Riemann–Liouville derivative

E899968

fractional derivative mathematical concept operator in fractional calculus

The Riemann–Liouville derivative is a fundamental definition of fractional-order differentiation in fractional calculus, generalizing the classical derivative to non-integer orders via integral transforms.

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Statements (48)

Predicate	Object
instanceOf	fractional derivative ⓘ mathematical concept ⓘ operator in fractional calculus ⓘ
application	anomalous diffusion modeling ⓘ control theory ⓘ signal processing ⓘ viscoelasticity ⓘ
belongsTo	analysis ⓘ operator theory ⓘ
contrastedWith	Caputo fractional derivative NERFINISHED ⓘ
definedBy	fractional integral followed by integer-order differentiation ⓘ integral transform ⓘ
dependsOn	entire past history of the function over an interval ⓘ
domain	complex-valued functions ⓘ real-valued functions ⓘ
field	fractional calculus ⓘ
generalizes	classical derivative ⓘ integer-order derivative ⓘ
hasAlternativeFormulation	Laplace transform representation ⓘ
hasIssue	initial conditions expressed in terms of fractional integrals ⓘ non-zero derivative of constants ⓘ
hasRepresentation	integral representation ⓘ
hasVariant	left-sided Riemann–Liouville derivative NERFINISHED ⓘ right-sided Riemann–Liouville derivative ⓘ
introducedIn	19th century ⓘ
isSpecialCaseOf	Riemann–Liouville fractional operator NERFINISHED ⓘ
mathematicalNature	non-local operator ⓘ
namedAfter	Bernhard Riemann NERFINISHED ⓘ Joseph Liouville NERFINISHED ⓘ
notation	D_{a+}^α f(x) ⓘ _{a}D_{x}^{α} f(x) ⓘ
orderType	fractional order ⓘ non-integer order ⓘ
parameter	lower limit a ⓘ order α ⓘ upper limit b ⓘ
reducesTo	nth derivative when order is integer n ⓘ
relatedTo	Caputo derivative NERFINISHED ⓘ Grünwald–Letnikov derivative ⓘ Riemann–Liouville integral NERFINISHED ⓘ
requires	sufficient function regularity ⓘ
satisfies	linearity ⓘ semigroup property for fractional integrals ⓘ
usedFor	modeling power-law memory kernels ⓘ
usedIn	fractional differential equations ⓘ memory-effect models ⓘ
usesConcept	Gamma function NERFINISHED ⓘ improper integral ⓘ

Referenced by (1)

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Caputo derivative → modifies → Riemann–Liouville derivative ⓘ