= Poisson's equation
{c}
{title2=$\Delta u=f$}
{wiki}
= Poisson equation
{c}
{synonym}
<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 $\Delta V=-\rho/\epsilon_0$. 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 $u''=1$ with endpoint values zero and $1/2$. On a <periodic domain> the source must have zero mean, by integrating the equation.
Back to article page
