# Partial differential equation

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention.

mko-partial-differential-equation · definition · v0.10.0



## Mathematical statement

This slice fixes bivariate quadratic polynomial candidates u and linear constant-coefficient operators and computes first- or second-order residuals exactly. The second-order principal part is a*u_xx+b*u_xy+c*u_yy, with discriminant b²−4*a*c. This convention must be explicit.





## Representative formula (exact TeX)

L(u)=g





## Assumptions


- Exercises handle only symbolic polynomial derivatives, point residuals, principal-part classification and polynomial identities.
- A zero point residual proves only that point check. Only an identically zero symbolic residual polynomial proves the specified polynomial satisfies the operator equation globally.





## Symbol scope


- L — linear PDE operator
- u — candidate unknown function





## Intuition, definition, and misconceptions

Operator coefficients and the mixed-term convention determine the classification formula.

A point residual is not a proof of a global solution.

A successful candidate check does not automatically supply a general solution, existence, uniqueness or well-posedness.

Do not infer a global solution from zero sampled residuals or mix the discriminants for b*u_xy and 2*b*u_xy conventions.





## Adopted derivation guide

[Partial differential equation: principal-part classification and symbolic residuals of quadratic polynomial candidates](https://ocme.evemisslab.com/en/derivations/derivation-mko-partial-differential-equation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Partial derivative](https://ocme.evemisslab.com/en/mko/mko-partial-derivative/) — PDE operators use first or higher partial derivatives.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — A PDE is an equation involving functions and partials.
- [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) — Polynomial partials follow from derivative rules.





## Verification boundary

Only the specified linear constant-coefficient polynomial slice is checked; general solution theory is not handled.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add boundary-value problems, weak solutions and well-posedness.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/partial-differential-equation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/index.md)