Calculator Input
Formula Used
The calculator uses the second order linear form: A uxx + B uxy + C uyy + D ux + E uy + F u = G.
The main classification formula is: Δ = B² - 4AC.
If Δ is positive, the equation is hyperbolic. If Δ is zero, it is parabolic. If Δ is negative, it is elliptic.
Real characteristic slopes are found from: A m² - B m + C = 0.
The rotation angle used to remove the mixed derivative is: θ = 0.5 atan2(B, A - C).
How to Use This Calculator
- Enter A, B, and C from the second derivative terms.
- Add D, E, F, and G if you want a fuller residual check.
- Enter optional derivative sample values for a test evaluation.
- Press the calculate button.
- Read the PDE type, discriminant, and canonical suggestion.
- Download the result as CSV or PDF if needed.
Example Data Table
| A | B | C | Discriminant | Type | Typical canonical direction |
|---|---|---|---|---|---|
| 1 | 0 | 1 | -4 | Elliptic | Uξξ + Uηη |
| 1 | 0 | -1 | 4 | Hyperbolic | Uξη |
| 1 | 2 | 1 | 0 | Parabolic | One repeated direction |
| 2 | 3 | 1 | 1 | Hyperbolic | Two real characteristic families |
Understanding Canonical PDE Form
A second order partial differential equation often looks complex at first. Its highest derivative terms carry the main geometric meaning. The calculator studies those terms first. It uses the coefficients A, B, and C from A uxx plus B uxy plus C uyy. These values create the discriminant B squared minus four A C. That single value separates many equations into elliptic, parabolic, or hyperbolic classes.
Why Classification Matters
Canonical form gives a cleaner version of the same principal behavior. It helps students see whether an equation acts like a wave problem, a heat style problem, or an equilibrium problem. Hyperbolic equations usually have two real characteristic families. Parabolic equations have one repeated family. Elliptic equations have no real characteristic family, so a rotation is more useful than real characteristic lines.
What This Tool Computes
This calculator accepts the principal coefficients and lower order coefficients. It reports the discriminant, PDE type, characteristic slopes where they exist, and a suggested variable transformation. For elliptic cases, it also gives a rotation angle and transformed principal coefficients. For parabolic cases, it gives the repeated characteristic direction. For hyperbolic cases, it gives two characteristic variables.
Using Results Carefully
The output is best used for equations with constant coefficients. Variable coefficient equations can still be studied at one selected point, but the classification may change from place to place. Always check the coefficient convention used in your class. This tool uses A uxx plus B uxy plus C uyy. Some books write the mixed term as two B uxy. That convention changes the discriminant formula.
Study Workflow
Enter the coefficients exactly as they appear in your equation. Add optional derivative sample values if you want a residual check. Press calculate. Read the type first. Then study the proposed canonical variables. Export the CSV for spreadsheet work. Export the PDF for notes, homework checks, or lesson records. The example table below shows how common coefficient sets lead to different canonical forms.
Practical Benefits
A clear canonical form reduces algebra mistakes. It also shows which numerical method may fit. Wave models need boundary data along suitable curves. Diffusion models need time direction care. Potential models need boundary region care during practice and review.
FAQs
What is a PDE canonical form?
It is a simplified form of a partial differential equation after a suitable variable change. It keeps the main second order behavior while making the equation easier to classify and study.
Which PDE convention does this calculator use?
It uses A uxx plus B uxy plus C uyy. Some textbooks use two B uxy instead. Check your source before comparing discriminants.
What does the discriminant show?
The discriminant B² - 4AC shows the main PDE type. Positive means hyperbolic. Zero means parabolic. Negative means elliptic.
Can I use variable coefficients?
Yes, but treat the result as local. Enter coefficient values at the chosen point. The PDE type may change at other points.
Why are lower order terms included?
Lower order terms do not decide the basic second order classification. They are included to support residual checks and fuller equation review.
What is a characteristic slope?
It is a real direction linked to the principal part. Hyperbolic equations have two real characteristic slopes. Parabolic equations have one repeated slope.
Why does an elliptic equation use rotation?
Elliptic equations have no real characteristic lines. A rotation can remove the mixed derivative and reveal the cleaner principal form.
Are CSV and PDF exports exact reports?
They export the calculated values shown on the page. Use them for records, notes, examples, or quick checking.