-2022-0001s
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