Laplace equation
E139492
elliptic partial differential equation
partial differential equation
second-order differential 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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Laplace equation canonical | 3 |
| Laplace’s equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1221703 — 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: Laplace equation Context triple: [Pierre-Simon Laplace, developedConcept, Laplace equation]
-
A.
Klein–Gordon equation
The Klein–Gordon equation is a relativistic wave equation that describes spin-0 (scalar) particles in quantum field theory.
-
B.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
-
C.
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.
-
D.
Maxwell's equations
Maxwell's equations are the fundamental set of four equations in classical electromagnetism that describe how electric and magnetic fields are generated and interact with charges and currents.
-
E.
Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
- 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: Laplace equation Target entity description: 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.
-
A.
Klein–Gordon equation
The Klein–Gordon equation is a relativistic wave equation that describes spin-0 (scalar) particles in quantum field theory.
-
B.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
-
C.
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.
-
D.
Maxwell's equations
Maxwell's equations are the fundamental set of four equations in classical electromagnetism that describe how electric and magnetic fields are generated and interact with charges and currents.
-
E.
Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
elliptic partial differential equation
ⓘ
partial differential equation ⓘ second-order differential equation ⓘ |
| appearsIn |
electrostatic potential problems
ⓘ
gravitational potential problems ⓘ steady-state heat distribution problems ⓘ |
| coordinateSystems |
Cartesian coordinates
ⓘ
cylindrical coordinates ⓘ spherical coordinates ⓘ |
| describes | steady-state phenomena ⓘ |
| dimension | can be defined in any spatial dimension ⓘ |
| domain | scalar fields ⓘ |
| hasAlternativeName |
Laplace equation
ⓘ
surface form:
Laplace’s equation
|
| hasMathematicalForm | ∇²u = 0 ⓘ |
| involvesOperator |
Laplace operator
ⓘ
surface form:
Laplacian
|
| isSpecialCaseOf | Poisson equation ⓘ |
| linearity | linear ⓘ |
| mathematicalField |
analysis
ⓘ
partial differential equations ⓘ potential theory ⓘ |
| namedAfter | Pierre-Simon Laplace ⓘ |
| order | 2 ⓘ |
| property | solutions are harmonic functions ⓘ |
| relatedConcept |
Laplacian operator
ⓘ
Poisson equation ⓘ harmonic function ⓘ |
| requires | boundary conditions ⓘ |
| solutionMethod |
Fourier series
ⓘ
Green’s functions ⓘ conformal mapping ⓘ finite difference method ⓘ finite element method ⓘ separation of variables ⓘ |
| solutionProperty |
satisfies maximum principle
ⓘ
satisfies mean value property ⓘ |
| symbolOfOperator |
Δ
ⓘ
∇² ⓘ |
| type | homogeneous ⓘ |
| typicalBoundaryCondition |
Dirichlet boundary conditions
ⓘ
surface form:
Dirichlet boundary condition
Neumann boundary conditions in potential theory ⓘ
surface form:
Neumann boundary condition
Robin boundary condition ⓘ |
| usedIn |
electrostatics
ⓘ
engineering ⓘ fluid dynamics ⓘ geophysics ⓘ gravitation ⓘ heat conduction ⓘ mathematical physics ⓘ potential theory ⓘ |
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: Laplace equation Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Laplace’s equation
subject surface form:
Partial Differential Equations