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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 : 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 : as an analog of the functional square root for the differentiation operator (an operator half iterated), 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 : 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 −''n''th power of ''J'' when ''n'' < 0. The motivation behind this extension to the differential operator is that the semigroup of powers ''D''''a'' 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 (also known as extraordinary differential equations) are a generalization of differential equations through the application of fractional calculus. ==Nature of the fractional derivative== An important point is that the fractional derivative at a point ''x'' is a ''local property'' only when ''a'' is an integer; in non-integer cases we cannot say that the fractional derivative at ''x'' of a function ''f'' depends only on values of ''f'' very near ''x'', in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. To use a metaphor, the fractional derivative requires some peripheral vision. As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832. The fractional derivative of a function to order ''a'' is often now defined by means of the Fourier or Mellin integral transforms.〔For the history of the subject, see the thesis (in French): Stéphane Dugowson, (''Les différentielles métaphysiques'' ) (''histoire et philosophie de la généralisation de l'ordre de dérivation''), Thèse, Université Paris Nord (1994)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fractional calculus」の詳細全文を読む スポンサード リンク
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