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Poisson's equation

Poisson's equation is the partial differential equation:

${\partial^2 \over \partial x^2 }\varphi(x,y,z) + {\partial^2 \over \partial y^2 }\varphi(x,y,z) + {\partial^2 \over \partial z^2 }\varphi(x,y,z) = f(x,y,z)$

Or alternately:

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

or

$\Delta\varphi=f,$

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.

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

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