Encyclopedia > Elementary fractional calculus

Article Content

Elementary fractional calculus

mathematics -> fractional calculus -> Elementary fractional calculus

Some fundamental properties of the differintegral Composition rule:

$\mathbb{D}^a\mathbb{D}^{b}x = \mathbb{D}^{a+b}x$
Zero rule:
$\mathbb{D}^{0}x=x$
Subclass rule:
$\mathbb{D}^{a \in N}x=d^{a}x$
Linear rule:
$\mathbb{D}^{q}(x+y)=\mathbb{D}^{q}(x)+\mathbb{D}^{q}(y)$
$\mathbb{D}^{q}(ax)=a\mathbb{D}^{q}(x)$
$\mathbb{D}^{q}(x^n)=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}x^{n-q}$
$\mathbb{D}^{q}(\sin(x))=sin(x+\frac{q\pi}{2})$
$\mathbb{D}^{q}(e^{ax})=a^{q}e^{ax}$

A note on inverses and initialization First off, a certain oddity about the differintegral should be pointed out. Although:

$\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}$

That is, Dq is the left inverse of D-q. On first glance, the converse is not neccessarily true.

$\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}$

But hold on, the ship isn't sunk yet! Let's take a look at integral calculus, to get a better idea of what's happening. First, let's integrate, then differentiate, using the arbitrary function 3x2+1:

$d\left[\int (3x^2+1)dx\right]/dx = d[x^3+x+c]/dx = 3x^2+1$

Well, that was pretty straightforward, and it worked. Now, what happens when we exchange the order of composition?

$\int [d(3x^2+1)/dx]dx = \int 6xdx = 3x^2+c$

Hmmm... I think it's fairly obvious what that integration constant is. Even if it wasn't obvious, we would simply use the initialization terms[?] such as f'(0) = c, f' '(0)= d, ect. If we neglected those initilization terms, the last equation would fail our test. This is exactly the problem that we encountered with the differintegral. If the differintegral is initialized properly, then the composition holds. The problem is that in differentation, we lose state information, as we lost the c in the first equation. (see dynamical systems).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complimentary function is needed, not just a constant or set of constants.

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

(Working with a properly initialized differintegral is the subject of initialized fractional calculus).

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Canadian Music Hall of Fame ... Paul Anka 1981 Joni Mitchell 1982 Neil Young 1983 Glenn Gould 1986 Gordon Lightfoot 1987 The Guess Who[?] 1989 The Band 1990 Maureen Forrester[?] 1991 ...