Polynomial accurate numerical fractional order integration and differentiation

In this paper we derive formulæ for composite numerical fractional integration and differentiation that are “polynomial accurate” in the sense that when applied to polynomials of a given degree they yield exact results. Initially, we develop a fractional equivalent to the trapezoidal rule, as well as an analytic error bound for when it is applied to arbitrary functions. Subsequently, we demonstrate how the formulæ are extended to higher order Lagrange Interpolating polynomials. Generally, we show how fractional integration is applied to piecewise defined functions, and hence the method can be extended, for example, to local Savitzky-Golay smoothing, or the numerical solution of Fractional Order Differential Equations. The methods are verified by applying the formulæ to both polynomial and non-polynomial functions.