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

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