Riemann-Liouville differintegral

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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly.

It is noted:

${}_a \mathbb{D}^q_t$

and is most generally defined as:

${}_a\mathbb{D}^q_t= \left\{\begin{matrix} \frac{d^q}{dx^q}, & \Re(q)>0 \\ 1, & \Re(q)=0 \\ \int^t_a(dx)^{-q}, & \Re(q)<0 \end{matrix}\right.$

The Riemann-Liouville differintegral (RL) is the simplest and easiest to use, and consequently it is the most often used.

[edit] Constructing the Riemann-Liouville differintegral

We first introduce the Riemann-Liouville fractional integral, which is a straightforward generalization of the Cauchy integral formula:

${}_a\mathbb{D}^{-q}_tf(x)=\frac{1}{\Gamma(q)} \int_{a}^{t}(t-\tau)^{q-1}f(\tau)d\tau$

This gives us integration to an arbitrary order. To get differentiation to an arbitrary order, we simply integrate to arbitrary order n − q, and differentiate the result to integer order n. (We choose n and q so that n is the smallest positive integer greater than or equal to q (that is, the ceiling of q)):

${}_a\mathbb{D}^q_tf(x)=\frac{d^n}{dx^n}{}_a\mathbb{D}^{-(n-q)}_tf(x)$

Thus, we have differentiated n − (n − q) = q times. The RL differintegral is thus defined as (the constant is brought to the front):

${}_a\mathbb{D}^q_tf(x)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dx^n}\int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau$ definition

When we are taking the differintegral at the upper bound (t), it is usually written:

${}_a\mathbb{D}^q_tf(t)=\frac{d^qf(t)}{d(t-a)^q}=\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau$ definition

And when we are assuming that the lower bound is zero, it is usually written:

$\mathbb{D}^q_tf(t)=\frac{d^qf(t)}{d(t)^q}=\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{0}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau$

That is, we are taking the differintegral of f(t) with respect to t.

[edit] Caputo fractional derivative

A change introduced by Caputo in 1967 produces a derivative that has different properties: it produces zero from constant functions and, more important, the initial value terms of the Laplace Transform are expressed by means of the values of the function and of its derivative of integer order rather than the derivatives of fractional order as in the Riemann-Liouville derivative.[1] Instead of integrating then differentiating

$D^q=D^{\lceil q\rceil}J^{\lceil q\rceil-q}$ *