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Gradual Variations and Partial Differential Equations

  • Li M. Chen
Chapter

Abstract

Numerically solving partial differential equations (PDE) has made a significant impact on science and engineering since the last century. The finite difference method and finite elements method are two main methods for numerical PDEs. This chapter presents a new method that uses gradually varied functions to solve partial differential equations, specifically in groundwater flow equations. In this chapter, we first introduce basic partial differential equations including elliptic, parabolic, and hyperbolic differential equations and the brief descriptions of their numerical solving methods. Second, we establish a mathematical model based on gradually varied functions for parabolic differential equations, then we use this method for groundwater data reconstruction. This model can also be used to solve elliptic differential equations. Lastly, we present a case study for solving hyperbolic differential equations using our new method.

Keywords

Finite Element Method Partial Differential Equation Sample Point Varied Function Finite Difference Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This research was supported by USGS seed grants. The author expresses thanks to Dr. William Hare and members of the UDC DC Water Resources Research Institute for their help. The lab data was provided by Professor Xunhong Chen at UNL. The author would also like to thank his student Mr. Travis L. Branham for his work in data collection. This chapter is based on the report entitled: “Gradual Variation Analysis for Groundwater Flow of DC, DC Water Resources Research Institute Final Report 2009”.

References

  1. 1.
    Bouwer H (1978) Groundwater hydrology. McGraw Hill, New YorkGoogle Scholar
  2. 2.
    Chen L (1990) The necessary and sufficient condition and the efficient algorithms for gradually varied fill. Chin Sci Bull 35:10Google Scholar
  3. 3.
    Chen L (2004) Discrete surfaces and manifolds: a theory of digital-discrete geometry and topology, Scientific and Practical Computing, RockvilleGoogle Scholar
  4. 4.
    Chen L (2009) Gradual variation analysis for groundwater flow of DC, DC water resources research institute final report. http://arxiv.org/abs/1001.3190
  5. 5.
    Chen L (2010) A digital-discrete method for smooth-continuous data reconstruction. J Wash Acad Sci 96(2):47–65Google Scholar
  6. 6.
    Chen L, Chen X (2011) Solving groundwater flow equations using gradually varied functions (Submitted for publication). http://arxiv.org/ftp/arxiv/papers/1210/1210.4213.pdf
  7. 7.
    Fausett L (2003) Numerical methods: algorithms and applications. Prentice Hall, Upper Saddle RiverGoogle Scholar
  8. 8.
    Haitjema HM (1995) Analytic Element modeling of ground water flow. Academic, San DiegoGoogle Scholar
  9. 9.
    Haitjema HM, Kelson VA, de Lange W (2001) Selecting MODFLOW cell sizes for accurate flow fields. Ground Water 39(6):931–938CrossRefGoogle Scholar
  10. 10.
    Hamming RW (1973) Numerical methods for scientists and engineers. Dover, New York (Ch2)Google Scholar
  11. 11.
    Harbaugh AW, Banta ER, Hill MC, McDonald MG (2000) MODFLOW-2000, the U.S. Geological Survey modular ground-water model – user guide to modularization concepts and the ground-water flow process: U.S. Geological survey open-file report 00–92, 121 p 2000. http://water.usgs.gov/nrp/gwsoftware/modflow2000/modflow2000.html
  12. 12.
    Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Studentlitteratur, LundzbMATHGoogle Scholar
  13. 13.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, New YorkzbMATHGoogle Scholar
  14. 14.
    Pruist GW, Gilding BH, Peters MJ (1993) A comparison of different numerical methods for solving the forward problem in EEG and MEG. Physiol Meas 14:A1–A9CrossRefGoogle Scholar
  15. 15.
    Sanford W (2002) Recharge and groundwater models: an overview. Hydrogeol J 10:110–120CrossRefGoogle Scholar
  16. 16.
    Stoer J, Bulirsch R (2002) Introduction to numerical analysis, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  17. 17.
    Yan S, Minsker B (2006) Optimal groundwater remediation design using an adaptive Neural Network Genetic Algorithm. Water Resour Res 42(5) W05407, doi:10.1029/2005WR004303CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Li M. Chen
    • 1
  1. 1.University of the District of ColumbiaWashingtonUSA

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