Elliptic, Parabolic, Hyperbolic

Take a quadratic curve Ax^2 + Bxy + Cy^2 = 1. A single number built from its coefficients — the discriminant B^2 - 4AC — tells you, before you plot a single point, whether the curve is an ellipse, a parabola, or a hyperbola. Negative discriminant: a closed ellipse. Zero: a parabola. Positive: a two-branched hyperbola. Three wildly different shapes, sorted by the sign of one expression.

Second-order linear PDEs have a discriminant of their own, built the very same way from the coefficients on their second derivatives — and it sorts every such equation into exactly three families with genuinely different behaviour: how their solutions look, what information you need to pin one down, and which techniques will actually solve them.

Building the discriminant

Nearly every second-order linear PDE in two variables can be written

A\,u_{xx} + B\,u_{xy} + C\,u_{yy} + (\text{lower-order terms}) = 0,

where A, B, C may themselves be functions of x and y, but at any point we can still read off their values and compute B^2 - 4AC just as we did for the conic. Only the coefficients of the highest-order derivatives matter for the classification — first-order terms, zeroth-order terms, and the right-hand side are along for the ride but don't affect the type.

For A u_{xx} + B u_{xy} + C u_{yy} + \cdots = 0, compute B^2 - 4AC:

< 0 → elliptic (steady state, e.g. Laplace).
= 0 → parabolic (diffusion, e.g. heat).
> 0 → hyperbolic (waves, e.g. the wave equation).

The three types, and what they mean

Elliptic (B^2 - 4AC < 0) — e.g. Laplace's equation u_{xx} + u_{yy} = 0. Models steady states; smooth everywhere; determined by boundary values all around a region.
Parabolic (B^2 - 4AC = 0) — e.g. the heat equation u_t - \alpha u_{xx} = 0. Models diffusion; smooths data out and marches forward in time.
Hyperbolic (B^2 - 4AC > 0) — e.g. the wave equation u_{tt} - c^2 u_{xx} = 0. Models propagation; carries signals at finite speed along characteristics, preserving sharp features.

The type is not bureaucratic labelling — it dictates what conditions you must supply, whether solutions stay smooth, and which solution method applies.

Three worked classifications

Read off A, B, C from the coefficients of u_{xx}, u_{xy}, and u_{yy} — treating time t as the second variable wherever the equation has one — then plug into B^2 - 4AC.

The heat equation u_t = \alpha u_{xx}: rewritten as \alpha u_{xx} - u_t = 0, the only second-order term is u_{xx}, so A = \alpha, B = 0, C = 0 (there is no u_{tt} term at all). Then B^2 - 4AC = 0^2 - 4\alpha \cdot 0 = 0 — parabolic, exactly matching its diffusion behaviour.

The wave equation u_{tt} - c^2 u_{xx} = 0: here A = -c^2 (the coefficient on u_{xx}), B = 0, C = 1 (the coefficient on u_{tt}). So B^2 - 4AC = 0 - 4(-c^2)(1) = 4c^2 > 0 for any real wave speed c \neq 0 — hyperbolic.

Laplace's equation u_{xx} + u_{yy} = 0: now both variables are spatial, with A = 1, B = 0, C = 1. So B^2 - 4AC = 0 - 4(1)(1) = -4 < 0 — elliptic. No surprise: Laplace's equation describes a steady state that has settled down, with no time variable left to propagate or diffuse along.

Why the classification matters

The three names aren't just a filing system — each family's mathematics matches a distinct kind of physical behaviour remarkably closely:

Parabolic equations model diffusion and smoothing. Heat spreads out, sharp temperature spikes soften over time, and information about the initial state gets blurred together as time runs forward — you can't run a parabolic equation backwards and recover a sharp starting profile.
Hyperbolic equations model propagation at finite speed. A plucked string's kink travels along it without smearing out; a sound wave carries a sharp click across a room intact. Disturbances have a definite "light cone" they cannot outrun.
Elliptic equations model equilibrium. There is no time variable at all — just a steady shape that has already settled, like the temperature distribution inside a metal plate long after any transient heating has died away.

The type also decides which solution techniques even apply. Elliptic problems are solved all at once, over the whole region, because every boundary point influences every interior point simultaneously. Parabolic and hyperbolic problems, by contrast, can be marched forward one time step at a time — but numerical schemes for them look very different, because a hyperbolic scheme must respect the finite propagation speed (take too large a time step and the numerical solution tries to send information faster than the true equation allows), while a parabolic scheme has no such speed limit to respect.

This match between abstract classification and concrete behaviour is exactly why, once you've settled a PDE's type, the next natural question is what data you need to supply to pin down a single solution — the subject of boundary and initial conditions, which turns out to depend directly on which of these three families you're in.

It's tempting to think "elliptic PDE" and "ellipse" must be deeply, secretly the same mathematical object — after all, they share both a name and the exact same discriminant test. They don't. The shared names come from an analogy: both a conic Ax^2 + Bxy + Cy^2 = 1 and a PDE Au_{xx} + Bu_{xy} + Cu_{yy} = 0 have the same algebraic skeleton — a quadratic form in two "directions" — so the same discriminant formula happens to sort both of them.

But that's where the resemblance ends. An ellipse is a closed curve you can draw with a piece of string and two pins; an elliptic PDE is a rule about how a function's second derivatives must balance across a whole region, with no curve to draw at all. Don't expect facts about foci, eccentricity, or asymptotes to carry over to the PDE side just because the names match — the analogy is in the algebra of the discriminant, not in the geometry of the shape.

Here's the genuinely startling part. Mathematicians didn't set out to build a theory that predicts physics — they borrowed three names from ancient conic-section geometry purely because the algebra matched. And yet it turns out that this one algebraic split, discovered by staring at coefficients, cleanly separates almost every classical PDE of physics into families whose real-world behaviour matches their type astonishingly well: the "parabolic" equations really do spread and smooth like diffusion, the "hyperbolic" ones really do carry sharp signals at a finite speed, and the "elliptic" ones really do describe things that have settled into equilibrium.

It's a rare and beautiful case of an abstract classification, built for entirely different curves two thousand years earlier, turning out to predict how the physical world actually behaves. Mixed-type equations exist too — the equation can even change type from region to region, as happens for the flow equations describing air moving past an aircraft wing as it crosses the speed of sound — which is a large part of why classification is one of the first questions asked of any new PDE.