Riemann–Liouville integral

In mathematics, the Riemann–Liouville integral associates with a real function another function Iα f of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, Iα f is an iterated antiderivative of f 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][2][3][4] The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions.[5] It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential.

Definition

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 f 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 I1 f is an antiderivative of f (of first order), and for positive integer values of α, Iα f is an antiderivative of order α by Cauchy formula for repeated integration. Another notation, which emphasizes the base point, is[6]

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

The fundamental relations hold

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 Iα f.

Properties

Fix a bounded interval (a,b). The operator Iα associates to each integrable function f on (a,b) the function Iα f on (a,b) which is also integrable by Fubini's theorem. 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 ‖ · ‖1 denotes the norm on L1(a,b).

More generally, by Hölder's inequality, it follows that if fLp(a, b), then Iα fLp(a, b) as well, and the analogous inequality holds

where ‖ · ‖p is the Lp norm on the interval (a,b). Thus we have a bounded linear operator Iα : Lp(a, b) → Lp(a, b). Furthermore, Iα ff 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 Iα ff holds pointwise almost everywhere.

The operator Iα is well-defined on the set of locally integrable function on the whole real line . It defines a bounded transformation on any of the Banach spaces of functions of exponential type consisting of locally integrable functions for which the norm

is finite. For fXσ, the Laplace transform of Iα f takes the particularly simple form

for Re(s) > σ. Here F(s) denotes the Laplace transform of f, and this property expresses that Iα is a Fourier multiplier.

Fractional derivatives

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

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,[7] 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.[8] The Caputo fractional derivative with base point x, is then:

Another representation is:

Fractional derivative of a basic power function

The half derivative (purple curve) of the function f(x) = x (blue curve) together with the first derivative (red curve).
The animation shows the derivative operator oscillating between the antiderivative (α = −1: y = 1/2x2) and the derivative (α = +1: y = 1) of the simple power function y = x continuously.

Let us assume that f(x) is a monomial of the form

The first derivative is as usual

Repeating this gives the more general result that

which, after replacing the factorials with the gamma function, leads to

For k = 1 and a = 1/2, we obtain the half-derivative of the function as

To demonstrate that this is, in fact, the "half derivative" (where H2f(x) = Df(x)), we repeat the process to get:

(because and Γ(1) = 1) which is indeed the expected result of

For negative integer power k, 1/ is 0, so it is convenient to use the following relation:[9]

This extension of the above differential operator need not be constrained only to real powers; it also applies for complex powers. For example, the (1 + i)-th derivative of the (1 − i)-th derivative yields the second derivative. Also setting negative values for a yields integrals.

For a general function f(x) and 0 < α < 1, the complete fractional derivative is

For arbitrary α, since the gamma function is infinite for negative (real) integers, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,

Laplace transform

We can also come at the question via the Laplace transform. Knowing that

and

and so on, we assert

.

For example,

as expected. Indeed, given the convolution rule

and shorthanding p(x) = xα − 1 for clarity, we find that

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.

Notes

  1. Lizorkin 2001
  2. Liouville, Joseph (1832), "Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions", Journal de l'École Polytechnique, Paris, 13: 1–69.
  3. Liouville, Joseph (1832), "Mémoire sur le calcul des différentielles à indices quelconques", Journal de l'École Polytechnique, Paris, 13: 71–162.
  4. Riemann, Georg Friedrich Bernhard (1896) [1847], "Versuch einer allgemeinen Auffassung der integration und differentiation", in Weber, H. (ed.), Gesammelte Mathematische Werke, Leipzig{{citation}}: CS1 maint: location missing publisher (link).
  5. Brychkov & Prudnikov 2001
  6. Miller & Ross 1993, p. 21
  7. Caputo 1967
  8. Loverro 2004
  9. Bologna, Mauro, Short Introduction to Fractional Calculus (PDF), Universidad de Tarapaca, Arica, Chile, archived from the original (PDF) on 2016-10-17, retrieved 2014-04-06

References

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