Examples of "differintegral"
includes the integer order differentiation and integration functions, and allows a continuous range of functions around them. The differintegral parameters are "a", "t", and "q". The parameters "a" and "t" describe the range over which to compute the result. The differintegral parameter "q" may be any real number or complex number. If "q" is greater than zero, the differintegral computes a derivative. If "q" is less than zero, the differintegral computes an integral.
This is the problem that with the differintegral. If the differintegral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, we lose information, as we lost the "c" in the first equation.
In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the "q"-differintegral of "f", here denoted by
Working with a properly initialized differintegral is the subject of initialized fractional calculus.
where formula_13 denotes the ceiling function. One also obtains a differintegral interpolating between differentiation and integration by defining
A certain counterintuitive property of the differintegral operator should be pointed out, namely the composition law. Although
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.
Letnikov's most renowned contribution is the creation of the Grunwald-Letnikov differintegral. He also published results about Analytic geometry, Ordinary differential equation and Non-Euclidean geometry.
This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator "D" is significantly more complex, but it can be shown that "D" is neither commutative nor additive in general.
With the advent of powerful personal computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald–Letnikov differintegral to evaluate the derivatives.
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.
The implementation differintegral calculation using fast fourier transform has certain benefits because it is easily combined with low pass quadratic filtering methods. This is very useful when cyclic voltammograms are recorded in high resistivity solvents like tetrahydrofuran or toluene, where feedback oscillations are a frequent problem.
A fractional-order integrator or just simply fractional integrator is an integrator device that calculates the fractional-order integral or derivative (usually called a differintegral) of an input. Differentiation or integration is a real or complex parameter. The fractional integrator is useful in fractional-order control where the history of the system under control is important to the control system output.
In fractional calculus, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of times. Integrating a fractional number of times with this formula is straightforward; one can use fractional "n" by interpreting ("n"-1)! as Γ("n") (see Gamma function). Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.
The integer order integration can be computed as a Riemann–Liouville differintegral, where the weight of each element in the sum is the constant unit value 1, which is equivalent to the Riemann sum. To compute an integer order derivative, the weights in the summation would be zero, with the exception of the most recent data points, where (in the case of the first unit derivative) the weight of the data point at "t" − 1 is −1 and the weight of the data point at "t" is 1. The sum of the points in the input function using these weights results in the difference of the most recent data points.