Riemann-Liouville differintegral

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Fractional calculus is a branch of Mathematical analysis that studies the possibility of taking Real number powers of the Differential operator In Mathematics, the differintegral is the combined differentiation / integration operator used in Fractional calculus.

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.

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. In Mathematics, Cauchy's integral formula, named after Augustin Louis Cauchy, is a central statement in Complex analysis. 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. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The 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.

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. In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic [1] Instead of integrating then differentiating

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

*Such operator notation reads right-to-left. J is commonly used for the integral instead of I, probably to save confusion with identities. In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of

the differentiating is done first

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

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