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 or composition, in the same sense that f 2(x) = f(f(x)).
For example, one may ask the question of meaningfully interpreting
as a square root of the differentiation operator (an operator half iterate), i.e., an expression for some operator that when applied twice to a function will have the same effect as differentiation. More generally, one can look at the question of defining
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 Da will form a continuous semigroup with parameter a, inside which the original discrete semigroup of Dn for integer n can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent a, since it need not be rational; the use of the term fractional calculus is merely conventional.
Fractional differential equations are a generalization of differential equations through the application of fractional calculus.
Other articles related to "fractional calculus, fractional":
... The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics ... The fractional Schrödinger equation discovered by Nick Laskin has the following form Here is a 3-dimensional vector, is the Planck constant, is the wavefunction ... where m is a particle mass), and the operator is the 3-dimensional fractional quantum Riesz derivative defined by Here the wave functions in the space and momentum representations ...
... Fractional Calculus and Applied Analysis is a peer-reviewed mathematics journal published by Springer Science+Business Media on behalf of the Institute of Mathematics Informatics of the Bulgarian ... It covers research on fractional calculus, special functions, integral transforms, and some closely related areas of applied analysis ...
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