Partial Differential Equations
Heat, waves, and how continuous space dictates motion
1. Building Intuition
An Ordinary Differential Equation (ODE) tracks one independent variable — usually time. A Partial Differential Equation (PDE) tracks how a quantity changes with respect to multiple independent variables simultaneously, such as both time and position.
Think of a metal rod heated at one end. The temperature u(x, t) depends on where you are on the rod (x) and when you look (t). The heat equation describes how that temperature profile evolves: heat flows from hot regions to cold ones, smoothing out sharp differences over time.
This same structure — a quantity depending on space and time — appears everywhere: sound waves propagating through air, quantum wavefunctions evolving, electrostatic potentials in a capacitor. All are PDEs.
A square-pulse temperature profile evolves over time. Each colored curve is a snapshot at a later instant. The sharp edges smooth out — heat diffuses from hot to cold until the temperature equalizes.
2. The Mathematics
The three classic PDEs of engineering physics each model a distinct physical phenomenon:
The heat equation says: the rate of change of temperature in time is proportional to the curvature in space. Where the profile is concave (hump), it decreases; where convex (valley), it increases. The constant α² is the thermal diffusivity.
The wave equation has two time derivatives instead of one. This gives it inertia — solutions oscillate indefinitely rather than decaying. The constant c is the wave speed. Every solution is a superposition of left- and right-traveling waves: u(x, t) = f(x − ct) + g(x + ct).
When there is no time dependence (steady state), the heat equation reduces to Laplace's equation. Solutions are called harmonic functions — they satisfy the mean-value property: the value at any point equals the average over any surrounding sphere.
Separation of Variables — the Core Solution Method
The most powerful technique for standard PDEs on simple domains is separation of variables. The key idea: assume the solution factors as a product of single-variable functions, substitute into the PDE, and the equation splits into independent ODEs — one for each variable.
For the heat equation ut = α²uxx, assume u(x, t) = X(x) · T(t):
Since the left side depends only on t and the right only on x, both must equal the same constant −λ. This gives two ODEs: X″ + λX = 0 and T′ + α²λT = 0. Applying boundary conditions (e.g. X(0) = X(L) = 0) quantizes λ into discrete eigenvalues λₙ = (nπ/L)², producing the eigenfunctions Xₙ(x) = sin(nπx/L). The general solution is then a superposition (Fourier series):
Each mode decays at a rate proportional to n². High-frequency oscillations smooth out rapidly; low-frequency components persist. The coefficients bₙ are determined from the Fourier series of the initial condition.
3. Applications
A plucked guitar string vibrates as a sum of harmonics (modes n=1, 2, 3, ...). Each mode oscillates at frequency nπc/L. The white dots are fixed endpoints (boundary conditions u=0).
Thermodynamics
Heat equation models temperature diffusion in microprocessors, heat exchangers, and thermal insulation. Finite element solvers discretize it into millions of ODEs.
Acoustics & Electromagnetics
Sound pressure and electromagnetic fields obey wave equations. Maxwell's equations in vacuum reduce to wave equations for E and B fields, predicting the speed of light c = 1/√(ε₀μ₀).
Structural Mechanics
Vibrating beams (Euler-Bernoulli equation) and membranes (2D wave equation) determine resonant frequencies. Engineers use these to prevent catastrophic resonance (e.g. Tacoma Narrows bridge).
Quantum Mechanics
The Schrödinger equation is a PDE for the wave function ψ(x,t). In a box with fixed walls, separation of variables gives energy eigenvalues and quantized states — the quantum analogue of standing waves.
4. Worked Examples
Example 1: 1D Heat Equation on [0, π]
Solve ut = uxx on 0 < x < π, with boundary conditions u(0,t) = u(π,t) = 0 and initial condition u(x,0) = sin(x) + 0.5 sin(3x).
Step 1: The eigenvalues are λₙ = n² and eigenfunctions sin(nx).
Step 2: Write u(x,t) = Σ bₙ sin(nx) e-n²t.
Step 3: Match the initial condition by inspection:
The solution decays: the n=3 mode fades 9× faster than n=1.
Example 2: 1D Wave Equation — Plucked String
Solve utt = c² uxx on 0 < x < L, with u(0,t) = u(L,t) = 0, initial displacement f(x), and zero initial velocity ut(x,0) = 0.
Separation gives X(x) = sin(nπx/L), and T(t) = cos(nπct/L). The general solution is:
The coefficients bₙ are just the Fourier sine series of f(x). Unlike the heat equation, no decay — the modes oscillate forever at frequencies ωₙ = nπc/L (musical harmonics).
Example 3: Laplace Equation — Steady-State Temperature in a Rectangle
Solve ∇²u = 0 on the rectangle 0 < x < a, 0 < y < b, with three sides at 0°C and the top edge at temperature g(x).
Separation of variables gives X(x) = sin(nπx/a) and Y(y) = sinh(nπy/a). The solution is:
Applying the top boundary condition u(x, b) = g(x) gives:
This pattern — sine series in one direction, hyperbolic sine in the other — appears in electrostatics (potential between plates) and fluid mechanics (stream functions).