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Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper. [ 17 ] In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions.
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 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.
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.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) {\textstyle \arctan(y,x)} .
Combining derivatives of different variables results in a notion of a partial differential operator. The linear operator which assigns to each function its derivative is an example of a differential operator on a function space. By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.
An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Vol. 198. Academic Press. ISBN 0-12-558840-2. Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag.
In mathematics, the Riemann–Liouville integral associates with a real function: another function I α f of the same kind for each value of the parameter α > 0.The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, I α f is an iterated antiderivative of f of order α.