Poisson's equation
|
|
Poisson's equation is the partial differential equation:
- <math>
\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) <math>
Or alternately:
- <math>{\nabla}^2 \varphi = f<math>
or
- <math>\Delta\varphi=f,<math>
i.e., it sets the Laplacian equal to f. The equation is named after the French mathematician, geometer and physicist Siméon-Denis Poisson.
Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution.
- <math>{\nabla}^2 V = - {\rho \over \epsilon_0}<math>
There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.
See also: Screened Poisson equation
External link
- Poisson Equation (http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf) at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.de:Poisson-Gleichung
fr:Équation de Poisson ja:ポアソン方程式 pl:Równanie różniczkowe Poissona sl:Poissonova enačba