Partial differential equations

The formula

A second-order linear PDE in two variables has the general shape:

A·u_xx + B·u_xy + C·u_yy + (lower-order terms) = 0

Here u_xx means the second partial derivative of u with respect to x, and so on. The subscripts are the variable you differentiate against. The whole character of the problem is decided by the top three coefficients A, B, C through one number — the discriminant:

D = B² − 4·A·C

D < 0 → elliptic. Steady-state, no time direction. Value at every interior point is a smoothed average of its neighbors.
D = 0 → parabolic. One-way diffusion in time. Sharp features smear out and never come back.
D > 0 → hyperbolic. Waves. Disturbances travel at finite speed along characteristic lines and keep their shape.

The three canonical equations, one per class:

Laplace (elliptic):    u_xx + u_yy = 0          steady heat, potential, seepage
Heat    (parabolic):   u_t = α·u_xx             transient conduction, diffusion
Wave    (hyperbolic):  u_tt = c²·u_xx           vibration, acoustics, EM in free space

Reading them: Laplace says a quantity with no source sits at the average of what surrounds it. Heat says the local value climbs or falls in proportion to how curved the profile is right there. Wave says acceleration of the value equals c² times its spatial curvature — the recipe for a signal that propagates at speed c.

Poisson is Laplace with a source term: u_xx + u_yy = −f, where f is a distributed load, charge density, or heat generation rate.

Where you meet it

Thermal survey on a test stand. A bracket bolted to a warm avionics box reaches a steady temperature map. That map is the solution of Laplace's equation (Poisson if the bracket dissipates its own heat) with the box temperature fixed on one face and convection on the others. Elliptic, so a steady-state solver is the right tool — no time-stepping needed.
Transient thermal in a design review. Somebody asks how long until a potted assembly reaches thermal equilibrium after power-on. That is the heat equation, parabolic, and it needs an initial condition plus a time-marching solver. The half-answer engineers reach for is the lumped-capacitance time constant; the PDE is what you fall back on when the part isn't thermally thin.
Modal and acoustic work. A panel ringing on a shaker table, a duct resonance, an antenna feed — all governed by the wave equation, hyperbolic. Finite speed of propagation is the tell: energy shows up at a sensor only after the wave has had time to arrive.
CFD and structural FEA at the RF/aero bench. Incompressible flow around a fairing mixes elliptic pressure behavior with parabolic viscous diffusion; the mesher and solver settings you pick are downstream of that classification whether the GUI tells you or not.

How it works

The classification is not academic bookkeeping — it dictates everything downstream. It tells you what boundary and initial conditions you're allowed to impose, whether the solution is smooth or can hold a shock, and which numerical scheme won't blow up.

Boundary conditions come in two basic flavors you'll see on every solver setup, plus a combination of the two:

Dirichlet fixes the value of u on the boundary. "This face is held at 85 °C." "This edge is clamped, displacement zero."
Neumann fixes the derivative — the flux — across the boundary. "This surface is insulated, so the normal temperature gradient is zero." "This edge carries a known heat flux." An insulated wall is Neumann with the gradient set to zero.
Robin (mixed) fixes a weighted combination of value and flux. The everyday case is convection: h·(T − T_inf) = −k·dT/dn on every film-coefficient face in a thermal deck — the flux leaving the surface depends on how hot the surface itself is. This is the condition behind the "convection on the others" in the test-stand example above.

Get the pairing wrong and the problem is ill-posed. Elliptic problems want conditions all the way around a closed boundary — miss a piece and the solution floats. A pure-Neumann Laplace problem is only solvable up to an additive constant, and only if the net flux balances; forget that and the solver either diverges or hands you garbage with a straight face. Parabolic problems want an initial condition plus boundary conditions for all later time. Hyperbolic problems want initial value and initial velocity — position and speed of the string at t = 0 — because the equation is second order in time.

The mistake people make is trusting a solver to guess the physics. A time-marching scheme tuned for the heat equation will smear a genuine wave into mush, because it has numerical diffusion baked in. Run an elliptic problem through a marcher and you're solving the wrong equation. Match the method to the class first.

Two closed-form solutions are worth carrying in your head because they're exact and they anchor intuition. For the 1-D wave equation, d'Alembert's solution is:

u(x,t) = f(x − c·t) + g(x + c·t)

— any shape f sliding right at speed c, plus any shape g sliding left, unchanged. Substitute it and the residual is exactly zero. For the heat equation, a single Fourier mode:

u(x,t) = e^(−α·k²·t) · sin(k·x)

decays in time, and the decay rate goes as k² — so the sharpest wiggles (largest k) die fastest. That k² is why diffusion smooths: high spatial frequency is punished hardest. Both of these check out to zero residual when you plug them back in.

Limits of validity: the discriminant test as stated is for second-order linear PDEs with constant-ish coefficients. When coefficients depend on the solution itself — transonic flow, shock formation, large-deformation structures — the same equation can be elliptic in one region and hyperbolic in another, and the boundary between them is where the hard physics (and the hard convergence) lives.

History

The first PDE in the record is the wave equation. Jean le Rond d'Alembert wrote it down in 1747 while chasing the motion of a vibrating string, and produced the traveling-wave solution that still carries his name ^[1]^[2]. His contemporary Leonhard Euler pushed on the same string problem, and the two argued for years over what counted as an allowable initial shape — the "vibrating string controversy" — a fight that was really about what a function is allowed to be, decades before anyone had a rigorous answer ^[1].

The heat equation came next, from Joseph Fourier. He read his memoir on heat propagation in solids to the Paris Institute on 21 December 1807, arguing that any temperature profile could be built from sines and cosines — the trigonometric series now called Fourier series ^[3]^[4]. The Academy's referees, Lagrange among them, were skeptical enough that the full treatment waited until his Théorie analytique de la chaleur appeared in 1822 ^[3]^[4]. That book is the parabolic anchor of the whole subject and the reason "Fourier" is stamped on both the transform and the conduction law.

The steady-state elliptic equation is tied to Pierre-Simon Laplace, who put it at the center of his work on gravitation and potential theory in the late eighteenth century. The two boundary conditions are named for later hands: Dirichlet's condition, fixing values on the boundary, after Peter Gustav Lejeune Dirichlet; and the flux condition after the German mathematician Carl Neumann (1832–1925), who worked on boundary-value problems for elliptic equations ^[5]. The naming is retroactive — engineers were imposing "hold this edge fixed" and "insulate that face" long before the labels stuck.

Related tools

/tools/heat-conduction
/tools/heat-exchanger-duty
/tools/thermal-expansion
/tools/vibration-natural-freq
/tools/beam-deflection
/tools/frequency-wavelength
/tools/isentropic-flow

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →