生物離子通道是一種多孔性的蛋白質。鈣離子通道則是透過控制鈣離子的濃度來取決於讓鈣離子或那裏子優先通過。而我用的模型，Poisson-Fermi Equation，是一個四階的非線性微分方程。而我們在解方程時在當中加入了空間位組(steric)和關聯效應。最後在透過中央有限差分法和牛頓法來解 Poisson-Fermi Equation。

Biological Ion channel is a kind of porous protein on cell membrane. Calcium channels preferentially conduct calcium ions. The Poisson-Fermi equation is a fourth-order nonlinear PDE that deals with both steric and correlation effects of all ions and solvent molecules involved in a model system. This equation is approximated by using the central finite difference method in three dimensions. A simplified matched interface and boundary method is used to treat jump conditions between solvent and molecular domains. Some numerical results of ionic concentration profiles are presented in this thesis.

1 Introduction .............................................3
2 The Poisson-Fermi Equation ...............................3
3 Numerical Methods and Results ............................8
3.1 The central …nite di¤erence .............................8
3.2 Matched interface and boundary method (MIB) .......... 11
3.3 Newton’s method ....................................... 15
3.4 Algorithm .............................................16
4 Unit Conversion and Physical Constants ..................18
5 Conclusions .............................................19

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