Fractional Calculus: Integration & Differentiation Explained

[deepurple]

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 integrations?

 

[sokrates]

I am also interested in this,

Welcome to PF forums deepurple.

I hope people'll take the time to respond.

 

[luisgml_2000]

Hello. As far as I know there is no geometric interpretation for a derivative of fractional order.

 

[ExactlySolved]

Hello. As far as I know there is no geometric interpretation for a derivative of fractional order.

It turns out that for every power of fractional derivative there is a fractal function whose fractional derivative at that order is a constant, equal to the fractal dimension of the curve!

By a fractal function I mean a Weistrauss type continuous function that is nowhere differentiable in the traditional sense and which has a non-integer dimension in the sense of Hausdorf.

Incidentally fractional-calculus should be called non-integer calculus, since the index of the differential operator can be any complex value! A further generalization is possible by considering functions of the differential operator that cannot be represented as polynomials, i.e. F(d/dx), the most common of which is exp(d/dx), which acts as translation by one unit on real-valued functions:

$$e^{\frac{d}{dx}}f(x) = f(x + 1)$$

I always vote for this one to go on the t-shirts instead of e^{i pi} + 1 = 0.

 

[luisgml_2000]

It turns out that for every power of fractional derivative there is a fractal function whose fractional derivative at that order is a constant, equal to the fractal dimension of the curve!

Wow. I would never have imagined that fractals and fractional (or 'complex order') calculus have anything to do. It's quite interesting.

 

[mnb96]

Hello,
I know this thread is very old, but perhaps http://www.maa.org/joma/Volume7/Podlubny/GIFI.html" might provide useful information related to geometric interpretations of fractional integration/derivative to anyone coming upon this page.