Quadratic Second Order PDE Solver
Enter coefficients for the equation A u_xx + B u_xy + C u_yy + D u_x + E u_y + F u = G.
Example Data Table
| Example | A | B | C | D | E | F | G | Classification |
|---|---|---|---|---|---|---|---|---|
| Wave style case | 1 | 4 | 1 | 0 | 0 | 0 | 0 | Hyperbolic |
| Heat style case | 1 | 2 | 1 | 1 | 0 | 0 | 3 | Parabolic |
| Laplace style case | 1 | 0 | 1 | 0 | 0 | 0 | 5 | Elliptic |
Plotly Graph
This graph shows how the discriminant changes as B varies, while A and C stay fixed.
Formula Used
The principal part is A uxx + B uxy + C uyy. The quadratic discriminant is Δ = B² − 4AC.
If Δ > 0, the PDE is hyperbolic. If Δ = 0, the PDE is parabolic. If Δ < 0, the PDE is elliptic.
Characteristic slopes come from A m² − Bm + C = 0. These roots help identify characteristic directions.
Canonical principal coefficients come from the rotated matrix eigenvalues: λ₁, λ₂ = ((A + C) ± √((A − C)² + B²)) / 2.
The rotation angle that removes the mixed term is θ = ½ atan2(B, A − C).
How to Use This Calculator
- Enter coefficients A, B, and C for the principal part.
- Add D, E, F, and G if lower terms exist.
- Optionally enter x, y, u, ux, and uy estimates.
- Click Solve PDE to classify the equation.
- Review discriminant, characteristic roots, and canonical coefficients.
- Use CSV or PDF buttons to export the result.
- Inspect the graph to see discriminant behavior clearly.
Frequently Asked Questions
1. What does this calculator solve?
It analyzes the quadratic principal part of a second order PDE. It classifies the equation, computes characteristic roots, estimates a rotation angle, and shows canonical principal coefficients.
2. What is the discriminant here?
The discriminant is B² − 4AC. It comes from the quadratic characteristic equation. Its sign determines whether the PDE is elliptic, parabolic, or hyperbolic.
3. Why are A, B, and C most important?
These coefficients define the principal second derivative terms. PDE type depends on them alone. Lower order terms affect the full model but not the standard classification.
4. What do characteristic roots mean?
They represent characteristic slopes from the quadratic relation A m² − Bm + C = 0. Real roots indicate real characteristic curves. Complex roots usually appear for elliptic equations.
5. What is the rotation angle used for?
The angle rotates coordinates to remove the mixed derivative term. This helps convert the principal part into canonical form and simplifies interpretation.
6. Does this produce a full analytic PDE solution?
No. It solves the quadratic characteristic structure and classification problem. A full analytic solution usually also requires boundary conditions, forcing details, and a chosen solving method.
7. Why can the roots be complex?
Complex roots occur when B² − 4AC is negative. That means the PDE is elliptic, so no real characteristic directions exist in the usual sense.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet style records. Use the PDF button to open a print-ready layout that you can save as a PDF.