泊松方程

（重定向自帕松方程式

方程的叙述

${\displaystyle \Delta \varphi =f}$

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

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

${\displaystyle \Delta \varphi =0.\!}$

数学表达

${\displaystyle -\Delta \varphi =f}$

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

${\displaystyle \Phi (x)={\begin{cases}-{\dfrac {1}{2\pi }}\ln |x|&n=2\\{\dfrac {1}{n(n-2)\omega _{n}}}{\dfrac {1}{|x|^{n-2}}}&n\geq 3\end{cases}}}$

${\displaystyle G(x,y)=\Phi (y-x)-\phi ^{x}(y)}$

${\displaystyle \phi ^{x}}$ 为一个校正函数，它满足

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

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

靜電學

${\displaystyle {\nabla }^{2}\Phi =-{\rho \over \epsilon _{0}}}$

${\displaystyle \Phi \!}$代表電勢（單位為伏特），${\displaystyle \rho \!}$電荷體密度（單位為庫侖/立方公尺），而${\displaystyle \epsilon _{0}\!}$真空電容率（單位為法拉/公尺）。

${\displaystyle \rho =0,\,}$

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

高斯電荷分佈的電場

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

${\displaystyle \Phi (r)={1 \over 4\pi \epsilon _{0}}{\frac {Q}{r}}\,{\mbox{erf}}\left({\frac {r}{{\sqrt {2}}\sigma }}\right)}$

erf(x)代表的是误差函数.

參考資料

1. ^ Jackson, Julia A.; Mehl, James P.; Neuendorf, Klaus K. E. (编), Glossary of Geology, American Geological Institute, Springer: 503, 2005, ISBN 9780922152766.
• 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