What is Fractional calculus: Definition and 17 Discussions
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
D
f
(
x
)
=
d
d
x
f
(
x
)
,
{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}
and of the integration operator J
J
f
(
x
)
=
∫
0
x
f
(
s
)
d
s
,
{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in
D
n
(
f
)
=
(
D
∘
D
∘
D
∘
⋯
∘
D
⏟
n
)
(
f
)
=
D
(
D
(
D
(
⋯
D
⏟
n
(
f
)
⋯
)
)
)
{\displaystyle D^{n}(f)=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots )))}
.
For example, one may ask for a meaningful interpretation of
D
=
D
1
2
{\displaystyle {\sqrt {D}}=D^{\frac {1}{2}}}
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear operator
D
a
{\displaystyle D^{a}}
for every real number a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual n-fold differentiation D if n > 0, and with the (−n)-th power of J when n < 0.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator D is that the sets of operator powers { Da | a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of { Dn | n ∈ ℤ } for integer n is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.
The note is entitled: Evaluation of a Class of n-fold Integrals by Means of Hadamard Fractional Integration. 4 pgs pdf format.
I assure you that you need not know anything about fractional calculus at all to understand this note that Howard Cohl helped me with. We only use a single...
Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
So the problem I’m attempting to solve is ##\lim_{x\to a} I_{\alpha}f(x)=\zeta (\alpha )## for f, and a, where ##\zeta (\cdot )## is the Riemann zeta function and ##I_{\alpha}## is the Riemann-Liouville left fractional integral operator, namely the integral equation
$$\lim_{x\to...
Hi i’m having problems with the following equations:
X(w)=2/(-1+iw)(-2+iw)(-3+iw)
This then becomes the following equation according the the tutorial, although there is no explanation as to how:
X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw
The commas indicated the end of each fraction to make it easier...
The Complete Idiot's Guide to Calculus
INTRODUCTION
I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are...
Okay, maybe not really fractional calculus but I don't know what this stands for. Its in the black circle (more like an ellipse though), what does the mu under the natural logarithm mean?
So, apparently, it's possible to generalize integration and derivation into non-integer orders. For instance, it's apparently possible to take the 0.5th derivative of a function.
What I'm wondering is what would be represented by such an equation? If a derivative represents how a function...
According to Fractional Calculus, the power rule can be written as
(dm/dzm) zn = n!/(n-m)! zn-m
For example,
(d1/2/dz1/2) z1/2 = (1/2)!/(1/2-1/2)! z0 = (1/2)√π
To find the residue of f(z) = f(z)/(z-z0)m at z→z0, the formula is Res(z→z0) f(z) = 1/(m-1)! dm-1/dzm-1 f(z).
For...
Fractional Calculus...? What??
I came across this Wiki article a couple of days ago:
http://en.wikipedia.org/wiki/Fractional_calculus
As a student who just finished an undergrad major in math, the idea of a "fractional derivative" or "fractional integral" is mind blowing! Up until I read...
Does anyone know any applications / uses for fractional derivatives and integrals?
I supposed the idea and asked a professor, he tried to explain fractional calculus to me, but I was in calculus 2 at the time... so it was way over my head back then. I asked him what it could be used for and...
Hello guys :),
I have a question which I think is very advanced and weird. But I need the answer for some signal analysis purpose.
As we know, the derivative of a sine function, per se, shifts the phase, by Pi/2; i.e.,
f(x) = A sin (w t)
df(x)/dt = A sin (w t + Pi/2) = A cos(w t)...
Hi everyone, this is my first post on PF (yaay!). I hope this is in the right forum, if not I don't mind a mod moving this. I'm an undergrad physics student and one of my professors has hired me as a research worker over the summer break.
Has anyone here done any work on fractional calculus...
I am interested in fractional Calculus which means integration and differentiation of an arbitrary or fractional order. But I am confused about the geometric meaning. We know that 1st derivative gives us a slope but what about 1/2th derivative. How can we describe this kind of derivatives or...
The fractional dynamics has been appearing in many phenomena such as the movement of protein in the cytoplasm. The Fractional calculus used to describe it.
Xuru's Website
Introductory Notes on Fractional Calculus
http://www.xuru.org/fc/toc.asp
----------------------------------...
So it is well-known that for n=2,3,... the following equation holds
\zeta(n)=\int_{x_{n}=0}^{1}\int_{x_{n-1}=0}^{1}\cdot\cdot\cdot\int_{x_{1}=0}^{1}\left(\frac{1}{1-\prod_{k=1}^{n}x_{k}}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}dx_{n}
My question is how can this relation be extended to...
Has any menaing the use of fractional calculus in..physics?.i have used it several time to quantizy non-polynomial hamiltonians in quantum mechanics...(such us H=p**e e not an integer).but does the fractional calculus have a meaning or utility in physics?..to cancel infinities to calculate...