Riemann–Liouville derivative
E899968
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Riemann–Liouville derivative canonical | 3 |
| Riemann–Liouville fractional derivative | 1 |
| Riemann–Liouville fractional operator | 1 |
| left-sided Riemann–Liouville derivative | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992194 — 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: Riemann–Liouville derivative Context triple: [Caputo derivative, modifies, Riemann–Liouville derivative]
-
A.
Riemann–Liouville integral
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.
-
B.
Grünwald–Letnikov derivative
The Grünwald–Letnikov derivative is a fundamental definition of fractional differentiation based on limit processes and finite differences, widely used as a foundation for fractional calculus.
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C.
Caputo derivative
The Caputo derivative is a commonly used definition of a fractional derivative that modifies the Riemann–Liouville approach to allow for more physically meaningful initial conditions in differential equations.
-
D.
Hadamard fractional integral
The Hadamard fractional integral is a generalization of the classical integral that defines fractional-order integration using logarithmic kernels, particularly suited to functions defined on multiplicative (e.g., positive real) domains.
-
E.
Weyl fractional integral
The Weyl fractional integral is a generalization of the classical integral to arbitrary (including non-integer) orders, defined on periodic functions or the whole real line and used in fractional calculus to model memory and hereditary properties in various systems.
- 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: Riemann–Liouville derivative Target entity description: 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.
-
A.
Riemann–Liouville integral
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.
-
B.
Grünwald–Letnikov derivative
The Grünwald–Letnikov derivative is a fundamental definition of fractional differentiation based on limit processes and finite differences, widely used as a foundation for fractional calculus.
-
C.
Caputo derivative
The Caputo derivative is a commonly used definition of a fractional derivative that modifies the Riemann–Liouville approach to allow for more physically meaningful initial conditions in differential equations.
-
D.
Hadamard fractional integral
The Hadamard fractional integral is a generalization of the classical integral that defines fractional-order integration using logarithmic kernels, particularly suited to functions defined on multiplicative (e.g., positive real) domains.
-
E.
Weyl fractional integral
The Weyl fractional integral is a generalization of the classical integral to arbitrary (including non-integer) orders, defined on periodic functions or the whole real line and used in fractional calculus to model memory and hereditary properties in various systems.
- F. None of above. chosen
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
ⓘ
linked to:
Caputo derivative
|
| 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
ⓘ
linked to:
Riemann–Liouville derivative
right-sided Riemann–Liouville derivative ⓘ |
| introducedIn | 19th century ⓘ |
| isSpecialCaseOf |
Riemann–Liouville fractional operator
ⓘ
linked to:
Riemann–Liouville derivative
|
| mathematicalNature | non-local operator ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Joseph Liouville ⓘ |
| 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
ⓘ
Grünwald–Letnikov derivative ⓘ Riemann–Liouville integral ⓘ |
| 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
ⓘ
improper integral ⓘ |
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: Riemann–Liouville derivative Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Riemann–Liouville derivative
linked to: Riemann–Liouville derivative
linked to: Riemann–Liouville derivative