Definition of a Fractional Derivative/ Integral

[SheikYerbouti]

The geometric and physical properties of derivatives and integrals to an integer order are easy to describe, but fractional calculus is obviously present in modern mathematics and physics. That being said, are there a generalizations of the definitions derivatives and integrals that include these operations to arbitrary orders? To be more specific, I am not looking for a mathematical representation of one; I have seen these before. I am curious as to how a qualitative definition could describe these mathematically and/or physically with regards to the original function. I guess what I am looking for is an interpretation.

 

[lurflurf]

The main trouble conceptualizing fractional calculus is that some properties are specific to integers and absent in the generalization. For example we come to think of derivatives as local properties, determinable from a neighborhood of a point, but in fractional calculus they are determined by a finite interval.

 

[JJacquelin]

Hi !
The paper "La dérivation fractionnaire" is in French, not translated yet. But on the last page there are several references of papers and documents in English dealing with partial differintegration. Through the link :
http://www.scribd.com/JJacquelin/documents

 

[mdo]

I work in Fractional Calculus, since 1994, firstly as a hobby, and in the last 10 years I've published papers regularly. The posed question has some interest and has motivated some attempts, although not completely satisfactory. I do not care about geometric interpretations. Most of the attempts were based on the integral representations of the fractional derivative. I suggest to look for the papers of Igor Podlubny or Tenreiro Machado. For surveys on FC I suggest my papers
Ortigueira, M. D., “An Introduction to the Fractional Continuous-Time Linear Systems” IEEE Circuits and Systems Magazine , third quarter 2008, pp 19-26

Magin, R., Ortigueira, M. D., Podlubny, I., and Trujillo, J., “On the Fractional Signals and Systems,” (invited paper), Signal Processing, 91(2011) 350–371.

 

[SheikYerbouti]

Thank you for all the responses, you have been very helpful. I have been studying this independently, but I am considering doing a focused inquiry with a professor to speed things along a little more. This would consist of weekly meetings with the professor for an hour working through the theory and then being given some relevant proof to solve. There are two professors who I think I could convince to participate. The first professor is an applied mathematician who specializes in mathematical modelling and has supported student research with the Gamma function. On the other hand, there is a number theorist who seems interested in going over the theory of the technique in greater depth to learn how to use it for classical problems. Which of the two do you think would be a better fit for such a project?

 

[mdo]

Thank you for all the responses, you have been very helpful. I have been studying this independently, but I am considering doing a focused inquiry with a professor to speed things along a little more. This would consist of weekly meetings with the professor for an hour working through the theory and then being given some relevant proof to solve. There are two professors who I think I could convince to participate. The first professor is an applied mathematician who specializes in mathematical modelling and has supported student research with the Gamma function. On the other hand, there is a number theorist who seems interested in going over the theory of the technique in greater depth to learn how to use it for classical problems. Which of the two do you think would be a better fit for such a project?

Of course, the first.

 

[mdo]

The main trouble conceptualizing fractional calculus is that some properties are specific to integers and absent in the generalization. For example we come to think of derivatives as local properties, determinable from a neighborhood of a point, but in fractional calculus they are determined by a finite interval.

Not exactly. The fractional derivatives are operators with long (infinite) memory. Even with short duration functions the corresponding derivatives have infinite duration. In fact, the local property is lost but all the others are valid, mainly the commutativity.

 

[SheikYerbouti]

Not exactly. The fractional derivatives are operators with long (infinite) memory. Even with short duration functions the corresponding derivatives have infinite duration. In fact, the local property is lost but all the others are valid, mainly the commutativity.

What exactly do you mean by mathematical memory? I haven't gotten into the higher level mathematics courses yet, but I am eager to learn. I'm assuming that this is likely to be something different from a probabilistic memory. I have seen an example where the operator was applied to a relativistic scenario, and the author stated that the operator contained some of the object's "history"; is this the same as the memory you are mentioning?

 

[mdo]

Well I suggest you to read a good book on Signals and Systems. There are lots, e.g. Oppenheim+Wilsky, Haykin+Van Veen, Lindner, Roberts. You'll see that linear systems are chartacterised by an impulse response and a transfer function. The transfer function of a fractional derivative is H(s) = s^α for Re(s) > 0 and the impulse response is a negative power function. This states the memory because it weights the influence of the past over the present.