Fractional calculus

It’s an awesome thing that I just found out about.

It’s a kind of mathematical analysis that explores the possibility of taking real or complex number powers or complex numbers powers of the differentiation operator (D = d/dx) and integration operator (J).

Powers refers to the iterative application of the function – also called function composition. It’s like if you want to write f(f(x)), you can write f²x.

In other words, fractional calculus is concerned with meaningfully interpreting fractional iterations of the differentiation and integration operators. In other words, one looks at defining D^a, for real-number values of a so that when a takes up an positive integer value ‘n’, the usual n-fold differentiation is recovered, and for negative integer values ‘n’ of of a, n-fold integration is recovered.

A major application of this is generalising differential equations by applying the concepts and notions of fractional calculus.

It can only be imagined that this concept will have a number of specialised applications outside of Mathematics – in Physics and Chemistry, even Biology.

In fluid mechanics, certain complex systems require the use of fractional powers of the differentiation operator in order to describe complex behaviour.

In order to give a geometrical sense of what’s going on with fractional powers of derivatives and all that, I’m going to reproduce an image that’s already there on the Wikipedia page about fractional calculus:

The blue function, y=x is the original function. The function in purple is when the power of the derivative is -1, i.e., the integration operator is applied +1 time, and the function in red is when the power of the derivative is §, i.e., the differentiation operator is applied +1 time.