Poisson's Equation - Statement of The Equation

Statement of The Equation

The Poisson Equation is

where is the Laplace operator, and f and φ are real or complex-valued functions on a manifold. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as

In three-dimensional Cartesian coordinates, it takes the form

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).

For vanishing f, this equation becomes Laplace's equation

The Poisson equation may be solved using a Green's function; a general exposition of the Green's function for the Poisson equation is given in the article on the screened Poisson equation. There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.

Read more about this topic:  Poisson's Equation

Other articles related to "statement of the equation, equation":

Screened Poisson Equation - Statement of The Equation
... In the homogenous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation ... In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets ...

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