What Is Fractional Calculus and How Does It Work?

[ssayani87]

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 the article, I pretty much thought that the differential and integral operators were "fixed."

Let f:R -> R be a fcn.

Geometrically speaking, if the derivative of a fcn f at a point p in its domain is the rate of change of the fcn at that point and the integral of a fcn over an interval in the domain is the "area under the curve," how can I interpret the fractional derivative and fractional integral?

 

[mnb96]

Hi ssayani87,

I have first of all to admit that I am not an expert in this topic. However when I posed the same question to myself I came across http://www.maa.org/joma/Volume7/Podlubny/GIFI.html" that looked quite interesting.

Hopefully someone more competent than me here in PF will add more useful information.

 

[pwsnafu]

Do a forum search and you will come across a couple of threads here.

Geometrically speaking, if the derivative of a fcn f at a point p in its domain is the rate of change of the fcn at that point and the integral of a fcn over an interval in the domain is the "area under the curve," how can I interpret the fractional derivative and fractional integral?

This is incorrect. The "rate of change at a point" is more the physical interpretation of the derivative. That's why velocity is defined as rate of change of position. The geometrical interpretation is "slope of the tangent at that point", and then we associate "rate of change" to that.

While I agree Podulbny's "shadows on the wall" is a good geometrical interpretation, still not a physical interpretation, which is what we want.

 

[chiro]

I remember vaguely reading about this before and how fractional calculus was used to explain a way of interpolation in analyzing frequency information in the way that Fourier analysis is done.

I can't remember the exact website, but perhaps you could look for websites talking about fractional calculus with respect to analyzing frequency information.

It had to do with idea of analyzing differentials of trigonometric functions, and if you think about d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x) but it's what happens in-between is what you need to pay attention to.