Caputo derivative
E259777
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Caputo fractional derivative | 3 |
| Caputo derivative canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364665 — 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: Caputo derivative Context triple: [Riemann–Liouville integral, relatedTo, Caputo 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.
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B.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
-
C.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
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D.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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E.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
- 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: Caputo derivative Target entity description: 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.
-
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.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
-
C.
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
-
D.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
E.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
fractional derivative
ⓘ
mathematical concept ⓘ operator ⓘ |
| advantageOver | Riemann–Liouville derivative in handling initial value problems ⓘ |
| allows | classical initial conditions in terms of integer-order derivatives ⓘ |
| alsoKnownAs |
Caputo derivative
ⓘ
surface form:
Caputo fractional derivative
|
| appearsIn |
fractional diffusion equations
ⓘ
fractional relaxation equations ⓘ fractional-order control systems ⓘ |
| basedOn |
Riemann–Liouville integral
ⓘ
surface form:
Riemann–Liouville derivative
|
| contrastWith | Riemann–Liouville derivative initial condition formulation ⓘ |
| domain |
complex-valued functions
ⓘ
real-valued functions ⓘ |
| field |
fractional calculus
ⓘ
mathematics ⓘ |
| generalizes | integer-order derivative ⓘ |
| hasVariant |
Atangana–Baleanu–Caputo derivative
ⓘ
Caputo–Fabrizio derivative ⓘ |
| kernelType | power-law kernel ⓘ |
| mathematicalArea |
analysis
ⓘ
differential equations ⓘ |
| modifies | Riemann–Liouville derivative ⓘ |
| namedAfter | Michele Caputo ⓘ |
| orderType |
fractional order between 0 and 1
ⓘ
non-integer order ⓘ real order ⓘ |
| property |
linear operator
ⓘ
nonlocal operator ⓘ |
| purpose | to define fractional derivatives with physically meaningful initial conditions ⓘ |
| relatedTo |
Grünwald–Letnikov derivative
ⓘ
Hadamard fractional derivative ⓘ Riemann–Liouville integral ⓘ |
| requires | sufficient smoothness of the function ⓘ |
| specialCase | coincides with classical derivative when order is integer ⓘ |
| typicalNotation |
D^α_C f(t)
ⓘ
^C D_t^α f(t) ⓘ |
| typicalOrderRange | 0 < α < 1 ⓘ |
| usedFor |
modeling hereditary phenomena
ⓘ
modeling memory effects ⓘ modeling power-law relaxation ⓘ |
| usedIn |
anomalous diffusion modeling
ⓘ
bioengineering ⓘ control theory ⓘ engineering ⓘ fractional differential equations ⓘ physics ⓘ signal processing ⓘ viscoelasticity modeling ⓘ |
| variable | time variable t ⓘ |
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: Caputo derivative Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Caputo fractional derivative
this entity surface form:
Caputo fractional derivative
this entity surface form:
Caputo fractional derivative