CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The one-dimensional value depends on the characteristic interval

Statement

Let $c > 0$ and let $u$ be the solution of d'Alembert's formula and uniqueness in one dimension for data $u_{0} \in C^{2} (R)$ , $u_{1} \in C^{1} (R)$ . For every $(x, t) \in R \times [0, \infty)$ , the value $u (x, t)$ is determined by the restrictions of $u_{0}$ and $u_{1}$ to the closed interval $[x - c t, x + c t]$ : if $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ are admissible data agreeing with $(u_{0}, u_{1})$ there, then the corresponding solution satisfies $\tilde{u} (x, t) = u (x, t)$ . In particular, changing the data outside $[x - c t, x + c t]$ does not change the value at $(x, t)$ .

Facts & Assumptions

Given: a speed $c > 0$ , data $u_{0} \in C^{2} (R)$ , $u_{1} \in C^{1} (R)$ , a point $(x, t) \in R \times [0, \infty)$ , and admissible data $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ with ${\tilde{u}}_{0} = u_{0}$ and ${\tilde{u}}_{1} = u_{1}$ on $[x - c t, x + c t]$ .

[F1]

The d'Alembert formula of d'Alembert's formula and uniqueness in one dimension reads $u (x, t) = \frac{1}{2} (u_{0} (x - c t) + u_{0} (x + c t)) + \frac{1}{2 c} \int_{x - c t}^{x + c t} u_{1} (y) d y$ .

[F2]

The d'Alembert expression attains both initial data and defines the unique classical solution of the corresponding Cauchy problem (The d'Alembert expression attains both initial data, d'Alembert's formula and uniqueness in one dimension).

Proof

1.1F1algebra

The formula of [F1] evaluates $u_{0}$ only at the two endpoints $x - c t$ and $x + c t$ of the interval and integrates $u_{1}$ only over that interval; hence replacing $(u_{0}, u_{1})$ by any admissible pair with the same restrictions to $[x - c t, x + c t]$ leaves the right-hand side unchanged, so the d'Alembert expression of the new data equals $u (x, t)$ at the given point.

2.1F2algebra∎

By [F2] both expressions are the solutions of their respective Cauchy problems, so the solution $\tilde{u}$ of the data $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ satisfies $\tilde{u} (x, t) = u (x, t)$ ; in particular the value at $(x, t)$ is unchanged by altering the data off $[x - c t, x + c t]$ .

Depends on

Used by

Displacement data versus velocity data in one dimension Example

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Per Kristen Jakobsen, An Introduction to Partial Differential Equations (arXiv:1901.03022) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #10: Introduction to the Wave Equation (Fall 2011) (standard reference, not scraped)