'Fractional Calculus I'

In summary, The conversation discusses the possibility of fractional calculus and its applications in physics, biology, and engineering. Reference is made to a paper that claims to show that fractional integration is possible. However, it is noted that certain results do not hold in fractional calculus, such as the exponential and cosine functions. The conversation also touches on the idea of fractional derivatives on transcendental functions and the use of gamma functions instead of factorials.
  • #1
973
3

I have located a paper claiming that it is possible to fractionally differentiate, called 'Fractional Calculus I'

Orion1 derivative integer factorial theorem:
[tex]\frac{d^n}{dx^n} (x^n) = n![/tex]

Is this paper correct? is 'Fractional Calculus' really possible?
http://nrich.maths.org/public/viewer.php?obj_id=1365&refpage=monthindex.php&part=index&nomenu=1

Fractional Integration?

Reference:
https://www.physicsforums.com/showpost.php?p=672326&postcount=1
 
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  • #2
Yep! It's not nearly as nice as ordinary calculus, but you can still do it, and can apparently do cool stuff with it.
 
  • #3
Fractional (and real and complex) order operators are possible and are used. Unfortunately several results that one might expect do not hold. For example
(D^n)exp(a x)=(a^n)exp(a x)
and
(D^n)cos(a x)=(a^n)cos(a x+n pi/2)
do not hold in fractional calculus.
 
  • #4
Yes fractional calculus is really useful tool for modeling problems in physics, biology and engineering. Actually fractional difference calculus is possible also.
 
  • #5
How does one perform a fractional derivative on a transcendental function? Although it seems quite trivial on algebraic functions.
 
  • #6
The first time the idea of fractional calculus occurred to me, not knowing it was a real thing, was when I was thinking about how to calculate, in quantum mechanics, something like [itex]<\psi |\hat{p}^n |\psi>[/itex] where [itex]\hat{p}=-i\hbar \frac{d}{dx}[/itex]. I was indeed quite surprised to find that fractional calculus was a real thing.
 
  • #7
I always wondered what would happen if you substituted values other than integers (and replacing factorials with gamma functions) in cauchy's differentiation formula. Would this give the fractional derivative in the sense you guys are talking about?
 
  • #8
matticus said:
I always wondered what would happen if you substituted values other than integers (and replacing factorials with gamma functions) in cauchy's differentiation formula. Would this give the fractional derivative in the sense you guys are talking about?

That's exactly it sir
 

1. What is Fractional Calculus I?

Fractional Calculus I is a branch of mathematics that deals with the study of integrals and derivatives of non-integer order. It extends the concept of traditional calculus to non-integer values, allowing for the analysis of complex systems and phenomena.

2. How is Fractional Calculus I different from traditional calculus?

Fractional Calculus I differs from traditional calculus in that it deals with non-integer orders of derivatives and integrals, while traditional calculus only deals with integer orders. This allows for the analysis of systems with fractal geometry and non-linear dynamics.

3. What are the applications of Fractional Calculus I?

Fractional Calculus I has various applications in fields such as physics, engineering, economics, and biology. It has been used to model complex phenomena such as viscoelasticity, diffusion processes, and chaotic systems.

4. What are some common techniques used in Fractional Calculus I?

Some common techniques used in Fractional Calculus I include Grunwald-Letnikov and Riemann-Liouville fractional derivatives, and Caputo and Hadamard fractional integrals. These techniques are used to solve fractional differential equations.

5. Is Fractional Calculus I difficult to learn?

Fractional Calculus I can be challenging to learn, as it requires a solid understanding of traditional calculus and a new way of thinking about derivatives and integrals. However, with dedication and practice, it can be mastered like any other branch of mathematics.

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