Section 26.3
Terminology & Classification
Just like conic sections, 2nd order linear PDEs fall into three distinct categories.
1
Introduction
General form: .
The "discriminant" tells us the type.
2
The 3 Types
- : Hyperbolic (Wave-like).
Example: Wave Equation (). - : Parabolic (Diffusion-like).
Example: Heat Equation (). - : Elliptic (Equilibrium).
Example: Laplace Equation ().
3
Worked Examples
Example 1: Wave Equation
.
(coeff of ).
.
(coeff of ).
. Hyperbolic.
Example 2: Laplace Equation
.
.
. Elliptic.
Example 3: Mixed Type
(Tricomi).
.
Disc: .
Elliptic for . Hyperbolic for .
This happens in trans-sonic flight physics.
4
Practice Quiz
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