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Concept
Fractional calculus
Summary
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
and of the integration operator
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator to a function , that is, repeatedly composing with itself, as in
For example, one may ask for a meaningful interpretation of
as an analogue of the functional square root for the differentiation operator, that is, 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 a linear operator
for every real number in such a way that, when takes an integer value , it coincides with the usual -fold differentiation if , and with the -th power of when .
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator is that the sets of operator powers defined in this way are continuous semigroups with parameter , of which the original discrete semigroup of for integer is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.
In applied mathematics and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695. Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions.
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