Plot the region where the Principal Symbol is zero for both the Wave Operator-Hyperbolic and the Diffusion Operator-Parabolic

The classification of Partial Differential Equations (PDEs)-Hyperbolic, Parabolic, and Elliptic-is fundamentally determined by the zero set of their Principal Symbol, $\sigma_m(x, \xi)$, which geometrically defines the Characteristic Directions (the paths along which information naturally flows or spreads). When the symbol is zero, the operator dictates wave propagation (Hyperbolic, with two distinct characteristic lines) or diffusion (Parabolic, with a single, repeated characteristic direction defining instantaneous spreading); conversely, where the symbol is nonzero ( $\sigma_m \neq 0$ ), the direction is non-characteristic, implying that solutions are exceptionally smooth and initial/boundary data can be freely prescribed. The Elliptic operator (Laplace), having virtually no real characteristic directions outside the origin, defines steady-state phenomena where the solution is globally determined by the boundary conditions.

🎬Narrated Video

https://youtu.be/dmjtkDHqAdA

🏗️Structural clarification of Condensed Notes

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🗒️Downloadable Files - Recursive updates

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