Classification of Second Order Pdes: Explained with Examples

Second-order linear PDEs of the form A·uxx + B·uxy + C·uyy + (lower-order terms) = 0 are classified by their discriminant D = B2 − 4AC into three types: hyperbolic (D > 0), parabolic (D = 0), and elliptic (D < 0) — the exact analogue of classifying conic sections.

The type is not bookkeeping; it dictates the physics and the mathematics. Hyperbolic equations propagate waves at finite speed, parabolic equations diffuse and smooth, elliptic equations describe steady equilibria. Crucially, each type demands different auxiliary data and different numerical methods — applying a wave-equation scheme to a diffusion problem produces nonsense.

Key Concepts

• Discriminant: D = B2 − 4AC computed from the second-derivative coefficients only; lower-order terms never affect the type.
• Hyperbolic (D > 0): prototype utt = c2uxx (wave equation); two families of real characteristic curves; disturbances travel at finite speed.
• Parabolic (D = 0): prototype ut = α2uxx (heat equation); one characteristic family; infinite propagation speed with smoothing.
• Elliptic (D < 0): prototype uxx + uyy = 0 (Laplace equation); no real characteristics; solutions are smooth and satisfy maximum principles.
• Characteristic curves: special curves along which the PDE degenerates; they carry signals (hyperbolic) or are absent (elliptic).
• Well-posedness: each type pairs with its natural auxiliary conditions — initial data for hyperbolic/parabolic, boundary data for elliptic.

How It Works

1. Identify A, B, C: A is the coefficient of uxx, B of uxy, C of uyy. Watch out: some texts write the mixed term as 2B·uxy, which changes the discriminant formula — confirm the convention first.
2. Compute D = B2 − 4AC at the point of interest (for variable coefficients the type can change across the domain).
3. Classify: D > 0 → hyperbolic; D = 0 → parabolic; D < 0 → elliptic.
4. Match the prototype: compare with the wave, heat, and Laplace equations to predict behaviour and choose solution methods.

Worked Example

Classify 3uxx + 5uxy + 2uyy + ux = 0. Read off A = 3, B = 5, C = 2 (the ux term is lower-order and ignored). D = 25 − 4·3·2 = 25 − 24 = 1 > 0, so the equation is hyperbolic — wave-like, with two characteristic families. For contrast: uxx + 2uxy + uyy = 0 gives D = 4 − 4 = 0, parabolic (diffusion-like); and uxx + uyy = 0 (Laplace) gives D = −4 < 0, elliptic (steady-state). Note how the classification used only A, B, C in each case.

Common Misconceptions

• Lower-order terms change the type. They never do — only the principal (second-derivative) part counts.
• Classification is just naming. It determines which initial/boundary conditions give a well-posed problem and which numerical schemes are stable.
• Nonlinear PDEs classify the same way. For nonlinear equations the type is local — it depends on the solution itself and can change (transonic flow is the classic example).

Real-World Applications

• Acoustics, vibrating strings, and electromagnetic waves are hyperbolic — modelled by the wave equation with finite signal speed.
• Heat conduction, chemical diffusion, and option pricing (Black-Scholes) are parabolic.
• Electrostatics, steady-state heat, and ideal fluid flow are elliptic (Laplace/Poisson).
• Numerical analysts choose finite-difference stencils based on type: explicit schemes suit hyperbolic problems, implicit schemes suit parabolic ones.

Second-order PDEs are classified by D = B2 − 4AC into hyperbolic, parabolic, and elliptic — the PDE versions of hyperbolas, parabolas, and ellipses. The type, set by the second-derivative coefficients alone, decides whether the physics is wave propagation, diffusion, or steady equilibrium, and dictates the correct auxiliary conditions and numerical methods. Always classify before solving.