Poisson's Equation
Definition
The following equation is called the Poisson’s equation.
$$ \Delta u = \nabla^{2} u = f $$
Here $\Delta = \nabla^{2}$ is the Laplacian, and $u, f : \mathbb{R}^{n} \to \mathbb{R}$.
Explanation
The special case $f = 0$ is called the Laplace equation. Physically, $f$ denotes the source, and $u$ denotes the result. For example, in electrostatics the potential $V$ and the charge density $\rho$ are expressed by the following formula.
$$ \Delta V = \nabla^{2} V = -\frac{1}{\epsilon_{0}} \rho $$