泊松能斯特普朗克(PNP)模型是一個在開放離子通道的離子流之連續的基本模
型模擬。靜電，密度分佈和擴散的影響有限粒徑離子溶液[8]的研究一直是長期存在的話題。泊松方程是推導庫侖律的靜電與在微積分中的高斯定理而得到的。能斯特普朗克方程是對流擴散的模型。一個熵的函數解釋了Borukhov[1]等人所提出在泊松波茲曼(PB)方程中，電解質離子在有限尺寸下的影響，並且由Lu 和Zhou [8]推廣至泊松能斯特普朗克(PNP)模型。我們的有限尺寸線性PNP 模型利用正解而得到二階收斂性結果。對於非線性有限尺寸PNP 模型利用正解，數值誤差幾乎為零。

The Poisson-Nernst-Planck (PNP) model is a basic continuum model for simulating ionic flows in an open ion channel. The effects of finite particle size on electrostatics, density profile, and diffusion have been a long existing topic in the study of ionic solution [8]. The Poisson equation is derived from Coulomb's law in electrostatics and Gauss's theorem in calculus. The Nernst-Planck equation is equivalent to the convection-diffussion model. An entropy functional that accounts for the finite size effects of ions in electrolytes proposed by Borukhov et al. [1] for the Poisson-Boltzmann (PB) equation has been generalized by Lu and Zhou [8] to the PNP model. We obtain second-order convergent results for the finite size linear PNP model with exact solutions.
For nonlinear finite size PNP model with exact solutions, the numerical errors are almost zero.

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