Partial Differential Equations: One Law for Every Point of a Field

Pluck a guitar string and it sings a note and a tangle of fainter overtones. Pull a hot pizza from the oven and the crust cools long before the middle. Hang a wire between two hot and cold ends and, after a while, the temperature settles into a smooth ramp that never changes again. Three everyday scenes, three very different behaviours: something that rings and travels, something that smooths out and forgets, something that has settled and stays.

The remarkable thing is that one idea writes all three down. Where an ordinary differential equation follows one quantity through time, a partial differential equation (PDE) follows a whole field: temperature at every point of the pizza, displacement at every point of the string. And the key quantity in all of them is the same: how the field curves in space. A peak with sharply curved top, a gentle ramp with none.

As of the Clay Mathematics Institute's list, announced in 2000, whether the Navier–Stokes equations of fluid flow always have smooth solutions in three dimensions is still one of the seven Millennium Prize problems, worth one million dollars. A PDE course will not solve it, but it will show why even a one-line equation can be this hard.

Heat spreads, waves travel, and three kinds of equation

Take a lump of heat on a long rod, or a lump of displacement on a long string, and ask what each equation does to it.

The heat equation is first order in time: its curvature tells the temperature how fast to change, so a peak sinks and a dip fills. The wave equation is second order in time: curvature tells the field how fast its velocity changes, like a spring, so a bump does not sink but sets off a ripple. Add a third, Laplace's equation ∂²u/∂x² + ∂²u/∂y² = 0, in which time is absent altogether: it describes what is left when everything has settled. These are called parabolic, hyperbolic and elliptic, after the shapes of the curves Ax² + Bxy + Cy² = 1 whose coefficients A, B, C appear in the equation. For second-order equations, the sign of B² − 4AC sorts them: positive hyperbolic (wave), zero parabolic (heat), negative elliptic (Laplace).

Proof sketch: d'Alembert's solution of the wave equation. Change to coordinates that ride with the waves: ξ = x − ct and η = x + ct. In those coordinates the equation becomes ∂²u/∂ξ∂η = 0. Nothing depends on ξ after differentiating by η, so the solution is a function of ξ plus a function of η, u = F(x − ct) + G(x + ct): a shape moving right plus a shape moving left. The real trick is the change of variables. Jean le Rond d'Alembert found it for the vibrating string in 1747. For a lump starting at rest, the two shapes are half-size copies, exactly what the gold curve shows. No such pair of copies exists for heat: the lump sinks and widens, and its area stays the same.

A plucked string is a chorus of standing waves

A guitar string is pinned at both ends, so not every shape can ring steadily. Only some can: one hump, two humps with a still point in the middle, three humps, and so on. These standing waves or modes have frequencies in the ratio 1 : 2 : 3 : …, the harmonic series of music. Any starting shape, however jagged, can be built from them.

Proof sketch: separation of variables. Guess a solution that is a product, u = X(x)T(t). Put it into the wave equation and divide by XT: the left side, X″/X, depends on x alone, and the right side, T″/(c²T), on t alone. A function of x alone can equal a function of t alone only if both are the same constant, say −k². Then X″ = −k²X, so X is a sine. Pinned ends force sin(kL) = 0, so k = nπ/L: only whole-number humps survive. The real trick is that last step: the boundary conditions pick out a discrete list of modes. A sum of solutions is again a solution, because the equation is linear, so you add the modes up with weights b_n to match the starting pluck.

Add harmonics one at a time and the rounded blob sharpens into the triangle of the pluck; one is not enough, and six already look close. Pluck at the exact middle and every even harmonic vanishes, because the string is held at a node of each; pluck near the end and many overtones appear, which is why a guitar sounds brighter, thinner and more nasal near the bridge. Move the time slider and each harmonic advances at its own speed, so after half a period the whole shape has flipped upside down.

This was not obvious to the people who first met it. D'Alembert and Euler, in the late 1740s, argued over which starting shapes the equation allowed, and Daniel Bernoulli proposed around 1750 that every motion was a sum of sines. The dispute ran for decades. Joseph Fourier's treatise on heat (1822) put sums of sines on firm ground as a general tool.

Diffusion forgets: you cannot run heat backwards

Put the same sine-mode recipe into the heat equation and a new thing happens. Each mode decays instead of oscillating, and mode n decays by a factor e^−n²π²t (for a rod of unit length and unit diffusivity). The factor involves n squared, so wiggles die far faster than broad shapes. After a short time only the smooth parts of the starting temperature remain; the sharp detail is gone.

Now try to go backwards: given the temperature today, what was it earlier? You must divide each mode by that decay factor, which means multiplying by e^n²π²t. For t = 0.01, mode 5 is boosted about 12 times, mode 10 about 19,000 times and mode 20 by about 10¹⁷. Any measurement error in the high modes is blown up by that factor. Proof sketch: there is nothing more than that exponential. The real trick is seeing that forward evolution is a smoothing operation that destroys information, and inverting it must therefore create enormous amplification of noise. Jacques Hadamard called a problem well-posed (1902) if a solution exists, is unique and depends continuously on the data; the backward heat problem fails the third test.

The starting temperature here is a hot rod with cold ends. With barely any noise and a short time, the recovery is close; wait longer or add a speck of noise, and the blue curve becomes a violent zigzag that bears no relation to the truth. This is the mathematical face of the arrow of time (see Entropy, 13/12): the forward direction is stable, and the backward is hopeless. It matters in practice, too: deblurring a photograph, or finding where a pollutant came from, are inverse problems with the same disease, which is why they need regularisation, a method of damping the highest modes on purpose.

At equilibrium every point is the average of its neighbours

Let the heat equation run for a long time with the edges of a plate held at fixed temperatures. The temperature stops changing, and at that point its time derivative is zero: what is left obeys Laplace's equation. Its meaning in plain words is that each point equals the average of the points around it. A point hotter than its neighbours would cool; a colder one would warm; at rest, none is out of line. The same equation rules the electrostatic potential in empty space, the flow of an ideal fluid and the shape of a soap film over a wire frame, when it is nearly flat.

Proof sketch: no hot spots inside. Suppose the largest value of u sat strictly inside the plate. That value is the average of its neighbours, none of which can exceed it, so all of them must equal it. Repeat for each neighbour and the maximum spreads out until it reaches the boundary. So the maximum (and likewise the minimum) lives on the edge: this is the maximum principle. The real trick is to use the averaging property twice. It also gives a second payoff: a square plate with four edges at four different temperatures has a centre that is exactly the average of the four, because rotating the plate by 90 degrees swaps the edges, and the solutions simply add.

The simulation does exactly what the equation says: it repeatedly replaces each interior cell by the average of its four neighbours, starting from a cold plate. Heat creeps in from the edges, slowly at first, and takes a few hundred sweeps to settle on this small grid; on a finer grid it takes many more, which is a hint of why large simulations need cleverer solvers.

A simulation must let information cross each cell

Put these equations on a computer and you replace the smooth field by values on a grid, stepping through time. 34/78 (Numerical Analysis) showed that stepping can explode; for PDEs it has a neat geometric explanation. Take the simplest wave: a shape sliding right at speed 1, ∂u/∂t + ∂u/∂x = 0. The upwind scheme updates each cell using its left neighbour:

The number ν is the Courant number: the fraction of a cell the real wave travels in one step. Proof sketch. The true value at a point at the next time depends on the value one wave-travel upstream, which is ν cells to the left. If ν > 1 the true source lies more than one cell away, but the scheme only ever looks one cell to the left: it ignores the very information it needs, and the result cannot converge. The real trick is the idea of a domain of dependence. Richard Courant, Kurt Friedrichs and Hans Lewy published this necessary condition in 1928, in Mathematische Annalen; it is now the CFL condition.

At ν = 1 the scheme is exact, each step moving the pulse one cell. Below 1 it is stable but the pulse smears: the scheme secretly solves the exact equation plus a little diffusion, with a coefficient of about ½(1 − ν) times the cell width. Above 1 the shortest wavelengths are multiplied by |1 − 2ν| each step, which exceeds 1, and the numbers grow without bound; by ν = 1.2 the highest value is above 10⁵ after 42 steps. The same kind of limit applied to the explicit heat equation in 34/03, where the time step had to stay below half the cell width squared.

The same trick quantises the electron in a box

Erwin Schrödinger's equation of 1926 looks different from the wave equation, with an i in it and only one time derivative:

But the method is identical. Separate variables, ψ = X(x)T(t), and the space part must again be a sine that vanishes at the walls, so only whole numbers of humps fit. Now the boundary condition does something stunning: it allows only a discrete ladder of energies, E_n = n²h²/8mL². This is quantisation, and it has the same proof as the guitar string's harmonics. The time part is just a spinning phase, so a single mode has a probability density |ψ|² that never changes: nothing visibly moves until you mix two modes, and then the density sloshes back and forth at the beat frequency set by their energy gap.

For an electron in a box 1 nm wide (about the size of a few atoms), the lowest energy is 0.376 eV, the next 1.504 eV, the third 3.384 eV: each is n² times the first, using CODATA values for Planck's constant and the electron mass. Mixing the first two makes the electron slosh from one side to the other every 3.67 femtoseconds, some 270 trillion times a second. Narrow the box and the energies, and so the sloshing speed, rise sharply: confinement costs energy, a first taste of why chips shrink into the quantum regime (Quantum Mechanics, 34/83). What ψ means is a matter of real disagreement between interpretations (Copenhagen, many-worlds, pilot-wave), but all of them use these same numbers.

What a graduate just sees

• A PDE says how a field changes at a point from how it is curved around the point.
• Heat smooths and forgets, waves carry shape without losing it, Laplace describes what is left at rest: the three families behave differently.
• Separation of variables turns a PDE into simpler pieces, and boundaries pick out discrete modes: harmonics of a string, energy levels of an atom.
• Linear problems add up: any shape is a sum of modes, and each evolves on its own.
• Diffusion destroys detail exponentially fast, so running it backwards amplifies noise: some problems are ill-posed.
• At equilibrium each point is the average of its neighbours, so extremes live on the boundary.
• A numerical scheme must look where the information comes from: step size and cell size are linked by CFL.
• Boundary and initial conditions matter as much as the equation; an equation alone has many solutions.

Where it shows up next

• Quantum Mechanics (34/83): the Schrödinger equation is a PDE and the particle in a box is the first example. Hydrogen's orbitals come from separating variables in three dimensions.
• Electromagnetics and Waves (34/38): Maxwell's equations imply the wave equation for light and Laplace's equation for static fields.
• Heat Transfer (34/16) and Fluid Mechanics (34/15): the heat equation and the Navier–Stokes equations are PDEs; wind tunnels and weather models are numerical PDE solvers.
• Finance: the Black–Scholes option-pricing equation is a heat equation in disguise.
• Computing and imaging: image blurring, medical-scanner reconstruction and the physics in animated films are all PDE solvers, with the stability caution above.

If you want to go deeper

• 📘 Textbook: Walter Strauss, Partial Differential Equations: An Introduction (2nd ed., Wiley, 2008): the standard first course, clear on waves, heat and Laplace together.
• 📘 Textbook: Lawrence C. Evans, Partial Differential Equations (2nd ed., American Mathematical Society, 2010): the rigorous graduate-level text for mathematicians.
• 📘 Gentler: Ian Stewart, In Pursuit of the Unknown: 17 Equations That Changed the World (Basic Books, 2012), which tells the stories of the wave, heat, Navier–Stokes and Schrödinger equations.
• 🌐 Free course: MIT OpenCourseWare 18.303 Linear Partial Differential Equations: Analysis and Numerics (Fall 2014, Steven G. Johnson), with notes and problem sets.