Poisson-Fermi理論是通過表徵電勢函數來理解雙電層（EDL）的有用工具。首先，本論文提出了線性Poisson-Fermi模型和非線性Poisson-Fermi模型。接下來，針對應用邊界條件導出一維線性Poisson-Fermi模型的解析解。此外，我們開發了有限差分方法來解決線性和非線性情形的四階Poissoin-Fermi模型。最後，我們通過模擬包含不同邊界條件和不同長度的各種電勢函數圖來歸納我們的結果。

The Poisson-Fermi theory is an useful tool to understand the electric double layer(EDL) by characterizing the electric potential function. First, the linear Poisson-Fermi Model and the nonlinear Poisson-Fermi Model are presented in this thesis. Next, the analytic solution of the one-dimensional linear Poisson-Fermi Model is derived for applied boundary conditions. In addition, we develop finite-difference method to solve the fourth-order Poissoin-Fermi Model with linear and nonlinear cases. Finally, we conclude our results by simulating the graphs that contain the various electric potential function for different boundary conditions and different length.

Contents
1 Introduction and Literature Review 1
2 One-Dimensional Poisson-Fermi Model 3
2.1 The Nonlinear Poisson-Fermi Model . . . . . . . . . . . . . . . . . . 3
2.2 The Linear Poisson-Fermi Model . . . . . . . . . . . . . . . . . . . 5
2.3 The exact solution of the linear potential function (x) . . . . . . . 8
3 Numerical methods for the Linear Poisson-Fermi Model 11
3.1 Finite-Dierential Method for Linear Problem Fourth-Order Boundary
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Introduction of Centered-Dierence Formula . . . . . . . . . 11
3.1.2 Boundary conditions for solving linear system . . . . . . . . 15
4 Numerical Methods for the Nonlinear Poisson-Fermi Model 20
4.1 Newton's method for Nonlinear Systems . . . . . . . . . . . . . . . 20
4.2 Finite-Dierence Method for Fourth-Order Nonlinear Boundary Value
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Simulations for Electrical Double Layer 25
5.1 Potential Function for Linear Poisson-Fermi Model . . . . . . . . . 25
5.2 Potential Function for Nonlinear Poisson-Fermi Model . . . . . . . . 29
6 Conclusions 34
7 References 35

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