Step-by-step integration for fractional operators

Abstract

In this paper, an approach based on the definition of the Riemann–Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grünwald–Letnikov operators. The proposed Riemann–Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function f(t). It has been shown that, as f(t) is discretised as stepwise or piecewise function, the Riemann–Liouville fractional integral and derivative are governing by operators very similar to the Grünwald–Letnikov operators.In order to show the accuracy and capabilities of the proposed Riemann–Liouville discretisation technique and the Grünwald–Letnikov discrete operators, both techniques have been applied to: unit step functions, exponential functions and sample functions of white noise.