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Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection

Received: 28 December 2022     Accepted: 16 January 2023     Published: 1 February 2023
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Abstract

Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.

Published in Mathematics Letters (Volume 8, Issue 3)
DOI 10.11648/j.ml.20220803.11
Page(s) 43-47
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Homogenization, Partial Differential Equation, Multi-scale Expansion

References
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[17] J. B. Keller, Effective behavior of heterogeneous media. Statistical mechanics and statistical methods in theory and application. Springer, Boston, MA, 1977. 631-644.
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    Tao Jiang. (2023). Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Mathematics Letters, 8(3), 43-47. https://doi.org/10.11648/j.ml.20220803.11

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    Tao Jiang. Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Math. Lett. 2023, 8(3), 43-47. doi: 10.11648/j.ml.20220803.11

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    AMA Style

    Tao Jiang. Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Math Lett. 2023;8(3):43-47. doi: 10.11648/j.ml.20220803.11

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  • @article{10.11648/j.ml.20220803.11,
      author = {Tao Jiang},
      title = {Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection},
      journal = {Mathematics Letters},
      volume = {8},
      number = {3},
      pages = {43-47},
      doi = {10.11648/j.ml.20220803.11},
      url = {https://doi.org/10.11648/j.ml.20220803.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20220803.11},
      abstract = {Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection
    AU  - Tao Jiang
    Y1  - 2023/02/01
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ml.20220803.11
    DO  - 10.11648/j.ml.20220803.11
    T2  - Mathematics Letters
    JF  - Mathematics Letters
    JO  - Mathematics Letters
    SP  - 43
    EP  - 47
    PB  - Science Publishing Group
    SN  - 2575-5056
    UR  - https://doi.org/10.11648/j.ml.20220803.11
    AB  - Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.
    VL  - 8
    IS  - 3
    ER  - 

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Author Information
  • Hubei Center for Data and Analysis, Hubei University of Economics, Wuhan, China

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