Monte Carlo method for fractional-order differentiation extended to higher orders

In this work the Monte Carlo method, introduced recently by the authors for orders of differentiation between zero and one, is further extended to differentiation of orders higher than one. Two approaches have been developed on this way. The first approach is based on interpreting the coefficients of the Grunwald-Letnikov fractional differences as so called signed probabilities, which in the case of orders higher than one can be negative or positive. We demonstrate how this situation can be processed and used for computations. The second approach uses the Monte Carlo method for orders between zero and one and the semi-group property of fractional-order differences. Both methods have been implemented in MATLAB and illustrated by several examples of fractional-order differentiation of several functions that typically appear in applications. Computational results of both methods were in mutual agreement and conform with the exact fractional-order derivatives of the functions used in the examples.