Gradual Variations and Partial Differential Equations
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.
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”.
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