Poisson's equation

Poisson's equation is $\Delta u=f$, where $\Delta$ is the Laplace operator and $f$ a prescribed source. It generalizes the Laplace equation, recovered when $f=0$. Boundary conditions are needed to determine a unique solution; Neumann boundary conditions can leave an additive constant.

For electrostatics in vacuum, the convention is $\Delta V=-\rho /{ϵ}_0$. For Newtonian gravity, $\Delta \Phi =4\pi G\rho$. The opposite signs reflect the specified force and potential conventions. A simple mathematical example is $u(x)=x^2/2$ on $[0,1]$, which solves $u^{\prime \prime }=1$ with endpoint values zero and $1/2$. On a periodic domain the source must have zero mean, by integrating the equation.

Generalization of Laplace's equation where the value is not necessarily 0.