# 泊松方程

$-\Delta\varphi=f$

${\nabla}^2 \varphi = f$

$-\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)\varphi(x,y,z) = f(x,y,z).$

$\Delta \varphi = 0. \!$

## 数学表达

$-\Delta\varphi=f$

$\begin{cases} -\Delta \varphi = f & \text{in} \ \Omega \\ \varphi = g & \text{auf} \ \partial\Omega \end{cases}$

$\Phi(x) := \begin{cases} -\dfrac{1}{2\pi}\ln |x| & n=2 \\ \dfrac{1}{n(n-2)\omega_n} \dfrac{1}{|x|^{n-2}} & n \ge 3 \end{cases}$

$G(x,y) := \Phi(y-x) - \phi^x(y)$

$\phi^x$ 为一个校正函数，它满足

$\begin{cases} \Delta \phi^x = 0 &\text{in} \ \Omega \\ \phi^x = \Phi(y-x) &\text{auf} \ \partial\Omega \end{cases}$

$u(x) = -\int_{\partial\Omega}g(y)\frac{\partial G}{\partial \nu}(x,y)\mathrm{d}\sigma(y) + \int_\Omega f(y) G(x,y) \mathrm{d}y$

## 靜電學

${\nabla}^2 \Phi = - {\rho \over \epsilon_0}$

$\Phi \!$代表電勢（單位為伏特），$\rho \!$電荷體密度（單位為庫侖/立方公尺），而$\epsilon_0 \!$真空電容率（單位為法拉/公尺）。

$\rho = 0, \,$

${\nabla}^2 \Phi = 0.$

## 高斯電荷分佈的電場

$\rho(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},$

$\Phi(r) = { 1 \over 4 \pi \epsilon_0 } \frac{Q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)$

erf(x)代表的是误差函数.

## 參考資料

• Poisson Equation at EqWorld: The World of Mathematical Equations.
• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9