Poisson equation
E559802
The Poisson equation is a fundamental partial differential equation in mathematical physics that relates the Laplacian of a potential field to a given source distribution, widely used in electrostatics, gravitation, and heat conduction.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Poisson equation canonical | 7 |
| Poisson's equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5973625 — 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: Poisson equation Context triple: [Siméon Denis Poisson, notableWork, Poisson equation]
-
A.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
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B.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
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C.
Helmholtz equation
The Helmholtz equation is a fundamental partial differential equation that describes time-harmonic wave propagation in fields such as acoustics, electromagnetism, and optics.
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D.
Laplace operator
The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
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E.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
- 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: Poisson equation Target entity description: The Poisson equation is a fundamental partial differential equation in mathematical physics that relates the Laplacian of a potential field to a given source distribution, widely used in electrostatics, gravitation, and heat conduction.
-
A.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
-
B.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
-
C.
Helmholtz equation
The Helmholtz equation is a fundamental partial differential equation that describes time-harmonic wave propagation in fields such as acoustics, electromagnetism, and optics.
-
D.
Laplace operator
The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
-
E.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
partial differential equation ⓘ |
| classification | elliptic for time-independent problems ⓘ |
| commonBoundaryConditions |
Dirichlet boundary conditions
ⓘ
Neumann boundary conditions NERFINISHED ⓘ Robin boundary conditions ⓘ |
| coordinateInvariance | invariant under Euclidean rotations ⓘ |
| definedOver | scalar field ⓘ |
| dimension | can be defined in any spatial dimension n ≥ 1 ⓘ |
| equationType |
inhomogeneous
ⓘ
linear ⓘ second-order ⓘ |
| field |
applied mathematics
ⓘ
electrostatics ⓘ gravitation ⓘ heat conduction ⓘ mathematical physics ⓘ potential theory ⓘ |
| generalizes | Laplace equation NERFINISHED ⓘ |
| governs | electrostatic potential in vacuum with charge density ρ ⓘ |
| historicalPeriod | 19th century ⓘ |
| involvesOperator | Laplacian NERFINISHED ⓘ |
| mathematicalArea |
analysis
ⓘ
mathematical physics ⓘ partial differential equations ⓘ |
| namedAfter | Siméon Denis Poisson NERFINISHED ⓘ |
| physicalForm |
∇²T = −q/k in steady-state heat conduction
ⓘ
∇²Φ = 4πGρ in Newtonian gravitation ⓘ ∇²φ = −ρ/ε₀ in electrostatics ⓘ |
| reducesTo | Laplace equation when f = 0 ⓘ |
| relatedConcept |
Green's function
ⓘ
fundamental solution of Laplacian ⓘ potential field ⓘ source distribution ⓘ |
| requires | boundary conditions for unique solution ⓘ |
| solutionMethod |
Fourier transform methods
ⓘ
Green's function methods ⓘ finite difference methods ⓘ finite element methods ⓘ separation of variables ⓘ spectral methods ⓘ |
| sourceTerm | function f ⓘ |
| standardForm | ∇²φ = f ⓘ |
| typicalDomain | subset of Euclidean space ⓘ |
| unknownFunction | potential φ ⓘ |
| usedFor |
modeling electrostatic potential from charge density
ⓘ
modeling gravitational potential from mass density ⓘ steady-state diffusion with sources ⓘ steady-state heat conduction with internal sources ⓘ |
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: Poisson equation Description of subject: The Poisson equation is a fundamental partial differential equation in mathematical physics that relates the Laplacian of a potential field to a given source distribution, widely used in electrostatics, gravitation, and heat conduction.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poisson's equation
Vlasov equation (for long-range interactions and negligible collisions)
→
combinedWith
→
Poisson equation
ⓘ
subject surface form:
Vlasov equation
subject surface form:
Partial Differential Equations