Riemann-Liouville differintegral
Encyclopedia
In mathematics
, the Riemann–Liouville integral associates with a real function
ƒ : R → R another function Iαƒ of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative
of ƒ in the sense that for positive integer values of α, Iαƒ is an iterated antiderivative of ƒ of order α. The Riemann–Liouville integral is named for Bernhard Riemann
and Joseph Liouville
, the latter of whom was the first to consider the possibility of fractional calculus
in 1832. The operator agrees with the Euler transform, after Leonhard Euler
, when applied to analytic function
s. It was generalized to arbitrary dimensions by Marcel Riesz
, who introduced the Riesz potential
.
The Riemann–Liouville integral is defined by
where Γ is the Gamma function
and a is an arbitrary but fixed base point. The integral is well-defined provided ƒ is a locally integrable function
, and α is a complex number
in the half-plane re(α) > 0. The dependence on the base-point a is often suppressed, and represents a freedom in constant of integration. Clearly I1ƒ is an antiderivative of ƒ (of first order), and for positive integer values of α Iαƒ is an antiderivative of order α by Cauchy formula for repeated integration
. Another notation, which emphasizes the basepoint, is
This also makes sense if a = −∞, with suitable restrictions on ƒ.
The fundamental relations hold
the latter of which is a semigroup
property. These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of Iαƒ.
. Thus Iα defines a linear operator on L1(a,b)
:
Fubini's theorem also shows that this operator is continuous
with respect to the Banach space
structure on L1, and that the following inequality holds:
Here
denotes the norm
on L1(a,b).
More generally, by Hölder's inequality
, it follows that if ƒ ∈ Lp(a,b), then Iαƒ ∈ Lp(a,b) as well, and the analogous inequality holds
where
is the Lp norm
on the interval (a,b). Thus Iα defines a bounded linear operator from Lp(a,b) to itself. Furthermore, Iαƒ tends to ƒ in the Lp sense as α → 0 along the real axis. That is
for all p ≥ 1. Moreover, by estimating the maximal function
of I, one can show that the limit Iαƒ → ƒ holds pointwise almost everywhere
.
The operator Iα is well-defined on the set of locally integrable function on the whole real line R. It defines a bounded transformation on any of the Banach space
s of functions of exponential type Xσ = L1(e-σ|t|dt), consisting of locally integrable functions for which the norm
is finite. For ƒ in Xσ, the Laplace transform of Iαƒ takes the particularly simple form
for re(s) > σ. Here F(s) denotes the Laplace transform of ƒ, and this property expresses that Iα is a Fourier multiplier.
where
denotes the ceiling function. One also obtains a differintegral interpolating between differentiation and integration by defining
An alternative fractional derivative was introduced by Caputo in 1967, and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the Laplace Transform are expressed by means of the values of that function and of its derivative of integer order rather than the derivatives of fractional order as in the Riemann–Liouville derivative.http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf The Caputo fractional derivative with base point
, is then:
Another representation is:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Riemann–Liouville integral associates with a real function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
ƒ : R → R another function Iαƒ of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
of ƒ in the sense that for positive integer values of α, Iαƒ is an iterated antiderivative of ƒ of order α. The Riemann–Liouville integral is named for Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
and Joseph Liouville
Joseph Liouville
- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
, the latter of whom was the first to consider the possibility of fractional calculus
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
in 1832. The operator agrees with the Euler transform, after Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
, when applied to analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s. It was generalized to arbitrary dimensions by Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...
, who introduced the Riesz potential
Riesz potential
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
.
The Riemann–Liouville integral is defined by
where Γ is the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and a is an arbitrary but fixed base point. The integral is well-defined provided ƒ is a locally integrable function
Locally integrable function
In mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.- Formal definition :...
, and α is a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
in the half-plane re(α) > 0. The dependence on the base-point a is often suppressed, and represents a freedom in constant of integration. Clearly I1ƒ is an antiderivative of ƒ (of first order), and for positive integer values of α Iαƒ is an antiderivative of order α by Cauchy formula for repeated integration
Cauchy formula for repeated integration
The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral .-Scalar case:Let ƒ be a continuous function on the real line...
. Another notation, which emphasizes the basepoint, is
This also makes sense if a = −∞, with suitable restrictions on ƒ.
The fundamental relations hold
the latter of which is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
property. These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of Iαƒ.
Properties
Fix a bounded interval (a,b). The operator Iα associates to each integrable function ƒ on (a,b) the function Iαƒ on (a,b) which is also integrable by Fubini's theoremFubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
. Thus Iα defines a linear operator on L1(a,b)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
:
Fubini's theorem also shows that this operator is continuous
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
with respect to the Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
structure on L1, and that the following inequality holds:
Here
denotes the normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on L1(a,b).
More generally, by Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
, it follows that if ƒ ∈ Lp(a,b), then Iαƒ ∈ Lp(a,b) as well, and the analogous inequality holds
where
is the Lp normLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
on the interval (a,b). Thus Iα defines a bounded linear operator from Lp(a,b) to itself. Furthermore, Iαƒ tends to ƒ in the Lp sense as α → 0 along the real axis. That is
for all p ≥ 1. Moreover, by estimating the maximal function
Maximal function
Maximal functions appear in many forms in harmonic analysis . One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations...
of I, one can show that the limit Iαƒ → ƒ holds pointwise almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
.
The operator Iα is well-defined on the set of locally integrable function on the whole real line R. It defines a bounded transformation on any of the Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s of functions of exponential type Xσ = L1(e-σ|t|dt), consisting of locally integrable functions for which the norm
is finite. For ƒ in Xσ, the Laplace transform of Iαƒ takes the particularly simple form
for re(s) > σ. Here F(s) denotes the Laplace transform of ƒ, and this property expresses that Iα is a Fourier multiplier.
Fractional derivatives
One can define fractional-order derivatives of ƒ as well bywhere
denotes the ceiling function. One also obtains a differintegral interpolating between differentiation and integration by definingAn alternative fractional derivative was introduced by Caputo in 1967, and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the Laplace Transform are expressed by means of the values of that function and of its derivative of integer order rather than the derivatives of fractional order as in the Riemann–Liouville derivative.http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf The Caputo fractional derivative with base point
, is then:Another representation is: