Fractional calculus - uses, examples?

In summary: The impedance of a network is a function of the properties of the network elements, and can be determined by solving a set of simultaneous equations.
  • #1
elegysix
406
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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 he didn't really give me an answer, just that it was too complex to explain lol.

Now, 3 years later, I'm looking back on it and wondering what a fractional derivative can be interpreted as, and what it could be used for. Anyone have some insight?
 
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  • #2
Way too many to count. What interests you?

We have three specialized journals:
  1. Fractional Calculus and Applied Analysis
  2. Fractional Differential Equations
  3. Communications in Fractional Calculus.
Of which the second is applications based.

I highly recommend Podlubny's book Fractional Differential Equations, or West's book Physics of Fractal Operators as they both have a chapter on applications. Also http://people.tuke.sk/igor.podlubny/fc_resources.html" is really good place to start.
 
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  • #3
Thanks, I'll check those out.

I think asking how to use it is too broad of a question,
is there like a specific physics problem where you need to use it?
 
  • #4
elegysix said:
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 he didn't really give me an answer, just that it was too complex to explain lol.

Now, 3 years later, I'm looking back on it and wondering what a fractional derivative can be interpreted as, and what it could be used for. Anyone have some insight?

Some references are provided page 5 in a paper written for general public "La dérivation fractionnaire" :
http://www.scribd.com/JJacquelin/documents
(Sorry, not translated yet).
There are a lot of applications for fractionnal calculus. One of them is the genertalization of the basic physical notions of Inductance, Resistance, Capacitance. A very useful generalization for the calculus of impedance of some electrical networks, especialy in case of presence of CPE in it (Constant Phase Element) :
 

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1. What is fractional calculus?

Fractional calculus is a branch of mathematics that deals with derivatives and integrals of non-integer orders. It extends the concept of traditional calculus to include fractional orders, which can have both fractional and non-integer values.

2. What are the uses of fractional calculus?

Fractional calculus has many practical applications in various fields such as physics, engineering, and economics. It can be used to model and analyze complex systems, such as materials with fractal properties, and to solve differential equations that cannot be solved with traditional calculus.

3. Can you give an example of a real-world application of fractional calculus?

One example is the use of fractional calculus in the modeling of viscoelastic materials, such as rubber or polymers. These materials exhibit time-dependent behavior and can be described using fractional derivatives. Fractional calculus is also used in signal processing, where it helps to analyze and process non-stationary signals.

4. What are some advantages of using fractional calculus?

One advantage is that fractional calculus allows for a more accurate description of real-world phenomena that cannot be fully captured by traditional calculus. It also provides a more flexible and powerful tool for analyzing and solving complex problems, particularly in areas where traditional calculus falls short.

5. How is fractional calculus related to traditional calculus?

Fractional calculus is an extension of traditional calculus, using fractional orders instead of integer orders. It follows the same basic principles and methods, but also introduces new concepts, such as the fractional derivative and integral. Traditional calculus can be seen as a special case of fractional calculus when the order is an integer.

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