# Poisson's equation

Poisson's equation is the partial differential equation:

[itex]

\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) [itex]

Or alternately:

[itex]{\nabla}^2 \varphi = f[itex]

or

[itex]\Delta\varphi=f,[itex]

i.e., it sets the Laplacian equal to f. The equation is named after the French mathematician, geometer and physicist Simon-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.

[itex]{\nabla}^2 V = - {\rho \over \epsilon_0}[itex]

There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.

• 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

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