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

Tao Jiang

doi:doi:10.11648/j.ml.20220803.11

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

Tao Jiang

Published in Mathematics Letters (Volume 8, Issue 3)

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), 2024. Published by Science Publishing Group

Keywords

Homogenization, Partial Differential Equation, Multi-scale Expansion

References

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APA Style

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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ACS 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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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

Tao Jiang

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

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APA Style

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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ACS 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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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

Copy | Download

@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}
}

Copy | Download

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  -

Copy | Download