Riemann–Liouville integral

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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.[1] The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions.[2] It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential.

Definition[edit]

The Riemann–Liouville integral is defined by

I^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^xf(t)(x-t)^{\alpha-1}\,dt

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[3]

{}_aD_x^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)}\int_a^x f(t)(x-t)^{\alpha-1}\,dt.

This also makes sense if a = −∞, with suitable restrictions on ƒ.

The fundamental relations hold

\frac{d}{dx}I^{\alpha+1} f(x) =  I^\alpha f(x),\quad I^\alpha(I^\beta f) = I^{\alpha+\beta}f,

the latter of which is a semigroup property.[1] These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of Iαƒ.

Properties[edit]

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 theorem. Thus Iα defines a linear operator on L1(a,b):

I^\alpha : L^1(a,b) \to L^1(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:

\|I^\alpha f\|_1 \le \frac{|b-a|^{\operatorname{re}(\alpha)}}{\operatorname{re}(\alpha)|\Gamma(\alpha)|}\|f\|_1.

Here \|\cdot\|_1 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

\|I^\alpha f\|_p \le \frac{|b-a|^{\operatorname{re}(\alpha)/p}}{\operatorname{re}(\alpha)|\Gamma(\alpha)|}\|f\|_p

where \|\cdot\|_p 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

\underset{\alpha > 0}{\lim_{\alpha\to 0}} \|I^\alpha f-f\|_p = 0

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 spaces of functions of exponential type Xσ = L1(e-σ|t|dt), consisting of locally integrable functions for which the norm

\|f\| = \int_{-\infty}^\infty |f(t)|e^{-\sigma|t|}\,dt

is finite. For ƒ in Xσ, the Laplace transform of Iαƒ takes the particularly simple form

(\mathcal{L}I^\alpha f)(s) = s^{-\alpha}F(s)

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[edit]

One can define fractional-order derivatives of ƒ as well by

\frac{d^\alpha}{dx^\alpha} f \overset{def}{=} \frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha\rceil}} I^{\lceil\alpha\rceil-\alpha}f

where \lceil\cdot\rceil denotes the ceiling function. One also obtains a differintegral interpolating between differentiation and integration by defining

D^\alpha_x f(x) =
\begin{cases}
\frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha\rceil}} I^{\lceil\alpha\rceil-\alpha}f(x)& \alpha>0\\
f(x) & \alpha=0\\
I^{-\alpha}f(x) & \alpha<0.
\end{cases}

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.[1] The Caputo fractional derivative with base point x, is then:

D_x^{\alpha}f(y)=\frac{1}{\Gamma(1-\alpha)}\int_x^y f'(y-u)(u-x)^{-\alpha}du.

Another representation is:

{}_a\tilde{D}^\alpha_x f(x)=I^{\lceil \alpha\rceil-\alpha}\left(\frac{d^{\lceil \alpha\rceil}f}{dx^{\lceil \alpha\rceil}}\right).

Notes[edit]

  1. ^ a b Lizorkin 2001
  2. ^ Brychkov & Prudnikov 2001
  3. ^ Miller & Ross 1993, p. 21

References[edit]

External links[edit]