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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Poisson's equation | 1 |
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 ⓘ |
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