PDE classification
Objective. Classify second-order linear PDEs.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
PDE classification sorts second-order linear partial differential equations into elliptic, parabolic or hyperbolic types based on the sign of a discriminant, mirroring the classification of conics. For Auₓₓ + Buₓᵧ + Cuᵧᵧ + ... = 0, the discriminant B²−4AC determines type: negative gives elliptic (like Laplace's equation), zero gives parabolic (like the heat equation), positive gives hyperbolic (like the wave equation). Example: uₜₜ = c²uₓₓ rewritten as uₜₜ − c²uₓₓ = 0 has A=1, C=−c², B=0, so B²−4AC=4c²>0, confirming it is hyperbolic. The type determines which solution methods and boundary/initial conditions are appropriate.
Worked example
Classify the PDE uₜₜ − 9uₓₓ = 0 using second-order PDE classification.
- 1Identify coefficients of second derivatives: A = 1 (coefficient of uₜₜ), B = 0, C = −9 (coefficient of uₓₓ)
- 2Compute discriminant: B² − 4AC = 0 − 4(1)(−9) = 36
- 3Since the discriminant is positive, the equation is hyperbolic
- 4This matches the known wave-equation type
Answer: Hyperbolic
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Start practisingCommon mistake. Students compute the discriminant using the equation's coefficients of u or its first derivatives instead of only the second-order derivative coefficients A, B, C; classification depends solely on the highest-order terms.
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