Laplace's equation
describes a steady state with no sources. Add a source — heat generated inside the
plate, electric charge filling a region — and the right-hand side is no longer zero. The result
is Poisson's equation:
\nabla^2 u = u_{xx} + u_{yy} = f(x, y),
where f is the source density. It is the inhomogeneous cousin of
Laplace's equation, and it is everywhere in physics: in electrostatics
\nabla^2 \varphi = -\rho/\varepsilon_0 ties the potential to the charge;
in gravity \nabla^2 \Phi = 4\pi G\rho ties the field to the mass.
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. The source sets the interior bulge; the harmonic part bends the
whole solution to fit its frame.
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.
- \nabla^2 u = f — Laplace with a source term f.
- General solution u = u_p + u_h: a particular solution plus a harmonic correction fitting the boundary.
- On all of space, u is the source convolved with the fundamental solution (Green's function).