Poisson's Equation

Laplace's equation describes a room, a metal plate, or a charged conductor once every source inside has been switched off and the system has settled into a purely boundary-driven steady state — no heater running in the middle of the room, no charge sitting inside the plate, nothing but the edges shaping the pattern. Real problems are rarely so tidy. Bury a heating coil inside a slab of metal, or pack a cloud of electric charge into a chunk of space, and the interior itself now drives the pattern, not just the boundary. That is Poisson's equation — Laplace's equation's slightly more general cousin, carrying a built-in source term:

\nabla^2 u = u_{xx} + u_{yy} = f(x, y),

where f is the source density: how much heat is being pumped in (or how much charge is packed in) at each point. It shows up wherever a field has sources sitting inside its own domain: in electrostatics \nabla^2 \varphi = -\rho/\varepsilon_0 ties the potential to the charge density \rho; in gravity \nabla^2 \Phi = 4\pi G\rho ties the gravitational potential to the mass density; in a heated solid with internal heating, \nabla^2 u = -q/k ties the steady temperature to the rate of heat generation q.

Turn the source off, and Laplace's equation reappears

Set f = 0 everywhere, and \nabla^2 u = f collapses to \nabla^2 u = 0 — genuine, word-for-word Laplace's equation. This is not a coincidence dressed up as a special case: it is the only way to get there. Laplace's equation is the source-free member of the Poisson family, sitting at f \equiv 0, and every harmonic function you already know (mean-value property, no interior maxima or minima, the Poisson-kernel-weighted average of the boundary) is really a fact about Poisson's equation restricted to that one special, sourceless case.

Turn the source back up from zero, however tiny, and none of those harmonic facts survive unchanged. A function with an interior bump that Laplace's equation would forbid is not a defect — it's exactly what a positive source is supposed to do: push the solution upward right where the source lives.

Worked example: a hidden heater bends the line

A thin rod lies along x \in [0, 1], held at temperature 0 at the left end and 1 at the right end. With no heating inside, the steady temperature solves Laplace's equation in one dimension, u'' = 0, and the only solution meeting both ends is the straight line u(x) = x.

Now bury a uniform heater of strength A along the whole rod, so the steady equation becomes Poisson's equation u'' = -A. Follow the particular-plus-homogeneous recipe below: a particular solution is u_p = -\tfrac{A}{2}x^2 (check: u_p'' = -A), and a harmonic correction u_h = Bx + C is added to force the boundary values back to 0 and 1. Matching u(0)=0 gives C=0; matching u(1)=1 gives B = 1 + A/2. The full solution is

u(x) = x + \frac{A}{2}\,x(1-x).

Slide the heater strength A below. At A = 0 the straight Laplace line reappears exactly — the parabola term vanishes identically. Crank A up and the buried heater pushes the rod's temperature into a bulge above the line; make A negative — a heat sink pulling heat out — and it bulges the other way. The boundary values 0 and 1 never move: only the interior, where the source actually lives, responds to it.

Worked example: charge filling a disk

On Laplace's equation on a disk, the interior is completely empty — not a speck of charge lives inside the disk, and the potential everywhere inside is dictated purely by values painted around the rim, recovered as a weighted average of the boundary data.

Now fill that same disk with a uniform electric charge density \rho — think of a lightly and evenly charged plastic disk — instead of leaving it empty. The interior pushes back: at every point inside the disk the potential obeys \nabla^2 \varphi = -\rho/\varepsilon_0, not Laplace's equation, and only outside the charged disk, in the surrounding empty space, does the potential settle back into ordinary harmonic behaviour. Solving this exactly means finding one particular solution that produces the right quadratic-shaped bulge inside the charged disk (the same flavour of parabola as the heated-rod example above, just in two dimensions), then patching on a harmonic piece outside so the potential and its slope agree smoothly across the disk's edge.

Particular plus homogeneous

Being linear, Poisson's equation yields to the standard split. Find any particular solution u_p with \nabla^2 u_p = f; then the general solution is

u = u_p + u_h, \qquad \nabla^2 u_h = 0,

where u_h is harmonic and chosen so that the total u meets the boundary conditions — exactly the particular-plus-homogeneous idea used for the heat equation, and the same trick both worked examples above just used by hand. The source sets the interior bulge; the harmonic part bends the whole solution to fit its frame. Poisson's equation is still linear throughout — this is a superposition of two solutions of two different equations, not a superposition within one equation.

On the whole plane, the solution is built from the fundamental solution (the field of a single point source) by superposing one copy for every bit of the source — a convolution with the Green's function. A single positive point source produces a logarithmic potential \frac{1}{2\pi}\ln r in 2-D, the harmonic-away-from-the-source field that diffusion and gravity both obey.

It's tempting to think Poisson's equation is "Laplace's equation, just with different boundary conditions." That mixes up two completely different things. Boundary conditions were always free to change, even for plain Laplace's equation — paint different temperatures around the rim of a disk and you get a different harmonic function, but it still solves \nabla^2 u = 0 everywhere inside. Poisson's equation changes something far deeper: the differential equation itself, at every interior point, not merely the data painted on the edges.

A genuinely harmonic function — one satisfying \nabla^2 u = 0 — can never also satisfy \nabla^2 u = f unless f is identically zero throughout the interior. No amount of boundary-fiddling can patch that up, because the mismatch lives inside the domain, not on its rim. The only way to turn a Poisson problem back into a genuine Laplace problem is to remove the source entirely — set f = 0, full stop.

Poisson's equation is named after Siméon Denis Poisson, and it turns out to be the mathematical foundation of not one but two of the most fundamental forces in the universe. Newtonian gravity obeys \nabla^2 \Phi = 4\pi G\rho, with mass density \rho as the source; classical electrostatics obeys \nabla^2 \varphi = -\rho/\varepsilon_0, with electric charge density as the source. Strip away the constants out front, and gravity and electricity are quietly obeying the exact same equation — one 19th-century piece of mathematics silently running both the fall of an apple and the crackle of static electricity.