Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems

Ise Ise Ekpoudom; Anamekere Ime Jacob; Umana Thompson Itaketo

doi:doi:10.11648/j.ijtam.20170305.12

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Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems

Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 5)

Received: 20 February 2017 Accepted: 14 March 2017 Published: 28 October 2017

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Abstract

This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 5)
DOI	10.11648/j.ijtam.20170305.12
Page(s)	163-166
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), 2017. Published by Science Publishing Group

Keywords

Richards’ Equation, Partial Differential Equations, Irrigation, Hydrology, Soil Moisture

References

[1]	Gardner, W. “Some Steady-State Solutions of the Unsaturated Moisture Low Equation with Application to Evaporation from a Water Table”, Soil Science, Vol. 85, 1958, pp. 228-232.
[2]	Kuraz, M., Mayer, P. and Pechm P. “Solving the Non-linear Richards’ Equation Model with Adaptive Domain Decomposition”, Journal of Computational and Applied Mathematics, Vol. 270, 2014, pp 2-11.
[3]	Bouchemella, S., Seridi, A. and Alimi-Ichola, I. “Numerical Solution of Water Flow in Unsaturated Soils: Comparative Study of Different Forms of Richards’ Equations” European Journal of Environmental and Civil Engineering, Vol. 9, 2014, pp 1.
[4]	Bhosale P. A. and Dixit V. V. ” Water Saving-irrigation Automatic Agricultural Controller”, International Journal of Science and Technology, Volume 1, Number 2, 2012, pp. 34-36.
[5]	Israelsen O. W. and Hansen V. E., Irrigation Principle and Practice, John Wiley and Son Inc., 3rd Edition, New Delhi, p 171, 2011.
[6]	Hillel, D., Introduction to Environmental Soil Physics, Academic Press, Hardcover, Boston, 2003.
[7]	Philip, J. R., “Hydrostatics and Hydrodynamics in Selling Soils”, Water Resources, Res. 5, 1969, pp 1070-1077.
[8]	Celia, M. A., Bouloutas, E. T. and Zarba, R. L. “A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation”, Water Resources Research, Volume 26, Number 7, 1990, pp. 1483-1696.
[9]	Majumdar D. K., Irrigation Water Management: Principles and Practice, Eastern Economy Edition, Prentice – Hall of India Private Limited. New Delhi, 2006.
[10]	Tracy, F. “Clean Two- And Three-Dimensional Analytical Solutions of Richards' Equation for Testing Numerical Solvers”, Water Resources Research, Vol. 42, Number 8, 2006, pp. 3106-109.
[11]	Tracy, F. “Three-Dimensional Analytical Solutions of Richards' Equation for a Box-Shaped Soil Sample with Piecewise-Constant Head Boundary Conditions on the Top”, Journal of Hydrology, Vol. 336, 2007, pp. 391-400.
[12]	Warrick, A., Soil Water Dynamics, Oxford University Press, Paperback, New York, 2003.
[13]	Genuchten, M. “A Closed-Form Equation for Producing the Hydraulic Conductivity of Unsaturated Soils”, Soil Science American Journal, Vol. 44, 1980, pp. 892-898.

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APA Style

Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. (2017). Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. International Journal of Theoretical and Applied Mathematics, 3(5), 163-166. https://doi.org/10.11648/j.ijtam.20170305.12

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ACS Style

Ise Ise Ekpoudom; Anamekere Ime Jacob; Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int. J. Theor. Appl. Math. 2017, 3(5), 163-166. doi: 10.11648/j.ijtam.20170305.12

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AMA Style

Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int J Theor Appl Math. 2017;3(5):163-166. doi: 10.11648/j.ijtam.20170305.12

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@article{10.11648/j.ijtam.20170305.12,
  author = {Ise Ise Ekpoudom and Anamekere Ime Jacob and Umana Thompson Itaketo},
  title = {Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {5},
  pages = {163-166},
  doi = {10.11648/j.ijtam.20170305.12},
  url = {https://doi.org/10.11648/j.ijtam.20170305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170305.12},
  abstract = {This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.},
 year = {2017}
}

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TY  - JOUR
T1  - Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems
AU  - Ise Ise Ekpoudom
AU  - Anamekere Ime Jacob
AU  - Umana Thompson Itaketo
Y1  - 2017/10/28
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170305.12
DO  - 10.11648/j.ijtam.20170305.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 163
EP  - 166
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170305.12
AB  - This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.
VL  - 3
IS  - 5
ER  -

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Author Information

Ise Ise Ekpoudom

Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria
Anamekere Ime Jacob

Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria
Umana Thompson Itaketo

Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria

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APA Style

Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. (2017). Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. International Journal of Theoretical and Applied Mathematics, 3(5), 163-166. https://doi.org/10.11648/j.ijtam.20170305.12

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ACS Style

Ise Ise Ekpoudom; Anamekere Ime Jacob; Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int. J. Theor. Appl. Math. 2017, 3(5), 163-166. doi: 10.11648/j.ijtam.20170305.12

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AMA Style

Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int J Theor Appl Math. 2017;3(5):163-166. doi: 10.11648/j.ijtam.20170305.12

Copy | Download

@article{10.11648/j.ijtam.20170305.12,
  author = {Ise Ise Ekpoudom and Anamekere Ime Jacob and Umana Thompson Itaketo},
  title = {Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {5},
  pages = {163-166},
  doi = {10.11648/j.ijtam.20170305.12},
  url = {https://doi.org/10.11648/j.ijtam.20170305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170305.12},
  abstract = {This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems
AU  - Ise Ise Ekpoudom
AU  - Anamekere Ime Jacob
AU  - Umana Thompson Itaketo
Y1  - 2017/10/28
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170305.12
DO  - 10.11648/j.ijtam.20170305.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 163
EP  - 166
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170305.12
AB  - This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.
VL  - 3
IS  - 5
ER  -

Copy | Download