Riemann–Liouville integral

E47352

concept in fractional calculus fractional integral operator mathematical operator

The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.

Observed surface forms (3)

Surface form	Occurrences
Riemann–Liouville derivative	1
left-sided Riemann–Liouville integral	1
right-sided Riemann–Liouville integral	1

Statements (46)

Predicate	Object
instanceOf	concept in fractional calculus ⓘ fractional integral operator ⓘ mathematical operator ⓘ
actsOn	locally integrable functions ⓘ suitable functions ⓘ
appearsIn	fractional Sobolev spaces ⓘ theory of fractional-order systems ⓘ
assumes	sufficient regularity of the integrand ⓘ
belongsTo	integral transforms with weakly singular kernels ⓘ
comparedWith	Hadamard fractional integral ⓘ Weyl fractional integral ⓘ
domain	interval [a,b] ⓘ
field	fractional calculus ⓘ mathematical analysis ⓘ
generalizes	Cauchy formula for repeated integration ⓘ n-fold repeated integral ⓘ
hasLimitingCase	identity operator when α → 0^+ ⓘ
hasNotation	I_{-b}^{α} ⓘ I_{a+}^{α} ⓘ
hasOrder	real order α > 0 ⓘ
hasVariant	Riemann–Liouville integral self-linksurface differs ⓘ surface form: left-sided Riemann–Liouville integral Riemann–Liouville integral self-linksurface differs ⓘ surface form: right-sided Riemann–Liouville integral
introducedIn	19th century ⓘ
isDefinedBy	convolution with power-law kernel ⓘ
isLinear	true ⓘ
isToolFor	defining fractional derivatives ⓘ solving fractional integral equations ⓘ
kernelType	power-law kernel (x-t)^{α-1} ⓘ
namedAfter	Bernhard Riemann ⓘ Joseph Liouville ⓘ
parameter	lower limit a ⓘ order α ⓘ
property	depends on entire past history from a to x ⓘ nonlocal operator ⓘ
reducesTo	n-fold classical integral when α is a positive integer n ⓘ
relatedTo	Caputo derivative ⓘ Riemann–Liouville integral self-linksurface differs ⓘ surface form: Riemann–Liouville derivative fractional differential equations ⓘ
satisfies	semigroup property in the order α under suitable conditions ⓘ
usedFor	defining fractional powers of operators ⓘ
usedIn	anomalous diffusion models ⓘ control theory ⓘ modeling memory effects ⓘ signal processing ⓘ viscoelasticity ⓘ
usesFunction	Gamma function ⓘ

Referenced by (4)

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

Riemann–Liouville integral → hasVariant → Riemann–Liouville integral self-linksurface differs ⓘ

this entity surface form: left-sided Riemann–Liouville integral

Riemann–Liouville integral → hasVariant → Riemann–Liouville integral self-linksurface differs ⓘ

this entity surface form: right-sided Riemann–Liouville integral

Bernhard Riemann → knownFor → Riemann–Liouville integral ⓘ

Riemann–Liouville integral → relatedTo → Riemann–Liouville integral self-linksurface differs ⓘ

this entity surface form: Riemann–Liouville derivative