تفاضل وتكامل كسري

من ويكيبيديا، الموسوعة الحرة
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قالب:Calculus

"المشتقة الكسرية" redirects to here.

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 or composition, in the same sense that f2(x) = f(f(x)). For example, one may ask the question of meaningfully interpreting

\sqrt{D} = D^{\frac{1}{2}} \,

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

D^a \,

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.

Nature of the fractional derivative[عدل]

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.[1]

Heuristics[عدل]

A fairly natural question to ask is whether there exists an operator H, or half-derivative, such that

H^2 f(x) = D f(x) = \dfrac{d}{dx} f(x) = f'(x) .

It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that

(P ^ a f)(x) = f'(x) \,,

or to put it another way, the definition of dny/dxn can be extended to all real values of n.


التفاضل و التكامل الكسري (Fractional Calculus ) يعتبر البعض هذا العلم جزءا من التحليل الرياضي و يتعامل مع تطبيقات التكامل و الاشتقاق في حالة الرتب الاختيارية، و هذا المجال يهتم بتعميم المشتقة لاقتران (دالة) ما لأي مشتقة ذات رتبة غير صحيحة، فمثلا: نحن في العادة نتعامل مع المشتقة الأولى و الثانية ... أما هذا المجال (التفاضل الكسري) فيفيدنا في ايجاد المشتقة رقم نصف أو 0.3 أو 0.7 ...الخ. البداية: بدات أصول هذا الاتجاه في القرن الـ17 حينما وضع نيوتن و لايبنز أساسات التفاضل و التكامل، فقد وضع لايبنز الرمز الشهير (d^n y/dx^n ) ليدل على المشتقة النونية للاقتران(الدالة) f ، فأرسل لاينز رسالة إلى لوبيتال يخبره بهذا الرمز الجديد لكن لوبيتال رد على الرسالة بسؤال محير: " ماذا لو كانت n=1/2 ؟" الرسالة كتبت عام 1695 و تعد اليوم أول ظهور للمشتقة الكسرية. بدأ العالم الرياضي "ليوفيل" بالتقصي و البحث في الموضوع و اصدر سلسلة ابحاث في الفترة 1832-1837 ، حيث عرف أول عامل(operator ) للتكامل الكسري(fractional integration )، و بعد أن ولج "ريمان" هذا الموضوع و طور عليه ظهر ما يعرف اليوم بتعريف "ريمان-ليوفيل" (Riemann-Liouville fractional operator ) تبع ذلك اهتمام غير مسبوق و تطوير كبير لهذا المجال.

مراجع[عدل]

  1. ^ 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)