Classification of Second-Order PDEs

PDEs · Axiom Academy

LESSON Classification of Second-Order PDEs Learn how to classify second-order linear partial differential equations using the discriminant, and understand how each type describes fundamentally different physical phenomena Every second-order linear PDE in two variables can be written in the canonical form: The coefficients A , B , and C multiply the second-order partial derivatives. These coefficients may be constants or functions of the independent variables x and y . Step 2: The Discriminant Formula The classification is determined by computing the discriminant : This single value tells us everything we need to know about the PDE's fundamental character. The sign of the discriminant reveals which of the three types we're dealing with. This discriminant is analogous to the discriminant for conic sections. Just as B²-4AC distinguishes between ellipses, parabolas, and hyperbolas in geometry, the same formula classifies PDEs based on their mathematical structure. Step 3: The Three Classifications The sign of the discriminant divides all second-order linear PDEs into three categories: Each PDE type exhibits fundamentally different solution behavior and physical interpretation: Describe equilibrium or steady-state problems. Solutions are infinitely smooth and determined by boundary conditions on all sides. Information propagates from all directions . Examples: electrostatics, steady-state heat distribution, minimal surfaces.

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