Partial Differential Equations

A partial differential equation (PDE) is an equation relating an unknown multivariable function to its partial derivatives. Key result: PDEs split into elliptic, parabolic, and hyperbolic types, each governing fundamentally different physical behavior (equilibrium, diffusion, waves). Applies to: heat flow, vibrations, electrostatics, fluid dynamics, quantum mechanics, finance. Foundations laid by: d'Alembert, Euler, Fourier, and Cauchy (18th–19th centuries).

A partial differential equation is an equation involving an unknown function $u (x_{1}, \dots, x_{n})$ of two or more independent variables and its partial derivatives. Unlike ordinary differential equations, whose solutions form finitely-parametrized families, the general solution of a PDE typically involves arbitrary functions, reflecting infinitely many degrees of freedom. PDEs are the central mathematical language of continuum physics: nearly every fundamental law — heat, sound, light, gravity, electricity — is expressed as one. A surprising fact is that the same second-order equation changes character entirely depending on the sign of its discriminant: the wave equation propagates sharp signals at finite speed, while the heat equation instantly smooths and spreads disturbances everywhere at once.

The Core Insight

A PDE is not solved by a single formula but by a function constrained by local relations among its rates of change. The type of the equation — elliptic, parabolic, or hyperbolic — is determined by the second-order terms alone and dictates which auxiliary conditions (boundary or initial data) make the problem well-posed. Geometry of the equation's characteristics tells you how information propagates.

Formal Statement

Partial Differential Equation

A PDE of order $k$ for an unknown function $u : Ω \subseteq R^{n} \to R$ is a relation

F (x, u, D u, D^{2} u, \dots, D^{k} u) = 0,

where

D^{j} u

denotes the collection of all partial derivatives of order

j

, and

k

is the highest order of derivative appearing.

$x = (x_{1}, \dots, x_{n}) \in Ω$ : independent variables.
$u$ : the unknown (dependent) function.
$D u = (u_{x_{1}}, \dots, u_{x_{n}})$ : the gradient.
$D^{2} u$ : the Hessian matrix of second partials $u_{x_{i} x_{j}}$ .
$F$ : a given function of all these arguments.

⟳This is simply the most general way to write "a relation among

u

and its derivatives"; specific PDEs fix the form of

F

Linearity and Classification by Form

A second-order PDE in two variables has the form

A u_{xx} + 2 B u_{x y} + C u_{y y} + D u_{x} + E u_{y} + F u = G,

where

A, B, C, D, E, F, G

may depend on

x, y

. It is:

linear if all coefficients depend only on $x, y$ ;
semilinear if $A, B, C$ depend only on $x, y$ but lower-order terms may depend on $u, u_{x}, u_{y}$ ;
quasilinear if $A, B, C$ may depend on $u, D u$ but the equation is linear in the highest derivatives;
fully nonlinear otherwise.

⟳Group terms by derivative order; "linear in highest derivatives" defines quasilinearity.

Classification by Discriminant

The second-order linear PDE [2] is classified at a point by the discriminant $Δ = B^{2} - A C$ :

⎩ ⎨ ⎧ Δ < 0 Δ = 0 Δ > 0 elliptic (e.g. Laplace u_{xx} + u_{y y} = 0) parabolic (e.g. heat u_{t} = u_{xx}) hyperbolic (e.g. wave u_{tt} = u_{xx})

The type may vary from point to point (e.g. the Tricomi equation

u_{xx} + x u_{y y} = 0

⟳The principal part

A u_{xx} + 2 B u_{x y} + C u_{y y}

is a quadratic form with matrix

(A B B C)

; the sign of its determinant

A C - B^{2} = - Δ

classifies the conic of characteristics, in analogy with classifying

A x^{2} + 2 B x y + C y^{2}

Why It Holds / Derivation

Reduction to Canonical Form via Characteristics

We show why the discriminant governs the equation's character by seeking characteristic curves $ϕ (x, y) = const$ along which the second-order part degenerates.

Seek a change of variables $ξ = ϕ (x, y), η = ψ (x, y)$ that simplifies [2]. Under this change, the coefficient of $u_{ξ ξ}$ becomes

\overset{ˉ}{A} = A ϕ_{x}^{2} + 2 B ϕ_{x} ϕ_{y} + C ϕ_{y}^{2} .

Setting $\overset{ˉ}{A} = 0$ (so $ξ$ is a characteristic coordinate) and writing $m = - ϕ_{x} / ϕ_{y} = d y / d x$ along the curve, divide by $ϕ_{y}^{2}$ :

A (\frac{d y}{d x})^{2} - 2 B \frac{d y}{d x} + C = 0.

Solving this quadratic for the slope of characteristics gives

\frac{d y}{d x} = \frac{B \pm B ^{2} - A C}{A} .

The number of real characteristic directions is governed by the radicand $B^{2} - A C = Δ$ :

Δ > 0

: two real families → hyperbolic; canonical form

u_{ξ η} = \dots

(wave-like). -

Δ = 0

: one real family → parabolic; canonical form

u_{η η} = \dots

(heat-like). -

Δ < 0

: no real characteristics → elliptic; canonical form

u_{ξ ξ} + u_{η η} = \dots

(Laplace-like).

Substituting characteristic coordinates eliminates cross terms and reduces each type to its standard model, completing the classification. $■$

⟳The characteristic equation [5] is exactly the condition that the principal symbol vanishes along

ϕ = const

; its discriminant equals

Δ

Joseph Fourier | 1768–1830

Fourier, studying heat conduction for his 1822 treatise Théorie analytique de la chaleur, derived the heat equation and introduced the revolutionary idea that any function on an interval could be represented as a sum of sines and cosines. His method of separating variables and superposing trigonometric solutions remains the prototype for solving linear PDEs on bounded domains. Initially controversial — Lagrange doubted the convergence of his series — Fourier's work founded harmonic analysis and reshaped both mathematics and physics.

Building Intuition

Think of a PDE as a set of local rules a function must obey at every point, like a giant interlocking constraint network.

The Laplace equation $u_{xx} + u_{y y} = 0$ says each value equals the average of its neighbors — like a stretched soap film or a steady temperature with no hot spots. Such functions are infinitely smooth and have no local maxima inside the region.
The heat equation $u_{t} = u_{xx}$ says the temperature rises where the profile is concave up (a valley fills in) and falls where concave down. It smears information out instantly and irreversibly — you cannot run it backward.
The wave equation $u_{tt} = u_{xx}$ says acceleration equals curvature: a taut string snaps back toward its average shape, producing oscillation. Signals travel at finite speed along characteristics and keep their shape.

The discriminant is just asking: do disturbances travel along definite paths (hyperbolic), spread everywhere at once (elliptic/parabolic), or sit in equilibrium (elliptic)?

Characteristic curves and the three PDE types in the (x,t) plane

Worked Examples

1D Heat Equation by Separation of Variables

Given: A rod of length

L = π

m with thermal diffusivity

α = 1 m^{2} / s

, insulated sides, ends held at

0^{\circ}

u (0, t) = u (π, t) = 0

. Initial temperature

u (x, 0) = 5 sin (2 x)^{\circ}

C. Solve

u_{t} = u_{xx}

. Steps:

Separate: $u = X (x) T (t)$ gives $\frac{T ^{'}}{T} = \frac{X ^{''}}{X} = - λ$ .
Spatial problem $X^{''} + λ X = 0$ , $X (0) = X (π) = 0$ gives eigenfunctions $X_{n} = sin (n x)$ , $λ_{n} = n^{2}$ .
Temporal: $T_{n}^{'} = - n^{2} T_{n} \Rightarrow T_{n} = e^{- n^{2} t}$ .
General solution: $u = \sum_{n} b_{n} sin (n x) e^{- n^{2} t}$ .
Match initial data $5 sin (2 x)$ : only the $n = 2$ term survives, $b_{2} = 5$ , all others zero.

u (x, t) = 5 sin (2 x) e^{- 4 t}^{\circ} C

Check: at

t = 0

it equals

5 sin 2 x

✓; decays with rate

n^{2} = 4 s^{- 1}

. After

t = ln 2/4 \approx 0.173

s the amplitude halves.

⟳Separation reduces the PDE to two ODEs; the boundary conditions quantize

n

, and orthogonality of sines picks coefficients.

d'Alembert Solution of the Wave Equation

Given: Infinite string, wave speed

c = 2 m/s

, governed by

u_{tt} = 4 u_{xx}

. Initial shape

u (x, 0) = e^{- x^{2}}

m, initially at rest

u_{t} (x, 0) = 0

. Steps:

General solution travels as two waves: $u (x, t) = F (x - c t) + G (x + c t)$ .
Rest condition $u_{t} (x, 0) = - c F^{'} + c G^{'} = 0 \Rightarrow F = G$ (up to constant).
Shape condition $F + G = e^{- x^{2}} \Rightarrow F = G = \frac{1}{2} e^{- x^{2}}$ .

u (x, t) = \frac{1}{2} e^{- (x - 2 t)^{2}} + \frac{1}{2} e^{- (x + 2 t)^{2}} m

Interpretation: the initial bump splits into two half-height bumps moving left and right at

2

m/s. At

t = 1

s, the peaks sit at

x = \pm 2

m, each of height

0.5

⟳Factor

\partial_{tt} - c^{2} \partial_{xx} = (\partial_{t} - c \partial_{x}) (\partial_{t} + c \partial_{x})

; each factor gives a one-way wave, yielding d'Alembert's formula.

Common Misconceptions

"The general solution of a PDE is a formula with constants, like an ODE." False. Because PDEs constrain functions of several variables, their general solutions involve arbitrary functions, not arbitrary constants. For example, the general solution of

u_{x} = 0

in two variables is

u (x, y) = f (y)

for any function

f

. Pinning down a unique solution requires boundary/initial data along curves or surfaces, not just numbers. "More data always makes a problem better-posed." False. Specifying too much (or the wrong kind of) data can make a problem ill-posed. The backward heat equation, or the Cauchy problem for Laplace's equation, can have solutions that fail to depend continuously on the data — tiny perturbations cause arbitrarily large changes.

Hadamard's Ill-Posed Cauchy Problem

Consider Laplace's equation $u_{xx} + u_{y y} = 0$ with Cauchy data

u (x, 0) = 0, u_{y} (x, 0) = \frac{1}{n} sin (n x) .

The solution is

u (x, y) = \frac{1}{n ^{2}} sin (n x) sinh (n y)

. As

n \to \infty

the data

\to 0

uniformly, yet for any

y > 0

the solution grows like

\frac{1}{n ^{2}} e^{n y} \to \infty

. Hence the elliptic Cauchy problem is ill-posed — small data changes produce unbounded solution changes. Elliptic equations want boundary data, not Cauchy data.

Connections

PDEs sit at a crossroads of analysis, geometry, and physics.

They generalize the Ordinary Differential Equations to many variables; separation of variables turns a PDE into ODEs.
Solving linear PDEs on intervals invokes Fourier Series and Sturm-Liouville Theory, whose eigenfunctions diagonalize the spatial operator.
The Laplace Equation and Harmonic Functions connect to complex analysis: real and imaginary parts of holomorphic functions are harmonic.
Distributional and energy methods rest on Functional Analysis and Sobolev Spaces, the modern setting for weak solutions.
Conservation laws and characteristics tie hyperbolic PDEs to Vector Calculus and the Divergence Theorem.

Real-World Applications

Heat & diffusion — the heat equation models thermal management in electronics, geothermal gradients, and chemical diffusion (Fick's law).
Acoustics & seismology — the wave equation governs sound propagation, musical instrument design, and seismic imaging of the Earth's interior.
Electromagnetism — Maxwell's equations are a coupled PDE system; their wave-equation consequences predicted light and underpin antennas and optics.
Fluid dynamics — the Navier–Stokes equations model weather, aerodynamics of aircraft, and blood flow.
Quantum mechanics — the Schrödinger equation $i ℏ ψ_{t} = - \frac{ℏ ^{2}}{2 m} ψ_{xx} + V ψ$ is a (complex) parabolic-type PDE determining atomic and molecular structure.
Finance — the Black–Scholes equation for option pricing is a backward parabolic PDE transformable to the heat equation.
Image processing & graphics — anisotropic diffusion PDEs denoise images; the Laplace/Poisson equations drive seamless image cloning ("Poisson editing").

Historical Development

1746 — Jean d'Alembert derives the 1D wave equation for a vibrating string and gives its traveling-wave solution $F (x - c t) + G (x + c t)$ .
1750s — Leonhard Euler extends wave studies; debate erupts over which "functions" are admissible as initial data (the "vibrating string controversy").
1752 — Euler writes down the continuity and Euler equations for ideal fluid flow.
1782 — Pierre-Simon Laplace studies the potential equation $\nabla^{2} u = 0$ in celestial mechanics.
1813 — Siméon Poisson generalizes to $\nabla^{2} u = - ρ$ , modeling gravitational and electrostatic potentials with sources.
1822 — Joseph Fourier publishes the heat equation and Fourier-series method in Théorie analytique de la chaleur.
1845–1850 — Navier and Stokes complete the Navier–Stokes equations for viscous fluids.
1860s — James Clerk Maxwell unifies electricity and magnetism into PDEs predicting electromagnetic waves.
1902 — Jacques Hadamard formalizes well-posedness, exposing the ill-posed elliptic Cauchy problem and reframing what "solving" a PDE means.

Subtleties & Limits

Navier–Stokes existence and smoothness is a Clay Millennium Prize Problem: it is unknown whether smooth, globally-defined solutions to the 3D incompressible Navier–Stokes equations always exist for smooth initial data, or whether they can form singularities ("blow up") in finite time. More broadly, the theory of nonlinear PDEs lacks a unified framework: regularity, uniqueness, and existence must often be established equation by equation. Even classification breaks down — many physically important equations are of mixed type or fully nonlinear, where characteristics, weak solutions, and entropy conditions (as in the theory of shock-forming conservation laws) become essential and delicate.