Abstract

Fractional-order derivatives and integrals are suitable tools for modelling in various fields (viscoelastic materials, hereditary effects, anomalous diffusion, electrochemistry, and other fields). The Monte Carlo method for numerical evaluation of fractional-order derivatives is presented. The main idea consists of using the Grünwald–Letnikov approximation and interpreting its coefficients as probabilities. Those probabilities turn to be signed probabilities: some are positive, some other are negative. First, the Monte Carlo method for fractional differentiation of orders between 0 and 1 is considered; in this case, all necessary probabilities are of the same sign. For orders above one, some probabilities become negative, and some are necessarily greater than one, and we demonstrate how to deal with them. The presented method allows further improvements and extensions (including parallel computations), and applications in fractional-order modelling. The talk is based on the joint results with Nikolai Leonenko (Cardiff University).