Module classification 

PDE classification algorithms

⚠️ IMPORTANT LIMITATIONS (Version 0.1.0):

This module currently uses simplified heuristics and is NOT mathematically rigorous. See DESIGN_PROPER_CLASSIFICATION.md for the planned proper implementation.

Current Implementation Status

❌ Known Issues:

1. No Coefficient Extraction: Does not extract actual coefficients A, B, C from equations
2. Heuristic Classification: Uses expression structure (Add vs Mul) as proxy for PDE type
3. Hardcoded Patterns: Only works for specific well-known PDEs (heat, wave, Laplace)
4. Invalid Discriminant: Returns hardcoded values instead of computing B² - 4AC

Recommended Usage:

• ✅ Educational purposes with standard PDEs only
• ❌ Do NOT use for arbitrary PDE classification
• ❌ Do NOT rely on results for mathematical correctness

Proper Implementation (Planned v0.2.0)

The proper implementation will:

1. Extract actual coefficients A, B, C from PDE equations
2. Compute true discriminant B² - 4AC
3. Use symbolic sign analysis for classification
4. Handle both constant and variable coefficient PDEs

See DESIGN_PROPER_CLASSIFICATION.md for complete design specification.

Mathematical Foundation (Reference)

For a second-order linear PDE: A·u_xx + B·u_xy + C·u_yy + ... = G

Classification by discriminant Δ = B² - 4AC:

• Elliptic (Δ < 0): Laplace equation, steady-state problems
• Parabolic (Δ = 0): Heat equation, diffusion processes
• Hyperbolic (Δ > 0): Wave equation, propagation phenomena

References:

• Evans, L. C. (2010). Partial Differential Equations. AMS.
• Strauss, W. A. (2007). Partial Differential Equations: An Introduction. Wiley.

Functions

classify_pde

Classify a PDE and return its type