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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 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 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 ) . {\displaystyle \arctan(y,x).}
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 algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
Any polynomial function is infinitely differentiable; taking derivatives repeatedly will eventually result in a constant function, and all subsequent derivatives of that function are zero. [ 35 ] One application of higher-order derivatives is in physics .
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.
Spline interpolation — interpolation by piecewise polynomials Spline (mathematics) — the piecewise polynomials used as interpolants; Perfect spline — polynomial spline of degree m whose mth derivate is ±1; Cubic Hermite spline. Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps