Cauchy problem
E239288
The Cauchy problem is a fundamental type of initial value problem in partial differential equations, where one seeks a solution satisfying prescribed data on a given initial surface.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cauchy problem canonical | 4 |
| Cauchy problem in differential equations | 1 |
| Hadamard well-posedness | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171649 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy problem Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy problem]
-
A.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
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B.
d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
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C.
ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
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D.
Hamilton–Jacobi equation
The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
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E.
local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy problem Target entity description: The Cauchy problem is a fundamental type of initial value problem in partial differential equations, where one seeks a solution satisfying prescribed data on a given initial surface.
-
A.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
B.
d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
-
C.
ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
-
D.
Hamilton–Jacobi equation
The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
-
E.
local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
initial value problem
ⓘ
mathematical concept ⓘ problem in partial differential equations ⓘ |
| appliesTo |
ordinary differential equations
ⓘ
partial differential equations ⓘ |
| concerns | propagation of information from initial data ⓘ |
| coreIdea | determine future and past evolution from initial state ⓘ |
| difficulty | may fail to be well-posed for elliptic equations ⓘ |
| example |
Schrödinger equation with initial wave function
ⓘ
heat equation with initial temperature distribution ⓘ wave equation with initial displacement and velocity ⓘ |
| field |
applied mathematics
ⓘ
mathematical analysis ⓘ partial differential equations ⓘ |
| hasComponent |
differential equation
ⓘ
initial data ⓘ initial surface ⓘ |
| hasDefinition | problem of finding a solution of a differential equation that satisfies prescribed initial data on a given set ⓘ |
| hasProperty | may be well-posed or ill-posed ⓘ |
| historicalPeriod | 19th century ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Cauchy–Kovalevskaya theorem
ⓘ
surface form:
Cauchy–Kowalevski theorem
local existence and uniqueness theorem ⓘ
surface form:
Cauchy–Lipschitz theorem
boundary value problem ⓘ elliptic partial differential equation ⓘ existence and uniqueness theorem ⓘ hyperbolic partial differential equation ⓘ parabolic partial differential equation ⓘ well-posed problem ⓘ |
| requires | specification of data on a non-characteristic surface for hyperbolic PDEs ⓘ |
| solvedBy |
Fourier transform methods
ⓘ
energy methods ⓘ fixed point theorems ⓘ semigroup theory ⓘ |
| specialCaseOf | initial value problem ⓘ |
| studiedIn |
Sobolev spaces
ⓘ
functional analysis ⓘ microlocal analysis ⓘ theory of distributions ⓘ |
| typicalFormulation | find u solving F(x,u,Du,…) = 0 with u|_S = f on an initial surface S ⓘ |
| usedIn |
control theory
ⓘ
engineering ⓘ fluid dynamics ⓘ general relativity ⓘ mathematical physics ⓘ |
| wellPosednessCriteria |
continuous dependence on initial data
ⓘ
existence of solution ⓘ uniqueness of solution ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy problem Description of subject: The Cauchy problem is a fundamental type of initial value problem in partial differential equations, where one seeks a solution satisfying prescribed data on a given initial surface.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hadamard well-posedness
subject surface form:
Augustin-Louis Cauchy
this entity surface form:
Cauchy problem in differential equations