Classification of PDEs: parabolic, hyperbolic, elliptic
A general second-order linear PDE Au_xx + Bu_xy + Cu_yy + ⋯ = 0 is classified by the discriminant B² − 4AC in exact analogy with a conic section: negative means elliptic (like Laplace's equation, describing steady states), zero means parabolic (like the heat equation, describing diffusion), and positive means hyperbolic (like the wave equation, describing propagation) — and each class genuinely needs a different kind of boundary or initial condition to have a well-posed solution.
Practise this conceptWrite the most general second-order linear PDE in two variables as Au_xx + Bu_xy + Cu_yy + (lower-order terms) = 0. The three coefficients A, B, C in front of the second-derivative terms determine everything about the equation's qualitative character, through the same discriminant that classifies a conic section: Δ = B² − 4AC. This isn't a coincidence — the characteristic curves of the PDE (directions along which information propagates) satisfy the same quadratic relation that defines an ellipse, parabola, or hyperbola, which is exactly where the names come from.
Δ < 0 is elliptic — Laplace's and Poisson's equations are the standard examples, and they describe equilibrium states with no preferred direction of propagation. An elliptic equation is well-posed with boundary conditions specified on the entire boundary of a closed region (a Dirichlet or Neumann problem), never with initial conditions on an open curve. Δ = 0 is parabolic — the heat equation is the standard example, describing diffusion, and it's well-posed with an initial condition at one time plus boundary conditions on the spatial edges, evolving forward. Δ > 0 is hyperbolic — the wave equation is the standard example, describing propagation at a finite speed, and it's well-posed with initial position and initial velocity at one time, with information travelling outward along characteristic curves.
Because A, B, C are the equation's own coefficients (not the solution), the type is a property of the equation itself and doesn't change from one solution to another. But if those coefficients depend on x and y (rather than being constants), the equation can genuinely change type from region to region — the Tricomi equation, elliptic in one region and hyperbolic in another, is the textbook example of exactly this.
Key ideas
Requires: Second-order differential equations, Partial derivatives
Formulas
| Symbol | Name | Unit |
|---|---|---|
| coefficient | 1 (dimensionless) | |
| coefficient | 1 (dimensionless) | |
| coefficient | 1 (dimensionless) | |
| discriminant | 1 (dimensionless) |
Common misconceptions
- Believes the classification depends on the specific solution u(x,y) being examined, rather than being a property of the equation's coefficients A, B, C alone — the same PDE has the same type everywhere those coefficients are evaluated, regardless of which solution you plug in.
- Thinks the well-posed setup (what kind of boundary or initial conditions to prescribe) is the same for every PDE type, so any equation can be handed initial conditions on an open boundary like the wave equation, or boundary conditions all around like Laplace's equation.
- Miscomputes the discriminant as B² + 4AC or A² − 4BC by pattern-matching loosely to the quadratic formula instead of the actual PDE discriminant B² − 4AC, which flips the classification for equations where the sign of that swap actually matters.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Equations of mathematical physicsLOFY.04.035