Английская Википедия:Differintegral

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Шаблон:Redirect-distinguish Шаблон:Calculus

In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by

<math>\mathbb{D}^q f</math>

is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.

Standard definitions

The four most common forms are:

{}^{RL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\ & =\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^t (t-\tau)^{n-q-1}f(\tau)d\tau \end{align}</math>

  • The Grunwald–Letnikov differintegralШаблон:PbThe Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot. <math display="block">\begin{align}

{}^{GL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\ & =\lim_{N \to \infty}\left[\frac{t-a}{N}\right]^{-q}\sum_{j=0}^{N-1}(-1)^j{q \choose j}f\left(t-j\left[\frac{t-a}{N}\right]\right) \end{align}</math>

  • The Weyl differintegralШаблон:Pb This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.
  • The Caputo differintegralШаблон:PbIn opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant <math>f(t)</math> is equal to zero. Moreover, a form of the Laplace transform allows to simply evaluate the initial conditions by computing finite, integer-order derivatives at point <math>a</math>. <math display="block">\begin{align}

{}^{C}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\ & =\frac{1}{\Gamma(n-q)} \int_{a}^t \frac{f^{(n)}(\tau)}{(t-\tau)^{q-n+1}}d\tau \end{align}</math>

Definitions via transforms

The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.[1] They can be represented via Laplace, Fourier transforms or via Newton series expansion.

Recall the continuous Fourier transform, here denoted <math> \mathcal{F}</math>: <math display="block"> F(\omega) = \mathcal{F}\{f(t)\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{- i\omega t}\,dt </math>

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication: <math display="block">\mathcal{F}\left[\frac{df(t)}{dt}\right] = i \omega \mathcal{F}[f(t)]</math>

So, <math display="block">\frac{d^nf(t)}{dt^n} = \mathcal{F}^{-1}\left\{(i \omega)^n\mathcal{F}[f(t)]\right\}</math> which generalizes to <math display="block">\mathbb{D}^qf(t) = \mathcal{F}^{-1}\left\{(i \omega)^q\mathcal{F}[f(t)]\right\}.</math>

Under the bilateral Laplace transform, here denoted by <math> \mathcal{L}</math> and defined as <math display="inline"> \mathcal{L}[f(t)] =\int_{-\infty}^\infty e^{-st} f(t)\, dt</math>, differentiation transforms into a multiplication <math display="block">\mathcal{L}\left[\frac{df(t)}{dt}\right] = s\mathcal{L}[f(t)].</math>

Generalizing to arbitrary order and solving for <math> \mathbb{D}^qf(t)</math>, one obtains <math display="block">\mathbb{D}^qf(t)=\mathcal{L}^{-1}\left\{s^q\mathcal{L}[f(t)]\right\}.</math>

Representation via Newton series is the Newton interpolation over consecutive integer orders:

<math display="block">\mathbb{D}^qf(t) =\sum_{m=0}^{\infty} \binom {q}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x).</math>

For fractional derivative definitions described in this section, the following identities hold:

<math>\mathbb{D}^q(t^n)=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}t^{n-q}</math>
<math>\mathbb{D}^q(\sin(t))=\sin \left( t+\frac{q\pi}{2} \right) </math>
<math>\mathbb{D}^q(e^{at})=a^q e^{at}</math>[2]

Basic formal properties

  • Linearity rules <math display="block">\mathbb{D}^q(f+g) = \mathbb{D}^q(f)+\mathbb{D}^q(g)</math>

<math display="block">\mathbb{D}^q(af) = a\mathbb{D}^q(f)</math>

  • Zero rule <math display="block">\mathbb{D}^0 f = f </math>
  • Product rule <math display="block">\mathbb{D}^q_t(fg) = \sum_{j=0}^{\infty} {q \choose j}\mathbb{D}^j_t(f)\mathbb{D}^{q-j}_t(g)</math>

In general, composition (or semigroup) rule is a desirable property, but is hard to achieve mathematically and hence is not always completely satisfied by each proposed operator;[3] this forms part of the decision making process on which one to choose:

  • <math display="inline">\mathbb{D}^a\mathbb{D}^{b}f = \mathbb{D}^{a+b}f</math> (ideally)
  • <math display="inline">\mathbb{D}^a\mathbb{D}^{b}f \neq \mathbb{D}^{a+b}f</math> (in practice)

See also

References

Шаблон:Reflist Шаблон:Refbegin

Шаблон:Refend

External links

Партнерские ресурсы
Криптовалюты
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Разное

  1. ↑ Шаблон:Cite book
  2. ↑ See Шаблон:Cite book
  3. ↑ See Шаблон:Cite book
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Differintegral&oldid=14307466