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By applying a Laplace transform to the LTI system above, the transfer function becomes = () = = =For general orders and this is a non-rational transfer function. Non-rational transfer functions cannot be written as an expansion in a finite number of terms (e.g., a binomial expansion would have an infinite number of terms) and in this sense fractional orders systems can be said to have the ...
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
In mathematics, the Caputo fractional derivative, also called Caputo-type fractional derivative, is a generalization of derivatives for non-integer orders named after Michele Caputo. Caputo first defined this form of fractional derivative in 1967.
In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague , in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the α th order derivative, the n th order derivative of the integral of order (n − α) is computed, where n is the smallest integer greater than α (that is, n = ⌈α⌉). The Riemann–Liouville fractional derivative and integral has ...
In addition to n th derivatives for any natural number n, there are various ways to define derivatives of fractional or negative orders, which are studied in fractional calculus. The −1 order derivative corresponds to the integral, whence the term differintegral.
Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A fractal measure t is scaled according to t α. Such a derivative is local, in contrast to the similarly applied fractional derivative. Fractal calculus is formulated as a generalization of ...
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 definitions of the differintegral.