Sturm–Liouville problem

E697758

boundary value problem concept in differential equations eigenvalue problem

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.

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Observed surface forms (2)

Surface form	Occurrences
Sturm–Liouville theory	2
Sturm–Liouville operators	1

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 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jacobi polynomials → areSolutionsOf → Sturm–Liouville problem ⓘ

Jacobi operator → relatedTo → Sturm–Liouville problem ⓘ

this entity surface form: Sturm–Liouville theory

Gelfand–Levitan theory → appliesTo → Sturm–Liouville problem ⓘ

this entity surface form: Sturm–Liouville operators

Boris Levitan → hasResearchArea → Sturm–Liouville problem ⓘ

this entity surface form: Sturm–Liouville theory