Partial Differential Equations — Mathematics Knowledge System

Overview

A partial differential equation (PDE) is an equation for an unknown function of several variables that relates the function to its partial derivatives. Where an ordinary differential equation (ODE) describes a quantity that varies in one variable (usually time), a PDE describes a field — temperature u(x,t) along a bar, deflection u(x,y) of a plate, hydraulic head h(x,y,z) in an aquifer — that varies in space and time at once. Almost every continuum model in civil, mechanical, and environmental engineering is a PDE: heat conduction, elastic deformation, groundwater seepage, and structural vibration are all governed by them.

Intuition

Think of a PDE as a local bookkeeping rule. At every point it says how the field's rate of change in time is tied to its curvature in space. The heat equation uₜ = α u_xx literally reads 'a point heats up when it is cooler than the average of its neighbors' — u_xx measures how much a point dips below its surroundings. That single local rule, applied everywhere and combined with what happens at the boundaries, dictates the entire evolution. Second-order linear PDEs come in three flavours — parabolic (diffusion/heat, things smooth out and decay), hyperbolic (waves, signals travel at finite speed), and elliptic (equilibrium/steady state, no time at all) — and knowing the type tells you what kind of behaviour and boundary data to expect before you solve anything.

Formal Definition

Definition

A general second-order linear PDE in two variables, and the three canonical equations engineers meet most often — the heat (diffusion), wave, and Laplace (potential) equations:

A u_{xx} + B u_{xy} + C u_{yy} + D u_x + E u_y + F u = G

General 2nd-order linear PDE

u_t = \alpha\, u_{xx}

Heat / diffusion equation (parabolic)

u_{tt} = c^2\, u_{xx}

Wave equation (hyperbolic)

u_{xx} + u_{yy} = 0

Laplace's equation (elliptic)

Notation

Notation	Meaning
$u_x = \dfrac{\partial u}{\partial x}$	First partial derivative of u with respect to x
$u_{xx} = \dfrac{\partial^2 u}{\partial x^2}$	Second partial derivative of u with respect to x
$\nabla^2 u = u_{xx}+u_{yy}+u_{zz}$	The Laplacian — sum of unmixed second partials; measures how a value differs from its neighbourhood average
$\alpha$	Thermal diffusivity (heat equation) or, more generally, a diffusion coefficient
$c$	Wave speed in the wave equation

Derivation

The classification of a second-order linear PDE comes from the discriminant B² − 4AC of its principal (highest-derivative) part — exactly the quantity that classifies conic sections, which is why the three types borrow the names parabolic, hyperbolic, and elliptic:

B^2 - 4AC < 0 \;\Rightarrow\; \text{elliptic (e.g. Laplace: } A=C=1,\,B=0)

Steady-state / equilibrium problems — boundary values only, no time evolution.

B^2 - 4AC = 0 \;\Rightarrow\; \text{parabolic (e.g. heat: } u_t=\alpha u_{xx})

Diffusion — disturbances smooth out and decay; needs one initial condition plus boundary conditions.

B^2 - 4AC > 0 \;\Rightarrow\; \text{hyperbolic (e.g. wave: } u_{tt}=c^2u_{xx})

Wave propagation at finite speed c; needs two initial conditions (shape and velocity).

Properties

Superposition (linear PDEs)

\text{If } u_1,u_2 \text{ solve a linear homogeneous PDE, so does } c_1u_1+c_2u_2

Condition: PDE is linear and homogeneous

Example: Fourier's method builds a solution as an infinite sum of simple product solutions.

Well-posedness (Hadamard)

\text{A solution should exist, be unique, and depend continuously on the data}

Condition: Correct number/type of boundary and initial conditions for the PDE class

Example: The heat equation is well-posed forward in time but ill-posed backward — you cannot reliably reconstruct the past temperature field.

Maximum principle (Laplace/heat)

\max_{\text{interior}} u \le \max_{\partial\Omega} u

Condition: u harmonic (Laplace) or a solution of the heat equation with no internal source

Example: A steadily-heated plate with no internal heater is hottest somewhere on its edge, never strictly inside.

Applications

Heat conduction through building walls and mass-concrete dams (heat equation) governs curing temperatures and thermal-crack control; the 1-D consolidation of saturated clay under a foundation load is a diffusion PDE (Terzaghi's theory) that predicts how fast a building settles.

Worked Examples

Read off the principal-part coefficients A, B, C.
$A=3,\; B=2,\; C=1$
Compute the discriminant B² − 4AC.
$B^2-4AC = 2^2 - 4(3)(1) = 4-12 = -8$
A negative discriminant means the equation is elliptic — an equilibrium-type problem.
$-8 < 0 \;\Rightarrow\; \text{elliptic}$

Answer: Elliptic (discriminant B² − 4AC = −8 < 0).

Practice Problems

Difficulty 5/10

The steady-state seepage of water under a concrete dam is modelled by h_xx + h_yy = 0 for the hydraulic head h. What type of PDE is this?

ParabolicHyperbolicEllipticFirst-order

Difficulty 6/10

Terzaghi's 1-D consolidation of a clay layer is ∂u/∂t = c_v ∂²u/∂z², where u is excess pore-water pressure and c_v the coefficient of consolidation. A clay layer drained on both faces has thickness 4 m and c_v = 2 m²/yr. The dimensionless time factor is T_v = c_v t / H_dr². Find the time factor after 1 year (drainage path H_dr = half thickness).

Difficulty 5/10

A steel bridge stay cable carries tension T = 40 kN and has mass per unit length ρ = 2.5 kg/m. Transverse waves obey u_tt = c² u_xx with c = √(T/ρ). Find the wave speed c.

Difficulty 6/10

Show that in steady state (uₜ = 0), the 1-D heat equation uₜ = α u_xx reduces to a linear temperature profile across a wall, and find u(x) for a wall from x=0 to x=L with u(0)=T₁ and u(L)=T₂.

Difficulty 4/10

A vibrating cable is modelled by the wave equation u_tt = c²u_xx. How many INITIAL conditions in time are needed for a unique solution?

ZeroOneTwoDepends on the boundary

Difficulty 7/10

For the heat equation on 0 ≤ x ≤ L with both ends held at 0°C, state the spatial eigenvalues λₙ and explain why negative separation constants are rejected.

Common Mistakes

Common Mistake

Treating a PDE like an ODE and expecting arbitrary constants of integration.

Integrating a PDE introduces arbitrary FUNCTIONS, not constants — e.g. u_x = 0 gives u = f(y) for any function f. The extra freedom is pinned down by boundary/initial conditions, not by a couple of numbers.

Common Mistake

Using the wrong number of conditions: one initial condition for the wave equation, or an initial condition for Laplace's equation.

Match the data to the class: heat (parabolic) needs one initial condition, the wave equation (hyperbolic) needs two, and Laplace (elliptic) is a pure boundary-value problem with no time data at all.

Common Mistake

Believing the heat equation runs equally well backwards in time.

Diffusion is irreversible: it is well-posed forward but ill-posed backward, because tiny errors in the present blow up exponentially when you try to reconstruct the past.

Quiz

Flashcards

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Summary

A PDE relates an unknown field of several variables to its partial derivatives; it is the language of every continuum model in engineering.
Second-order linear PDEs are classified by B² − 4AC into elliptic (equilibrium), parabolic (diffusion/heat), and hyperbolic (waves).
The three canonical equations are the heat equation uₜ = α u_xx, the wave equation u_tt = c² u_xx, and Laplace's equation ∇²u = 0.
Separation of variables plus Fourier series solves many linear PDEs on simple domains by superposing decaying or oscillating modes.
Matching the correct number of boundary/initial conditions to the PDE type is what makes a problem well-posed.

References

BookStrauss, W. A. Partial Differential Equations: An Introduction, 2nd ed. Wiley, 2007.
BookFarlow, S. J. Partial Differential Equations for Scientists and Engineers. Dover, 1993.
BookBoyce, W. & DiPrima, R. Elementary Differential Equations and Boundary Value Problems, 11th ed., Ch. 10–11.
BookDas, B. M. & Sobhan, K. Principles of Geotechnical Engineering, 9th ed. — Terzaghi 1-D consolidation as a diffusion PDE.
WebsiteWikipedia — Partial differential equation