電雙層(EDL)結構的模擬是各種研究領域中的一個有趣現象。電荷守恆Poisson-Boltzmann(CCPB)方程是描述靜電勢分佈的重要數學模型之一。該CCPB模型是非局部二階偏微分方程。本文提出了二維歐氏空間的特定有界域，通過有限差分方法逼近靜電勢。為了研究邊界層的漸近行為，還討論了各種參數值。最後，我們給出了數值近似，並透過CCPB模型了解電雙層的行為。

Simulation for the structure of electrical double layer (EDL) is an interesting phenomena in various research fields. The charge-conserving Poisson–Boltzmann (CCPB) equation is one of the important mathematical model to describe the electrostatic potential profiles. This CCPB model is a non-local second order partial differential equation. In this thesis, a specific bounded domain of two dimensional Euclidean space is presented to approach the electrostatic potential via the finite difference method. Various values of parameters are also discussed for investigating the asymptotic behavior of the boundary layer. As a conclusion, we give a numerical approximations to understand the behavior of electrical double layer through the CCPB model.

Abstract
Acknowledgements
Contents
1 Introduction--------------------------------------------------------------1
1.1 Background and problem formulation
1.2 Specific Boundary Conditions and Parameters
2 Numerical Methods for the CCPB Equation-------------------------------4
2.1 Finite-Difference Method for Second-Order Nonlinear Problems
2.2 Composite Simpson’s Rule for Two Dimensional Domain
2.3 Approximation for CCPB equation with Boundary Conditions
3 Numerical Simulations---------------------------------------------------15
3.1 The Effects of Partition Numbers n and m
3.2 The Effect of Singular Perturbation Parameter epsilon
3.3 The Effects of Various Values of A and B
3.4 The Effects of Various Values of p and q
4 Conclusions------------------------------------------------------------28
Reference----------------------------------------------------------------29

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