Sibuya probability distribution and numerical evaluation of fractional-order operators

Abstract In this work we explore the Sibuya probability distribution, which serves as the basis and the main instrument for numerical simulations of Grünwald–Letnikov fractional derivatives by the Monte Carlo method. We provide three methods for simulating the Sibuya distribution. We also introduce the Sibuya-like sieved probability distributions, and apply them to numerical fractional-order differentiation. Additionally, we use the Monte Carlo method for evaluating fractional-order integrals. The developed methods and tools are illustrated by examples of computation. We provide the MATLAB toolboxes for simulation of the Sibuya probability distribution, and for the numerical examples.