What is "fractional calculus"

n. (context calculus English) The study of fractional differentiation.

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator

$$D = \dfrac{d}{dx},$$

and the integration operator J. (Usually J is used instead of I to avoid confusion with other I-like glyphs and identities.)

In this context, the term powers refers to iterative application of a linear operator acting on a function, in some analogy to function composition acting on a variable, e.g., . For example, one may ask the question of meaningfully interpreting

$$\sqrt{D} = D^{\frac{1}{2}}$$
as an analog of the functional square root for the differentiation operator, i.e., an expression for some linear operator that when applied twice to any function will have the same effect as differentiation.

More generally, one can look at the question of defining the linear functional

D
for real-number values of a in such a way that when a takes an integer value, n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of J when n < 0.

The motivation behind this extension to the differential operator is that the semigroup of powers D will form a continuous semigroup with parameter a, inside which the original discrete semigroup of D for integer n can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory.

Fractional differential equations (also known as extraordinary differential equations) are a generalization of differential equations through the application of fractional calculus.