Sturm–Liouville problem
E697758
The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Sturm–Liouville theory | 2 |
| Sturm–Liouville operators | 1 |
| Sturm–Liouville problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871809 — 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: Sturm–Liouville problem Context triple: [Jacobi polynomials, areSolutionsOf, Sturm–Liouville problem]
-
A.
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.
-
B.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
-
C.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
-
D.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
-
E.
Poisson equation
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.
- 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: Sturm–Liouville problem Target entity description: The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.
-
A.
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.
-
B.
Gelfand–Levitan theory
Gelfand–Levitan theory is a foundational framework in inverse spectral theory that reconstructs differential operators or potentials from their spectral data using integral equations.
-
C.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
-
D.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
-
E.
Poisson equation
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
boundary value problem
ⓘ
concept in differential equations ⓘ eigenvalue problem ⓘ |
| assumesConditionOn |
p(x) > 0 on (a,b)
ⓘ
w(x) > 0 on (a,b) ⓘ |
| definedOnInterval | [a,b] ⓘ |
| hasBoundaryConditions | linear homogeneous boundary conditions at x=a and x=b ⓘ |
| hasBoundaryConditionType |
Dirichlet boundary conditions
NERFINISHED
ⓘ
Neumann boundary conditions NERFINISHED ⓘ Robin boundary conditions ⓘ periodic boundary conditions ⓘ |
| hasCoefficientFunction |
p(x)
ⓘ
q(x) ⓘ |
| hasEigenvalueEquation | L[y] = λ w(x) y(x) ⓘ |
| hasGeneralForm | -(d/dx)[p(x) y'(x)] + q(x) y(x) = λ w(x) y(x) ⓘ |
| hasHistoricalDevelopmentPeriod | 19th century ⓘ |
| hasKeyResult | Sturm–Liouville theory of eigenfunction expansions NERFINISHED ⓘ |
| hasKeyTheorem |
Sturm comparison theorem
ⓘ
Sturm oscillation theorem NERFINISHED ⓘ Sturm separation theorem NERFINISHED ⓘ |
| hasOperator | L[y] = -(d/dx)[p(x) y'(x)] + q(x) y(x) ⓘ |
| hasOrder | second-order ⓘ |
| hasOrthogonalityRelation | ∫_a^b w(x) y_m(x) y_n(x) dx = 0 for m ≠ n ⓘ |
| hasProperty |
eigenfunctions form a complete set under suitable conditions
ⓘ
eigenfunctions form an orthogonal set with respect to w(x) ⓘ real eigenvalues ⓘ self-adjoint differential operator ⓘ |
| hasSpecialCase |
Bessel differential equation
NERFINISHED
ⓘ
Hermite differential equation NERFINISHED ⓘ Laguerre differential equation NERFINISHED ⓘ Legendre differential equation NERFINISHED ⓘ spherical harmonics eigenvalue problem ⓘ |
| hasSpectrum | discrete under regular boundary conditions ⓘ |
| hasType | linear differential equation ⓘ |
| hasWeightFunction | w(x) ⓘ |
| namedAfter |
Jacques Charles François Sturm
NERFINISHED
ⓘ
Joseph Liouville NERFINISHED ⓘ |
| relatedTo |
Hilbert space theory
ⓘ
orthogonal polynomials ⓘ spectral theory of linear operators ⓘ |
| usedFor |
Fourier-type series expansions
ⓘ
separation of variables in partial differential equations ⓘ |
| usedIn |
engineering
ⓘ
heat conduction problems ⓘ mathematical physics ⓘ quantum mechanics ⓘ vibrations and acoustics ⓘ |
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: Sturm–Liouville problem Description of subject: The Sturm–Liouville problem is a class of second-order linear differential equations with boundary conditions that yield real eigenvalues and orthogonal eigenfunctions forming a basis for function expansions in mathematical physics and engineering.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Sturm–Liouville theory
this entity surface form:
Sturm–Liouville operators
this entity surface form:
Sturm–Liouville theory