One definition, three answers
The previous guide built the idea in the abstract: for a linear operator L, the fundamental solution E is the response to a single, infinitely concentrated point source, the distribution that satisfies L E = delta. Once you have E, superposition hands you the answer to any source f by convolution, u(x) = integral of E(x - y) f(y) dy — you smear the one-point response across the source. That story was honest but bloodless. This guide gives it a body by computing E for the three model operators of the whole subject, and the punchline is that the three E's look completely different.
Why care about exactly three? Because the operators u_xx + u_yy = 0 (Laplace), u_t = k u_xx (heat), and u_tt = c^2 u_xx (wave) are the elliptic, parabolic, and hyperbolic representatives — the three slots of the B^2 - A C classification you met two rungs down. A fundamental solution is the sharpest possible X-ray of an operator's character: it is what the operator does to the single most violent input imaginable. So when Laplace returns a smooth bump, heat a melting bell, and the wave a sharp moving spike, you are seeing the elliptic / parabolic / hyperbolic personalities laid bare in one picture each.
Laplace: a steady, smooth potential
Start with the elliptic operator, the Laplacian, and ask: what steady field does a single point charge sitting at the origin produce? That is exactly L E = delta for L = -Laplacian, and you already know the answer from physics — it is the field of a unit point charge. In three dimensions the fundamental solution is the Newtonian potential E(x) = 1/(4 pi |x|): a smooth, radially symmetric bump that is infinite at the source and falls off like one-over-distance everywhere else. In two dimensions it is instead the logarithmic potential, proportional to -log|x| / (2 pi).
Two features deserve a slow look. First, away from the single source point E is perfectly smooth and solves Laplacian E = 0 — it is a harmonic function everywhere except at the spike, just as you would hope from an elliptic equation, which has no preferred direction and no notion of time. Second, the only blemish, the singularity, sits exactly at the source; the delta lives entirely there. Convolving with a spread-out charge density f then gives the familiar potential u(x) = integral of f(y) / (4 pi |x - y|) dy — the gravitational or electrostatic potential of the density f, with no boundaries in sight.
Heat: a Gaussian that melts and spreads
Now switch on time. For the parabolic heat operator we ask: concentrate all the heat into one point at one instant, then watch. The answer is the heat kernel, and on the whole line it is the Gaussian G(x,t) = (1 / sqrt(4 pi k t)) e^(-x^2 / (4 k t)) for t > 0. Picture it as a movie: at the first instant it is an impossibly tall, thin spike (that is the delta), and as t grows it melts into an ever wider, ever lower bell. Its width grows like sqrt(4 k t) — diffusion crawls, spreading like the square root of time, not linearly — while its area stays exactly 1 forever, which is just conservation of the total heat you started with.
Two traits of the Gaussian carry the entire personality of diffusion, and both are worth naming honestly. Because the Gaussian is infinitely smooth, convolving any rough initial temperature f against it, u(x,t) = integral of G(x - y, t) f(y) dy, instantly turns f into something perfectly smooth — the smoothing effect you saw governing the Fourier modes a rung ago, now visible in a single formula. And because the Gaussian's tail is tiny but never exactly zero, every point feels the source immediately: heat has infinite propagation speed. That is physically a fib — nothing truly travels faster than light — and it is a known, accepted idealization of the heat model, not a literal claim.
There is a sharp asymmetry hiding in the t > 0. The kernel only exists going forward; try to run the same formula toward negative t and 1/sqrt(4 k t) goes imaginary and the Gaussian inverts into a blow-up. That is the same ill-posedness as the backward heat equation: diffusion has a direction. Note too the honest caveat — this clean Gaussian is the free-space heat kernel on the whole line. The moment you put the rod between two walls, the kernel changes shape (it gets rebuilt from eigenfunctions, or by the method of images in the next two guides), even though it plays the very same role.
Wave: a sharp shell that travels and stays sharp
Last comes the hyperbolic wave operator, and it could not be more different. Flash a source on for an instant at one point and watch what happens in three-dimensional air. The retarded fundamental solution is E_ret = delta(t - |x|/c) / (4 pi |x|): the entire response lives exactly on the expanding sphere |x| = c t. Nothing happens before the flash (causality — that is why it is called retarded), the disturbance races outward at the wave speed c, and crucially the spike stays a spike. The wave does not smooth at all; it carries the sharp delta along with it, undimmed in shape.
Compare the three side by side and the classification comes alive. Laplace's E is smooth and timeless, spread over all of space at once — elliptic, no direction. Heat's E is smooth, one-directional in time, and instantly everywhere — parabolic, infinite speed, smoothing. The wave's E is sharp, one-directional in time, and reaches you only at the exact moment c t = |x| — hyperbolic, finite speed, no smoothing. The sharp shell of the wave is finite propagation speed at its cleanest: data outside the sphere of radius c t simply has not arrived. It is the precise opposite of the heat kernel's instant, everywhere influence.
Three honest cautions before you use one
These three objects are the workhorses of the rest of the subject, but they come with fine print. First, none of them is unique. To any fundamental solution you can add any solution of the homogeneous equation L v = 0 and still have L(E + v) = delta. What pins E down is a side condition: decay at infinity for Laplace, forward-in-time for heat, and causality (the retarded rather than the advanced choice) for the wave. The 'fundamental solution' is really 'the one we selected by a physically sensible side condition'.
Second, every one of these lives in free space — the whole line, plane, or space, with no walls. They solve the unbounded problem. The instant a boundary appears, the fundamental solution must be corrected: you add a homogeneous-equation term to make it also satisfy the boundary condition, and the corrected object is the Green's function. That correction is exactly the subject of the next guide, and the cleverest way to build it — placing imaginary mirror sources — is the method of images two guides on.
operator fundamental solution E (free space) character -------- ---------------------------------- --------- -Laplacian 1/(4 pi |x|) (3D Newtonian potential) smooth, timeless u_t - k u_xx (1/sqrt(4 pi k t)) e^(-x^2/(4 k t)) smooth, forward, infinite speed wave (3D) delta(t - |x|/c) / (4 pi |x|) sharp shell, finite speed general source: u = E * f (convolve the point-response with f)
Third, the convolution trick u = E * f handles a source f, but a real problem usually also has initial or boundary data to satisfy at the same time. Splitting a driven problem into a piece that handles the source (via E) and a piece that handles the data, then adding, is the spirit of the representation formulas and Duhamel's principle that close out this rung. For now, the single thing to carry forward is this: each of the three model operators has one explicit point-response, and its shape — smooth bump, melting bell, or travelling shell — already tells you, before you compute anything, how that equation will behave.